## table of contents

complex16OTHERauxiliary(3) | LAPACK | complex16OTHERauxiliary(3) |

# NAME¶

complex16OTHERauxiliary - complex16# SYNOPSIS¶

## Functions¶

subroutine

**clag2z**(M, N, SA, LDSA, A, LDA, INFO)

**CLAG2Z**converts a complex single precision matrix to a complex double precision matrix. double precision function

**dzsum1**(N, CX, INCX)

**DZSUM1**forms the 1-norm of the complex vector using the true absolute value. integer function

**ilazlc**(M, N, A, LDA)

**ILAZLC**scans a matrix for its last non-zero column. integer function

**ilazlr**(M, N, A, LDA)

**ILAZLR**scans a matrix for its last non-zero row. subroutine

**zdrscl**(N, SA, SX, INCX)

**ZDRSCL**multiplies a vector by the reciprocal of a real scalar. subroutine

**zlabrd**(M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY)

**ZLABRD**reduces the first nb rows and columns of a general matrix to a bidiagonal form. subroutine

**zlacgv**(N, X, INCX)

**ZLACGV**conjugates a complex vector. subroutine

**zlacn2**(N, V, X, EST, KASE, ISAVE)

**ZLACN2**estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products. subroutine

**zlacon**(N, V, X, EST, KASE)

**ZLACON**estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products. subroutine

**zlacp2**(UPLO, M, N, A, LDA, B, LDB)

**ZLACP2**copies all or part of a real two-dimensional array to a complex array. subroutine

**zlacpy**(UPLO, M, N, A, LDA, B, LDB)

**ZLACPY**copies all or part of one two-dimensional array to another. subroutine

**zlacrm**(M, N, A, LDA, B, LDB, C, LDC, RWORK)

**ZLACRM**multiplies a complex matrix by a square real matrix. subroutine

**zlacrt**(N, CX, INCX, CY, INCY, C, S)

**ZLACRT**performs a linear transformation of a pair of complex vectors. complex *16 function

**zladiv**(X, Y)

**ZLADIV**performs complex division in real arithmetic, avoiding unnecessary overflow. subroutine

**zlaein**(RIGHTV, NOINIT, N, H, LDH, W, V, B, LDB, RWORK, EPS3, SMLNUM, INFO)

**ZLAEIN**computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration. subroutine

**zlaev2**(A, B, C, RT1, RT2, CS1, SN1)

**ZLAEV2**computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix. subroutine

**zlag2c**(M, N, A, LDA, SA, LDSA, INFO)

**ZLAG2C**converts a complex double precision matrix to a complex single precision matrix. subroutine

**zlags2**(UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV, SNV, CSQ, SNQ)

**ZLAGS2**subroutine

**zlagtm**(TRANS, N, NRHS, ALPHA, DL, D, DU, X, LDX, BETA, B, LDB)

**ZLAGTM**performs a matrix-matrix product of the form C = αAB+βC, where A is a tridiagonal matrix, B and C are rectangular matrices, and α and β are scalars, which may be 0, 1, or -1. subroutine

**zlahqr**(WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z, LDZ, INFO)

**ZLAHQR**computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm. subroutine

**zlahr2**(N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)

**ZLAHR2**reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A. subroutine

**zlaic1**(JOB, J, X, SEST, W, GAMMA, SESTPR, S, C)

**ZLAIC1**applies one step of incremental condition estimation. double precision function

**zlangt**(NORM, N, DL, D, DU)

**ZLANGT**returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general tridiagonal matrix. double precision function

**zlanhb**(NORM, UPLO, N, K, AB, LDAB, WORK)

**ZLANHB**returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hermitian band matrix. double precision function

**zlanhp**(NORM, UPLO, N, AP, WORK)

**ZLANHP**returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex Hermitian matrix supplied in packed form. double precision function

**zlanhs**(NORM, N, A, LDA, WORK)

**ZLANHS**returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of an upper Hessenberg matrix. double precision function

**zlanht**(NORM, N, D, E)

**ZLANHT**returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex Hermitian tridiagonal matrix. double precision function

**zlansb**(NORM, UPLO, N, K, AB, LDAB, WORK)

**ZLANSB**returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric band matrix. double precision function

**zlansp**(NORM, UPLO, N, AP, WORK)

**ZLANSP**returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric matrix supplied in packed form. double precision function

**zlantb**(NORM, UPLO, DIAG, N, K, AB, LDAB, WORK)

**ZLANTB**returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular band matrix. double precision function

**zlantp**(NORM, UPLO, DIAG, N, AP, WORK)

**ZLANTP**returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular matrix supplied in packed form. double precision function

**zlantr**(NORM, UPLO, DIAG, M, N, A, LDA, WORK)

**ZLANTR**returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix. subroutine

**zlapll**(N, X, INCX, Y, INCY, SSMIN)

**ZLAPLL**measures the linear dependence of two vectors. subroutine

**zlapmr**(FORWRD, M, N, X, LDX, K)

**ZLAPMR**rearranges rows of a matrix as specified by a permutation vector. subroutine

**zlapmt**(FORWRD, M, N, X, LDX, K)

**ZLAPMT**performs a forward or backward permutation of the columns of a matrix. subroutine

**zlaqhb**(UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED)

**ZLAQHB**scales a Hermitian band matrix, using scaling factors computed by cpbequ. subroutine

**zlaqhp**(UPLO, N, AP, S, SCOND, AMAX, EQUED)

**ZLAQHP**scales a Hermitian matrix stored in packed form. subroutine

**zlaqp2**(M, N, OFFSET, A, LDA, JPVT, TAU, VN1, VN2, WORK)

**ZLAQP2**computes a QR factorization with column pivoting of the matrix block. subroutine

**zlaqps**(M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1, VN2, AUXV, F, LDF)

**ZLAQPS**computes a step of QR factorization with column pivoting of a real m-by-n matrix A by using BLAS level 3. subroutine

**zlaqr0**(WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO)

**ZLAQR0**computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition. subroutine

**zlaqr1**(N, H, LDH, S1, S2, V)

**ZLAQR1**sets a scalar multiple of the first column of the product of 2-by-2 or 3-by-3 matrix H and specified shifts. subroutine

**zlaqr2**(WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, IHIZ, Z, LDZ, NS, ND, SH, V, LDV, NH, T, LDT, NV, WV, LDWV, WORK, LWORK)

**ZLAQR2**performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation). subroutine

**zlaqr3**(WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, IHIZ, Z, LDZ, NS, ND, SH, V, LDV, NH, T, LDT, NV, WV, LDWV, WORK, LWORK)

**ZLAQR3**performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation). subroutine

**zlaqr4**(WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO)

**ZLAQR4**computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition. subroutine

**zlaqr5**(WANTT, WANTZ, KACC22, N, KTOP, KBOT, NSHFTS, S, H, LDH, ILOZ, IHIZ, Z, LDZ, V, LDV, U, LDU, NV, WV, LDWV, NH, WH, LDWH)

**ZLAQR5**performs a single small-bulge multi-shift QR sweep. subroutine

**zlaqsb**(UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED)

**ZLAQSB**scales a symmetric/Hermitian band matrix, using scaling factors computed by spbequ. subroutine

**zlaqsp**(UPLO, N, AP, S, SCOND, AMAX, EQUED)

**ZLAQSP**scales a symmetric/Hermitian matrix in packed storage, using scaling factors computed by sppequ. subroutine

**zlar1v**(N, B1, BN, LAMBDA, D, L, LD, LLD, PIVMIN, GAPTOL, Z, WANTNC, NEGCNT, ZTZ, MINGMA, R, ISUPPZ, NRMINV, RESID, RQCORR, WORK)

**ZLAR1V**computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI. subroutine

**zlar2v**(N, X, Y, Z, INCX, C, S, INCC)

**ZLAR2V**applies a vector of plane rotations with real cosines and complex sines from both sides to a sequence of 2-by-2 symmetric/Hermitian matrices. subroutine

**zlarcm**(M, N, A, LDA, B, LDB, C, LDC, RWORK)

**ZLARCM**copies all or part of a real two-dimensional array to a complex array. subroutine

**zlarf**(SIDE, M, N, V, INCV, TAU, C, LDC, WORK)

**ZLARF**applies an elementary reflector to a general rectangular matrix. subroutine

**zlarfb**(SIDE, TRANS, DIRECT, STOREV, M, N, K, V, LDV, T, LDT, C, LDC, WORK, LDWORK)

**ZLARFB**applies a block reflector or its conjugate-transpose to a general rectangular matrix. subroutine

**zlarfb_gett**(IDENT, M, N, K, T, LDT, A, LDA, B, LDB, WORK, LDWORK)

**ZLARFB_GETT**subroutine

**zlarfg**(N, ALPHA, X, INCX, TAU)

**ZLARFG**generates an elementary reflector (Householder matrix). subroutine

**zlarfgp**(N, ALPHA, X, INCX, TAU)

**ZLARFGP**generates an elementary reflector (Householder matrix) with non-negative beta. subroutine

**zlarft**(DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT)

**ZLARFT**forms the triangular factor T of a block reflector H = I - vtvH subroutine

**zlarfx**(SIDE, M, N, V, TAU, C, LDC, WORK)

**ZLARFX**applies an elementary reflector to a general rectangular matrix, with loop unrolling when the reflector has order ≤ 10. subroutine

**zlarfy**(UPLO, N, V, INCV, TAU, C, LDC, WORK)

**ZLARFY**subroutine

**zlargv**(N, X, INCX, Y, INCY, C, INCC)

**ZLARGV**generates a vector of plane rotations with real cosines and complex sines. subroutine

**zlarnv**(IDIST, ISEED, N, X)

**ZLARNV**returns a vector of random numbers from a uniform or normal distribution. subroutine

**zlarrv**(N, VL, VU, D, L, PIVMIN, ISPLIT, M, DOL, DOU, MINRGP, RTOL1, RTOL2, W, WERR, WGAP, IBLOCK, INDEXW, GERS, Z, LDZ, ISUPPZ, WORK, IWORK, INFO)

**ZLARRV**computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT. subroutine

**zlartv**(N, X, INCX, Y, INCY, C, S, INCC)

**ZLARTV**applies a vector of plane rotations with real cosines and complex sines to the elements of a pair of vectors. subroutine

**zlascl**(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)

**ZLASCL**multiplies a general rectangular matrix by a real scalar defined as cto/cfrom. subroutine

**zlaset**(UPLO, M, N, ALPHA, BETA, A, LDA)

**ZLASET**initializes the off-diagonal elements and the diagonal elements of a matrix to given values. subroutine

**zlasr**(SIDE, PIVOT, DIRECT, M, N, C, S, A, LDA)

**ZLASR**applies a sequence of plane rotations to a general rectangular matrix. subroutine

**zlaswp**(N, A, LDA, K1, K2, IPIV, INCX)

**ZLASWP**performs a series of row interchanges on a general rectangular matrix. subroutine

**zlat2c**(UPLO, N, A, LDA, SA, LDSA, INFO)

**ZLAT2C**converts a double complex triangular matrix to a complex triangular matrix. subroutine

**zlatbs**(UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X, SCALE, CNORM, INFO)

**ZLATBS**solves a triangular banded system of equations. subroutine

**zlatdf**(IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV, JPIV)

**ZLATDF**uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate. subroutine

**zlatps**(UPLO, TRANS, DIAG, NORMIN, N, AP, X, SCALE, CNORM, INFO)

**ZLATPS**solves a triangular system of equations with the matrix held in packed storage. subroutine

**zlatrd**(UPLO, N, NB, A, LDA, E, TAU, W, LDW)

**ZLATRD**reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation. subroutine

**zlatrs**(UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE, CNORM, INFO)

**ZLATRS**solves a triangular system of equations with the scale factor set to prevent overflow. subroutine

**zlauu2**(UPLO, N, A, LDA, INFO)

**ZLAUU2**computes the product UUH or LHL, where U and L are upper or lower triangular matrices (unblocked algorithm). subroutine

**zlauum**(UPLO, N, A, LDA, INFO)

**ZLAUUM**computes the product UUH or LHL, where U and L are upper or lower triangular matrices (blocked algorithm). subroutine

**zrot**(N, CX, INCX, CY, INCY, C, S)

**ZROT**applies a plane rotation with real cosine and complex sine to a pair of complex vectors. subroutine

**zspmv**(UPLO, N, ALPHA, AP, X, INCX, BETA, Y, INCY)

**ZSPMV**computes a matrix-vector product for complex vectors using a complex symmetric packed matrix subroutine

**zspr**(UPLO, N, ALPHA, X, INCX, AP)

**ZSPR**performs the symmetrical rank-1 update of a complex symmetric packed matrix. subroutine

**ztprfb**(SIDE, TRANS, DIRECT, STOREV, M, N, K, L, V, LDV, T, LDT, A, LDA, B, LDB, WORK, LDWORK)

**ZTPRFB**applies a real or complex 'triangular-pentagonal' blocked reflector to a real or complex matrix, which is composed of two blocks.

# Detailed Description¶

This is the group of complex16 other auxiliary routines# Function Documentation¶

## subroutine clag2z (integer M, integer N, complex, dimension( ldsa, * ) SA, integer LDSA, complex*16, dimension( lda, * ) A, integer LDA, integer INFO)¶

**CLAG2Z**converts a complex single precision matrix to a complex double precision matrix.

**Purpose:**

CLAG2Z converts a COMPLEX matrix, SA, to a COMPLEX*16 matrix, A. Note that while it is possible to overflow while converting from double to single, it is not possible to overflow when converting from single to double. This is an auxiliary routine so there is no argument checking.

**Parameters**

*M*

M is INTEGER The number of lines of the matrix A. M >= 0.

*N*

N is INTEGER The number of columns of the matrix A. N >= 0.

*SA*

SA is COMPLEX array, dimension (LDSA,N) On entry, the M-by-N coefficient matrix SA.

*LDSA*

LDSA is INTEGER The leading dimension of the array SA. LDSA >= max(1,M).

*A*

A is COMPLEX*16 array, dimension (LDA,N) On exit, the M-by-N coefficient matrix A.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).

*INFO*

INFO is INTEGER = 0: successful exit

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## double precision function dzsum1 (integer N, complex*16, dimension( * ) CX, integer INCX)¶

**DZSUM1**forms the 1-norm of the complex vector using the true absolute value.

**Purpose:**

DZSUM1 takes the sum of the absolute values of a complex vector and returns a double precision result. Based on DZASUM from the Level 1 BLAS. The change is to use the 'genuine' absolute value.

**Parameters**

*N*

N is INTEGER The number of elements in the vector CX.

*CX*

CX is COMPLEX*16 array, dimension (N) The vector whose elements will be summed.

*INCX*

INCX is INTEGER The spacing between successive values of CX. INCX > 0.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

## integer function ilazlc (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA)¶

**ILAZLC**scans a matrix for its last non-zero column.

**Purpose:**

ILAZLC scans A for its last non-zero column.

**Parameters**

*M*

M is INTEGER The number of rows of the matrix A.

*N*

N is INTEGER The number of columns of the matrix A.

*A*

A is COMPLEX*16 array, dimension (LDA,N) The m by n matrix A.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## integer function ilazlr (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA)¶

**ILAZLR**scans a matrix for its last non-zero row.

**Purpose:**

ILAZLR scans A for its last non-zero row.

**Parameters**

*M*

M is INTEGER The number of rows of the matrix A.

*N*

N is INTEGER The number of columns of the matrix A.

*A*

A is COMPLEX*16 array, dimension (LDA,N) The m by n matrix A.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zdrscl (integer N, double precision SA, complex*16, dimension( * ) SX, integer INCX)¶

**ZDRSCL**multiplies a vector by the reciprocal of a real scalar.

**Purpose:**

ZDRSCL multiplies an n-element complex vector x by the real scalar 1/a. This is done without overflow or underflow as long as the final result x/a does not overflow or underflow.

**Parameters**

*N*

N is INTEGER The number of components of the vector x.

*SA*

SA is DOUBLE PRECISION The scalar a which is used to divide each component of x. SA must be >= 0, or the subroutine will divide by zero.

*SX*

SX is COMPLEX*16 array, dimension (1+(N-1)*abs(INCX)) The n-element vector x.

*INCX*

INCX is INTEGER The increment between successive values of the vector SX. > 0: SX(1) = X(1) and SX(1+(i-1)*INCX) = x(i), 1< i<= n

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlabrd (integer M, integer N, integer NB, complex*16, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) D, double precision, dimension( * ) E, complex*16, dimension( * ) TAUQ, complex*16, dimension( * ) TAUP, complex*16, dimension( ldx, * ) X, integer LDX, complex*16, dimension( ldy, * ) Y, integer LDY)¶

**ZLABRD**reduces the first nb rows and columns of a general matrix to a bidiagonal form.

**Purpose:**

ZLABRD reduces the first NB rows and columns of a complex general m by n matrix A to upper or lower real bidiagonal form by a unitary transformation Q**H * A * P, and returns the matrices X and Y which are needed to apply the transformation to the unreduced part of A. If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower bidiagonal form. This is an auxiliary routine called by ZGEBRD

**Parameters**

*M*

M is INTEGER The number of rows in the matrix A.

*N*

N is INTEGER The number of columns in the matrix A.

*NB*

NB is INTEGER The number of leading rows and columns of A to be reduced.

*A*

A is COMPLEX*16 array, dimension (LDA,N) On entry, the m by n general matrix to be reduced. On exit, the first NB rows and columns of the matrix are overwritten; the rest of the array is unchanged. If m >= n, elements on and below the diagonal in the first NB columns, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors; and elements above the diagonal in the first NB rows, with the array TAUP, represent the unitary matrix P as a product of elementary reflectors. If m < n, elements below the diagonal in the first NB columns, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors, and elements on and above the diagonal in the first NB rows, with the array TAUP, represent the unitary matrix P as a product of elementary reflectors. See Further Details.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).

*D*

D is DOUBLE PRECISION array, dimension (NB) The diagonal elements of the first NB rows and columns of the reduced matrix. D(i) = A(i,i).

*E*

E is DOUBLE PRECISION array, dimension (NB) The off-diagonal elements of the first NB rows and columns of the reduced matrix.

*TAUQ*

TAUQ is COMPLEX*16 array, dimension (NB) The scalar factors of the elementary reflectors which represent the unitary matrix Q. See Further Details.

*TAUP*

TAUP is COMPLEX*16 array, dimension (NB) The scalar factors of the elementary reflectors which represent the unitary matrix P. See Further Details.

*X*

X is COMPLEX*16 array, dimension (LDX,NB) The m-by-nb matrix X required to update the unreduced part of A.

*LDX*

LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).

*Y*

Y is COMPLEX*16 array, dimension (LDY,NB) The n-by-nb matrix Y required to update the unreduced part of A.

*LDY*

LDY is INTEGER The leading dimension of the array Y. LDY >= max(1,N).

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

The matrices Q and P are represented as products of elementary reflectors: Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb) Each H(i) and G(i) has the form: H(i) = I - tauq * v * v**H and G(i) = I - taup * u * u**H where tauq and taup are complex scalars, and v and u are complex vectors. If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i). If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i). The elements of the vectors v and u together form the m-by-nb matrix V and the nb-by-n matrix U**H which are needed, with X and Y, to apply the transformation to the unreduced part of the matrix, using a block update of the form: A := A - V*Y**H - X*U**H. The contents of A on exit are illustrated by the following examples with nb = 2: m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 ) ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 ) ( v1 v2 a a a ) ( v1 1 a a a a ) ( v1 v2 a a a ) ( v1 v2 a a a a ) ( v1 v2 a a a ) ( v1 v2 a a a a ) ( v1 v2 a a a ) where a denotes an element of the original matrix which is unchanged, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i).

## subroutine zlacgv (integer N, complex*16, dimension( * ) X, integer INCX)¶

**ZLACGV**conjugates a complex vector.

**Purpose:**

ZLACGV conjugates a complex vector of length N.

**Parameters**

*N*

N is INTEGER The length of the vector X. N >= 0.

*X*

X is COMPLEX*16 array, dimension (1+(N-1)*abs(INCX)) On entry, the vector of length N to be conjugated. On exit, X is overwritten with conjg(X).

*INCX*

INCX is INTEGER The spacing between successive elements of X.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlacn2 (integer N, complex*16, dimension( * ) V, complex*16, dimension( * ) X, double precision EST, integer KASE, integer, dimension( 3 ) ISAVE)¶

**ZLACN2**estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products.

**Purpose:**

ZLACN2 estimates the 1-norm of a square, complex matrix A. Reverse communication is used for evaluating matrix-vector products.

**Parameters**

*N*

N is INTEGER The order of the matrix. N >= 1.

*V*

V is COMPLEX*16 array, dimension (N) On the final return, V = A*W, where EST = norm(V)/norm(W) (W is not returned).

*X*

X is COMPLEX*16 array, dimension (N) On an intermediate return, X should be overwritten by A * X, if KASE=1, A**H * X, if KASE=2, where A**H is the conjugate transpose of A, and ZLACN2 must be re-called with all the other parameters unchanged.

*EST*

EST is DOUBLE PRECISION On entry with KASE = 1 or 2 and ISAVE(1) = 3, EST should be unchanged from the previous call to ZLACN2. On exit, EST is an estimate (a lower bound) for norm(A).

*KASE*

KASE is INTEGER On the initial call to ZLACN2, KASE should be 0. On an intermediate return, KASE will be 1 or 2, indicating whether X should be overwritten by A * X or A**H * X. On the final return from ZLACN2, KASE will again be 0.

*ISAVE*

ISAVE is INTEGER array, dimension (3) ISAVE is used to save variables between calls to ZLACN2

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

Originally named CONEST, dated March 16, 1988. Last modified: April, 1999 This is a thread safe version of ZLACON, which uses the array ISAVE in place of a SAVE statement, as follows: ZLACON ZLACN2 JUMP ISAVE(1) J ISAVE(2) ITER ISAVE(3)

**Contributors:**

**References:**

## subroutine zlacon (integer N, complex*16, dimension( n ) V, complex*16, dimension( n ) X, double precision EST, integer KASE)¶

**ZLACON**estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products.

**Purpose:**

ZLACON estimates the 1-norm of a square, complex matrix A. Reverse communication is used for evaluating matrix-vector products.

**Parameters**

*N*

N is INTEGER The order of the matrix. N >= 1.

