## table of contents

realOTHEReigen(3) | LAPACK | realOTHEReigen(3) |

# NAME¶

realOTHEReigen

# SYNOPSIS¶

## Functions¶

subroutine **sbdsvdx** (UPLO, JOBZ, RANGE, N, D, E, VL, VU, IL,
IU, NS, S, Z, LDZ, WORK, IWORK, INFO)

**SBDSVDX** subroutine **sggglm** (N, M, P, A, LDA, B, LDB, D, X, Y,
WORK, LWORK, INFO)

**SGGGLM** subroutine **ssbev** (JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ,
WORK, INFO)

** SSBEV computes the eigenvalues and, optionally, the left and/or right
eigenvectors for OTHER matrices** subroutine **ssbev_2stage** (JOBZ,
UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, LWORK, INFO)

** SSBEV_2STAGE computes the eigenvalues and, optionally, the left and/or
right eigenvectors for OTHER matrices** subroutine **ssbevd** (JOBZ,
UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO)

** SSBEVD computes the eigenvalues and, optionally, the left and/or right
eigenvectors for OTHER matrices** subroutine **ssbevd_2stage** (JOBZ,
UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO)

** SSBEVD_2STAGE computes the eigenvalues and, optionally, the left and/or
right eigenvectors for OTHER matrices** subroutine **ssbevx** (JOBZ,
RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ,
WORK, IWORK, IFAIL, INFO)

** SSBEVX computes the eigenvalues and, optionally, the left and/or right
eigenvectors for OTHER matrices** subroutine **ssbevx_2stage** (JOBZ,
RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ,
WORK, LWORK, IWORK, IFAIL, INFO)

** SSBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or
right eigenvectors for OTHER matrices** subroutine **ssbgv** (JOBZ,
UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z, LDZ, WORK, INFO)

**SSBGV** subroutine **ssbgvd** (JOBZ, UPLO, N, KA, KB, AB, LDAB, BB,
LDBB, W, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO)

**SSBGVD** subroutine **ssbgvx** (JOBZ, RANGE, UPLO, N, KA, KB, AB,
LDAB, BB, LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK,
IFAIL, INFO)

**SSBGVX** subroutine **sspev** (JOBZ, UPLO, N, AP, W, Z, LDZ, WORK,
INFO)

** SSPEV computes the eigenvalues and, optionally, the left and/or right
eigenvectors for OTHER matrices** subroutine **sspevd** (JOBZ, UPLO, N,
AP, W, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO)

** SSPEVD computes the eigenvalues and, optionally, the left and/or right
eigenvectors for OTHER matrices** subroutine **sspevx** (JOBZ, RANGE,
UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO)

** SSPEVX computes the eigenvalues and, optionally, the left and/or right
eigenvectors for OTHER matrices** subroutine **sspgv** (ITYPE, JOBZ,
UPLO, N, AP, BP, W, Z, LDZ, WORK, INFO)

**SSPGV** subroutine **sspgvd** (ITYPE, JOBZ, UPLO, N, AP, BP, W, Z,
LDZ, WORK, LWORK, IWORK, LIWORK, INFO)

**SSPGVD** subroutine **sspgvx** (ITYPE, JOBZ, RANGE, UPLO, N, AP, BP,
VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO)

**SSPGVX** subroutine **sstev** (JOBZ, N, D, E, Z, LDZ, WORK, INFO)

** SSTEV computes the eigenvalues and, optionally, the left and/or right
eigenvectors for OTHER matrices** subroutine **sstevd** (JOBZ, N, D, E,
Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO)

** SSTEVD computes the eigenvalues and, optionally, the left and/or right
eigenvectors for OTHER matrices** subroutine **sstevr** (JOBZ, RANGE,
N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK,
LIWORK, INFO)

** SSTEVR computes the eigenvalues and, optionally, the left and/or right
eigenvectors for OTHER matrices** subroutine **sstevx** (JOBZ, RANGE,
N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO)

** SSTEVX computes the eigenvalues and, optionally, the left and/or right
eigenvectors for OTHER matrices**

# Detailed Description¶

This is the group of real Other Eigenvalue routines

# Function Documentation¶

## subroutine sbdsvdx (character UPLO, character JOBZ, character RANGE, integer N, real, dimension( * ) D, real, dimension( * ) E, real VL, real VU, integer IL, integer IU, integer NS, real, dimension( * ) S, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)¶

**SBDSVDX**

**Purpose:**

SBDSVDX computes the singular value decomposition (SVD) of a real

N-by-N (upper or lower) bidiagonal matrix B, B = U * S * VT,

where S is a diagonal matrix with non-negative diagonal elements

(the singular values of B), and U and VT are orthogonal matrices

of left and right singular vectors, respectively.