*V*

V is COMPLEX*16 array, dimension (N) On the final return, V = A*W, where EST = norm(V)/norm(W) (W is not returned).

*X*

X is COMPLEX*16 array, dimension (N) On an intermediate return, X should be overwritten by A * X, if KASE=1, A**H * X, if KASE=2, where A**H is the conjugate transpose of A, and ZLACON must be re-called with all the other parameters unchanged.

*EST*

EST is DOUBLE PRECISION On entry with KASE = 1 or 2 and JUMP = 3, EST should be unchanged from the previous call to ZLACON. On exit, EST is an estimate (a lower bound) for norm(A).

*KASE*

KASE is INTEGER On the initial call to ZLACON, KASE should be 0. On an intermediate return, KASE will be 1 or 2, indicating whether X should be overwritten by A * X or A**H * X. On the final return from ZLACON, KASE will again be 0.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

**Contributors:**

**References:**

## subroutine zlacp2 (character UPLO, integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB)¶

**ZLACP2**copies all or part of a real two-dimensional array to a complex array.

**Purpose:**

ZLACP2 copies all or part of a real two-dimensional matrix A to a complex matrix B.

**Parameters**

*UPLO*

UPLO is CHARACTER*1 Specifies the part of the matrix A to be copied to B. = 'U': Upper triangular part = 'L': Lower triangular part Otherwise: All of the matrix A

*M*

M is INTEGER The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER The number of columns of the matrix A. N >= 0.

*A*

A is DOUBLE PRECISION array, dimension (LDA,N) The m by n matrix A. If UPLO = 'U', only the upper trapezium is accessed; if UPLO = 'L', only the lower trapezium is accessed.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).

*B*

B is COMPLEX*16 array, dimension (LDB,N) On exit, B = A in the locations specified by UPLO.

*LDB*

LDB is INTEGER The leading dimension of the array B. LDB >= max(1,M).

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlacpy (character UPLO, integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB)¶

**ZLACPY**copies all or part of one two-dimensional array to another.

**Purpose:**

ZLACPY copies all or part of a two-dimensional matrix A to another matrix B.

**Parameters**

*UPLO*

UPLO is CHARACTER*1 Specifies the part of the matrix A to be copied to B. = 'U': Upper triangular part = 'L': Lower triangular part Otherwise: All of the matrix A

*M*

M is INTEGER The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER The number of columns of the matrix A. N >= 0.

*A*

A is COMPLEX*16 array, dimension (LDA,N) The m by n matrix A. If UPLO = 'U', only the upper trapezium is accessed; if UPLO = 'L', only the lower trapezium is accessed.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).

*B*

B is COMPLEX*16 array, dimension (LDB,N) On exit, B = A in the locations specified by UPLO.

*LDB*

LDB is INTEGER The leading dimension of the array B. LDB >= max(1,M).

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlacrm (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldc, * ) C, integer LDC, double precision, dimension( * ) RWORK)¶

**ZLACRM**multiplies a complex matrix by a square real matrix.

**Purpose:**

ZLACRM performs a very simple matrix-matrix multiplication: C := A * B, where A is M by N and complex; B is N by N and real; C is M by N and complex.

**Parameters**

*M*

M is INTEGER The number of rows of the matrix A and of the matrix C. M >= 0.

*N*

N is INTEGER The number of columns and rows of the matrix B and the number of columns of the matrix C. N >= 0.

*A*

A is COMPLEX*16 array, dimension (LDA, N) On entry, A contains the M by N matrix A.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >=max(1,M).

*B*

B is DOUBLE PRECISION array, dimension (LDB, N) On entry, B contains the N by N matrix B.

*LDB*

LDB is INTEGER The leading dimension of the array B. LDB >=max(1,N).

*C*

C is COMPLEX*16 array, dimension (LDC, N) On exit, C contains the M by N matrix C.

*LDC*

LDC is INTEGER The leading dimension of the array C. LDC >=max(1,N).

*RWORK*

RWORK is DOUBLE PRECISION array, dimension (2*M*N)

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlacrt (integer N, complex*16, dimension( * ) CX, integer INCX, complex*16, dimension( * ) CY, integer INCY, complex*16 C, complex*16 S)¶

**ZLACRT**performs a linear transformation of a pair of complex vectors.

**Purpose:**

ZLACRT performs the operation ( c s )( x ) ==> ( x ) ( -s c )( y ) ( y ) where c and s are complex and the vectors x and y are complex.

**Parameters**

*N*

N is INTEGER The number of elements in the vectors CX and CY.

*CX*

CX is COMPLEX*16 array, dimension (N) On input, the vector x. On output, CX is overwritten with c*x + s*y.

*INCX*

INCX is INTEGER The increment between successive values of CX. INCX <> 0.

*CY*

CY is COMPLEX*16 array, dimension (N) On input, the vector y. On output, CY is overwritten with -s*x + c*y.

*INCY*

INCY is INTEGER The increment between successive values of CY. INCY <> 0.

*C*

C is COMPLEX*16

*S*

S is COMPLEX*16 C and S define the matrix [ C S ]. [ -S C ]

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## complex*16 function zladiv (complex*16 X, complex*16 Y)¶

**ZLADIV**performs complex division in real arithmetic, avoiding unnecessary overflow.

**Purpose:**

ZLADIV := X / Y, where X and Y are complex. The computation of X / Y will not overflow on an intermediary step unless the results overflows.

**Parameters**

*X*

X is COMPLEX*16

*Y*

Y is COMPLEX*16 The complex scalars X and Y.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlaein (logical RIGHTV, logical NOINIT, integer N, complex*16, dimension( ldh, * ) H, integer LDH, complex*16 W, complex*16, dimension( * ) V, complex*16, dimension( ldb, * ) B, integer LDB, double precision, dimension( * ) RWORK, double precision EPS3, double precision SMLNUM, integer INFO)¶

**ZLAEIN**computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration.

**Purpose:**

ZLAEIN uses inverse iteration to find a right or left eigenvector corresponding to the eigenvalue W of a complex upper Hessenberg matrix H.

**Parameters**

*RIGHTV*

RIGHTV is LOGICAL = .TRUE. : compute right eigenvector; = .FALSE.: compute left eigenvector.

*NOINIT*

NOINIT is LOGICAL = .TRUE. : no initial vector supplied in V = .FALSE.: initial vector supplied in V.

*N*

N is INTEGER The order of the matrix H. N >= 0.

*H*

H is COMPLEX*16 array, dimension (LDH,N) The upper Hessenberg matrix H.

*LDH*

LDH is INTEGER The leading dimension of the array H. LDH >= max(1,N).

*W*

W is COMPLEX*16 The eigenvalue of H whose corresponding right or left eigenvector is to be computed.

*V*

V is COMPLEX*16 array, dimension (N) On entry, if NOINIT = .FALSE., V must contain a starting vector for inverse iteration; otherwise V need not be set. On exit, V contains the computed eigenvector, normalized so that the component of largest magnitude has magnitude 1; here the magnitude of a complex number (x,y) is taken to be |x| + |y|.

*B*

B is COMPLEX*16 array, dimension (LDB,N)

*LDB*

LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*RWORK*

RWORK is DOUBLE PRECISION array, dimension (N)

*EPS3*

EPS3 is DOUBLE PRECISION A small machine-dependent value which is used to perturb close eigenvalues, and to replace zero pivots.

*SMLNUM*

SMLNUM is DOUBLE PRECISION A machine-dependent value close to the underflow threshold.

*INFO*

INFO is INTEGER = 0: successful exit = 1: inverse iteration did not converge; V is set to the last iterate.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlaev2 (complex*16 A, complex*16 B, complex*16 C, double precision RT1, double precision RT2, double precision CS1, complex*16 SN1)¶

**ZLAEV2**computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix.

**Purpose:**

ZLAEV2 computes the eigendecomposition of a 2-by-2 Hermitian matrix [ A B ] [ CONJG(B) C ]. On return, RT1 is the eigenvalue of larger absolute value, RT2 is the eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right eigenvector for RT1, giving the decomposition [ CS1 CONJG(SN1) ] [ A B ] [ CS1 -CONJG(SN1) ] = [ RT1 0 ] [-SN1 CS1 ] [ CONJG(B) C ] [ SN1 CS1 ] [ 0 RT2 ].

**Parameters**

*A*

A is COMPLEX*16 The (1,1) element of the 2-by-2 matrix.

*B*

B is COMPLEX*16 The (1,2) element and the conjugate of the (2,1) element of the 2-by-2 matrix.

*C*

C is COMPLEX*16 The (2,2) element of the 2-by-2 matrix.

*RT1*

RT1 is DOUBLE PRECISION The eigenvalue of larger absolute value.

*RT2*

RT2 is DOUBLE PRECISION The eigenvalue of smaller absolute value.

*CS1*

CS1 is DOUBLE PRECISION

*SN1*

SN1 is COMPLEX*16 The vector (CS1, SN1) is a unit right eigenvector for RT1.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

RT1 is accurate to a few ulps barring over/underflow. RT2 may be inaccurate if there is massive cancellation in the determinant A*C-B*B; higher precision or correctly rounded or correctly truncated arithmetic would be needed to compute RT2 accurately in all cases. CS1 and SN1 are accurate to a few ulps barring over/underflow. Overflow is possible only if RT1 is within a factor of 5 of overflow. Underflow is harmless if the input data is 0 or exceeds underflow_threshold / macheps.

## subroutine zlag2c (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex, dimension( ldsa, * ) SA, integer LDSA, integer INFO)¶

**ZLAG2C**converts a complex double precision matrix to a complex single precision matrix.

**Purpose:**

ZLAG2C converts a COMPLEX*16 matrix, SA, to a COMPLEX matrix, A. RMAX is the overflow for the SINGLE PRECISION arithmetic ZLAG2C checks that all the entries of A are between -RMAX and RMAX. If not the conversion is aborted and a flag is raised. This is an auxiliary routine so there is no argument checking.

**Parameters**

*M*

M is INTEGER The number of lines of the matrix A. M >= 0.

*N*

N is INTEGER The number of columns of the matrix A. N >= 0.

*A*

A is COMPLEX*16 array, dimension (LDA,N) On entry, the M-by-N coefficient matrix A.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).

*SA*

SA is COMPLEX array, dimension (LDSA,N) On exit, if INFO=0, the M-by-N coefficient matrix SA; if INFO>0, the content of SA is unspecified.

*LDSA*

LDSA is INTEGER The leading dimension of the array SA. LDSA >= max(1,M).

*INFO*

INFO is INTEGER = 0: successful exit. = 1: an entry of the matrix A is greater than the SINGLE PRECISION overflow threshold, in this case, the content of SA in exit is unspecified.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlags2 (logical UPPER, double precision A1, complex*16 A2, double precision A3, double precision B1, complex*16 B2, double precision B3, double precision CSU, complex*16 SNU, double precision CSV, complex*16 SNV, double precision CSQ, complex*16 SNQ)¶

**ZLAGS2**

**Purpose:**

ZLAGS2 computes 2-by-2 unitary matrices U, V and Q, such that if ( UPPER ) then U**H *A*Q = U**H *( A1 A2 )*Q = ( x 0 ) ( 0 A3 ) ( x x ) and V**H*B*Q = V**H *( B1 B2 )*Q = ( x 0 ) ( 0 B3 ) ( x x ) or if ( .NOT.UPPER ) then U**H *A*Q = U**H *( A1 0 )*Q = ( x x ) ( A2 A3 ) ( 0 x ) and V**H *B*Q = V**H *( B1 0 )*Q = ( x x ) ( B2 B3 ) ( 0 x ) where U = ( CSU SNU ), V = ( CSV SNV ), ( -SNU**H CSU ) ( -SNV**H CSV ) Q = ( CSQ SNQ ) ( -SNQ**H CSQ ) The rows of the transformed A and B are parallel. Moreover, if the input 2-by-2 matrix A is not zero, then the transformed (1,1) entry of A is not zero. If the input matrices A and B are both not zero, then the transformed (2,2) element of B is not zero, except when the first rows of input A and B are parallel and the second rows are zero.

**Parameters**

*UPPER*

UPPER is LOGICAL = .TRUE.: the input matrices A and B are upper triangular. = .FALSE.: the input matrices A and B are lower triangular.

*A1*

A1 is DOUBLE PRECISION

*A2*

A2 is COMPLEX*16

*A3*

A3 is DOUBLE PRECISION On entry, A1, A2 and A3 are elements of the input 2-by-2 upper (lower) triangular matrix A.

*B1*

B1 is DOUBLE PRECISION

*B2*

B2 is COMPLEX*16

*B3*

B3 is DOUBLE PRECISION On entry, B1, B2 and B3 are elements of the input 2-by-2 upper (lower) triangular matrix B.

*CSU*

CSU is DOUBLE PRECISION

*SNU*

SNU is COMPLEX*16 The desired unitary matrix U.

*CSV*

CSV is DOUBLE PRECISION

*SNV*

SNV is COMPLEX*16 The desired unitary matrix V.

*CSQ*

CSQ is DOUBLE PRECISION

*SNQ*

SNQ is COMPLEX*16 The desired unitary matrix Q.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlagtm (character TRANS, integer N, integer NRHS, double precision ALPHA, complex*16, dimension( * ) DL, complex*16, dimension( * ) D, complex*16, dimension( * ) DU, complex*16, dimension( ldx, * ) X, integer LDX, double precision BETA, complex*16, dimension( ldb, * ) B, integer LDB)¶

**ZLAGTM**performs a matrix-matrix product of the form C = αAB+βC, where A is a tridiagonal matrix, B and C are rectangular matrices, and α and β are scalars, which may be 0, 1, or -1.

**Purpose:**

ZLAGTM performs a matrix-vector product of the form B := alpha * A * X + beta * B where A is a tridiagonal matrix of order N, B and X are N by NRHS matrices, and alpha and beta are real scalars, each of which may be 0., 1., or -1.

**Parameters**

*TRANS*

TRANS is CHARACTER*1 Specifies the operation applied to A. = 'N': No transpose, B := alpha * A * X + beta * B = 'T': Transpose, B := alpha * A**T * X + beta * B = 'C': Conjugate transpose, B := alpha * A**H * X + beta * B

*N*

N is INTEGER The order of the matrix A. N >= 0.

*NRHS*

NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices X and B.

*ALPHA*

ALPHA is DOUBLE PRECISION The scalar alpha. ALPHA must be 0., 1., or -1.; otherwise, it is assumed to be 0.

*DL*

DL is COMPLEX*16 array, dimension (N-1) The (n-1) sub-diagonal elements of T.

*D*

D is COMPLEX*16 array, dimension (N) The diagonal elements of T.

*DU*

DU is COMPLEX*16 array, dimension (N-1) The (n-1) super-diagonal elements of T.

*X*

X is COMPLEX*16 array, dimension (LDX,NRHS) The N by NRHS matrix X.

*LDX*

LDX is INTEGER The leading dimension of the array X. LDX >= max(N,1).

*BETA*

BETA is DOUBLE PRECISION The scalar beta. BETA must be 0., 1., or -1.; otherwise, it is assumed to be 1.

*B*

B is COMPLEX*16 array, dimension (LDB,NRHS) On entry, the N by NRHS matrix B. On exit, B is overwritten by the matrix expression B := alpha * A * X + beta * B.

*LDB*

LDB is INTEGER The leading dimension of the array B. LDB >= max(N,1).

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlahqr (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI, complex*16, dimension( ldh, * ) H, integer LDH, complex*16, dimension( * ) W, integer ILOZ, integer IHIZ, complex*16, dimension( ldz, * ) Z, integer LDZ, integer INFO)¶

**ZLAHQR**computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.

**Purpose:**

ZLAHQR is an auxiliary routine called by CHSEQR to update the eigenvalues and Schur decomposition already computed by CHSEQR, by dealing with the Hessenberg submatrix in rows and columns ILO to IHI.

**Parameters**

*WANTT*

WANTT is LOGICAL = .TRUE. : the full Schur form T is required; = .FALSE.: only eigenvalues are required.

*WANTZ*

WANTZ is LOGICAL = .TRUE. : the matrix of Schur vectors Z is required; = .FALSE.: Schur vectors are not required.

*N*

N is INTEGER The order of the matrix H. N >= 0.

*ILO*

ILO is INTEGER

*IHI*

IHI is INTEGER It is assumed that H is already upper triangular in rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless ILO = 1). ZLAHQR works primarily with the Hessenberg submatrix in rows and columns ILO to IHI, but applies transformations to all of H if WANTT is .TRUE.. 1 <= ILO <= max(1,IHI); IHI <= N.

*H*

H is COMPLEX*16 array, dimension (LDH,N) On entry, the upper Hessenberg matrix H. On exit, if INFO is zero and if WANTT is .TRUE., then H is upper triangular in rows and columns ILO:IHI. If INFO is zero and if WANTT is .FALSE., then the contents of H are unspecified on exit. The output state of H in case INF is positive is below under the description of INFO.

*LDH*

LDH is INTEGER The leading dimension of the array H. LDH >= max(1,N).

*W*

W is COMPLEX*16 array, dimension (N) The computed eigenvalues ILO to IHI are stored in the corresponding elements of W. If WANTT is .TRUE., the eigenvalues are stored in the same order as on the diagonal of the Schur form returned in H, with W(i) = H(i,i).

*ILOZ*

ILOZ is INTEGER

*IHIZ*

IHIZ is INTEGER Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 <= ILOZ <= ILO; IHI <= IHIZ <= N.

*Z*

Z is COMPLEX*16 array, dimension (LDZ,N) If WANTZ is .TRUE., on entry Z must contain the current matrix Z of transformations accumulated by CHSEQR, and on exit Z has been updated; transformations are applied only to the submatrix Z(ILOZ:IHIZ,ILO:IHI). If WANTZ is .FALSE., Z is not referenced.

*LDZ*

LDZ is INTEGER The leading dimension of the array Z. LDZ >= max(1,N).

*INFO*

INFO is INTEGER = 0: successful exit > 0: if INFO = i, ZLAHQR failed to compute all the eigenvalues ILO to IHI in a total of 30 iterations per eigenvalue; elements i+1:ihi of W contain those eigenvalues which have been successfully computed. If INFO > 0 and WANTT is .FALSE., then on exit, the remaining unconverged eigenvalues are the eigenvalues of the upper Hessenberg matrix rows and columns ILO through INFO of the final, output value of H. If INFO > 0 and WANTT is .TRUE., then on exit (*) (initial value of H)*U = U*(final value of H) where U is an orthogonal matrix. The final value of H is upper Hessenberg and triangular in rows and columns INFO+1 through IHI. If INFO > 0 and WANTZ is .TRUE., then on exit (final value of Z) = (initial value of Z)*U where U is the orthogonal matrix in (*) (regardless of the value of WANTT.)

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

02-96 Based on modifications by David Day, Sandia National Laboratory, USA 12-04 Further modifications by Ralph Byers, University of Kansas, USA This is a modified version of ZLAHQR from LAPACK version 3.0. It is (1) more robust against overflow and underflow and (2) adopts the more conservative Ahues & Tisseur stopping criterion (LAWN 122, 1997).

## subroutine zlahr2 (integer N, integer K, integer NB, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( nb ) TAU, complex*16, dimension( ldt, nb ) T, integer LDT, complex*16, dimension( ldy, nb ) Y, integer LDY)¶

**ZLAHR2**reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.

**Purpose:**

ZLAHR2 reduces the first NB columns of A complex general n-BY-(n-k+1) matrix A so that elements below the k-th subdiagonal are zero. The reduction is performed by an unitary similarity transformation Q**H * A * Q. The routine returns the matrices V and T which determine Q as a block reflector I - V*T*V**H, and also the matrix Y = A * V * T. This is an auxiliary routine called by ZGEHRD.

**Parameters**

*N*

N is INTEGER The order of the matrix A.

*K*

K is INTEGER The offset for the reduction. Elements below the k-th subdiagonal in the first NB columns are reduced to zero. K < N.

*NB*

NB is INTEGER The number of columns to be reduced.

*A*

A is COMPLEX*16 array, dimension (LDA,N-K+1) On entry, the n-by-(n-k+1) general matrix A. On exit, the elements on and above the k-th subdiagonal in the first NB columns are overwritten with the corresponding elements of the reduced matrix; the elements below the k-th subdiagonal, with the array TAU, represent the matrix Q as a product of elementary reflectors. The other columns of A are unchanged. See Further Details.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*TAU*

TAU is COMPLEX*16 array, dimension (NB) The scalar factors of the elementary reflectors. See Further Details.

*T*

T is COMPLEX*16 array, dimension (LDT,NB) The upper triangular matrix T.

*LDT*

LDT is INTEGER The leading dimension of the array T. LDT >= NB.

*Y*

Y is COMPLEX*16 array, dimension (LDY,NB) The n-by-nb matrix Y.

*LDY*

LDY is INTEGER The leading dimension of the array Y. LDY >= N.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

The matrix Q is represented as a product of nb elementary reflectors Q = H(1) H(2) . . . H(nb). Each H(i) has the form H(i) = I - tau * v * v**H where tau is a complex scalar, and v is a complex vector with v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in A(i+k+1:n,i), and tau in TAU(i). The elements of the vectors v together form the (n-k+1)-by-nb matrix V which is needed, with T and Y, to apply the transformation to the unreduced part of the matrix, using an update of the form: A := (I - V*T*V**H) * (A - Y*V**H). The contents of A on exit are illustrated by the following example with n = 7, k = 3 and nb = 2: ( a a a a a ) ( a a a a a ) ( a a a a a ) ( h h a a a ) ( v1 h a a a ) ( v1 v2 a a a ) ( v1 v2 a a a ) where a denotes an element of the original matrix A, h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i). This subroutine is a slight modification of LAPACK-3.0's ZLAHRD incorporating improvements proposed by Quintana-Orti and Van de Gejin. Note that the entries of A(1:K,2:NB) differ from those returned by the original LAPACK-3.0's ZLAHRD routine. (This subroutine is not backward compatible with LAPACK-3.0's ZLAHRD.)

**References:**

## subroutine zlaic1 (integer JOB, integer J, complex*16, dimension( j ) X, double precision SEST, complex*16, dimension( j ) W, complex*16 GAMMA, double precision SESTPR, complex*16 S, complex*16 C)¶

**ZLAIC1**applies one step of incremental condition estimation.

**Purpose:**

ZLAIC1 applies one step of incremental condition estimation in its simplest version: Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j lower triangular matrix L, such that twonorm(L*x) = sest Then ZLAIC1 computes sestpr, s, c such that the vector [ s*x ] xhat = [ c ] is an approximate singular vector of [ L 0 ] Lhat = [ w**H gamma ] in the sense that twonorm(Lhat*xhat) = sestpr. Depending on JOB, an estimate for the largest or smallest singular value is computed. Note that [s c]**H and sestpr**2 is an eigenpair of the system diag(sest*sest, 0) + [alpha gamma] * [ conjg(alpha) ] [ conjg(gamma) ] where alpha = x**H * w.

**Parameters**

*JOB*

JOB is INTEGER = 1: an estimate for the largest singular value is computed. = 2: an estimate for the smallest singular value is computed.

*J*

J is INTEGER Length of X and W

*X*

X is COMPLEX*16 array, dimension (J) The j-vector x.

*SEST*

SEST is DOUBLE PRECISION Estimated singular value of j by j matrix L

*W*

W is COMPLEX*16 array, dimension (J) The j-vector w.

*GAMMA*

GAMMA is COMPLEX*16 The diagonal element gamma.

*SESTPR*

SESTPR is DOUBLE PRECISION Estimated singular value of (j+1) by (j+1) matrix Lhat.

*S*

S is COMPLEX*16 Sine needed in forming xhat.

*C*

C is COMPLEX*16 Cosine needed in forming xhat.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## double precision function zlangt (character NORM, integer N, complex*16, dimension( * ) DL, complex*16, dimension( * ) D, complex*16, dimension( * ) DU)¶

**ZLANGT**returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general tridiagonal matrix.

**Purpose:**

ZLANGT returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex tridiagonal matrix A.

**Returns**

ZLANGT = ( max(abs(A(i,j))), NORM = 'M' or 'm' ( ( norm1(A), NORM = '1', 'O' or 'o' ( ( normI(A), NORM = 'I' or 'i' ( ( normF(A), NORM = 'F', 'f', 'E' or 'e' where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.

**Parameters**

*NORM*

NORM is CHARACTER*1 Specifies the value to be returned in ZLANGT as described above.

*N*

N is INTEGER The order of the matrix A. N >= 0. When N = 0, ZLANGT is set to zero.

*DL*

DL is COMPLEX*16 array, dimension (N-1) The (n-1) sub-diagonal elements of A.

*D*

D is COMPLEX*16 array, dimension (N) The diagonal elements of A.

*DU*

DU is COMPLEX*16 array, dimension (N-1) The (n-1) super-diagonal elements of A.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## double precision function zlanhb (character NORM, character UPLO, integer N, integer K, complex*16, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) WORK)¶

**ZLANHB**returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hermitian band matrix.

**Purpose:**

ZLANHB returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of an n by n hermitian band matrix A, with k super-diagonals.

**Returns**

ZLANHB = ( max(abs(A(i,j))), NORM = 'M' or 'm' ( ( norm1(A), NORM = '1', 'O' or 'o' ( ( normI(A), NORM = 'I' or 'i' ( ( normF(A), NORM = 'F', 'f', 'E' or 'e' where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.

**Parameters**

*NORM*

NORM is CHARACTER*1 Specifies the value to be returned in ZLANHB as described above.

*UPLO*

UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the band matrix A is supplied. = 'U': Upper triangular = 'L': Lower triangular

*N*

N is INTEGER The order of the matrix A. N >= 0. When N = 0, ZLANHB is set to zero.

*K*

K is INTEGER The number of super-diagonals or sub-diagonals of the band matrix A. K >= 0.

*AB*

AB is COMPLEX*16 array, dimension (LDAB,N) The upper or lower triangle of the hermitian band matrix A, stored in the first K+1 rows of AB. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k). Note that the imaginary parts of the diagonal elements need not be set and are assumed to be zero.

*LDAB*

LDAB is INTEGER The leading dimension of the array AB. LDAB >= K+1.

*WORK*

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)), where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise, WORK is not referenced.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## double precision function zlanhp (character NORM, character UPLO, integer N, complex*16, dimension( * ) AP, double precision, dimension( * ) WORK)¶

**ZLANHP**returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex Hermitian matrix supplied in packed form.

**Purpose:**

ZLANHP returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex hermitian matrix A, supplied in packed form.

**Returns**

ZLANHP = ( max(abs(A(i,j))), NORM = 'M' or 'm' ( ( norm1(A), NORM = '1', 'O' or 'o' ( ( normI(A), NORM = 'I' or 'i' ( ( normF(A), NORM = 'F', 'f', 'E' or 'e' where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.

**Parameters**

*NORM*

NORM is CHARACTER*1 Specifies the value to be returned in ZLANHP as described above.

*UPLO*

UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the hermitian matrix A is supplied. = 'U': Upper triangular part of A is supplied = 'L': Lower triangular part of A is supplied

*N*

N is INTEGER The order of the matrix A. N >= 0. When N = 0, ZLANHP is set to zero.

*AP*

AP is COMPLEX*16 array, dimension (N*(N+1)/2) The upper or lower triangle of the hermitian matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. Note that the imaginary parts of the diagonal elements need not be set and are assumed to be zero.

*WORK*

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)), where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise, WORK is not referenced.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## double precision function zlanhs (character NORM, integer N, complex*16, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) WORK)¶

**ZLANHS**returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of an upper Hessenberg matrix.

**Purpose:**

ZLANHS returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hessenberg matrix A.

**Returns**

ZLANHS = ( max(abs(A(i,j))), NORM = 'M' or 'm' ( ( norm1(A), NORM = '1', 'O' or 'o' ( ( normI(A), NORM = 'I' or 'i' ( ( normF(A), NORM = 'F', 'f', 'E' or 'e' where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.

**Parameters**

*NORM*

NORM is CHARACTER*1 Specifies the value to be returned in ZLANHS as described above.

*N*

N is INTEGER The order of the matrix A. N >= 0. When N = 0, ZLANHS is set to zero.

*A*

A is COMPLEX*16 array, dimension (LDA,N) The n by n upper Hessenberg matrix A; the part of A below the first sub-diagonal is not referenced.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(N,1).

*WORK*

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)), where LWORK >= N when NORM = 'I'; otherwise, WORK is not referenced.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## double precision function zlanht (character NORM, integer N, double precision, dimension( * ) D, complex*16, dimension( * ) E)¶

**ZLANHT**returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex Hermitian tridiagonal matrix.

**Purpose:**

ZLANHT returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex Hermitian tridiagonal matrix A.

**Returns**

ZLANHT = ( max(abs(A(i,j))), NORM = 'M' or 'm' ( ( norm1(A), NORM = '1', 'O' or 'o' ( ( normI(A), NORM = 'I' or 'i' ( ( normF(A), NORM = 'F', 'f', 'E' or 'e' where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.

**Parameters**

*NORM*

NORM is CHARACTER*1 Specifies the value to be returned in ZLANHT as described above.

*N*

N is INTEGER The order of the matrix A. N >= 0. When N = 0, ZLANHT is set to zero.

*D*

D is DOUBLE PRECISION array, dimension (N) The diagonal elements of A.

*E*

E is COMPLEX*16 array, dimension (N-1) The (n-1) sub-diagonal or super-diagonal elements of A.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## double precision function zlansb (character NORM, character UPLO, integer N, integer K, complex*16, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) WORK)¶

**ZLANSB**returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric band matrix.

**Purpose:**

ZLANSB returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of an n by n symmetric band matrix A, with k super-diagonals.

**Returns**

ZLANSB = ( max(abs(A(i,j))), NORM = 'M' or 'm' ( ( norm1(A), NORM = '1', 'O' or 'o' ( ( normI(A), NORM = 'I' or 'i' ( ( normF(A), NORM = 'F', 'f', 'E' or 'e' where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.

**Parameters**

*NORM*

NORM is CHARACTER*1 Specifies the value to be returned in ZLANSB as described above.

*UPLO*

UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the band matrix A is supplied. = 'U': Upper triangular part is supplied = 'L': Lower triangular part is supplied

*N*

N is INTEGER The order of the matrix A. N >= 0. When N = 0, ZLANSB is set to zero.

*K*

K is INTEGER The number of super-diagonals or sub-diagonals of the band matrix A. K >= 0.

*AB*

AB is COMPLEX*16 array, dimension (LDAB,N) The upper or lower triangle of the symmetric band matrix A, stored in the first K+1 rows of AB. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k).

*LDAB*

LDAB is INTEGER The leading dimension of the array AB. LDAB >= K+1.

*WORK*

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)), where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise, WORK is not referenced.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## double precision function zlansp (character NORM, character UPLO, integer N, complex*16, dimension( * ) AP, double precision, dimension( * ) WORK)¶

**ZLANSP**returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric matrix supplied in packed form.

**Purpose:**

ZLANSP returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex symmetric matrix A, supplied in packed form.

**Returns**

ZLANSP = ( max(abs(A(i,j))), NORM = 'M' or 'm' ( ( norm1(A), NORM = '1', 'O' or 'o' ( ( normI(A), NORM = 'I' or 'i' ( ( normF(A), NORM = 'F', 'f', 'E' or 'e' where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.

**Parameters**

*NORM*

NORM is CHARACTER*1 Specifies the value to be returned in ZLANSP as described above.

*UPLO*

UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is supplied. = 'U': Upper triangular part of A is supplied = 'L': Lower triangular part of A is supplied

*N*

N is INTEGER The order of the matrix A. N >= 0. When N = 0, ZLANSP is set to zero.

*AP*

AP is COMPLEX*16 array, dimension (N*(N+1)/2) The upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

*WORK*

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## double precision function zlantb (character NORM, character UPLO, character DIAG, integer N, integer K, complex*16, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) WORK)¶

**ZLANTB**returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular band matrix.

**Purpose:**

ZLANTB returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of an n by n triangular band matrix A, with ( k + 1 ) diagonals.

**Returns**

ZLANTB = ( max(abs(A(i,j))), NORM = 'M' or 'm' ( ( norm1(A), NORM = '1', 'O' or 'o' ( ( normI(A), NORM = 'I' or 'i' ( ( normF(A), NORM = 'F', 'f', 'E' or 'e' where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.

**Parameters**

*NORM*

NORM is CHARACTER*1 Specifies the value to be returned in ZLANTB as described above.

*UPLO*

UPLO is CHARACTER*1 Specifies whether the matrix A is upper or lower triangular. = 'U': Upper triangular = 'L': Lower triangular

*DIAG*

DIAG is CHARACTER*1 Specifies whether or not the matrix A is unit triangular. = 'N': Non-unit triangular = 'U': Unit triangular

*N*

N is INTEGER The order of the matrix A. N >= 0. When N = 0, ZLANTB is set to zero.

*K*

K is INTEGER The number of super-diagonals of the matrix A if UPLO = 'U', or the number of sub-diagonals of the matrix A if UPLO = 'L'. K >= 0.

*AB*

AB is COMPLEX*16 array, dimension (LDAB,N) The upper or lower triangular band matrix A, stored in the first k+1 rows of AB. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k). Note that when DIAG = 'U', the elements of the array AB corresponding to the diagonal elements of the matrix A are not referenced, but are assumed to be one.

*LDAB*

LDAB is INTEGER The leading dimension of the array AB. LDAB >= K+1.

*WORK*

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)), where LWORK >= N when NORM = 'I'; otherwise, WORK is not referenced.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## double precision function zlantp (character NORM, character UPLO, character DIAG, integer N, complex*16, dimension( * ) AP, double precision, dimension( * ) WORK)¶

**ZLANTP**returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular matrix supplied in packed form.

**Purpose:**

ZLANTP returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular matrix A, supplied in packed form.

**Returns**

ZLANTP = ( max(abs(A(i,j))), NORM = 'M' or 'm' ( ( norm1(A), NORM = '1', 'O' or 'o' ( ( normI(A), NORM = 'I' or 'i' ( ( normF(A), NORM = 'F', 'f', 'E' or 'e' where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.

**Parameters**

*NORM*

NORM is CHARACTER*1 Specifies the value to be returned in ZLANTP as described above.

*UPLO*

UPLO is CHARACTER*1 Specifies whether the matrix A is upper or lower triangular. = 'U': Upper triangular = 'L': Lower triangular

*DIAG*

DIAG is CHARACTER*1 Specifies whether or not the matrix A is unit triangular. = 'N': Non-unit triangular = 'U': Unit triangular

*N*

N is INTEGER The order of the matrix A. N >= 0. When N = 0, ZLANTP is set to zero.

*AP*

AP is COMPLEX*16 array, dimension (N*(N+1)/2) The upper or lower triangular matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. Note that when DIAG = 'U', the elements of the array AP corresponding to the diagonal elements of the matrix A are not referenced, but are assumed to be one.

*WORK*

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)), where LWORK >= N when NORM = 'I'; otherwise, WORK is not referenced.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## double precision function zlantr (character NORM, character UPLO, character DIAG, integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) WORK)¶

**ZLANTR**returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix.

**Purpose:**

ZLANTR returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix A.

**Returns**

ZLANTR = ( max(abs(A(i,j))), NORM = 'M' or 'm' ( ( norm1(A), NORM = '1', 'O' or 'o' ( ( normI(A), NORM = 'I' or 'i' ( ( normF(A), NORM = 'F', 'f', 'E' or 'e' where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.

**Parameters**

*NORM*

NORM is CHARACTER*1 Specifies the value to be returned in ZLANTR as described above.

*UPLO*

UPLO is CHARACTER*1 Specifies whether the matrix A is upper or lower trapezoidal. = 'U': Upper trapezoidal = 'L': Lower trapezoidal Note that A is triangular instead of trapezoidal if M = N.

*DIAG*

DIAG is CHARACTER*1 Specifies whether or not the matrix A has unit diagonal. = 'N': Non-unit diagonal = 'U': Unit diagonal

*M*

M is INTEGER The number of rows of the matrix A. M >= 0, and if UPLO = 'U', M <= N. When M = 0, ZLANTR is set to zero.

*N*

N is INTEGER The number of columns of the matrix A. N >= 0, and if UPLO = 'L', N <= M. When N = 0, ZLANTR is set to zero.

*A*

A is COMPLEX*16 array, dimension (LDA,N) The trapezoidal matrix A (A is triangular if M = N). If UPLO = 'U', the leading m by n upper trapezoidal part of the array A contains the upper trapezoidal matrix, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading m by n lower trapezoidal part of the array A contains the lower trapezoidal matrix, and the strictly upper triangular part of A is not referenced. Note that when DIAG = 'U', the diagonal elements of A are not referenced and are assumed to be one.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(M,1).

*WORK*

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)), where LWORK >= M when NORM = 'I'; otherwise, WORK is not referenced.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlapll (integer N, complex*16, dimension( * ) X, integer INCX, complex*16, dimension( * ) Y, integer INCY, double precision SSMIN)¶

**ZLAPLL**measures the linear dependence of two vectors.

**Purpose:**

Given two column vectors X and Y, let A = ( X Y ). The subroutine first computes the QR factorization of A = Q*R, and then computes the SVD of the 2-by-2 upper triangular matrix R. The smaller singular value of R is returned in SSMIN, which is used as the measurement of the linear dependency of the vectors X and Y.

**Parameters**

*N*

N is INTEGER The length of the vectors X and Y.

*X*

X is COMPLEX*16 array, dimension (1+(N-1)*INCX) On entry, X contains the N-vector X. On exit, X is overwritten.

*INCX*

INCX is INTEGER The increment between successive elements of X. INCX > 0.

*Y*

Y is COMPLEX*16 array, dimension (1+(N-1)*INCY) On entry, Y contains the N-vector Y. On exit, Y is overwritten.

*INCY*

INCY is INTEGER The increment between successive elements of Y. INCY > 0.

*SSMIN*

SSMIN is DOUBLE PRECISION The smallest singular value of the N-by-2 matrix A = ( X Y ).

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlapmr (logical FORWRD, integer M, integer N, complex*16, dimension( ldx, * ) X, integer LDX, integer, dimension( * ) K)¶

**ZLAPMR**rearranges rows of a matrix as specified by a permutation vector.

**Purpose:**

ZLAPMR rearranges the rows of the M by N matrix X as specified by the permutation K(1),K(2),...,K(M) of the integers 1,...,M. If FORWRD = .TRUE., forward permutation: X(K(I),*) is moved X(I,*) for I = 1,2,...,M. If FORWRD = .FALSE., backward permutation: X(I,*) is moved to X(K(I),*) for I = 1,2,...,M.

**Parameters**

*FORWRD*

FORWRD is LOGICAL = .TRUE., forward permutation = .FALSE., backward permutation

*M*

M is INTEGER The number of rows of the matrix X. M >= 0.

*N*

N is INTEGER The number of columns of the matrix X. N >= 0.

*X*

X is COMPLEX*16 array, dimension (LDX,N) On entry, the M by N matrix X. On exit, X contains the permuted matrix X.

*LDX*

LDX is INTEGER The leading dimension of the array X, LDX >= MAX(1,M).

*K*

K is INTEGER array, dimension (M) On entry, K contains the permutation vector. K is used as internal workspace, but reset to its original value on output.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlapmt (logical FORWRD, integer M, integer N, complex*16, dimension( ldx, * ) X, integer LDX, integer, dimension( * ) K)¶

**ZLAPMT**performs a forward or backward permutation of the columns of a matrix.

**Purpose:**

ZLAPMT rearranges the columns of the M by N matrix X as specified by the permutation K(1),K(2),...,K(N) of the integers 1,...,N. If FORWRD = .TRUE., forward permutation: X(*,K(J)) is moved X(*,J) for J = 1,2,...,N. If FORWRD = .FALSE., backward permutation: X(*,J) is moved to X(*,K(J)) for J = 1,2,...,N.

**Parameters**

*FORWRD*

FORWRD is LOGICAL = .TRUE., forward permutation = .FALSE., backward permutation

*M*

M is INTEGER The number of rows of the matrix X. M >= 0.

*N*

N is INTEGER The number of columns of the matrix X. N >= 0.

*X*

X is COMPLEX*16 array, dimension (LDX,N) On entry, the M by N matrix X. On exit, X contains the permuted matrix X.

*LDX*

LDX is INTEGER The leading dimension of the array X, LDX >= MAX(1,M).

*K*

K is INTEGER array, dimension (N) On entry, K contains the permutation vector. K is used as internal workspace, but reset to its original value on output.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlaqhb (character UPLO, integer N, integer KD, complex*16, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) S, double precision SCOND, double precision AMAX, character EQUED)¶

**ZLAQHB**scales a Hermitian band matrix, using scaling factors computed by cpbequ.

**Purpose:**

ZLAQHB equilibrates a Hermitian band matrix A using the scaling factors in the vector S.

**Parameters**

*UPLO*

UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored. = 'U': Upper triangular = 'L': Lower triangular

*N*

N is INTEGER The order of the matrix A. N >= 0.

*KD*

KD is INTEGER The number of super-diagonals of the matrix A if UPLO = 'U', or the number of sub-diagonals if UPLO = 'L'. KD >= 0.

*AB*

AB is COMPLEX*16 array, dimension (LDAB,N) On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). On exit, if INFO = 0, the triangular factor U or L from the Cholesky factorization A = U**H *U or A = L*L**H of the band matrix A, in the same storage format as A.

*LDAB*

LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD+1.

*S*

S is DOUBLE PRECISION array, dimension (N) The scale factors for A.

*SCOND*

SCOND is DOUBLE PRECISION Ratio of the smallest S(i) to the largest S(i).

*AMAX*

AMAX is DOUBLE PRECISION Absolute value of largest matrix entry.

*EQUED*

EQUED is CHARACTER*1 Specifies whether or not equilibration was done. = 'N': No equilibration. = 'Y': Equilibration was done, i.e., A has been replaced by diag(S) * A * diag(S).

**Internal Parameters:**

THRESH is a threshold value used to decide if scaling should be done based on the ratio of the scaling factors. If SCOND < THRESH, scaling is done. LARGE and SMALL are threshold values used to decide if scaling should be done based on the absolute size of the largest matrix element. If AMAX > LARGE or AMAX < SMALL, scaling is done.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlaqhp (character UPLO, integer N, complex*16, dimension( * ) AP, double precision, dimension( * ) S, double precision SCOND, double precision AMAX, character EQUED)¶

**ZLAQHP**scales a Hermitian matrix stored in packed form.

**Purpose:**

ZLAQHP equilibrates a Hermitian matrix A using the scaling factors in the vector S.

**Parameters**

*UPLO*

UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored. = 'U': Upper triangular = 'L': Lower triangular

*N*

N is INTEGER The order of the matrix A. N >= 0.

*AP*

AP is COMPLEX*16 array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. On exit, the equilibrated matrix: diag(S) * A * diag(S), in the same storage format as A.

*S*

S is DOUBLE PRECISION array, dimension (N) The scale factors for A.

*SCOND*

SCOND is DOUBLE PRECISION Ratio of the smallest S(i) to the largest S(i).

*AMAX*

AMAX is DOUBLE PRECISION Absolute value of largest matrix entry.

*EQUED*

EQUED is CHARACTER*1 Specifies whether or not equilibration was done. = 'N': No equilibration. = 'Y': Equilibration was done, i.e., A has been replaced by diag(S) * A * diag(S).

**Internal Parameters:**

THRESH is a threshold value used to decide if scaling should be done based on the ratio of the scaling factors. If SCOND < THRESH, scaling is done. LARGE and SMALL are threshold values used to decide if scaling should be done based on the absolute size of the largest matrix element. If AMAX > LARGE or AMAX < SMALL, scaling is done.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlaqp2 (integer M, integer N, integer OFFSET, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) JPVT, complex*16, dimension( * ) TAU, double precision, dimension( * ) VN1, double precision, dimension( * ) VN2, complex*16, dimension( * ) WORK)¶

**ZLAQP2**computes a QR factorization with column pivoting of the matrix block.

**Purpose:**

ZLAQP2 computes a QR factorization with column pivoting of the block A(OFFSET+1:M,1:N). The block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.

**Parameters**

*M*

M is INTEGER The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER The number of columns of the matrix A. N >= 0.

*OFFSET*

OFFSET is INTEGER The number of rows of the matrix A that must be pivoted but no factorized. OFFSET >= 0.