Given an upper bidiagonal B with diagonal D = [ d_1 d_2 ... d_N ]

and superdiagonal E = [ e_1 e_2 ... e_N-1 ], SBDSVDX computes the

singular value decompositon of B through the eigenvalues and

eigenvectors of the N*2-by-N*2 tridiagonal matrix

| 0 d_1 |

| d_1 0 e_1 |

TGK = | e_1 0 d_2 |

| d_2 . . |

| . . . |

If (s,u,v) is a singular triplet of B with ||u|| = ||v|| = 1, then

(+/-s,q), ||q|| = 1, are eigenpairs of TGK, with q = P * ( u' +/-v' ) /

sqrt(2) = ( v_1 u_1 v_2 u_2 ... v_n u_n ) / sqrt(2), and

P = [ e_{n+1} e_{1} e_{n+2} e_{2} ... ].

Given a TGK matrix, one can either a) compute -s,-v and change signs

so that the singular values (and corresponding vectors) are already in

descending order (as in SGESVD/SGESDD) or b) compute s,v and reorder

the values (and corresponding vectors). SBDSVDX implements a) by

calling SSTEVX (bisection plus inverse iteration, to be replaced

with a version of the Multiple Relative Robust Representation

algorithm. (See P. Willems and B. Lang, A framework for the MR^3

algorithm: theory and implementation, SIAM J. Sci. Comput.,

35:740-766, 2013.)

**Parameters**

*UPLO*

UPLO is CHARACTER*1

= 'U': B is upper bidiagonal;

= 'L': B is lower bidiagonal.

*JOBZ*

JOBZ is CHARACTER*1

= 'N': Compute singular values only;

= 'V': Compute singular values and singular vectors.

*RANGE*

RANGE is CHARACTER*1

= 'A': all singular values will be found.

= 'V': all singular values in the half-open interval [VL,VU)

will be found.

= 'I': the IL-th through IU-th singular values will be found.

*N*

N is INTEGER

The order of the bidiagonal matrix. N >= 0.

*D*

D is REAL array, dimension (N)

The n diagonal elements of the bidiagonal matrix B.

*E*

E is REAL array, dimension (max(1,N-1))

The (n-1) superdiagonal elements of the bidiagonal matrix

B in elements 1 to N-1.

*VL*

VL is REAL

If RANGE='V', the lower bound of the interval to

be searched for singular values. VU > VL.

Not referenced if RANGE = 'A' or 'I'.

*VU*

VU is REAL

If RANGE='V', the upper bound of the interval to

be searched for singular values. VU > VL.

Not referenced if RANGE = 'A' or 'I'.

*IL*

IL is INTEGER

If RANGE='I', the index of the

smallest singular value to be returned.

1 <= IL <= IU <= min(M,N), if min(M,N) > 0.

Not referenced if RANGE = 'A' or 'V'.

*IU*

IU is INTEGER

If RANGE='I', the index of the

largest singular value to be returned.

1 <= IL <= IU <= min(M,N), if min(M,N) > 0.

Not referenced if RANGE = 'A' or 'V'.

*NS*

NS is INTEGER

The total number of singular values found. 0 <= NS <= N.

If RANGE = 'A', NS = N, and if RANGE = 'I', NS = IU-IL+1.

*S*

S is REAL array, dimension (N)

The first NS elements contain the selected singular values in

ascending order.

*Z*

Z is REAL array, dimension (2*N,K)

If JOBZ = 'V', then if INFO = 0 the first NS columns of Z

contain the singular vectors of the matrix B corresponding to

the selected singular values, with U in rows 1 to N and V

in rows N+1 to N*2, i.e.

Z = [ U ]

[ V ]

If JOBZ = 'N', then Z is not referenced.

Note: The user must ensure that at least K = NS+1 columns are

supplied in the array Z; if RANGE = 'V', the exact value of

NS is not known in advance and an upper bound must be used.

*LDZ*

LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if

JOBZ = 'V', LDZ >= max(2,N*2).