*A*

A is COMPLEX*16 array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the upper triangle of block A(OFFSET+1:M,1:N) is the triangular factor obtained; the elements in block A(OFFSET+1:M,1:N) below the diagonal, together with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. Block A(1:OFFSET,1:N) has been accordingly pivoted, but no factorized.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).

*JPVT*

JPVT is INTEGER array, dimension (N) On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted to the front of A*P (a leading column); if JPVT(i) = 0, the i-th column of A is a free column. On exit, if JPVT(i) = k, then the i-th column of A*P was the k-th column of A.

*TAU*

TAU is COMPLEX*16 array, dimension (min(M,N)) The scalar factors of the elementary reflectors.

*VN1*

VN1 is DOUBLE PRECISION array, dimension (N) The vector with the partial column norms.

*VN2*

VN2 is DOUBLE PRECISION array, dimension (N) The vector with the exact column norms.

*WORK*

WORK is COMPLEX*16 array, dimension (N)

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

**References:**

## subroutine zlaqps (integer M, integer N, integer OFFSET, integer NB, integer KB, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) JPVT, complex*16, dimension( * ) TAU, double precision, dimension( * ) VN1, double precision, dimension( * ) VN2, complex*16, dimension( * ) AUXV, complex*16, dimension( ldf, * ) F, integer LDF)¶

**ZLAQPS**computes a step of QR factorization with column pivoting of a real m-by-n matrix A by using BLAS level 3.

**Purpose:**

ZLAQPS computes a step of QR factorization with column pivoting of a complex M-by-N matrix A by using Blas-3. It tries to factorize NB columns from A starting from the row OFFSET+1, and updates all of the matrix with Blas-3 xGEMM. In some cases, due to catastrophic cancellations, it cannot factorize NB columns. Hence, the actual number of factorized columns is returned in KB. Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.

**Parameters**

*M*

M is INTEGER The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER The number of columns of the matrix A. N >= 0

*OFFSET*

OFFSET is INTEGER The number of rows of A that have been factorized in previous steps.

*NB*

NB is INTEGER The number of columns to factorize.

*KB*

KB is INTEGER The number of columns actually factorized.

*A*

A is COMPLEX*16 array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, block A(OFFSET+1:M,1:KB) is the triangular factor obtained and block A(1:OFFSET,1:N) has been accordingly pivoted, but no factorized. The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has been updated.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).

*JPVT*

JPVT is INTEGER array, dimension (N) JPVT(I) = K <==> Column K of the full matrix A has been permuted into position I in AP.

*TAU*

TAU is COMPLEX*16 array, dimension (KB) The scalar factors of the elementary reflectors.

*VN1*

VN1 is DOUBLE PRECISION array, dimension (N) The vector with the partial column norms.

*VN2*

VN2 is DOUBLE PRECISION array, dimension (N) The vector with the exact column norms.

*AUXV*

AUXV is COMPLEX*16 array, dimension (NB) Auxiliary vector.

*F*

F is COMPLEX*16 array, dimension (LDF,NB) Matrix F**H = L * Y**H * A.

*LDF*

LDF is INTEGER The leading dimension of the array F. LDF >= max(1,N).

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

**References:**

## subroutine zlaqr0 (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI, complex*16, dimension( ldh, * ) H, integer LDH, complex*16, dimension( * ) W, integer ILOZ, integer IHIZ, complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)¶

**ZLAQR0**computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

**Purpose:**

ZLAQR0 computes the eigenvalues of a Hessenberg matrix H and, optionally, the matrices T and Z from the Schur decomposition H = Z T Z**H, where T is an upper triangular matrix (the Schur form), and Z is the unitary matrix of Schur vectors. Optionally Z may be postmultiplied into an input unitary matrix Q so that this routine can give the Schur factorization of a matrix A which has been reduced to the Hessenberg form H by the unitary matrix Q: A = Q*H*Q**H = (QZ)*H*(QZ)**H.

**Parameters**

*WANTT*

WANTT is LOGICAL = .TRUE. : the full Schur form T is required; = .FALSE.: only eigenvalues are required.

*WANTZ*

WANTZ is LOGICAL = .TRUE. : the matrix of Schur vectors Z is required; = .FALSE.: Schur vectors are not required.

*N*

N is INTEGER The order of the matrix H. N >= 0.

*ILO*

ILO is INTEGER

*IHI*

IHI is INTEGER It is assumed that H is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N and, if ILO > 1, H(ILO,ILO-1) is zero. ILO and IHI are normally set by a previous call to ZGEBAL, and then passed to ZGEHRD when the matrix output by ZGEBAL is reduced to Hessenberg form. Otherwise, ILO and IHI should be set to 1 and N, respectively. If N > 0, then 1 <= ILO <= IHI <= N. If N = 0, then ILO = 1 and IHI = 0.

*H*

H is COMPLEX*16 array, dimension (LDH,N) On entry, the upper Hessenberg matrix H. On exit, if INFO = 0 and WANTT is .TRUE., then H contains the upper triangular matrix T from the Schur decomposition (the Schur form). If INFO = 0 and WANT is .FALSE., then the contents of H are unspecified on exit. (The output value of H when INFO > 0 is given under the description of INFO below.) This subroutine may explicitly set H(i,j) = 0 for i > j and j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.

*LDH*

LDH is INTEGER The leading dimension of the array H. LDH >= max(1,N).

*W*

W is COMPLEX*16 array, dimension (N) The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored in W(ILO:IHI). If WANTT is .TRUE., then the eigenvalues are stored in the same order as on the diagonal of the Schur form returned in H, with W(i) = H(i,i).

*ILOZ*

ILOZ is INTEGER

*IHIZ*

IHIZ is INTEGER Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 <= ILOZ <= ILO; IHI <= IHIZ <= N.

*Z*

Z is COMPLEX*16 array, dimension (LDZ,IHI) If WANTZ is .FALSE., then Z is not referenced. If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the orthogonal Schur factor of H(ILO:IHI,ILO:IHI). (The output value of Z when INFO > 0 is given under the description of INFO below.)

*LDZ*

LDZ is INTEGER The leading dimension of the array Z. if WANTZ is .TRUE. then LDZ >= MAX(1,IHIZ). Otherwise, LDZ >= 1.

*WORK*

WORK is COMPLEX*16 array, dimension LWORK On exit, if LWORK = -1, WORK(1) returns an estimate of the optimal value for LWORK.

*LWORK*

LWORK is INTEGER The dimension of the array WORK. LWORK >= max(1,N) is sufficient, but LWORK typically as large as 6*N may be required for optimal performance. A workspace query to determine the optimal workspace size is recommended. If LWORK = -1, then ZLAQR0 does a workspace query. In this case, ZLAQR0 checks the input parameters and estimates the optimal workspace size for the given values of N, ILO and IHI. The estimate is returned in WORK(1). No error message related to LWORK is issued by XERBLA. Neither H nor Z are accessed.

*INFO*

INFO is INTEGER = 0: successful exit > 0: if INFO = i, ZLAQR0 failed to compute all of the eigenvalues. Elements 1:ilo-1 and i+1:n of WR and WI contain those eigenvalues which have been successfully computed. (Failures are rare.) If INFO > 0 and WANT is .FALSE., then on exit, the remaining unconverged eigenvalues are the eigen- values of the upper Hessenberg matrix rows and columns ILO through INFO of the final, output value of H. If INFO > 0 and WANTT is .TRUE., then on exit (*) (initial value of H)*U = U*(final value of H) where U is a unitary matrix. The final value of H is upper Hessenberg and triangular in rows and columns INFO+1 through IHI. If INFO > 0 and WANTZ is .TRUE., then on exit (final value of Z(ILO:IHI,ILOZ:IHIZ) = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U where U is the unitary matrix in (*) (regard- less of the value of WANTT.) If INFO > 0 and WANTZ is .FALSE., then Z is not accessed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

**References:**

K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume 23, pages 929--947, 2002.

K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.

## subroutine zlaqr1 (integer N, complex*16, dimension( ldh, * ) H, integer LDH, complex*16 S1, complex*16 S2, complex*16, dimension( * ) V)¶

**ZLAQR1**sets a scalar multiple of the first column of the product of 2-by-2 or 3-by-3 matrix H and specified shifts.

**Purpose:**

Given a 2-by-2 or 3-by-3 matrix H, ZLAQR1 sets v to a scalar multiple of the first column of the product (*) K = (H - s1*I)*(H - s2*I) scaling to avoid overflows and most underflows. This is useful for starting double implicit shift bulges in the QR algorithm.

**Parameters**

*N*

N is INTEGER Order of the matrix H. N must be either 2 or 3.

*H*

H is COMPLEX*16 array, dimension (LDH,N) The 2-by-2 or 3-by-3 matrix H in (*).

*LDH*

LDH is INTEGER The leading dimension of H as declared in the calling procedure. LDH >= N

*S1*

S1 is COMPLEX*16

*S2*

S2 is COMPLEX*16 S1 and S2 are the shifts defining K in (*) above.

*V*

V is COMPLEX*16 array, dimension (N) A scalar multiple of the first column of the matrix K in (*).

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

## subroutine zlaqr2 (logical WANTT, logical WANTZ, integer N, integer KTOP, integer KBOT, integer NW, complex*16, dimension( ldh, * ) H, integer LDH, integer ILOZ, integer IHIZ, complex*16, dimension( ldz, * ) Z, integer LDZ, integer NS, integer ND, complex*16, dimension( * ) SH, complex*16, dimension( ldv, * ) V, integer LDV, integer NH, complex*16, dimension( ldt, * ) T, integer LDT, integer NV, complex*16, dimension( ldwv, * ) WV, integer LDWV, complex*16, dimension( * ) WORK, integer LWORK)¶

**ZLAQR2**performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).

**Purpose:**

ZLAQR2 is identical to ZLAQR3 except that it avoids recursion by calling ZLAHQR instead of ZLAQR4. Aggressive early deflation: ZLAQR2 accepts as input an upper Hessenberg matrix H and performs an unitary similarity transformation designed to detect and deflate fully converged eigenvalues from a trailing principal submatrix. On output H has been over- written by a new Hessenberg matrix that is a perturbation of an unitary similarity transformation of H. It is to be hoped that the final version of H has many zero subdiagonal entries.

**Parameters**

*WANTT*

WANTT is LOGICAL If .TRUE., then the Hessenberg matrix H is fully updated so that the triangular Schur factor may be computed (in cooperation with the calling subroutine). If .FALSE., then only enough of H is updated to preserve the eigenvalues.

*WANTZ*

WANTZ is LOGICAL If .TRUE., then the unitary matrix Z is updated so so that the unitary Schur factor may be computed (in cooperation with the calling subroutine). If .FALSE., then Z is not referenced.

*N*

N is INTEGER The order of the matrix H and (if WANTZ is .TRUE.) the order of the unitary matrix Z.

*KTOP*

KTOP is INTEGER It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0. KBOT and KTOP together determine an isolated block along the diagonal of the Hessenberg matrix.

*KBOT*

KBOT is INTEGER It is assumed without a check that either KBOT = N or H(KBOT+1,KBOT)=0. KBOT and KTOP together determine an isolated block along the diagonal of the Hessenberg matrix.

*NW*

NW is INTEGER Deflation window size. 1 <= NW <= (KBOT-KTOP+1).

*H*

H is COMPLEX*16 array, dimension (LDH,N) On input the initial N-by-N section of H stores the Hessenberg matrix undergoing aggressive early deflation. On output H has been transformed by a unitary similarity transformation, perturbed, and the returned to Hessenberg form that (it is to be hoped) has some zero subdiagonal entries.

*LDH*

LDH is INTEGER Leading dimension of H just as declared in the calling subroutine. N <= LDH

*ILOZ*

ILOZ is INTEGER

*IHIZ*

IHIZ is INTEGER Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.

*Z*

Z is COMPLEX*16 array, dimension (LDZ,N) IF WANTZ is .TRUE., then on output, the unitary similarity transformation mentioned above has been accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right. If WANTZ is .FALSE., then Z is unreferenced.

*LDZ*

LDZ is INTEGER The leading dimension of Z just as declared in the calling subroutine. 1 <= LDZ.

*NS*

NS is INTEGER The number of unconverged (ie approximate) eigenvalues returned in SR and SI that may be used as shifts by the calling subroutine.

*ND*

ND is INTEGER The number of converged eigenvalues uncovered by this subroutine.

*SH*

SH is COMPLEX*16 array, dimension (KBOT) On output, approximate eigenvalues that may be used for shifts are stored in SH(KBOT-ND-NS+1) through SR(KBOT-ND). Converged eigenvalues are stored in SH(KBOT-ND+1) through SH(KBOT).

*V*

V is COMPLEX*16 array, dimension (LDV,NW) An NW-by-NW work array.

*LDV*

LDV is INTEGER The leading dimension of V just as declared in the calling subroutine. NW <= LDV

*NH*

NH is INTEGER The number of columns of T. NH >= NW.

*T*

T is COMPLEX*16 array, dimension (LDT,NW)

*LDT*

LDT is INTEGER The leading dimension of T just as declared in the calling subroutine. NW <= LDT

*NV*

NV is INTEGER The number of rows of work array WV available for workspace. NV >= NW.

*WV*

WV is COMPLEX*16 array, dimension (LDWV,NW)

*LDWV*

LDWV is INTEGER The leading dimension of W just as declared in the calling subroutine. NW <= LDV

*WORK*

WORK is COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) is set to an estimate of the optimal value of LWORK for the given values of N, NW, KTOP and KBOT.

*LWORK*

LWORK is INTEGER The dimension of the work array WORK. LWORK = 2*NW suffices, but greater efficiency may result from larger values of LWORK. If LWORK = -1, then a workspace query is assumed; ZLAQR2 only estimates the optimal workspace size for the given values of N, NW, KTOP and KBOT. The estimate is returned in WORK(1). No error message related to LWORK is issued by XERBLA. Neither H nor Z are accessed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

## subroutine zlaqr3 (logical WANTT, logical WANTZ, integer N, integer KTOP, integer KBOT, integer NW, complex*16, dimension( ldh, * ) H, integer LDH, integer ILOZ, integer IHIZ, complex*16, dimension( ldz, * ) Z, integer LDZ, integer NS, integer ND, complex*16, dimension( * ) SH, complex*16, dimension( ldv, * ) V, integer LDV, integer NH, complex*16, dimension( ldt, * ) T, integer LDT, integer NV, complex*16, dimension( ldwv, * ) WV, integer LDWV, complex*16, dimension( * ) WORK, integer LWORK)¶

**ZLAQR3**performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).

**Purpose:**

Aggressive early deflation: ZLAQR3 accepts as input an upper Hessenberg matrix H and performs an unitary similarity transformation designed to detect and deflate fully converged eigenvalues from a trailing principal submatrix. On output H has been over- written by a new Hessenberg matrix that is a perturbation of an unitary similarity transformation of H. It is to be hoped that the final version of H has many zero subdiagonal entries.

**Parameters**

*WANTT*

WANTT is LOGICAL If .TRUE., then the Hessenberg matrix H is fully updated so that the triangular Schur factor may be computed (in cooperation with the calling subroutine). If .FALSE., then only enough of H is updated to preserve the eigenvalues.

*WANTZ*

WANTZ is LOGICAL If .TRUE., then the unitary matrix Z is updated so so that the unitary Schur factor may be computed (in cooperation with the calling subroutine). If .FALSE., then Z is not referenced.

*N*

N is INTEGER The order of the matrix H and (if WANTZ is .TRUE.) the order of the unitary matrix Z.

*KTOP*

KTOP is INTEGER It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0. KBOT and KTOP together determine an isolated block along the diagonal of the Hessenberg matrix.

*KBOT*

KBOT is INTEGER It is assumed without a check that either KBOT = N or H(KBOT+1,KBOT)=0. KBOT and KTOP together determine an isolated block along the diagonal of the Hessenberg matrix.

*NW*

NW is INTEGER Deflation window size. 1 <= NW <= (KBOT-KTOP+1).

*H*

H is COMPLEX*16 array, dimension (LDH,N) On input the initial N-by-N section of H stores the Hessenberg matrix undergoing aggressive early deflation. On output H has been transformed by a unitary similarity transformation, perturbed, and the returned to Hessenberg form that (it is to be hoped) has some zero subdiagonal entries.

*LDH*

LDH is INTEGER Leading dimension of H just as declared in the calling subroutine. N <= LDH

*ILOZ*

ILOZ is INTEGER

*IHIZ*

IHIZ is INTEGER Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.

*Z*

Z is COMPLEX*16 array, dimension (LDZ,N) IF WANTZ is .TRUE., then on output, the unitary similarity transformation mentioned above has been accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right. If WANTZ is .FALSE., then Z is unreferenced.

*LDZ*

LDZ is INTEGER The leading dimension of Z just as declared in the calling subroutine. 1 <= LDZ.

*NS*

NS is INTEGER The number of unconverged (ie approximate) eigenvalues returned in SR and SI that may be used as shifts by the calling subroutine.

*ND*

ND is INTEGER The number of converged eigenvalues uncovered by this subroutine.

*SH*

SH is COMPLEX*16 array, dimension (KBOT) On output, approximate eigenvalues that may be used for shifts are stored in SH(KBOT-ND-NS+1) through SR(KBOT-ND). Converged eigenvalues are stored in SH(KBOT-ND+1) through SH(KBOT).

*V*

V is COMPLEX*16 array, dimension (LDV,NW) An NW-by-NW work array.

*LDV*

LDV is INTEGER The leading dimension of V just as declared in the calling subroutine. NW <= LDV

*NH*

NH is INTEGER The number of columns of T. NH >= NW.

*T*

T is COMPLEX*16 array, dimension (LDT,NW)

*LDT*

LDT is INTEGER The leading dimension of T just as declared in the calling subroutine. NW <= LDT

*NV*

NV is INTEGER The number of rows of work array WV available for workspace. NV >= NW.

*WV*

WV is COMPLEX*16 array, dimension (LDWV,NW)

*LDWV*

LDWV is INTEGER The leading dimension of W just as declared in the calling subroutine. NW <= LDV

*WORK*

WORK is COMPLEX*16 array, dimension (LWORK) On exit, WORK(1) is set to an estimate of the optimal value of LWORK for the given values of N, NW, KTOP and KBOT.

*LWORK*

LWORK is INTEGER The dimension of the work array WORK. LWORK = 2*NW suffices, but greater efficiency may result from larger values of LWORK. If LWORK = -1, then a workspace query is assumed; ZLAQR3 only estimates the optimal workspace size for the given values of N, NW, KTOP and KBOT. The estimate is returned in WORK(1). No error message related to LWORK is issued by XERBLA. Neither H nor Z are accessed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

## subroutine zlaqr4 (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI, complex*16, dimension( ldh, * ) H, integer LDH, complex*16, dimension( * ) W, integer ILOZ, integer IHIZ, complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)¶

**ZLAQR4**computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

**Purpose:**

ZLAQR4 implements one level of recursion for ZLAQR0. It is a complete implementation of the small bulge multi-shift QR algorithm. It may be called by ZLAQR0 and, for large enough deflation window size, it may be called by ZLAQR3. This subroutine is identical to ZLAQR0 except that it calls ZLAQR2 instead of ZLAQR3. ZLAQR4 computes the eigenvalues of a Hessenberg matrix H and, optionally, the matrices T and Z from the Schur decomposition H = Z T Z**H, where T is an upper triangular matrix (the Schur form), and Z is the unitary matrix of Schur vectors. Optionally Z may be postmultiplied into an input unitary matrix Q so that this routine can give the Schur factorization of a matrix A which has been reduced to the Hessenberg form H by the unitary matrix Q: A = Q*H*Q**H = (QZ)*H*(QZ)**H.

**Parameters**

*WANTT*

WANTT is LOGICAL = .TRUE. : the full Schur form T is required; = .FALSE.: only eigenvalues are required.

*WANTZ*

WANTZ is LOGICAL = .TRUE. : the matrix of Schur vectors Z is required; = .FALSE.: Schur vectors are not required.

*N*

N is INTEGER The order of the matrix H. N >= 0.

*ILO*

ILO is INTEGER

*IHI*

IHI is INTEGER It is assumed that H is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N and, if ILO > 1, H(ILO,ILO-1) is zero. ILO and IHI are normally set by a previous call to ZGEBAL, and then passed to ZGEHRD when the matrix output by ZGEBAL is reduced to Hessenberg form. Otherwise, ILO and IHI should be set to 1 and N, respectively. If N > 0, then 1 <= ILO <= IHI <= N. If N = 0, then ILO = 1 and IHI = 0.

*H*

H is COMPLEX*16 array, dimension (LDH,N) On entry, the upper Hessenberg matrix H. On exit, if INFO = 0 and WANTT is .TRUE., then H contains the upper triangular matrix T from the Schur decomposition (the Schur form). If INFO = 0 and WANT is .FALSE., then the contents of H are unspecified on exit. (The output value of H when INFO > 0 is given under the description of INFO below.) This subroutine may explicitly set H(i,j) = 0 for i > j and j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.

*LDH*

LDH is INTEGER The leading dimension of the array H. LDH >= max(1,N).

*W*

W is COMPLEX*16 array, dimension (N) The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored in W(ILO:IHI). If WANTT is .TRUE., then the eigenvalues are stored in the same order as on the diagonal of the Schur form returned in H, with W(i) = H(i,i).

*ILOZ*

ILOZ is INTEGER

*IHIZ*

IHIZ is INTEGER Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 <= ILOZ <= ILO; IHI <= IHIZ <= N.

*Z*

Z is COMPLEX*16 array, dimension (LDZ,IHI) If WANTZ is .FALSE., then Z is not referenced. If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the orthogonal Schur factor of H(ILO:IHI,ILO:IHI). (The output value of Z when INFO > 0 is given under the description of INFO below.)

*LDZ*

LDZ is INTEGER The leading dimension of the array Z. if WANTZ is .TRUE. then LDZ >= MAX(1,IHIZ). Otherwise, LDZ >= 1.

*WORK*

WORK is COMPLEX*16 array, dimension LWORK On exit, if LWORK = -1, WORK(1) returns an estimate of the optimal value for LWORK.