*WORK*

WORK is REAL array, dimension (14*N)

*IWORK*

IWORK is INTEGER array, dimension (12*N)

If JOBZ = 'V', then if INFO = 0, the first NS elements of

IWORK are zero. If INFO > 0, then IWORK contains the indices

of the eigenvectors that failed to converge in DSTEVX.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, then i eigenvectors failed to converge

in SSTEVX. The indices of the eigenvectors

(as returned by SSTEVX) are stored in the

array IWORK.

if INFO = N*2 + 1, an internal error occurred.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine sggglm (integer N, integer M, integer P, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( * ) D, real, dimension( * ) X, real, dimension( * ) Y, real, dimension( * ) WORK, integer LWORK, integer INFO)¶

**SGGGLM**

**Purpose:**

SGGGLM solves a general Gauss-Markov linear model (GLM) problem:

minimize || y ||_2 subject to d = A*x + B*y

x

where A is an N-by-M matrix, B is an N-by-P matrix, and d is a

given N-vector. It is assumed that M <= N <= M+P, and

rank(A) = M and rank( A B ) = N.

Under these assumptions, the constrained equation is always

consistent, and there is a unique solution x and a minimal 2-norm

solution y, which is obtained using a generalized QR factorization

of the matrices (A, B) given by

A = Q*(R), B = Q*T*Z.

(0)

In particular, if matrix B is square nonsingular, then the problem

GLM is equivalent to the following weighted linear least squares

problem

minimize || inv(B)*(d-A*x) ||_2

x

where inv(B) denotes the inverse of B.

**Parameters**

*N*

N is INTEGER

The number of rows of the matrices A and B. N >= 0.

*M*

M is INTEGER

The number of columns of the matrix A. 0 <= M <= N.

*P*

P is INTEGER

The number of columns of the matrix B. P >= N-M.

*A*

A is REAL array, dimension (LDA,M)

On entry, the N-by-M matrix A.

On exit, the upper triangular part of the array A contains

the M-by-M upper triangular matrix R.

*LDA*

LDA is INTEGER

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

*B*

B is REAL array, dimension (LDB,P)

On entry, the N-by-P matrix B.

On exit, if N <= P, the upper triangle of the subarray

B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;

if N > P, the elements on and above the (N-P)th subdiagonal

contain the N-by-P upper trapezoidal matrix T.

*LDB*

LDB is INTEGER

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

*D*

D is REAL array, dimension (N)

On entry, D is the left hand side of the GLM equation.

On exit, D is destroyed.

*X*

X is REAL array, dimension (M)

*Y*

Y is REAL array, dimension (P)

On exit, X and Y are the solutions of the GLM problem.

*WORK*

WORK is REAL array, dimension (MAX(1,LWORK))

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK is INTEGER

The dimension of the array WORK. LWORK >= max(1,N+M+P).

For optimum performance, LWORK >= M+min(N,P)+max(N,P)*NB,

where NB is an upper bound for the optimal blocksizes for

SGEQRF, SGERQF, SORMQR and SORMRQ.

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit.

< 0: if INFO = -i, the i-th argument had an illegal value.

= 1: the upper triangular factor R associated with A in the

generalized QR factorization of the pair (A, B) is

singular, so that rank(A) < M; the least squares

solution could not be computed.

= 2: the bottom (N-M) by (N-M) part of the upper trapezoidal

factor T associated with B in the generalized QR

factorization of the pair (A, B) is singular, so that

rank( A B ) < N; the least squares solution could not

be computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine ssbev (character JOBZ, character UPLO, integer N, integer KD, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) W, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer INFO)¶

** SSBEV computes the eigenvalues and, optionally, the left
and/or right eigenvectors for OTHER matrices**

**Purpose:**

SSBEV computes all the eigenvalues and, optionally, eigenvectors of

a real symmetric band matrix A.

**Parameters**

*JOBZ*

JOBZ is CHARACTER*1

= 'N': Compute eigenvalues only;

= 'V': Compute eigenvalues and eigenvectors.

*UPLO*

UPLO is CHARACTER*1

= 'U': Upper triangle of A is stored;

= 'L': Lower triangle of A is stored.

*N*

N is INTEGER

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

*KD*

KD is INTEGER

The number of superdiagonals of the matrix A if UPLO = 'U',

or the number of subdiagonals if UPLO = 'L'. KD >= 0.

*AB*

AB is REAL 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, AB is overwritten by values generated during the

reduction to tridiagonal form. If UPLO = 'U', the first

superdiagonal and the diagonal of the tridiagonal matrix T

are returned in rows KD and KD+1 of AB, and if UPLO = 'L',

the diagonal and first subdiagonal of T are returned in the

first two rows of AB.

*LDAB*

LDAB is INTEGER

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

*W*

W is REAL array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

*Z*

Z is REAL array, dimension (LDZ, N)

If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal

eigenvectors of the matrix A, with the i-th column of Z

holding the eigenvector associated with W(i).

If JOBZ = 'N', then Z is not referenced.

*LDZ*

LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if

JOBZ = 'V', LDZ >= max(1,N).

*WORK*

WORK is REAL array, dimension (max(1,3*N-2))

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, the algorithm failed to converge; i

off-diagonal elements of an intermediate tridiagonal

form did not converge to zero.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine ssbev_2stage (character JOBZ, character UPLO, integer N, integer KD, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) W, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer LWORK, integer INFO)¶

** SSBEV_2STAGE computes the eigenvalues and, optionally, the
left and/or right eigenvectors for OTHER matrices**

**Purpose:**

SSBEV_2STAGE computes all the eigenvalues and, optionally, eigenvectors of

a real symmetric band matrix A using the 2stage technique for

the reduction to tridiagonal.

**Parameters**

*JOBZ*

JOBZ is CHARACTER*1

= 'N': Compute eigenvalues only;

= 'V': Compute eigenvalues and eigenvectors.

Not available in this release.

*UPLO*

UPLO is CHARACTER*1

= 'U': Upper triangle of A is stored;

= 'L': Lower triangle of A is stored.

*N*

N is INTEGER

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

*KD*

KD is INTEGER

The number of superdiagonals of the matrix A if UPLO = 'U',

or the number of subdiagonals if UPLO = 'L'. KD >= 0.

*AB*

AB is REAL 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, AB is overwritten by values generated during the

reduction to tridiagonal form. If UPLO = 'U', the first

superdiagonal and the diagonal of the tridiagonal matrix T

are returned in rows KD and KD+1 of AB, and if UPLO = 'L',

the diagonal and first subdiagonal of T are returned in the

first two rows of AB.

*LDAB*

LDAB is INTEGER

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

*W*

W is REAL array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

*Z*

Z is REAL array, dimension (LDZ, N)

If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal

eigenvectors of the matrix A, with the i-th column of Z

holding the eigenvector associated with W(i).

If JOBZ = 'N', then Z is not referenced.

*LDZ*

LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if

JOBZ = 'V', LDZ >= max(1,N).

*WORK*

WORK is REAL array, dimension LWORK

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK is INTEGER

The length of the array WORK. LWORK >= 1, when N <= 1;

otherwise

If JOBZ = 'N' and N > 1, LWORK must be queried.

LWORK = MAX(1, dimension) where

dimension = (2KD+1)*N + KD*NTHREADS + N

where KD is the size of the band.

NTHREADS is the number of threads used when

openMP compilation is enabled, otherwise =1.

If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available.

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, the algorithm failed to converge; i

off-diagonal elements of an intermediate tridiagonal

form did not converge to zero.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

All details about the 2stage techniques are available in:

Azzam Haidar, Hatem Ltaief, and Jack Dongarra.

Parallel reduction to condensed forms for symmetric eigenvalue problems

using aggregated fine-grained and memory-aware kernels. In Proceedings

of 2011 International Conference for High Performance Computing,

Networking, Storage and Analysis (SC '11), New York, NY, USA,

Article 8 , 11 pages.

http://doi.acm.org/10.1145/2063384.2063394

A. Haidar, J. Kurzak, P. Luszczek, 2013.

An improved parallel singular value algorithm and its implementation

for multicore hardware, In Proceedings of 2013 International Conference

for High Performance Computing, Networking, Storage and Analysis (SC '13).

Denver, Colorado, USA, 2013.

Article 90, 12 pages.

http://doi.acm.org/10.1145/2503210.2503292

A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.

A novel hybrid CPU-GPU generalized eigensolver for electronic structure

calculations based on fine-grained memory aware tasks.

International Journal of High Performance Computing Applications.

Volume 28 Issue 2, Pages 196-209, May 2014.

http://hpc.sagepub.com/content/28/2/196

## subroutine ssbevd (character JOBZ, character UPLO, integer N, integer KD, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) W, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)¶

** SSBEVD computes the eigenvalues and, optionally, the left
and/or right eigenvectors for OTHER matrices**

**Purpose:**

SSBEVD computes all the eigenvalues and, optionally, eigenvectors of

a real symmetric band matrix A. If eigenvectors are desired, it uses

a divide and conquer algorithm.

The divide and conquer algorithm makes very mild assumptions about

floating point arithmetic. It will work on machines with a guard

digit in add/subtract, or on those binary machines without guard

digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or

Cray-2. It could conceivably fail on hexadecimal or decimal machines

without guard digits, but we know of none.