*LWORK*

LWORK is INTEGER The dimension of the array WORK. LWORK >= max(1,N) is sufficient, but LWORK typically as large as 6*N may be required for optimal performance. A workspace query to determine the optimal workspace size is recommended. If LWORK = -1, then ZLAQR4 does a workspace query. In this case, ZLAQR4 checks the input parameters and estimates the optimal workspace size for the given values of N, ILO and IHI. The estimate is returned in WORK(1). No error message related to LWORK is issued by XERBLA. Neither H nor Z are accessed.

*INFO*

INFO is INTEGER = 0: successful exit > 0: if INFO = i, ZLAQR4 failed to compute all of the eigenvalues. Elements 1:ilo-1 and i+1:n of WR and WI contain those eigenvalues which have been successfully computed. (Failures are rare.) If INFO > 0 and WANT is .FALSE., then on exit, the remaining unconverged eigenvalues are the eigen- values of the upper Hessenberg matrix rows and columns ILO through INFO of the final, output value of H. If INFO > 0 and WANTT is .TRUE., then on exit (*) (initial value of H)*U = U*(final value of H) where U is a unitary matrix. The final value of H is upper Hessenberg and triangular in rows and columns INFO+1 through IHI. If INFO > 0 and WANTZ is .TRUE., then on exit (final value of Z(ILO:IHI,ILOZ:IHIZ) = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U where U is the unitary matrix in (*) (regard- less of the value of WANTT.) If INFO > 0 and WANTZ is .FALSE., then Z is not accessed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

**References:**

K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume 23, pages 929--947, 2002.

K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.

## subroutine zlaqr5 (logical WANTT, logical WANTZ, integer KACC22, integer N, integer KTOP, integer KBOT, integer NSHFTS, complex*16, dimension( * ) S, complex*16, dimension( ldh, * ) H, integer LDH, integer ILOZ, integer IHIZ, complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( ldv, * ) V, integer LDV, complex*16, dimension( ldu, * ) U, integer LDU, integer NV, complex*16, dimension( ldwv, * ) WV, integer LDWV, integer NH, complex*16, dimension( ldwh, * ) WH, integer LDWH)¶

**ZLAQR5**performs a single small-bulge multi-shift QR sweep.

**Purpose:**

ZLAQR5, called by ZLAQR0, performs a single small-bulge multi-shift QR sweep.

**Parameters**

*WANTT*

WANTT is LOGICAL WANTT = .true. if the triangular Schur factor is being computed. WANTT is set to .false. otherwise.

*WANTZ*

WANTZ is LOGICAL WANTZ = .true. if the unitary Schur factor is being computed. WANTZ is set to .false. otherwise.

*KACC22*

KACC22 is INTEGER with value 0, 1, or 2. Specifies the computation mode of far-from-diagonal orthogonal updates. = 0: ZLAQR5 does not accumulate reflections and does not use matrix-matrix multiply to update far-from-diagonal matrix entries. = 1: ZLAQR5 accumulates reflections and uses matrix-matrix multiply to update the far-from-diagonal matrix entries. = 2: Same as KACC22 = 1. This option used to enable exploiting the 2-by-2 structure during matrix multiplications, but this is no longer supported.

*N*

N is INTEGER N is the order of the Hessenberg matrix H upon which this subroutine operates.

*KTOP*

KTOP is INTEGER

*KBOT*

KBOT is INTEGER These are the first and last rows and columns of an isolated diagonal block upon which the QR sweep is to be applied. It is assumed without a check that either KTOP = 1 or H(KTOP,KTOP-1) = 0 and either KBOT = N or H(KBOT+1,KBOT) = 0.

*NSHFTS*

NSHFTS is INTEGER NSHFTS gives the number of simultaneous shifts. NSHFTS must be positive and even.

*S*

S is COMPLEX*16 array, dimension (NSHFTS) S contains the shifts of origin that define the multi- shift QR sweep. On output S may be reordered.

*H*

H is COMPLEX*16 array, dimension (LDH,N) On input H contains a Hessenberg matrix. On output a multi-shift QR sweep with shifts SR(J)+i*SI(J) is applied to the isolated diagonal block in rows and columns KTOP through KBOT.

*LDH*

LDH is INTEGER LDH is the leading dimension of H just as declared in the calling procedure. LDH >= MAX(1,N).

*ILOZ*

ILOZ is INTEGER

*IHIZ*

IHIZ is INTEGER Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N

*Z*

Z is COMPLEX*16 array, dimension (LDZ,IHIZ) If WANTZ = .TRUE., then the QR Sweep unitary similarity transformation is accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right. If WANTZ = .FALSE., then Z is unreferenced.

*LDZ*

LDZ is INTEGER LDA is the leading dimension of Z just as declared in the calling procedure. LDZ >= N.

*V*

V is COMPLEX*16 array, dimension (LDV,NSHFTS/2)

*LDV*

LDV is INTEGER LDV is the leading dimension of V as declared in the calling procedure. LDV >= 3.

*U*

U is COMPLEX*16 array, dimension (LDU,2*NSHFTS)

*LDU*

LDU is INTEGER LDU is the leading dimension of U just as declared in the in the calling subroutine. LDU >= 2*NSHFTS.

*NV*

NV is INTEGER NV is the number of rows in WV agailable for workspace. NV >= 1.

*WV*

WV is COMPLEX*16 array, dimension (LDWV,2*NSHFTS)

*LDWV*

LDWV is INTEGER LDWV is the leading dimension of WV as declared in the in the calling subroutine. LDWV >= NV.

*NH*

NH is INTEGER NH is the number of columns in array WH available for workspace. NH >= 1.

*WH*

WH is COMPLEX*16 array, dimension (LDWH,NH)

*LDWH*

LDWH is INTEGER Leading dimension of WH just as declared in the calling procedure. LDWH >= 2*NSHFTS.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

Lars Karlsson, Daniel Kressner, and Bruno Lang

Thijs Steel, Department of Computer science, KU Leuven, Belgium

**References:**

Lars Karlsson, Daniel Kressner, and Bruno Lang, Optimally packed chains of bulges in multishift QR algorithms. ACM Trans. Math. Softw. 40, 2, Article 12 (February 2014).

## subroutine zlaqsb (character UPLO, integer N, integer KD, complex*16, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) S, double precision SCOND, double precision AMAX, character EQUED)¶

**ZLAQSB**scales a symmetric/Hermitian band matrix, using scaling factors computed by spbequ.

**Purpose:**

ZLAQSB equilibrates a symmetric band matrix A using the scaling factors in the vector S.

**Parameters**

*UPLO*

UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored. = 'U': Upper triangular = 'L': Lower triangular

*N*

N is INTEGER The order of the matrix A. N >= 0.

*KD*

KD is INTEGER The number of super-diagonals of the matrix A if UPLO = 'U', or the number of sub-diagonals if UPLO = 'L'. KD >= 0.

*AB*

AB is COMPLEX*16 array, dimension (LDAB,N) On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). On exit, if INFO = 0, the triangular factor U or L from the Cholesky factorization A = U**H *U or A = L*L**H of the band matrix A, in the same storage format as A.

*LDAB*

LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD+1.

*S*

S is DOUBLE PRECISION array, dimension (N) The scale factors for A.

*SCOND*

SCOND is DOUBLE PRECISION Ratio of the smallest S(i) to the largest S(i).

*AMAX*

AMAX is DOUBLE PRECISION Absolute value of largest matrix entry.

*EQUED*

EQUED is CHARACTER*1 Specifies whether or not equilibration was done. = 'N': No equilibration. = 'Y': Equilibration was done, i.e., A has been replaced by diag(S) * A * diag(S).

**Internal Parameters:**

THRESH is a threshold value used to decide if scaling should be done based on the ratio of the scaling factors. If SCOND < THRESH, scaling is done. LARGE and SMALL are threshold values used to decide if scaling should be done based on the absolute size of the largest matrix element. If AMAX > LARGE or AMAX < SMALL, scaling is done.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlaqsp (character UPLO, integer N, complex*16, dimension( * ) AP, double precision, dimension( * ) S, double precision SCOND, double precision AMAX, character EQUED)¶

**ZLAQSP**scales a symmetric/Hermitian matrix in packed storage, using scaling factors computed by sppequ.

**Purpose:**

ZLAQSP equilibrates a symmetric matrix A using the scaling factors in the vector S.

**Parameters**

*UPLO*

UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored. = 'U': Upper triangular = 'L': Lower triangular

*N*

N is INTEGER The order of the matrix A. N >= 0.

*AP*

AP is COMPLEX*16 array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. On exit, the equilibrated matrix: diag(S) * A * diag(S), in the same storage format as A.

*S*

S is DOUBLE PRECISION array, dimension (N) The scale factors for A.

*SCOND*

SCOND is DOUBLE PRECISION Ratio of the smallest S(i) to the largest S(i).

*AMAX*

AMAX is DOUBLE PRECISION Absolute value of largest matrix entry.

*EQUED*

**Internal Parameters:**

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlar1v (integer N, integer B1, integer BN, double precision LAMBDA, double precision, dimension( * ) D, double precision, dimension( * ) L, double precision, dimension( * ) LD, double precision, dimension( * ) LLD, double precision PIVMIN, double precision GAPTOL, complex*16, dimension( * ) Z, logical WANTNC, integer NEGCNT, double precision ZTZ, double precision MINGMA, integer R, integer, dimension( * ) ISUPPZ, double precision NRMINV, double precision RESID, double precision RQCORR, double precision, dimension( * ) WORK)¶

**ZLAR1V**computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI.

**Purpose:**

ZLAR1V computes the (scaled) r-th column of the inverse of the sumbmatrix in rows B1 through BN of the tridiagonal matrix L D L**T - sigma I. When sigma is close to an eigenvalue, the computed vector is an accurate eigenvector. Usually, r corresponds to the index where the eigenvector is largest in magnitude. The following steps accomplish this computation : (a) Stationary qd transform, L D L**T - sigma I = L(+) D(+) L(+)**T, (b) Progressive qd transform, L D L**T - sigma I = U(-) D(-) U(-)**T, (c) Computation of the diagonal elements of the inverse of L D L**T - sigma I by combining the above transforms, and choosing r as the index where the diagonal of the inverse is (one of the) largest in magnitude. (d) Computation of the (scaled) r-th column of the inverse using the twisted factorization obtained by combining the top part of the the stationary and the bottom part of the progressive transform.

**Parameters**

*N*

N is INTEGER The order of the matrix L D L**T.

*B1*

B1 is INTEGER First index of the submatrix of L D L**T.

*BN*

BN is INTEGER Last index of the submatrix of L D L**T.

*LAMBDA*

LAMBDA is DOUBLE PRECISION The shift. In order to compute an accurate eigenvector, LAMBDA should be a good approximation to an eigenvalue of L D L**T.

*L*

L is DOUBLE PRECISION array, dimension (N-1) The (n-1) subdiagonal elements of the unit bidiagonal matrix L, in elements 1 to N-1.

*D*

D is DOUBLE PRECISION array, dimension (N) The n diagonal elements of the diagonal matrix D.

*LD*

LD is DOUBLE PRECISION array, dimension (N-1) The n-1 elements L(i)*D(i).

*LLD*

LLD is DOUBLE PRECISION array, dimension (N-1) The n-1 elements L(i)*L(i)*D(i).

*PIVMIN*

PIVMIN is DOUBLE PRECISION The minimum pivot in the Sturm sequence.

*GAPTOL*

GAPTOL is DOUBLE PRECISION Tolerance that indicates when eigenvector entries are negligible w.r.t. their contribution to the residual.

*Z*

Z is COMPLEX*16 array, dimension (N) On input, all entries of Z must be set to 0. On output, Z contains the (scaled) r-th column of the inverse. The scaling is such that Z(R) equals 1.

*WANTNC*

WANTNC is LOGICAL Specifies whether NEGCNT has to be computed.

*NEGCNT*

NEGCNT is INTEGER If WANTNC is .TRUE. then NEGCNT = the number of pivots < pivmin in the matrix factorization L D L**T, and NEGCNT = -1 otherwise.

*ZTZ*

ZTZ is DOUBLE PRECISION The square of the 2-norm of Z.

*MINGMA*

MINGMA is DOUBLE PRECISION The reciprocal of the largest (in magnitude) diagonal element of the inverse of L D L**T - sigma I.

*R*

R is INTEGER The twist index for the twisted factorization used to compute Z. On input, 0 <= R <= N. If R is input as 0, R is set to the index where (L D L**T - sigma I)^{-1} is largest in magnitude. If 1 <= R <= N, R is unchanged. On output, R contains the twist index used to compute Z. Ideally, R designates the position of the maximum entry in the eigenvector.

*ISUPPZ*

ISUPPZ is INTEGER array, dimension (2) The support of the vector in Z, i.e., the vector Z is nonzero only in elements ISUPPZ(1) through ISUPPZ( 2 ).

*NRMINV*

NRMINV is DOUBLE PRECISION NRMINV = 1/SQRT( ZTZ )

*RESID*

RESID is DOUBLE PRECISION The residual of the FP vector. RESID = ABS( MINGMA )/SQRT( ZTZ )

*RQCORR*

RQCORR is DOUBLE PRECISION The Rayleigh Quotient correction to LAMBDA. RQCORR = MINGMA*TMP

*WORK*

WORK is DOUBLE PRECISION array, dimension (4*N)

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

## subroutine zlar2v (integer N, complex*16, dimension( * ) X, complex*16, dimension( * ) Y, complex*16, dimension( * ) Z, integer INCX, double precision, dimension( * ) C, complex*16, dimension( * ) S, integer INCC)¶

**ZLAR2V**applies a vector of plane rotations with real cosines and complex sines from both sides to a sequence of 2-by-2 symmetric/Hermitian matrices.

**Purpose:**

ZLAR2V applies a vector of complex plane rotations with real cosines from both sides to a sequence of 2-by-2 complex Hermitian matrices, defined by the elements of the vectors x, y and z. For i = 1,2,...,n ( x(i) z(i) ) := ( conjg(z(i)) y(i) ) ( c(i) conjg(s(i)) ) ( x(i) z(i) ) ( c(i) -conjg(s(i)) ) ( -s(i) c(i) ) ( conjg(z(i)) y(i) ) ( s(i) c(i) )

**Parameters**

*N*

N is INTEGER The number of plane rotations to be applied.

*X*

X is COMPLEX*16 array, dimension (1+(N-1)*INCX) The vector x; the elements of x are assumed to be real.

*Y*

Y is COMPLEX*16 array, dimension (1+(N-1)*INCX) The vector y; the elements of y are assumed to be real.

*Z*

Z is COMPLEX*16 array, dimension (1+(N-1)*INCX) The vector z.

*INCX*

INCX is INTEGER The increment between elements of X, Y and Z. INCX > 0.

*C*

C is DOUBLE PRECISION array, dimension (1+(N-1)*INCC) The cosines of the plane rotations.

*S*

S is COMPLEX*16 array, dimension (1+(N-1)*INCC) The sines of the plane rotations.

*INCC*

INCC is INTEGER The increment between elements of C and S. INCC > 0.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlarcm (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldc, * ) C, integer LDC, double precision, dimension( * ) RWORK)¶

**ZLARCM**copies all or part of a real two-dimensional array to a complex array.

**Purpose:**

ZLARCM performs a very simple matrix-matrix multiplication: C := A * B, where A is M by M and real; B is M by N and complex; C is M by N and complex.

**Parameters**

*M*

M is INTEGER The number of rows of the matrix A and of the matrix C. M >= 0.

*N*

N is INTEGER The number of columns and rows of the matrix B and the number of columns of the matrix C. N >= 0.

*A*

A is DOUBLE PRECISION array, dimension (LDA, M) On entry, A contains the M by M matrix A.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >=max(1,M).

*B*

B is COMPLEX*16 array, dimension (LDB, N) On entry, B contains the M by N matrix B.

*LDB*

LDB is INTEGER The leading dimension of the array B. LDB >=max(1,M).

*C*

C is COMPLEX*16 array, dimension (LDC, N) On exit, C contains the M by N matrix C.

*LDC*

LDC is INTEGER The leading dimension of the array C. LDC >=max(1,M).

*RWORK*

RWORK is DOUBLE PRECISION array, dimension (2*M*N)

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlarf (character SIDE, integer M, integer N, complex*16, dimension( * ) V, integer INCV, complex*16 TAU, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( * ) WORK)¶

**ZLARF**applies an elementary reflector to a general rectangular matrix.

**Purpose:**

ZLARF applies a complex elementary reflector H to a complex M-by-N matrix C, from either the left or the right. H is represented in the form H = I - tau * v * v**H where tau is a complex scalar and v is a complex vector. If tau = 0, then H is taken to be the unit matrix. To apply H**H, supply conjg(tau) instead tau.

**Parameters**

*SIDE*

SIDE is CHARACTER*1 = 'L': form H * C = 'R': form C * H

*M*

M is INTEGER The number of rows of the matrix C.

*N*

N is INTEGER The number of columns of the matrix C.

*V*

V is COMPLEX*16 array, dimension (1 + (M-1)*abs(INCV)) if SIDE = 'L' or (1 + (N-1)*abs(INCV)) if SIDE = 'R' The vector v in the representation of H. V is not used if TAU = 0.

*INCV*

INCV is INTEGER The increment between elements of v. INCV <> 0.

*TAU*

TAU is COMPLEX*16 The value tau in the representation of H.

*C*

C is COMPLEX*16 array, dimension (LDC,N) On entry, the M-by-N matrix C. On exit, C is overwritten by the matrix H * C if SIDE = 'L', or C * H if SIDE = 'R'.

*LDC*

LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).

*WORK*

WORK is COMPLEX*16 array, dimension (N) if SIDE = 'L' or (M) if SIDE = 'R'

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlarfb (character SIDE, character TRANS, character DIRECT, character STOREV, integer M, integer N, integer K, complex*16, dimension( ldv, * ) V, integer LDV, complex*16, dimension( ldt, * ) T, integer LDT, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( ldwork, * ) WORK, integer LDWORK)¶

**ZLARFB**applies a block reflector or its conjugate-transpose to a general rectangular matrix.

**Purpose:**

ZLARFB applies a complex block reflector H or its transpose H**H to a complex M-by-N matrix C, from either the left or the right.

**Parameters**

*SIDE*

SIDE is CHARACTER*1 = 'L': apply H or H**H from the Left = 'R': apply H or H**H from the Right

*TRANS*

TRANS is CHARACTER*1 = 'N': apply H (No transpose) = 'C': apply H**H (Conjugate transpose)

*DIRECT*

DIRECT is CHARACTER*1 Indicates how H is formed from a product of elementary reflectors = 'F': H = H(1) H(2) . . . H(k) (Forward) = 'B': H = H(k) . . . H(2) H(1) (Backward)

*STOREV*

STOREV is CHARACTER*1 Indicates how the vectors which define the elementary reflectors are stored: = 'C': Columnwise = 'R': Rowwise

*M*

M is INTEGER The number of rows of the matrix C.

*N*

N is INTEGER The number of columns of the matrix C.

*K*

K is INTEGER The order of the matrix T (= the number of elementary reflectors whose product defines the block reflector). If SIDE = 'L', M >= K >= 0; if SIDE = 'R', N >= K >= 0.

*V*

V is COMPLEX*16 array, dimension (LDV,K) if STOREV = 'C' (LDV,M) if STOREV = 'R' and SIDE = 'L' (LDV,N) if STOREV = 'R' and SIDE = 'R' See Further Details.

*LDV*

LDV is INTEGER The leading dimension of the array V. If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M); if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N); if STOREV = 'R', LDV >= K.

*T*

T is COMPLEX*16 array, dimension (LDT,K) The triangular K-by-K matrix T in the representation of the block reflector.

*LDT*

LDT is INTEGER The leading dimension of the array T. LDT >= K.

*C*

C is COMPLEX*16 array, dimension (LDC,N) On entry, the M-by-N matrix C. On exit, C is overwritten by H*C or H**H*C or C*H or C*H**H.

*LDC*

LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).

*WORK*

WORK is COMPLEX*16 array, dimension (LDWORK,K)

*LDWORK*

LDWORK is INTEGER The leading dimension of the array WORK. If SIDE = 'L', LDWORK >= max(1,N); if SIDE = 'R', LDWORK >= max(1,M).

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

The shape of the matrix V and the storage of the vectors which define the H(i) is best illustrated by the following example with n = 5 and k = 3. The elements equal to 1 are not stored; the corresponding array elements are modified but restored on exit. The rest of the array is not used. DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R': V = ( 1 ) V = ( 1 v1 v1 v1 v1 ) ( v1 1 ) ( 1 v2 v2 v2 ) ( v1 v2 1 ) ( 1 v3 v3 ) ( v1 v2 v3 ) ( v1 v2 v3 ) DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R': V = ( v1 v2 v3 ) V = ( v1 v1 1 ) ( v1 v2 v3 ) ( v2 v2 v2 1 ) ( 1 v2 v3 ) ( v3 v3 v3 v3 1 ) ( 1 v3 ) ( 1 )

## subroutine zlarfb_gett (character IDENT, integer M, integer N, integer K, complex*16, dimension( ldt, * ) T, integer LDT, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldwork, * ) WORK, integer LDWORK)¶

**ZLARFB_GETT**

**Purpose:**

ZLARFB_GETT applies a complex Householder block reflector H from the left to a complex (K+M)-by-N "triangular-pentagonal" matrix composed of two block matrices: an upper trapezoidal K-by-N matrix A stored in the array A, and a rectangular M-by-(N-K) matrix B, stored in the array B. The block reflector H is stored in a compact WY-representation, where the elementary reflectors are in the arrays A, B and T. See Further Details section.

**Parameters**

*IDENT*

IDENT is CHARACTER*1 If IDENT = not 'I', or not 'i', then V1 is unit lower-triangular and stored in the left K-by-K block of the input matrix A, If IDENT = 'I' or 'i', then V1 is an identity matrix and not stored. See Further Details section.

*M*

M is INTEGER The number of rows of the matrix B. M >= 0.

*N*

N is INTEGER The number of columns of the matrices A and B. N >= 0.

*K*

K is INTEGER The number or rows of the matrix A. K is also order of the matrix T, i.e. the number of elementary reflectors whose product defines the block reflector. 0 <= K <= N.

*T*

T is COMPLEX*16 array, dimension (LDT,K) The upper-triangular K-by-K matrix T in the representation of the block reflector.

*LDT*

LDT is INTEGER The leading dimension of the array T. LDT >= K.