**Parameters**

*JOBZ*

JOBZ is CHARACTER*1

= 'N': Compute eigenvalues only;

= 'V': Compute eigenvalues and eigenvectors.

*UPLO*

UPLO is CHARACTER*1

= 'U': Upper triangle of A is stored;

= 'L': Lower triangle of A is stored.

*N*

N is INTEGER

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

*KD*

KD is INTEGER

The number of superdiagonals of the matrix A if UPLO = 'U',

or the number of subdiagonals if UPLO = 'L'. KD >= 0.

*AB*

AB is REAL 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, AB is overwritten by values generated during the

reduction to tridiagonal form. If UPLO = 'U', the first

superdiagonal and the diagonal of the tridiagonal matrix T

are returned in rows KD and KD+1 of AB, and if UPLO = 'L',

the diagonal and first subdiagonal of T are returned in the

first two rows of AB.

*LDAB*

LDAB is INTEGER

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

*W*

W is REAL array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

*Z*

Z is REAL array, dimension (LDZ, N)

If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal

eigenvectors of the matrix A, with the i-th column of Z

holding the eigenvector associated with W(i).

If JOBZ = 'N', then Z is not referenced.

*LDZ*

LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if

JOBZ = 'V', LDZ >= max(1,N).

*WORK*

WORK is REAL array,

dimension (LWORK)

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK is INTEGER

The dimension of the array WORK.

IF N <= 1, LWORK must be at least 1.

If JOBZ = 'N' and N > 2, LWORK must be at least 2*N.

If JOBZ = 'V' and N > 2, LWORK must be at least

( 1 + 5*N + 2*N**2 ).

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the optimal sizes of the WORK and IWORK

arrays, returns these values as the first entries of the WORK

and IWORK arrays, and no error message related to LWORK or

LIWORK is issued by XERBLA.

*IWORK*

IWORK is INTEGER array, dimension (MAX(1,LIWORK))

On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

*LIWORK*

LIWORK is INTEGER

The dimension of the array IWORK.

If JOBZ = 'N' or N <= 1, LIWORK must be at least 1.

If JOBZ = 'V' and N > 2, LIWORK must be at least 3 + 5*N.

If LIWORK = -1, then a workspace query is assumed; the

routine only calculates the optimal sizes of the WORK and

IWORK arrays, returns these values as the first entries of

the WORK and IWORK arrays, and no error message related to

LWORK or LIWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, the algorithm failed to converge; i

off-diagonal elements of an intermediate tridiagonal

form did not converge to zero.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine ssbevd_2stage (character JOBZ, character UPLO, integer N, integer KD, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) W, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)¶

** SSBEVD_2STAGE computes the eigenvalues and, optionally, the
left and/or right eigenvectors for OTHER matrices**

**Purpose:**

SSBEVD_2STAGE computes all the eigenvalues and, optionally, eigenvectors of

a real symmetric band matrix A using the 2stage technique for

the reduction to tridiagonal. If eigenvectors are desired, it uses

a divide and conquer algorithm.

The divide and conquer algorithm makes very mild assumptions about

floating point arithmetic. It will work on machines with a guard

digit in add/subtract, or on those binary machines without guard

digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or

Cray-2. It could conceivably fail on hexadecimal or decimal machines

without guard digits, but we know of none.

**Parameters**

*JOBZ*

JOBZ is CHARACTER*1

= 'N': Compute eigenvalues only;

= 'V': Compute eigenvalues and eigenvectors.

Not available in this release.

*UPLO*

UPLO is CHARACTER*1

= 'U': Upper triangle of A is stored;

= 'L': Lower triangle of A is stored.

*N*

N is INTEGER

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

*KD*

KD is INTEGER

The number of superdiagonals of the matrix A if UPLO = 'U',

or the number of subdiagonals if UPLO = 'L'. KD >= 0.

*AB*

AB is REAL 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, AB is overwritten by values generated during the

reduction to tridiagonal form. If UPLO = 'U', the first

superdiagonal and the diagonal of the tridiagonal matrix T

are returned in rows KD and KD+1 of AB, and if UPLO = 'L',

the diagonal and first subdiagonal of T are returned in the

first two rows of AB.

*LDAB*

LDAB is INTEGER

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

*W*

W is REAL array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

*Z*

Z is REAL array, dimension (LDZ, N)

If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal

eigenvectors of the matrix A, with the i-th column of Z

holding the eigenvector associated with W(i).

If JOBZ = 'N', then Z is not referenced.

*LDZ*

LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if

JOBZ = 'V', LDZ >= max(1,N).

*WORK*

WORK is REAL array, dimension LWORK

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK is INTEGER

The length of the array WORK. LWORK >= 1, when N <= 1;

otherwise

If JOBZ = 'N' and N > 1, LWORK must be queried.

LWORK = MAX(1, dimension) where

dimension = (2KD+1)*N + KD*NTHREADS + N

where KD is the size of the band.

NTHREADS is the number of threads used when

openMP compilation is enabled, otherwise =1.

If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available.

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the optimal sizes of the WORK and IWORK

arrays, returns these values as the first entries of the WORK

and IWORK arrays, and no error message related to LWORK or

LIWORK is issued by XERBLA.

*IWORK*

IWORK is INTEGER array, dimension (MAX(1,LIWORK))

On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

*LIWORK*

LIWORK is INTEGER

The dimension of the array IWORK.

If JOBZ = 'N' or N <= 1, LIWORK must be at least 1.

If JOBZ = 'V' and N > 2, LIWORK must be at least 3 + 5*N.

If LIWORK = -1, then a workspace query is assumed; the

routine only calculates the optimal sizes of the WORK and

IWORK arrays, returns these values as the first entries of

the WORK and IWORK arrays, and no error message related to

LWORK or LIWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, the algorithm failed to converge; i

off-diagonal elements of an intermediate tridiagonal

form did not converge to zero.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

All details about the 2stage techniques are available in:

Azzam Haidar, Hatem Ltaief, and Jack Dongarra.

Parallel reduction to condensed forms for symmetric eigenvalue problems

using aggregated fine-grained and memory-aware kernels. In Proceedings

of 2011 International Conference for High Performance Computing,

Networking, Storage and Analysis (SC '11), New York, NY, USA,

Article 8 , 11 pages.

http://doi.acm.org/10.1145/2063384.2063394

A. Haidar, J. Kurzak, P. Luszczek, 2013.

An improved parallel singular value algorithm and its implementation

for multicore hardware, In Proceedings of 2013 International Conference

for High Performance Computing, Networking, Storage and Analysis (SC '13).

Denver, Colorado, USA, 2013.

Article 90, 12 pages.

http://doi.acm.org/10.1145/2503210.2503292

A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.

A novel hybrid CPU-GPU generalized eigensolver for electronic structure

calculations based on fine-grained memory aware tasks.

International Journal of High Performance Computing Applications.

Volume 28 Issue 2, Pages 196-209, May 2014.

http://hpc.sagepub.com/content/28/2/196

## subroutine ssbevx (character JOBZ, character RANGE, character UPLO, integer N, integer KD, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( ldq, * ) Q, integer LDQ, real VL, real VU, integer IL, integer IU, real ABSTOL, integer M, real, dimension( * ) W, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)¶

** SSBEVX computes the eigenvalues and, optionally, the left
and/or right eigenvectors for OTHER matrices**

**Purpose:**

SSBEVX computes selected eigenvalues and, optionally, eigenvectors

of a real symmetric band matrix A. Eigenvalues and eigenvectors can

be selected by specifying either a range of values or a range of

indices for the desired eigenvalues.

**Parameters**

*JOBZ*

JOBZ is CHARACTER*1

= 'N': Compute eigenvalues only;

= 'V': Compute eigenvalues and eigenvectors.

*RANGE*

RANGE is CHARACTER*1

= 'A': all eigenvalues will be found;

= 'V': all eigenvalues in the half-open interval (VL,VU]

will be found;

= 'I': the IL-th through IU-th eigenvalues will be found.

*UPLO*

UPLO is CHARACTER*1

= 'U': Upper triangle of A is stored;

= 'L': Lower triangle of A is stored.

*N*

N is INTEGER

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

*KD*

KD is INTEGER

The number of superdiagonals of the matrix A if UPLO = 'U',

or the number of subdiagonals if UPLO = 'L'. KD >= 0.

*AB*

AB is REAL 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, AB is overwritten by values generated during the

reduction to tridiagonal form. If UPLO = 'U', the first

superdiagonal and the diagonal of the tridiagonal matrix T

are returned in rows KD and KD+1 of AB, and if UPLO = 'L',

the diagonal and first subdiagonal of T are returned in the

first two rows of AB.

*LDAB*

LDAB is INTEGER

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

*Q*

Q is REAL array, dimension (LDQ, N)

If JOBZ = 'V', the N-by-N orthogonal matrix used in the

reduction to tridiagonal form.