*A*

A is COMPLEX*16 array, dimension (LDA,N) On entry: a) In the K-by-N upper-trapezoidal part A: input matrix A. b) In the columns below the diagonal: columns of V1 (ones are not stored on the diagonal). On exit: A is overwritten by rectangular K-by-N product H*A. See Further Details section.

*LDA*

LDB is INTEGER The leading dimension of the array A. LDA >= max(1,K).

*B*

B is COMPLEX*16 array, dimension (LDB,N) On entry: a) In the M-by-(N-K) right block: input matrix B. b) In the M-by-N left block: columns of V2. On exit: B is overwritten by rectangular M-by-N product H*B. See Further Details section.

*LDB*

LDB is INTEGER The leading dimension of the array B. LDB >= max(1,M).

*WORK*

WORK is COMPLEX*16 array, dimension (LDWORK,max(K,N-K))

*LDWORK*

LDWORK is INTEGER The leading dimension of the array WORK. LDWORK>=max(1,K).

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

November 2020, Igor Kozachenko, Computer Science Division, University of California, Berkeley

**Further Details:**

(1) Description of the Algebraic Operation. The matrix A is a K-by-N matrix composed of two column block matrices, A1, which is K-by-K, and A2, which is K-by-(N-K): A = ( A1, A2 ). The matrix B is an M-by-N matrix composed of two column block matrices, B1, which is M-by-K, and B2, which is M-by-(N-K): B = ( B1, B2 ). Perform the operation: ( A_out ) := H * ( A_in ) = ( I - V * T * V**H ) * ( A_in ) = ( B_out ) ( B_in ) ( B_in ) = ( I - ( V1 ) * T * ( V1**H, V2**H ) ) * ( A_in ) ( V2 ) ( B_in ) On input: a) ( A_in ) consists of two block columns: ( B_in ) ( A_in ) = (( A1_in ) ( A2_in )) = (( A1_in ) ( A2_in )) ( B_in ) (( B1_in ) ( B2_in )) (( 0 ) ( B2_in )), where the column blocks are: ( A1_in ) is a K-by-K upper-triangular matrix stored in the upper triangular part of the array A(1:K,1:K). ( B1_in ) is an M-by-K rectangular ZERO matrix and not stored. ( A2_in ) is a K-by-(N-K) rectangular matrix stored in the array A(1:K,K+1:N). ( B2_in ) is an M-by-(N-K) rectangular matrix stored in the array B(1:M,K+1:N). b) V = ( V1 ) ( V2 ) where: 1) if IDENT == 'I',V1 is a K-by-K identity matrix, not stored; 2) if IDENT != 'I',V1 is a K-by-K unit lower-triangular matrix, stored in the lower-triangular part of the array A(1:K,1:K) (ones are not stored), and V2 is an M-by-K rectangular stored the array B(1:M,1:K), (because on input B1_in is a rectangular zero matrix that is not stored and the space is used to store V2). c) T is a K-by-K upper-triangular matrix stored in the array T(1:K,1:K). On output: a) ( A_out ) consists of two block columns: ( B_out ) ( A_out ) = (( A1_out ) ( A2_out )) ( B_out ) (( B1_out ) ( B2_out )), where the column blocks are: ( A1_out ) is a K-by-K square matrix, or a K-by-K upper-triangular matrix, if V1 is an identity matrix. AiOut is stored in the array A(1:K,1:K). ( B1_out ) is an M-by-K rectangular matrix stored in the array B(1:M,K:N). ( A2_out ) is a K-by-(N-K) rectangular matrix stored in the array A(1:K,K+1:N). ( B2_out ) is an M-by-(N-K) rectangular matrix stored in the array B(1:M,K+1:N). The operation above can be represented as the same operation on each block column: ( A1_out ) := H * ( A1_in ) = ( I - V * T * V**H ) * ( A1_in ) ( B1_out ) ( 0 ) ( 0 ) ( A2_out ) := H * ( A2_in ) = ( I - V * T * V**H ) * ( A2_in ) ( B2_out ) ( B2_in ) ( B2_in ) If IDENT != 'I': The computation for column block 1: A1_out: = A1_in - V1*T*(V1**H)*A1_in B1_out: = - V2*T*(V1**H)*A1_in The computation for column block 2, which exists if N > K: A2_out: = A2_in - V1*T*( (V1**H)*A2_in + (V2**H)*B2_in ) B2_out: = B2_in - V2*T*( (V1**H)*A2_in + (V2**H)*B2_in ) If IDENT == 'I': The operation for column block 1: A1_out: = A1_in - V1*T*A1_in B1_out: = - V2*T*A1_in The computation for column block 2, which exists if N > K: A2_out: = A2_in - T*( A2_in + (V2**H)*B2_in ) B2_out: = B2_in - V2*T*( A2_in + (V2**H)*B2_in ) (2) Description of the Algorithmic Computation. In the first step, we compute column block 2, i.e. A2 and B2. Here, we need to use the K-by-(N-K) rectangular workspace matrix W2 that is of the same size as the matrix A2. W2 is stored in the array WORK(1:K,1:(N-K)). In the second step, we compute column block 1, i.e. A1 and B1. Here, we need to use the K-by-K square workspace matrix W1 that is of the same size as the as the matrix A1. W1 is stored in the array WORK(1:K,1:K). NOTE: Hence, in this routine, we need the workspace array WORK only of size WORK(1:K,1:max(K,N-K)) so it can hold both W2 from the first step and W1 from the second step. Case (A), when V1 is unit lower-triangular, i.e. IDENT != 'I', more computations than in the Case (B). if( IDENT != 'I' ) then if ( N > K ) then (First Step - column block 2) col2_(1) W2: = A2 col2_(2) W2: = (V1**H) * W2 = (unit_lower_tr_of_(A1)**H) * W2 col2_(3) W2: = W2 + (V2**H) * B2 = W2 + (B1**H) * B2 col2_(4) W2: = T * W2 col2_(5) B2: = B2 - V2 * W2 = B2 - B1 * W2 col2_(6) W2: = V1 * W2 = unit_lower_tr_of_(A1) * W2 col2_(7) A2: = A2 - W2 else (Second Step - column block 1) col1_(1) W1: = A1 col1_(2) W1: = (V1**H) * W1 = (unit_lower_tr_of_(A1)**H) * W1 col1_(3) W1: = T * W1 col1_(4) B1: = - V2 * W1 = - B1 * W1 col1_(5) square W1: = V1 * W1 = unit_lower_tr_of_(A1) * W1 col1_(6) square A1: = A1 - W1 end if end if Case (B), when V1 is an identity matrix, i.e. IDENT == 'I', less computations than in the Case (A) if( IDENT == 'I' ) then if ( N > K ) then (First Step - column block 2) col2_(1) W2: = A2 col2_(3) W2: = W2 + (V2**H) * B2 = W2 + (B1**H) * B2 col2_(4) W2: = T * W2 col2_(5) B2: = B2 - V2 * W2 = B2 - B1 * W2 col2_(7) A2: = A2 - W2 else (Second Step - column block 1) col1_(1) W1: = A1 col1_(3) W1: = T * W1 col1_(4) B1: = - V2 * W1 = - B1 * W1 col1_(6) upper-triangular_of_(A1): = A1 - W1 end if end if Combine these cases (A) and (B) together, this is the resulting algorithm: if ( N > K ) then (First Step - column block 2) col2_(1) W2: = A2 if( IDENT != 'I' ) then col2_(2) W2: = (V1**H) * W2 = (unit_lower_tr_of_(A1)**H) * W2 end if col2_(3) W2: = W2 + (V2**H) * B2 = W2 + (B1**H) * B2] col2_(4) W2: = T * W2 col2_(5) B2: = B2 - V2 * W2 = B2 - B1 * W2 if( IDENT != 'I' ) then col2_(6) W2: = V1 * W2 = unit_lower_tr_of_(A1) * W2 end if col2_(7) A2: = A2 - W2 else (Second Step - column block 1) col1_(1) W1: = A1 if( IDENT != 'I' ) then col1_(2) W1: = (V1**H) * W1 = (unit_lower_tr_of_(A1)**H) * W1 end if col1_(3) W1: = T * W1 col1_(4) B1: = - V2 * W1 = - B1 * W1 if( IDENT != 'I' ) then col1_(5) square W1: = V1 * W1 = unit_lower_tr_of_(A1) * W1 col1_(6_a) below_diag_of_(A1): = - below_diag_of_(W1) end if col1_(6_b) up_tr_of_(A1): = up_tr_of_(A1) - up_tr_of_(W1) end if

## subroutine zlarfg (integer N, complex*16 ALPHA, complex*16, dimension( * ) X, integer INCX, complex*16 TAU)¶

**ZLARFG**generates an elementary reflector (Householder matrix).

**Purpose:**

ZLARFG generates a complex elementary reflector H of order n, such that H**H * ( alpha ) = ( beta ), H**H * H = I. ( x ) ( 0 ) where alpha and beta are scalars, with beta real, and x is an (n-1)-element complex vector. H is represented in the form H = I - tau * ( 1 ) * ( 1 v**H ) , ( v ) where tau is a complex scalar and v is a complex (n-1)-element vector. Note that H is not hermitian. If the elements of x are all zero and alpha is real, then tau = 0 and H is taken to be the unit matrix. Otherwise 1 <= real(tau) <= 2 and abs(tau-1) <= 1 .

**Parameters**

*N*

N is INTEGER The order of the elementary reflector.

*ALPHA*

ALPHA is COMPLEX*16 On entry, the value alpha. On exit, it is overwritten with the value beta.

*X*

X is COMPLEX*16 array, dimension (1+(N-2)*abs(INCX)) On entry, the vector x. On exit, it is overwritten with the vector v.

*INCX*

INCX is INTEGER The increment between elements of X. INCX > 0.

*TAU*

TAU is COMPLEX*16 The value tau.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlarfgp (integer N, complex*16 ALPHA, complex*16, dimension( * ) X, integer INCX, complex*16 TAU)¶

**ZLARFGP**generates an elementary reflector (Householder matrix) with non-negative beta.

**Purpose:**

ZLARFGP generates a complex elementary reflector H of order n, such that H**H * ( alpha ) = ( beta ), H**H * H = I. ( x ) ( 0 ) where alpha and beta are scalars, beta is real and non-negative, and x is an (n-1)-element complex vector. H is represented in the form H = I - tau * ( 1 ) * ( 1 v**H ) , ( v ) where tau is a complex scalar and v is a complex (n-1)-element vector. Note that H is not hermitian. If the elements of x are all zero and alpha is real, then tau = 0 and H is taken to be the unit matrix.

**Parameters**

*N*

N is INTEGER The order of the elementary reflector.

*ALPHA*

ALPHA is COMPLEX*16 On entry, the value alpha. On exit, it is overwritten with the value beta.

*X*

X is COMPLEX*16 array, dimension (1+(N-2)*abs(INCX)) On entry, the vector x. On exit, it is overwritten with the vector v.

*INCX*

INCX is INTEGER The increment between elements of X. INCX > 0.

*TAU*

TAU is COMPLEX*16 The value tau.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlarft (character DIRECT, character STOREV, integer N, integer K, complex*16, dimension( ldv, * ) V, integer LDV, complex*16, dimension( * ) TAU, complex*16, dimension( ldt, * ) T, integer LDT)¶

**ZLARFT**forms the triangular factor T of a block reflector H = I - vtvH

**Purpose:**

ZLARFT forms the triangular factor T of a complex block reflector H of order n, which is defined as a product of k elementary reflectors. If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular; If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular. If STOREV = 'C', the vector which defines the elementary reflector H(i) is stored in the i-th column of the array V, and H = I - V * T * V**H If STOREV = 'R', the vector which defines the elementary reflector H(i) is stored in the i-th row of the array V, and H = I - V**H * T * V

**Parameters**

*DIRECT*

DIRECT is CHARACTER*1 Specifies the order in which the elementary reflectors are multiplied to form the block reflector: = 'F': H = H(1) H(2) . . . H(k) (Forward) = 'B': H = H(k) . . . H(2) H(1) (Backward)

*STOREV*

STOREV is CHARACTER*1 Specifies how the vectors which define the elementary reflectors are stored (see also Further Details): = 'C': columnwise = 'R': rowwise

*N*

N is INTEGER The order of the block reflector H. N >= 0.

*K*

K is INTEGER The order of the triangular factor T (= the number of elementary reflectors). K >= 1.

*V*

V is COMPLEX*16 array, dimension (LDV,K) if STOREV = 'C' (LDV,N) if STOREV = 'R' The matrix V. See further details.

*LDV*

LDV is INTEGER The leading dimension of the array V. If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.

*TAU*

TAU is COMPLEX*16 array, dimension (K) TAU(i) must contain the scalar factor of the elementary reflector H(i).

*T*

T is COMPLEX*16 array, dimension (LDT,K) The k by k triangular factor T of the block reflector. If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is lower triangular. The rest of the array is not used.

*LDT*

LDT is INTEGER The leading dimension of the array T. LDT >= K.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

The shape of the matrix V and the storage of the vectors which define the H(i) is best illustrated by the following example with n = 5 and k = 3. The elements equal to 1 are not stored. DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R': V = ( 1 ) V = ( 1 v1 v1 v1 v1 ) ( v1 1 ) ( 1 v2 v2 v2 ) ( v1 v2 1 ) ( 1 v3 v3 ) ( v1 v2 v3 ) ( v1 v2 v3 ) DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R': V = ( v1 v2 v3 ) V = ( v1 v1 1 ) ( v1 v2 v3 ) ( v2 v2 v2 1 ) ( 1 v2 v3 ) ( v3 v3 v3 v3 1 ) ( 1 v3 ) ( 1 )

## subroutine zlarfx (character SIDE, integer M, integer N, complex*16, dimension( * ) V, complex*16 TAU, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( * ) WORK)¶

**ZLARFX**applies an elementary reflector to a general rectangular matrix, with loop unrolling when the reflector has order ≤ 10.

**Purpose:**

ZLARFX applies a complex elementary reflector H to a complex m by n matrix C, from either the left or the right. H is represented in the form H = I - tau * v * v**H where tau is a complex scalar and v is a complex vector. If tau = 0, then H is taken to be the unit matrix This version uses inline code if H has order < 11.

**Parameters**

*SIDE*

SIDE is CHARACTER*1 = 'L': form H * C = 'R': form C * H

*M*

M is INTEGER The number of rows of the matrix C.

*N*

N is INTEGER The number of columns of the matrix C.

*V*

V is COMPLEX*16 array, dimension (M) if SIDE = 'L' or (N) if SIDE = 'R' The vector v in the representation of H.

*TAU*

TAU is COMPLEX*16 The value tau in the representation of H.

*C*

C is COMPLEX*16 array, dimension (LDC,N) On entry, the m by n matrix C. On exit, C is overwritten by the matrix H * C if SIDE = 'L', or C * H if SIDE = 'R'.

*LDC*

LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M).

*WORK*

WORK is COMPLEX*16 array, dimension (N) if SIDE = 'L' or (M) if SIDE = 'R' WORK is not referenced if H has order < 11.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlarfy (character UPLO, integer N, complex*16, dimension( * ) V, integer INCV, complex*16 TAU, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( * ) WORK)¶

**ZLARFY**

**Purpose:**

ZLARFY applies an elementary reflector, or Householder matrix, H, to an n x n Hermitian matrix C, from both the left and the right. H is represented in the form H = I - tau * v * v' where tau is a scalar and v is a vector. If tau is zero, then H is taken to be the unit matrix.

**Parameters**

*UPLO*

UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the Hermitian matrix C is stored. = 'U': Upper triangle = 'L': Lower triangle

*N*

N is INTEGER The number of rows and columns of the matrix C. N >= 0.

*V*

V is COMPLEX*16 array, dimension (1 + (N-1)*abs(INCV)) The vector v as described above.

*INCV*

INCV is INTEGER The increment between successive elements of v. INCV must not be zero.

*TAU*

TAU is COMPLEX*16 The value tau as described above.

*C*

C is COMPLEX*16 array, dimension (LDC, N) On entry, the matrix C. On exit, C is overwritten by H * C * H'.

*LDC*

LDC is INTEGER The leading dimension of the array C. LDC >= max( 1, N ).

*WORK*

WORK is COMPLEX*16 array, dimension (N)

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlargv (integer N, complex*16, dimension( * ) X, integer INCX, complex*16, dimension( * ) Y, integer INCY, double precision, dimension( * ) C, integer INCC)¶

**ZLARGV**generates a vector of plane rotations with real cosines and complex sines.

**Purpose:**

ZLARGV generates a vector of complex plane rotations with real cosines, determined by elements of the complex vectors x and y. For i = 1,2,...,n ( c(i) s(i) ) ( x(i) ) = ( r(i) ) ( -conjg(s(i)) c(i) ) ( y(i) ) = ( 0 ) where c(i)**2 + ABS(s(i))**2 = 1 The following conventions are used (these are the same as in ZLARTG, but differ from the BLAS1 routine ZROTG): If y(i)=0, then c(i)=1 and s(i)=0. If x(i)=0, then c(i)=0 and s(i) is chosen so that r(i) is real.

**Parameters**

*N*

N is INTEGER The number of plane rotations to be generated.

*X*

X is COMPLEX*16 array, dimension (1+(N-1)*INCX) On entry, the vector x. On exit, x(i) is overwritten by r(i), for i = 1,...,n.

*INCX*

INCX is INTEGER The increment between elements of X. INCX > 0.

*Y*

Y is COMPLEX*16 array, dimension (1+(N-1)*INCY) On entry, the vector y. On exit, the sines of the plane rotations.

*INCY*

INCY is INTEGER The increment between elements of Y. INCY > 0.

*C*

C is DOUBLE PRECISION array, dimension (1+(N-1)*INCC) The cosines of the plane rotations.

*INCC*

INCC is INTEGER The increment between elements of C. INCC > 0.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

6-6-96 - Modified with a new algorithm by W. Kahan and J. Demmel This version has a few statements commented out for thread safety (machine parameters are computed on each entry). 10 feb 03, SJH.

## subroutine zlarnv (integer IDIST, integer, dimension( 4 ) ISEED, integer N, complex*16, dimension( * ) X)¶

**ZLARNV**returns a vector of random numbers from a uniform or normal distribution.

**Purpose:**

ZLARNV returns a vector of n random complex numbers from a uniform or normal distribution.

**Parameters**

*IDIST*

IDIST is INTEGER Specifies the distribution of the random numbers: = 1: real and imaginary parts each uniform (0,1) = 2: real and imaginary parts each uniform (-1,1) = 3: real and imaginary parts each normal (0,1) = 4: uniformly distributed on the disc abs(z) < 1 = 5: uniformly distributed on the circle abs(z) = 1

*ISEED*

ISEED is INTEGER array, dimension (4) On entry, the seed of the random number generator; the array elements must be between 0 and 4095, and ISEED(4) must be odd. On exit, the seed is updated.

*N*

N is INTEGER The number of random numbers to be generated.

*X*

X is COMPLEX*16 array, dimension (N) The generated random numbers.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

This routine calls the auxiliary routine DLARUV to generate random real numbers from a uniform (0,1) distribution, in batches of up to 128 using vectorisable code. The Box-Muller method is used to transform numbers from a uniform to a normal distribution.

## subroutine zlarrv (integer N, double precision VL, double precision VU, double precision, dimension( * ) D, double precision, dimension( * ) L, double precision PIVMIN, integer, dimension( * ) ISPLIT, integer M, integer DOL, integer DOU, double precision MINRGP, double precision RTOL1, double precision RTOL2, double precision, dimension( * ) W, double precision, dimension( * ) WERR, double precision, dimension( * ) WGAP, integer, dimension( * ) IBLOCK, integer, dimension( * ) INDEXW, double precision, dimension( * ) GERS, complex*16, dimension( ldz, * ) Z, integer LDZ, integer, dimension( * ) ISUPPZ, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)¶

**ZLARRV**computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.

**Purpose:**

ZLARRV computes the eigenvectors of the tridiagonal matrix T = L D L**T given L, D and APPROXIMATIONS to the eigenvalues of L D L**T. The input eigenvalues should have been computed by DLARRE.

**Parameters**

*N*

N is INTEGER The order of the matrix. N >= 0.

*VL*

VL is DOUBLE PRECISION Lower bound of the interval that contains the desired eigenvalues. VL < VU. Needed to compute gaps on the left or right end of the extremal eigenvalues in the desired RANGE.

*VU*

VU is DOUBLE PRECISION Upper bound of the interval that contains the desired eigenvalues. VL < VU. Needed to compute gaps on the left or right end of the extremal eigenvalues in the desired RANGE.

*D*

D is DOUBLE PRECISION array, dimension (N) On entry, the N diagonal elements of the diagonal matrix D. On exit, D may be overwritten.

*L*

L is DOUBLE PRECISION array, dimension (N) On entry, the (N-1) subdiagonal elements of the unit bidiagonal matrix L are in elements 1 to N-1 of L (if the matrix is not split.) At the end of each block is stored the corresponding shift as given by DLARRE. On exit, L is overwritten.

*PIVMIN*

PIVMIN is DOUBLE PRECISION The minimum pivot allowed in the Sturm sequence.

*ISPLIT*

ISPLIT is INTEGER array, dimension (N) The splitting points, at which T breaks up into blocks. The first block consists of rows/columns 1 to ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1 through ISPLIT( 2 ), etc.

*M*

M is INTEGER The total number of input eigenvalues. 0 <= M <= N.

*DOL*

DOL is INTEGER

*DOU*

DOU is INTEGER If the user wants to compute only selected eigenvectors from all the eigenvalues supplied, he can specify an index range DOL:DOU. Or else the setting DOL=1, DOU=M should be applied. Note that DOL and DOU refer to the order in which the eigenvalues are stored in W. If the user wants to compute only selected eigenpairs, then the columns DOL-1 to DOU+1 of the eigenvector space Z contain the computed eigenvectors. All other columns of Z are set to zero.

*MINRGP*

MINRGP is DOUBLE PRECISION

*RTOL1*

RTOL1 is DOUBLE PRECISION

*RTOL2*

RTOL2 is DOUBLE PRECISION Parameters for bisection. An interval [LEFT,RIGHT] has converged if RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )

*W*

W is DOUBLE PRECISION array, dimension (N) The first M elements of W contain the APPROXIMATE eigenvalues for which eigenvectors are to be computed. The eigenvalues should be grouped by split-off block and ordered from smallest to largest within the block ( The output array W from DLARRE is expected here ). Furthermore, they are with respect to the shift of the corresponding root representation for their block. On exit, W holds the eigenvalues of the UNshifted matrix.