If JOBZ = 'N', the array Q is not referenced.

*LDQ*

LDQ is INTEGER

The leading dimension of the array Q. If JOBZ = 'V', then

LDQ >= max(1,N).

*VL*

VL is REAL

If RANGE='V', the lower bound of the interval to

be searched for eigenvalues. VL < VU.

Not referenced if RANGE = 'A' or 'I'.

*VU*

VU is REAL

If RANGE='V', the upper bound of the interval to

be searched for eigenvalues. VL < VU.

Not referenced if RANGE = 'A' or 'I'.

*IL*

IL is INTEGER

If RANGE='I', the index of the

smallest eigenvalue to be returned.

1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.

Not referenced if RANGE = 'A' or 'V'.

*IU*

IU is INTEGER

If RANGE='I', the index of the

largest eigenvalue to be returned.

1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.

Not referenced if RANGE = 'A' or 'V'.

*ABSTOL*

ABSTOL is REAL

The absolute error tolerance for the eigenvalues.

An approximate eigenvalue is accepted as converged

when it is determined to lie in an interval [a,b]

of width less than or equal to

ABSTOL + EPS * max( |a|,|b| ) ,

where EPS is the machine precision. If ABSTOL is less than

or equal to zero, then EPS*|T| will be used in its place,

where |T| is the 1-norm of the tridiagonal matrix obtained

by reducing AB to tridiagonal form.

Eigenvalues will be computed most accurately when ABSTOL is

set to twice the underflow threshold 2*SLAMCH('S'), not zero.

If this routine returns with INFO>0, indicating that some

eigenvectors did not converge, try setting ABSTOL to

2*SLAMCH('S').

See "Computing Small Singular Values of Bidiagonal Matrices

with Guaranteed High Relative Accuracy," by Demmel and

Kahan, LAPACK Working Note #3.

*M*

M is INTEGER

The total number of eigenvalues found. 0 <= M <= N.

If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

*W*

W is REAL array, dimension (N)

The first M elements contain the selected eigenvalues in

ascending order.

*Z*

Z is REAL array, dimension (LDZ, max(1,M))

If JOBZ = 'V', then if INFO = 0, the first M columns of Z

contain the orthonormal eigenvectors of the matrix A

corresponding to the selected eigenvalues, with the i-th

column of Z holding the eigenvector associated with W(i).

If an eigenvector fails to converge, then that column of Z

contains the latest approximation to the eigenvector, and the

index of the eigenvector is returned in IFAIL.

If JOBZ = 'N', then Z is not referenced.

Note: the user must ensure that at least max(1,M) columns are

supplied in the array Z; if RANGE = 'V', the exact value of M

is not known in advance and an upper bound must be used.

*LDZ*

LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if

JOBZ = 'V', LDZ >= max(1,N).

*WORK*

WORK is REAL array, dimension (7*N)

*IWORK*

IWORK is INTEGER array, dimension (5*N)

*IFAIL*

IFAIL is INTEGER array, dimension (N)

If JOBZ = 'V', then if INFO = 0, the first M elements of

IFAIL are zero. If INFO > 0, then IFAIL contains the

indices of the eigenvectors that failed to converge.

If JOBZ = 'N', then IFAIL is not referenced.

*INFO*

INFO is INTEGER

= 0: successful exit.

< 0: if INFO = -i, the i-th argument had an illegal value.

> 0: if INFO = i, then i eigenvectors failed to converge.

Their indices are stored in array IFAIL.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine ssbevx_2stage (character JOBZ, character RANGE, character UPLO, integer N, integer KD, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( ldq, * ) Q, integer LDQ, real VL, real VU, integer IL, integer IU, real ABSTOL, integer M, real, dimension( * ) W, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)¶

** SSBEVX_2STAGE computes the eigenvalues and, optionally, the
left and/or right eigenvectors for OTHER matrices**

**Purpose:**

SSBEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors

of a real symmetric band matrix A using the 2stage technique for

the reduction to tridiagonal. Eigenvalues and eigenvectors can

be selected by specifying either a range of values or a range of

indices for the desired eigenvalues.

**Parameters**

*JOBZ*

JOBZ is CHARACTER*1

= 'N': Compute eigenvalues only;

= 'V': Compute eigenvalues and eigenvectors.

Not available in this release.

*RANGE*

RANGE is CHARACTER*1

= 'A': all eigenvalues will be found;

= 'V': all eigenvalues in the half-open interval (VL,VU]

will be found;

= 'I': the IL-th through IU-th eigenvalues will be found.