*WERR*

WERR is DOUBLE PRECISION array, dimension (N) The first M elements contain the semiwidth of the uncertainty interval of the corresponding eigenvalue in W

*WGAP*

WGAP is DOUBLE PRECISION array, dimension (N) The separation from the right neighbor eigenvalue in W.

*IBLOCK*

IBLOCK is INTEGER array, dimension (N) The indices of the blocks (submatrices) associated with the corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to the first block from the top, =2 if W(i) belongs to the second block, etc.

*INDEXW*

INDEXW is INTEGER array, dimension (N) The indices of the eigenvalues within each block (submatrix); for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the i-th eigenvalue W(i) is the 10-th eigenvalue in the second block.

*GERS*

GERS is DOUBLE PRECISION array, dimension (2*N) The N Gerschgorin intervals (the i-th Gerschgorin interval is (GERS(2*i-1), GERS(2*i)). The Gerschgorin intervals should be computed from the original UNshifted matrix.

*Z*

Z is COMPLEX*16 array, dimension (LDZ, max(1,M) ) If INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix T corresponding to the input eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). Note: the user must ensure that at least max(1,M) columns are supplied in the array Z.

*LDZ*

LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).

*ISUPPZ*

ISUPPZ is INTEGER array, dimension ( 2*max(1,M) ) The support of the eigenvectors in Z, i.e., the indices indicating the nonzero elements in Z. The I-th eigenvector is nonzero only in elements ISUPPZ( 2*I-1 ) through ISUPPZ( 2*I ).

*WORK*

WORK is DOUBLE PRECISION array, dimension (12*N)

*IWORK*

IWORK is INTEGER array, dimension (7*N)

*INFO*

INFO is INTEGER = 0: successful exit > 0: A problem occurred in ZLARRV. < 0: One of the called subroutines signaled an internal problem. Needs inspection of the corresponding parameter IINFO for further information. =-1: Problem in DLARRB when refining a child's eigenvalues. =-2: Problem in DLARRF when computing the RRR of a child. When a child is inside a tight cluster, it can be difficult to find an RRR. A partial remedy from the user's point of view is to make the parameter MINRGP smaller and recompile. However, as the orthogonality of the computed vectors is proportional to 1/MINRGP, the user should be aware that he might be trading in precision when he decreases MINRGP. =-3: Problem in DLARRB when refining a single eigenvalue after the Rayleigh correction was rejected. = 5: The Rayleigh Quotient Iteration failed to converge to full accuracy in MAXITR steps.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

## subroutine zlartv (integer N, complex*16, dimension( * ) X, integer INCX, complex*16, dimension( * ) Y, integer INCY, double precision, dimension( * ) C, complex*16, dimension( * ) S, integer INCC)¶

**ZLARTV**applies a vector of plane rotations with real cosines and complex sines to the elements of a pair of vectors.

**Purpose:**

ZLARTV applies a vector of complex plane rotations with real cosines to elements of the complex vectors x and y. For i = 1,2,...,n ( x(i) ) := ( c(i) s(i) ) ( x(i) ) ( y(i) ) ( -conjg(s(i)) c(i) ) ( y(i) )

**Parameters**

*N*

N is INTEGER The number of plane rotations to be applied.

*X*

X is COMPLEX*16 array, dimension (1+(N-1)*INCX) The vector x.

*INCX*

INCX is INTEGER The increment between elements of X. INCX > 0.

*Y*

Y is COMPLEX*16 array, dimension (1+(N-1)*INCY) The vector y.

*INCY*

INCY is INTEGER The increment between elements of Y. INCY > 0.

*C*

C is DOUBLE PRECISION array, dimension (1+(N-1)*INCC) The cosines of the plane rotations.

*S*

S is COMPLEX*16 array, dimension (1+(N-1)*INCC) The sines of the plane rotations.

*INCC*

INCC is INTEGER The increment between elements of C and S. INCC > 0.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlascl (character TYPE, integer KL, integer KU, double precision CFROM, double precision CTO, integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer INFO)¶

**ZLASCL**multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.

**Purpose:**

ZLASCL multiplies the M by N complex matrix A by the real scalar CTO/CFROM. This is done without over/underflow as long as the final result CTO*A(I,J)/CFROM does not over/underflow. TYPE specifies that A may be full, upper triangular, lower triangular, upper Hessenberg, or banded.

**Parameters**

*TYPE*

TYPE is CHARACTER*1 TYPE indices the storage type of the input matrix. = 'G': A is a full matrix. = 'L': A is a lower triangular matrix. = 'U': A is an upper triangular matrix. = 'H': A is an upper Hessenberg matrix. = 'B': A is a symmetric band matrix with lower bandwidth KL and upper bandwidth KU and with the only the lower half stored. = 'Q': A is a symmetric band matrix with lower bandwidth KL and upper bandwidth KU and with the only the upper half stored. = 'Z': A is a band matrix with lower bandwidth KL and upper bandwidth KU. See ZGBTRF for storage details.

*KL*

KL is INTEGER The lower bandwidth of A. Referenced only if TYPE = 'B', 'Q' or 'Z'.

*KU*

KU is INTEGER The upper bandwidth of A. Referenced only if TYPE = 'B', 'Q' or 'Z'.

*CFROM*

CFROM is DOUBLE PRECISION

*CTO*

CTO is DOUBLE PRECISION The matrix A is multiplied by CTO/CFROM. A(I,J) is computed without over/underflow if the final result CTO*A(I,J)/CFROM can be represented without over/underflow. CFROM must be nonzero.

*M*

M is INTEGER The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER The number of columns of the matrix A. N >= 0.

*A*

A is COMPLEX*16 array, dimension (LDA,N) The matrix to be multiplied by CTO/CFROM. See TYPE for the storage type.

*LDA*

LDA is INTEGER The leading dimension of the array A. If TYPE = 'G', 'L', 'U', 'H', LDA >= max(1,M); TYPE = 'B', LDA >= KL+1; TYPE = 'Q', LDA >= KU+1; TYPE = 'Z', LDA >= 2*KL+KU+1.

*INFO*

INFO is INTEGER 0 - successful exit <0 - if INFO = -i, the i-th argument had an illegal value.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlaset (character UPLO, integer M, integer N, complex*16 ALPHA, complex*16 BETA, complex*16, dimension( lda, * ) A, integer LDA)¶

**ZLASET**initializes the off-diagonal elements and the diagonal elements of a matrix to given values.

**Purpose:**

ZLASET initializes a 2-D array A to BETA on the diagonal and ALPHA on the offdiagonals.

**Parameters**

*UPLO*

UPLO is CHARACTER*1 Specifies the part of the matrix A to be set. = 'U': Upper triangular part is set. The lower triangle is unchanged. = 'L': Lower triangular part is set. The upper triangle is unchanged. Otherwise: All of the matrix A is set.

*M*

M is INTEGER On entry, M specifies the number of rows of A.

*N*

N is INTEGER On entry, N specifies the number of columns of A.

*ALPHA*

ALPHA is COMPLEX*16 All the offdiagonal array elements are set to ALPHA.

*BETA*

BETA is COMPLEX*16 All the diagonal array elements are set to BETA.

*A*

A is COMPLEX*16 array, dimension (LDA,N) On entry, the m by n matrix A. On exit, A(i,j) = ALPHA, 1 <= i <= m, 1 <= j <= n, i.ne.j; A(i,i) = BETA , 1 <= i <= min(m,n)

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlasr (character SIDE, character PIVOT, character DIRECT, integer M, integer N, double precision, dimension( * ) C, double precision, dimension( * ) S, complex*16, dimension( lda, * ) A, integer LDA)¶

**ZLASR**applies a sequence of plane rotations to a general rectangular matrix.

**Purpose:**

ZLASR applies a sequence of real plane rotations to a complex matrix A, from either the left or the right. When SIDE = 'L', the transformation takes the form A := P*A and when SIDE = 'R', the transformation takes the form A := A*P**T where P is an orthogonal matrix consisting of a sequence of z plane rotations, with z = M when SIDE = 'L' and z = N when SIDE = 'R', and P**T is the transpose of P. When DIRECT = 'F' (Forward sequence), then P = P(z-1) * ... * P(2) * P(1) and when DIRECT = 'B' (Backward sequence), then P = P(1) * P(2) * ... * P(z-1) where P(k) is a plane rotation matrix defined by the 2-by-2 rotation R(k) = ( c(k) s(k) ) = ( -s(k) c(k) ). When PIVOT = 'V' (Variable pivot), the rotation is performed for the plane (k,k+1), i.e., P(k) has the form P(k) = ( 1 ) ( ... ) ( 1 ) ( c(k) s(k) ) ( -s(k) c(k) ) ( 1 ) ( ... ) ( 1 ) where R(k) appears as a rank-2 modification to the identity matrix in rows and columns k and k+1. When PIVOT = 'T' (Top pivot), the rotation is performed for the plane (1,k+1), so P(k) has the form P(k) = ( c(k) s(k) ) ( 1 ) ( ... ) ( 1 ) ( -s(k) c(k) ) ( 1 ) ( ... ) ( 1 ) where R(k) appears in rows and columns 1 and k+1. Similarly, when PIVOT = 'B' (Bottom pivot), the rotation is performed for the plane (k,z), giving P(k) the form P(k) = ( 1 ) ( ... ) ( 1 ) ( c(k) s(k) ) ( 1 ) ( ... ) ( 1 ) ( -s(k) c(k) ) where R(k) appears in rows and columns k and z. The rotations are performed without ever forming P(k) explicitly.

**Parameters**

*SIDE*

SIDE is CHARACTER*1 Specifies whether the plane rotation matrix P is applied to A on the left or the right. = 'L': Left, compute A := P*A = 'R': Right, compute A:= A*P**T

*PIVOT*

PIVOT is CHARACTER*1 Specifies the plane for which P(k) is a plane rotation matrix. = 'V': Variable pivot, the plane (k,k+1) = 'T': Top pivot, the plane (1,k+1) = 'B': Bottom pivot, the plane (k,z)

*DIRECT*

DIRECT is CHARACTER*1 Specifies whether P is a forward or backward sequence of plane rotations. = 'F': Forward, P = P(z-1)*...*P(2)*P(1) = 'B': Backward, P = P(1)*P(2)*...*P(z-1)

*M*

M is INTEGER The number of rows of the matrix A. If m <= 1, an immediate return is effected.

*N*

N is INTEGER The number of columns of the matrix A. If n <= 1, an immediate return is effected.

*C*

C is DOUBLE PRECISION array, dimension (M-1) if SIDE = 'L' (N-1) if SIDE = 'R' The cosines c(k) of the plane rotations.

*S*

S is DOUBLE PRECISION array, dimension (M-1) if SIDE = 'L' (N-1) if SIDE = 'R' The sines s(k) of the plane rotations. The 2-by-2 plane rotation part of the matrix P(k), R(k), has the form R(k) = ( c(k) s(k) ) ( -s(k) c(k) ).

*A*

A is COMPLEX*16 array, dimension (LDA,N) The M-by-N matrix A. On exit, A is overwritten by P*A if SIDE = 'R' or by A*P**T if SIDE = 'L'.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlaswp (integer N, complex*16, dimension( lda, * ) A, integer LDA, integer K1, integer K2, integer, dimension( * ) IPIV, integer INCX)¶

**ZLASWP**performs a series of row interchanges on a general rectangular matrix.

**Purpose:**

ZLASWP performs a series of row interchanges on the matrix A. One row interchange is initiated for each of rows K1 through K2 of A.

**Parameters**

*N*

N is INTEGER The number of columns of the matrix A.

*A*

A is COMPLEX*16 array, dimension (LDA,N) On entry, the matrix of column dimension N to which the row interchanges will be applied. On exit, the permuted matrix.

*LDA*

LDA is INTEGER The leading dimension of the array A.

*K1*

K1 is INTEGER The first element of IPIV for which a row interchange will be done.

*K2*

K2 is INTEGER (K2-K1+1) is the number of elements of IPIV for which a row interchange will be done.

*IPIV*

IPIV is INTEGER array, dimension (K1+(K2-K1)*abs(INCX)) The vector of pivot indices. Only the elements in positions K1 through K1+(K2-K1)*abs(INCX) of IPIV are accessed. IPIV(K1+(K-K1)*abs(INCX)) = L implies rows K and L are to be interchanged.

*INCX*

INCX is INTEGER The increment between successive values of IPIV. If INCX is negative, the pivots are applied in reverse order.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

Modified by R. C. Whaley, Computer Science Dept., Univ. of Tenn., Knoxville, USA

## subroutine zlat2c (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex, dimension( ldsa, * ) SA, integer LDSA, integer INFO)¶

**ZLAT2C**converts a double complex triangular matrix to a complex triangular matrix.

**Purpose:**

ZLAT2C converts a COMPLEX*16 triangular matrix, SA, to a COMPLEX triangular matrix, A. RMAX is the overflow for the SINGLE PRECISION arithmetic ZLAT2C checks that all the entries of A are between -RMAX and RMAX. If not the conversion is aborted and a flag is raised. This is an auxiliary routine so there is no argument checking.

**Parameters**

*UPLO*

UPLO is CHARACTER*1 = 'U': A is upper triangular; = 'L': A is lower triangular.

*N*

N is INTEGER The number of rows and columns of the matrix A. N >= 0.

*A*

A is COMPLEX*16 array, dimension (LDA,N) On entry, the N-by-N triangular coefficient matrix A.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*SA*

SA is COMPLEX array, dimension (LDSA,N) Only the UPLO part of SA is referenced. On exit, if INFO=0, the N-by-N coefficient matrix SA; if INFO>0, the content of the UPLO part of SA is unspecified.

*LDSA*

LDSA is INTEGER The leading dimension of the array SA. LDSA >= max(1,M).

*INFO*

INFO is INTEGER = 0: successful exit. = 1: an entry of the matrix A is greater than the SINGLE PRECISION overflow threshold, in this case, the content of the UPLO part of SA in exit is unspecified.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlatbs (character UPLO, character TRANS, character DIAG, character NORMIN, integer N, integer KD, complex*16, dimension( ldab, * ) AB, integer LDAB, complex*16, dimension( * ) X, double precision SCALE, double precision, dimension( * ) CNORM, integer INFO)¶

**ZLATBS**solves a triangular banded system of equations.

**Purpose:**

ZLATBS solves one of the triangular systems A * x = s*b, A**T * x = s*b, or A**H * x = s*b, with scaling to prevent overflow, where A is an upper or lower triangular band matrix. Here A**T denotes the transpose of A, x and b are n-element vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine ZTBSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a non-trivial solution to A*x = 0 is returned.

**Parameters**

*UPLO*

UPLO is CHARACTER*1 Specifies whether the matrix A is upper or lower triangular. = 'U': Upper triangular = 'L': Lower triangular

*TRANS*

TRANS is CHARACTER*1 Specifies the operation applied to A. = 'N': Solve A * x = s*b (No transpose) = 'T': Solve A**T * x = s*b (Transpose) = 'C': Solve A**H * x = s*b (Conjugate transpose)

*DIAG*

DIAG is CHARACTER*1 Specifies whether or not the matrix A is unit triangular. = 'N': Non-unit triangular = 'U': Unit triangular

*NORMIN*

NORMIN is CHARACTER*1 Specifies whether CNORM has been set or not. = 'Y': CNORM contains the column norms on entry = 'N': CNORM is not set on entry. On exit, the norms will be computed and stored in CNORM.

*N*

N is INTEGER The order of the matrix A. N >= 0.

*KD*

KD is INTEGER The number of subdiagonals or superdiagonals in the triangular matrix A. KD >= 0.

*AB*

AB is COMPLEX*16 array, dimension (LDAB,N) The upper or lower triangular band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).

*LDAB*

LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD+1.

*X*

X is COMPLEX*16 array, dimension (N) On entry, the right hand side b of the triangular system. On exit, X is overwritten by the solution vector x.

*SCALE*

SCALE is DOUBLE PRECISION The scaling factor s for the triangular system A * x = s*b, A**T * x = s*b, or A**H * x = s*b. If SCALE = 0, the matrix A is singular or badly scaled, and the vector x is an exact or approximate solution to A*x = 0.

*CNORM*

CNORM is DOUBLE PRECISION array, dimension (N) If NORMIN = 'Y', CNORM is an input argument and CNORM(j) contains the norm of the off-diagonal part of the j-th column of A. If TRANS = 'N', CNORM(j) must be greater than or equal to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j) must be greater than or equal to the 1-norm. If NORMIN = 'N', CNORM is an output argument and CNORM(j) returns the 1-norm of the offdiagonal part of the j-th column of A.

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

A rough bound on x is computed; if that is less than overflow, ZTBSV is called, otherwise, specific code is used which checks for possible overflow or divide-by-zero at every operation. A columnwise scheme is used for solving A*x = b. The basic algorithm if A is lower triangular is x[1:n] := b[1:n] for j = 1, ..., n x(j) := x(j) / A(j,j) x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j] end Define bounds on the components of x after j iterations of the loop: M(j) = bound on x[1:j] G(j) = bound on x[j+1:n] Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}. Then for iteration j+1 we have M(j+1) <= G(j) / | A(j+1,j+1) | G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] | <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | ) where CNORM(j+1) is greater than or equal to the infinity-norm of column j+1 of A, not counting the diagonal. Hence G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | ) 1<=i<=j and |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| ) 1<=i< j Since |x(j)| <= M(j), we use the Level 2 BLAS routine ZTBSV if the reciprocal of the largest M(j), j=1,..,n, is larger than max(underflow, 1/overflow). The bound on x(j) is also used to determine when a step in the columnwise method can be performed without fear of overflow. If the computed bound is greater than a large constant, x is scaled to prevent overflow, but if the bound overflows, x is set to 0, x(j) to 1, and scale to 0, and a non-trivial solution to A*x = 0 is found. Similarly, a row-wise scheme is used to solve A**T *x = b or A**H *x = b. The basic algorithm for A upper triangular is for j = 1, ..., n x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j) end We simultaneously compute two bounds G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j M(j) = bound on x(i), 1<=i<=j The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1. Then the bound on x(j) is M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) | <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| ) 1<=i<=j and we can safely call ZTBSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow).

## subroutine zlatdf (integer IJOB, integer N, complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) RHS, double precision RDSUM, double precision RDSCAL, integer, dimension( * ) IPIV, integer, dimension( * ) JPIV)¶

**ZLATDF**uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate.

**Purpose:**

ZLATDF computes the contribution to the reciprocal Dif-estimate by solving for x in Z * x = b, where b is chosen such that the norm of x is as large as possible. It is assumed that LU decomposition of Z has been computed by ZGETC2. On entry RHS = f holds the contribution from earlier solved sub-systems, and on return RHS = x. The factorization of Z returned by ZGETC2 has the form Z = P * L * U * Q, where P and Q are permutation matrices. L is lower triangular with unit diagonal elements and U is upper triangular.

**Parameters**

*IJOB*

IJOB is INTEGER IJOB = 2: First compute an approximative null-vector e of Z using ZGECON, e is normalized and solve for Zx = +-e - f with the sign giving the greater value of 2-norm(x). About 5 times as expensive as Default. IJOB .ne. 2: Local look ahead strategy where all entries of the r.h.s. b is chosen as either +1 or -1. Default.

*N*

N is INTEGER The number of columns of the matrix Z.

*Z*

Z is COMPLEX*16 array, dimension (LDZ, N) On entry, the LU part of the factorization of the n-by-n matrix Z computed by ZGETC2: Z = P * L * U * Q

*LDZ*

LDZ is INTEGER The leading dimension of the array Z. LDA >= max(1, N).

*RHS*

RHS is COMPLEX*16 array, dimension (N). On entry, RHS contains contributions from other subsystems. On exit, RHS contains the solution of the subsystem with entries according to the value of IJOB (see above).

*RDSUM*

RDSUM is DOUBLE PRECISION On entry, the sum of squares of computed contributions to the Dif-estimate under computation by ZTGSYL, where the scaling factor RDSCAL (see below) has been factored out. On exit, the corresponding sum of squares updated with the contributions from the current sub-system. If TRANS = 'T' RDSUM is not touched. NOTE: RDSUM only makes sense when ZTGSY2 is called by CTGSYL.

*RDSCAL*

RDSCAL is DOUBLE PRECISION On entry, scaling factor used to prevent overflow in RDSUM. On exit, RDSCAL is updated w.r.t. the current contributions in RDSUM. If TRANS = 'T', RDSCAL is not touched. NOTE: RDSCAL only makes sense when ZTGSY2 is called by ZTGSYL.

*IPIV*

IPIV is INTEGER array, dimension (N). The pivot indices; for 1 <= i <= N, row i of the matrix has been interchanged with row IPIV(i).

*JPIV*

JPIV is INTEGER array, dimension (N). The pivot indices; for 1 <= j <= N, column j of the matrix has been interchanged with column JPIV(j).

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

**Contributors:**

**References:**

## subroutine zlatps (character UPLO, character TRANS, character DIAG, character NORMIN, integer N, complex*16, dimension( * ) AP, complex*16, dimension( * ) X, double precision SCALE, double precision, dimension( * ) CNORM, integer INFO)¶

**ZLATPS**solves a triangular system of equations with the matrix held in packed storage.

**Purpose:**

ZLATPS solves one of the triangular systems A * x = s*b, A**T * x = s*b, or A**H * x = s*b, with scaling to prevent overflow, where A is an upper or lower triangular matrix stored in packed form. Here A**T denotes the transpose of A, A**H denotes the conjugate transpose of A, x and b are n-element vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine ZTPSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a non-trivial solution to A*x = 0 is returned.

**Parameters**

*UPLO*

*TRANS*

TRANS is CHARACTER*1 Specifies the operation applied to A. = 'N': Solve A * x = s*b (No transpose) = 'T': Solve A**T * x = s*b (Transpose) = 'C': Solve A**H * x = s*b (Conjugate transpose)

*DIAG*

*NORMIN*

NORMIN is CHARACTER*1 Specifies whether CNORM has been set or not. = 'Y': CNORM contains the column norms on entry = 'N': CNORM is not set on entry. On exit, the norms will be computed and stored in CNORM.

*N*

N is INTEGER The order of the matrix A. N >= 0.

*AP*

AP is COMPLEX*16 array, dimension (N*(N+1)/2) The upper or lower triangular matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

*X*

X is COMPLEX*16 array, dimension (N) On entry, the right hand side b of the triangular system. On exit, X is overwritten by the solution vector x.

*SCALE*

SCALE is DOUBLE PRECISION The scaling factor s for the triangular system A * x = s*b, A**T * x = s*b, or A**H * x = s*b. If SCALE = 0, the matrix A is singular or badly scaled, and the vector x is an exact or approximate solution to A*x = 0.

*CNORM*

CNORM is DOUBLE PRECISION array, dimension (N) If NORMIN = 'Y', CNORM is an input argument and CNORM(j) contains the norm of the off-diagonal part of the j-th column of A. If TRANS = 'N', CNORM(j) must be greater than or equal to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j) must be greater than or equal to the 1-norm. If NORMIN = 'N', CNORM is an output argument and CNORM(j) returns the 1-norm of the offdiagonal part of the j-th column of A.

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

A rough bound on x is computed; if that is less than overflow, ZTPSV is called, otherwise, specific code is used which checks for possible overflow or divide-by-zero at every operation. A columnwise scheme is used for solving A*x = b. The basic algorithm if A is lower triangular is x[1:n] := b[1:n] for j = 1, ..., n x(j) := x(j) / A(j,j) x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j] end Define bounds on the components of x after j iterations of the loop: M(j) = bound on x[1:j] G(j) = bound on x[j+1:n] Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}. Then for iteration j+1 we have M(j+1) <= G(j) / | A(j+1,j+1) | G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] | <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | ) where CNORM(j+1) is greater than or equal to the infinity-norm of column j+1 of A, not counting the diagonal. Hence G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | ) 1<=i<=j and |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| ) 1<=i< j Since |x(j)| <= M(j), we use the Level 2 BLAS routine ZTPSV if the reciprocal of the largest M(j), j=1,..,n, is larger than max(underflow, 1/overflow). The bound on x(j) is also used to determine when a step in the columnwise method can be performed without fear of overflow. If the computed bound is greater than a large constant, x is scaled to prevent overflow, but if the bound overflows, x is set to 0, x(j) to 1, and scale to 0, and a non-trivial solution to A*x = 0 is found. Similarly, a row-wise scheme is used to solve A**T *x = b or A**H *x = b. The basic algorithm for A upper triangular is for j = 1, ..., n x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j) end We simultaneously compute two bounds G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j M(j) = bound on x(i), 1<=i<=j The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1. Then the bound on x(j) is M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) | <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| ) 1<=i<=j and we can safely call ZTPSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow).

## subroutine zlatrd (character UPLO, integer N, integer NB, complex*16, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) E, complex*16, dimension( * ) TAU, complex*16, dimension( ldw, * ) W, integer LDW)¶

**ZLATRD**reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation.

**Purpose:**

ZLATRD reduces NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal form by a unitary similarity transformation Q**H * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A. If UPLO = 'U', ZLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied; if UPLO = 'L', ZLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied. This is an auxiliary routine called by ZHETRD.

**Parameters**

*UPLO*

UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored: = 'U': Upper triangular = 'L': Lower triangular

*N*

N is INTEGER The order of the matrix A.

*NB*

NB is INTEGER The number of rows and columns to be reduced.

*A*

A is COMPLEX*16 array, dimension (LDA,N) On entry, the Hermitian matrix A. If UPLO = 'U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit: if UPLO = 'U', the last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements above the diagonal with the array TAU, represent the unitary matrix Q as a product of elementary reflectors; if UPLO = 'L', the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements below the diagonal with the array TAU, represent the unitary matrix Q as a product of elementary reflectors. See Further Details.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*E*

E is DOUBLE PRECISION array, dimension (N-1) If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal elements of the last NB columns of the reduced matrix; if UPLO = 'L', E(1:nb) contains the subdiagonal elements of the first NB columns of the reduced matrix.

*TAU*

TAU is COMPLEX*16 array, dimension (N-1) The scalar factors of the elementary reflectors, stored in TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'. See Further Details.

*W*

W is COMPLEX*16 array, dimension (LDW,NB) The n-by-nb matrix W required to update the unreduced part of A.

*LDW*

LDW is INTEGER The leading dimension of the array W. LDW >= max(1,N).

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors Q = H(n) H(n-1) . . . H(n-nb+1). Each H(i) has the form H(i) = I - tau * v * v**H where tau is a complex scalar, and v is a complex vector with v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), and tau in TAU(i-1). If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors Q = H(1) H(2) . . . H(nb). Each H(i) has the form H(i) = I - tau * v * v**H where tau is a complex scalar, and v is a complex vector with v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and tau in TAU(i). The elements of the vectors v together form the n-by-nb matrix V which is needed, with W, to apply the transformation to the unreduced part of the matrix, using a Hermitian rank-2k update of the form: A := A - V*W**H - W*V**H. The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2: if UPLO = 'U': if UPLO = 'L': ( a a a v4 v5 ) ( d ) ( a a v4 v5 ) ( 1 d ) ( a 1 v5 ) ( v1 1 a ) ( d 1 ) ( v1 v2 a a ) ( d ) ( v1 v2 a a a ) where d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that is unchanged, and vi denotes an element of the vector defining H(i).

## subroutine zlatrs (character UPLO, character TRANS, character DIAG, character NORMIN, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) X, double precision SCALE, double precision, dimension( * ) CNORM, integer INFO)¶

**ZLATRS**solves a triangular system of equations with the scale factor set to prevent overflow.

**Purpose:**

ZLATRS solves one of the triangular systems A * x = s*b, A**T * x = s*b, or A**H * x = s*b, with scaling to prevent overflow. Here A is an upper or lower triangular matrix, A**T denotes the transpose of A, A**H denotes the conjugate transpose of A, x and b are n-element vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine ZTRSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a non-trivial solution to A*x = 0 is returned.

**Parameters**

*UPLO*

*TRANS*

TRANS is CHARACTER*1 Specifies the operation applied to A. = 'N': Solve A * x = s*b (No transpose) = 'T': Solve A**T * x = s*b (Transpose) = 'C': Solve A**H * x = s*b (Conjugate transpose)

*DIAG*

*NORMIN*

NORMIN is CHARACTER*1 Specifies whether CNORM has been set or not. = 'Y': CNORM contains the column norms on entry = 'N': CNORM is not set on entry. On exit, the norms will be computed and stored in CNORM.

*N*

N is INTEGER The order of the matrix A. N >= 0.

*A*

A is COMPLEX*16 array, dimension (LDA,N) The triangular matrix A. If UPLO = 'U', the leading n by n upper triangular part of the array A contains the upper triangular matrix, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading n by n lower triangular part of the array A contains the lower triangular matrix, and the strictly upper triangular part of A is not referenced. If DIAG = 'U', the diagonal elements of A are also not referenced and are assumed to be 1.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max (1,N).

*X*

X is COMPLEX*16 array, dimension (N) On entry, the right hand side b of the triangular system. On exit, X is overwritten by the solution vector x.

*SCALE*

SCALE is DOUBLE PRECISION The scaling factor s for the triangular system A * x = s*b, A**T * x = s*b, or A**H * x = s*b. If SCALE = 0, the matrix A is singular or badly scaled, and the vector x is an exact or approximate solution to A*x = 0.

*CNORM*

CNORM is DOUBLE PRECISION array, dimension (N) If NORMIN = 'Y', CNORM is an input argument and CNORM(j) contains the norm of the off-diagonal part of the j-th column of A. If TRANS = 'N', CNORM(j) must be greater than or equal to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j) must be greater than or equal to the 1-norm. If NORMIN = 'N', CNORM is an output argument and CNORM(j) returns the 1-norm of the offdiagonal part of the j-th column of A.

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

A rough bound on x is computed; if that is less than overflow, ZTRSV is called, otherwise, specific code is used which checks for possible overflow or divide-by-zero at every operation. A columnwise scheme is used for solving A*x = b. The basic algorithm if A is lower triangular is x[1:n] := b[1:n] for j = 1, ..., n x(j) := x(j) / A(j,j) x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j] end Define bounds on the components of x after j iterations of the loop: M(j) = bound on x[1:j] G(j) = bound on x[j+1:n] Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}. Then for iteration j+1 we have M(j+1) <= G(j) / | A(j+1,j+1) | G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] | <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | ) where CNORM(j+1) is greater than or equal to the infinity-norm of column j+1 of A, not counting the diagonal. Hence G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | ) 1<=i<=j and |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| ) 1<=i< j Since |x(j)| <= M(j), we use the Level 2 BLAS routine ZTRSV if the reciprocal of the largest M(j), j=1,..,n, is larger than max(underflow, 1/overflow). The bound on x(j) is also used to determine when a step in the columnwise method can be performed without fear of overflow. If the computed bound is greater than a large constant, x is scaled to prevent overflow, but if the bound overflows, x is set to 0, x(j) to 1, and scale to 0, and a non-trivial solution to A*x = 0 is found. Similarly, a row-wise scheme is used to solve A**T *x = b or A**H *x = b. The basic algorithm for A upper triangular is for j = 1, ..., n x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j) end We simultaneously compute two bounds G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j M(j) = bound on x(i), 1<=i<=j The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1. Then the bound on x(j) is M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) | <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| ) 1<=i<=j and we can safely call ZTRSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow).

## subroutine zlauu2 (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer INFO)¶

**ZLAUU2**computes the product UUH or LHL, where U and L are upper or lower triangular matrices (unblocked algorithm).

**Purpose:**

ZLAUU2 computes the product U * U**H or L**H * L, where the triangular factor U or L is stored in the upper or lower triangular part of the array A. If UPLO = 'U' or 'u' then the upper triangle of the result is stored, overwriting the factor U in A. If UPLO = 'L' or 'l' then the lower triangle of the result is stored, overwriting the factor L in A. This is the unblocked form of the algorithm, calling Level 2 BLAS.

**Parameters**

*UPLO*

UPLO is CHARACTER*1 Specifies whether the triangular factor stored in the array A is upper or lower triangular: = 'U': Upper triangular = 'L': Lower triangular

*N*

N is INTEGER The order of the triangular factor U or L. N >= 0.

*A*

A is COMPLEX*16 array, dimension (LDA,N) On entry, the triangular factor U or L. On exit, if UPLO = 'U', the upper triangle of A is overwritten with the upper triangle of the product U * U**H; if UPLO = 'L', the lower triangle of A is overwritten with the lower triangle of the product L**H * L.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zlauum (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer INFO)¶

**ZLAUUM**computes the product UUH or LHL, where U and L are upper or lower triangular matrices (blocked algorithm).

**Purpose:**

ZLAUUM computes the product U * U**H or L**H * L, where the triangular factor U or L is stored in the upper or lower triangular part of the array A. If UPLO = 'U' or 'u' then the upper triangle of the result is stored, overwriting the factor U in A. If UPLO = 'L' or 'l' then the lower triangle of the result is stored, overwriting the factor L in A. This is the blocked form of the algorithm, calling Level 3 BLAS.

**Parameters**

*UPLO*

UPLO is CHARACTER*1 Specifies whether the triangular factor stored in the array A is upper or lower triangular: = 'U': Upper triangular = 'L': Lower triangular

*N*

N is INTEGER The order of the triangular factor U or L. N >= 0.

*A*

A is COMPLEX*16 array, dimension (LDA,N) On entry, the triangular factor U or L. On exit, if UPLO = 'U', the upper triangle of A is overwritten with the upper triangle of the product U * U**H; if UPLO = 'L', the lower triangle of A is overwritten with the lower triangle of the product L**H * L.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zrot (integer N, complex*16, dimension( * ) CX, integer INCX, complex*16, dimension( * ) CY, integer INCY, double precision C, complex*16 S)¶

**ZROT**applies a plane rotation with real cosine and complex sine to a pair of complex vectors.

**Purpose:**

ZROT applies a plane rotation, where the cos (C) is real and the sin (S) is complex, and the vectors CX and CY are complex.

**Parameters**

*N*

N is INTEGER The number of elements in the vectors CX and CY.

*CX*

CX is COMPLEX*16 array, dimension (N) On input, the vector X. On output, CX is overwritten with C*X + S*Y.

*INCX*

INCX is INTEGER The increment between successive values of CX. INCX <> 0.

*CY*

CY is COMPLEX*16 array, dimension (N) On input, the vector Y. On output, CY is overwritten with -CONJG(S)*X + C*Y.

*INCY*

INCY is INTEGER The increment between successive values of CY. INCX <> 0.

*C*

C is DOUBLE PRECISION

*S*

S is COMPLEX*16 C and S define a rotation [ C S ] [ -conjg(S) C ] where C*C + S*CONJG(S) = 1.0.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zspmv (character UPLO, integer N, complex*16 ALPHA, complex*16, dimension( * ) AP, complex*16, dimension( * ) X, integer INCX, complex*16 BETA, complex*16, dimension( * ) Y, integer INCY)¶

**ZSPMV**computes a matrix-vector product for complex vectors using a complex symmetric packed matrix

**Purpose:**

ZSPMV performs the matrix-vector operation y := alpha*A*x + beta*y, where alpha and beta are scalars, x and y are n element vectors and A is an n by n symmetric matrix, supplied in packed form.

**Parameters**

*UPLO*

UPLO is CHARACTER*1 On entry, UPLO specifies whether the upper or lower triangular part of the matrix A is supplied in the packed array AP as follows: UPLO = 'U' or 'u' The upper triangular part of A is supplied in AP. UPLO = 'L' or 'l' The lower triangular part of A is supplied in AP. Unchanged on exit.

*N*

N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. Unchanged on exit.

*ALPHA*

ALPHA is COMPLEX*16 On entry, ALPHA specifies the scalar alpha. Unchanged on exit.

*AP*

AP is COMPLEX*16 array, dimension at least ( ( N*( N + 1 ) )/2 ). Before entry, with UPLO = 'U' or 'u', the array AP must contain the upper triangular part of the symmetric matrix packed sequentially, column by column, so that AP( 1 ) contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 1, 2 ) and a( 2, 2 ) respectively, and so on. Before entry, with UPLO = 'L' or 'l', the array AP must contain the lower triangular part of the symmetric matrix packed sequentially, column by column, so that AP( 1 ) contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 2, 1 ) and a( 3, 1 ) respectively, and so on. Unchanged on exit.

*X*

X is COMPLEX*16 array, dimension at least ( 1 + ( N - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the N- element vector x. Unchanged on exit.

*INCX*

INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. Unchanged on exit.

*BETA*

BETA is COMPLEX*16 On entry, BETA specifies the scalar beta. When BETA is supplied as zero then Y need not be set on input. Unchanged on exit.

*Y*

Y is COMPLEX*16 array, dimension at least ( 1 + ( N - 1 )*abs( INCY ) ). Before entry, the incremented array Y must contain the n element vector y. On exit, Y is overwritten by the updated vector y.

*INCY*

INCY is INTEGER On entry, INCY specifies the increment for the elements of Y. INCY must not be zero. Unchanged on exit.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine zspr (character UPLO, integer N, complex*16 ALPHA, complex*16, dimension( * ) X, integer INCX, complex*16, dimension( * ) AP)¶

**ZSPR**performs the symmetrical rank-1 update of a complex symmetric packed matrix.

**Purpose:**

ZSPR performs the symmetric rank 1 operation A := alpha*x*x**H + A, where alpha is a complex scalar, x is an n element vector and A is an n by n symmetric matrix, supplied in packed form.

**Parameters**

*UPLO*

UPLO is CHARACTER*1 On entry, UPLO specifies whether the upper or lower triangular part of the matrix A is supplied in the packed array AP as follows: UPLO = 'U' or 'u' The upper triangular part of A is supplied in AP. UPLO = 'L' or 'l' The lower triangular part of A is supplied in AP. Unchanged on exit.

*N*

N is INTEGER On entry, N specifies the order of the matrix A. N must be at least zero. Unchanged on exit.

*ALPHA*

ALPHA is COMPLEX*16 On entry, ALPHA specifies the scalar alpha. Unchanged on exit.

*X*

X is COMPLEX*16 array, dimension at least ( 1 + ( N - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the N- element vector x. Unchanged on exit.

*INCX*

INCX is INTEGER On entry, INCX specifies the increment for the elements of X. INCX must not be zero. Unchanged on exit.

*AP*

AP is COMPLEX*16 array, dimension at least ( ( N*( N + 1 ) )/2 ). Before entry, with UPLO = 'U' or 'u', the array AP must contain the upper triangular part of the symmetric matrix packed sequentially, column by column, so that AP( 1 ) contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 1, 2 ) and a( 2, 2 ) respectively, and so on. On exit, the array AP is overwritten by the upper triangular part of the updated matrix. Before entry, with UPLO = 'L' or 'l', the array AP must contain the lower triangular part of the symmetric matrix packed sequentially, column by column, so that AP( 1 ) contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 2, 1 ) and a( 3, 1 ) respectively, and so on. On exit, the array AP is overwritten by the lower triangular part of the updated matrix. Note that the imaginary parts of the diagonal elements need not be set, they are assumed to be zero, and on exit they are set to zero.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine ztprfb (character SIDE, character TRANS, character DIRECT, character STOREV, integer M, integer N, integer K, integer L, complex*16, dimension( ldv, * ) V, integer LDV, complex*16, dimension( ldt, * ) T, integer LDT, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldwork, * ) WORK, integer LDWORK)¶

**ZTPRFB**applies a real or complex 'triangular-pentagonal' blocked reflector to a real or complex matrix, which is composed of two blocks.

**Purpose:**

ZTPRFB applies a complex "triangular-pentagonal" block reflector H or its conjugate transpose H**H to a complex matrix C, which is composed of two blocks A and B, either from the left or right.

**Parameters**

*SIDE*

SIDE is CHARACTER*1 = 'L': apply H or H**H from the Left = 'R': apply H or H**H from the Right

*TRANS*

TRANS is CHARACTER*1 = 'N': apply H (No transpose) = 'C': apply H**H (Conjugate transpose)

*DIRECT*

DIRECT is CHARACTER*1 Indicates how H is formed from a product of elementary reflectors = 'F': H = H(1) H(2) . . . H(k) (Forward) = 'B': H = H(k) . . . H(2) H(1) (Backward)

*STOREV*

STOREV is CHARACTER*1 Indicates how the vectors which define the elementary reflectors are stored: = 'C': Columns = 'R': Rows

*M*

M is INTEGER The number of rows of the matrix B. M >= 0.

*N*

N is INTEGER The number of columns of the matrix B. N >= 0.

*K*

K is INTEGER The order of the matrix T, i.e. the number of elementary reflectors whose product defines the block reflector. K >= 0.

*L*

L is INTEGER The order of the trapezoidal part of V. K >= L >= 0. See Further Details.

*V*

V is COMPLEX*16 array, dimension (LDV,K) if STOREV = 'C' (LDV,M) if STOREV = 'R' and SIDE = 'L' (LDV,N) if STOREV = 'R' and SIDE = 'R' The pentagonal matrix V, which contains the elementary reflectors H(1), H(2), ..., H(K). See Further Details.

*LDV*

LDV is INTEGER The leading dimension of the array V. If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M); if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N); if STOREV = 'R', LDV >= K.

*T*

T is COMPLEX*16 array, dimension (LDT,K) The triangular K-by-K matrix T in the representation of the block reflector.

*LDT*

LDT is INTEGER The leading dimension of the array T. LDT >= K.

*A*

A is COMPLEX*16 array, dimension (LDA,N) if SIDE = 'L' or (LDA,K) if SIDE = 'R' On entry, the K-by-N or M-by-K matrix A. On exit, A is overwritten by the corresponding block of H*C or H**H*C or C*H or C*H**H. See Further Details.

*LDA*

LDA is INTEGER The leading dimension of the array A. If SIDE = 'L', LDA >= max(1,K); If SIDE = 'R', LDA >= max(1,M).

*B*

B is COMPLEX*16 array, dimension (LDB,N) On entry, the M-by-N matrix B. On exit, B is overwritten by the corresponding block of H*C or H**H*C or C*H or C*H**H. See Further Details.

*LDB*

LDB is INTEGER The leading dimension of the array B. LDB >= max(1,M).

*WORK*

WORK is COMPLEX*16 array, dimension (LDWORK,N) if SIDE = 'L', (LDWORK,K) if SIDE = 'R'.

*LDWORK*

LDWORK is INTEGER The leading dimension of the array WORK. If SIDE = 'L', LDWORK >= K; if SIDE = 'R', LDWORK >= M.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

The matrix C is a composite matrix formed from blocks A and B. The block B is of size M-by-N; if SIDE = 'R', A is of size M-by-K, and if SIDE = 'L', A is of size K-by-N. If SIDE = 'R' and DIRECT = 'F', C = [A B]. If SIDE = 'L' and DIRECT = 'F', C = [A] [B]. If SIDE = 'R' and DIRECT = 'B', C = [B A]. If SIDE = 'L' and DIRECT = 'B', C = [B] [A]. The pentagonal matrix V is composed of a rectangular block V1 and a trapezoidal block V2. The size of the trapezoidal block is determined by the parameter L, where 0<=L<=K. If L=K, the V2 block of V is triangular; if L=0, there is no trapezoidal block, thus V = V1 is rectangular. If DIRECT = 'F' and STOREV = 'C': V = [V1] [V2] - V2 is upper trapezoidal (first L rows of K-by-K upper triangular) If DIRECT = 'F' and STOREV = 'R': V = [V1 V2] - V2 is lower trapezoidal (first L columns of K-by-K lower triangular) If DIRECT = 'B' and STOREV = 'C': V = [V2] [V1] - V2 is lower trapezoidal (last L rows of K-by-K lower triangular) If DIRECT = 'B' and STOREV = 'R': V = [V2 V1] - V2 is upper trapezoidal (last L columns of K-by-K upper triangular) If STOREV = 'C' and SIDE = 'L', V is M-by-K with V2 L-by-K. If STOREV = 'C' and SIDE = 'R', V is N-by-K with V2 L-by-K. If STOREV = 'R' and SIDE = 'L', V is K-by-M with V2 K-by-L. If STOREV = 'R' and SIDE = 'R', V is K-by-N with V2 K-by-L.

# Author¶

Generated automatically by Doxygen for LAPACK from the source code.Wed Sep 1 2021 | Version 3.10.0 |