*UPLO*

UPLO is CHARACTER*1

= 'U': Upper triangle of A is stored;

= 'L': Lower triangle of A is stored.

*N*

N is INTEGER

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

*KD*

KD is INTEGER

The number of superdiagonals of the matrix A if UPLO = 'U',

or the number of subdiagonals if UPLO = 'L'. KD >= 0.

*AB*

AB is REAL 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, AB is overwritten by values generated during the

reduction to tridiagonal form. If UPLO = 'U', the first

superdiagonal and the diagonal of the tridiagonal matrix T

are returned in rows KD and KD+1 of AB, and if UPLO = 'L',

the diagonal and first subdiagonal of T are returned in the

first two rows of AB.

*LDAB*

LDAB is INTEGER

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

*Q*

Q is REAL array, dimension (LDQ, N)

If JOBZ = 'V', the N-by-N orthogonal matrix used in the

reduction to tridiagonal form.

If JOBZ = 'N', the array Q is not referenced.

*LDQ*

LDQ is INTEGER

The leading dimension of the array Q. If JOBZ = 'V', then

LDQ >= max(1,N).

*VL*

VL is REAL

If RANGE='V', the lower bound of the interval to

be searched for eigenvalues. VL < VU.

Not referenced if RANGE = 'A' or 'I'.

*VU*

VU is REAL

If RANGE='V', the upper bound of the interval to

be searched for eigenvalues. VL < VU.

Not referenced if RANGE = 'A' or 'I'.

*IL*

IL is INTEGER

If RANGE='I', the index of the

smallest eigenvalue to be returned.

1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.

Not referenced if RANGE = 'A' or 'V'.

*IU*

IU is INTEGER

If RANGE='I', the index of the

largest eigenvalue to be returned.

1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.

Not referenced if RANGE = 'A' or 'V'.

*ABSTOL*

ABSTOL is REAL

The absolute error tolerance for the eigenvalues.

An approximate eigenvalue is accepted as converged

when it is determined to lie in an interval [a,b]

of width less than or equal to

ABSTOL + EPS * max( |a|,|b| ) ,

where EPS is the machine precision. If ABSTOL is less than

or equal to zero, then EPS*|T| will be used in its place,

where |T| is the 1-norm of the tridiagonal matrix obtained

by reducing AB to tridiagonal form.

Eigenvalues will be computed most accurately when ABSTOL is

set to twice the underflow threshold 2*SLAMCH('S'), not zero.

If this routine returns with INFO>0, indicating that some

eigenvectors did not converge, try setting ABSTOL to

2*SLAMCH('S').

See "Computing Small Singular Values of Bidiagonal Matrices

with Guaranteed High Relative Accuracy," by Demmel and

Kahan, LAPACK Working Note #3.

*M*

M is INTEGER

The total number of eigenvalues found. 0 <= M <= N.

If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

*W*

W is REAL array, dimension (N)

The first M elements contain the selected eigenvalues in

ascending order.

*Z*

Z is REAL array, dimension (LDZ, max(1,M))

If JOBZ = 'V', then if INFO = 0, the first M columns of Z

contain the orthonormal eigenvectors of the matrix A

corresponding to the selected eigenvalues, with the i-th

column of Z holding the eigenvector associated with W(i).

If an eigenvector fails to converge, then that column of Z

contains the latest approximation to the eigenvector, and the

index of the eigenvector is returned in IFAIL.

If JOBZ = 'N', then Z is not referenced.

Note: the user must ensure that at least max(1,M) columns are

supplied in the array Z; if RANGE = 'V', the exact value of M

is not known in advance and an upper bound must be used.

*LDZ*

LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if

JOBZ = 'V', LDZ >= max(1,N).

*WORK*

WORK is REAL array, dimension (LWORK)

*LWORK*

LWORK is INTEGER

The length of the array WORK. LWORK >= 1, when N <= 1;

otherwise

If JOBZ = 'N' and N > 1, LWORK must be queried.

LWORK = MAX(1, 7*N, dimension) where

dimension = (2KD+1)*N + KD*NTHREADS + 2*N

where KD is the size of the band.

NTHREADS is the number of threads used when

openMP compilation is enabled, otherwise =1.

If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the optimal size of the WORK array, returns

this value as the first entry of the WORK array, and no error

message related to LWORK is issued by XERBLA.

*IWORK*

IWORK is INTEGER array, dimension (5*N)

*IFAIL*

IFAIL is INTEGER array, dimension (N)

If JOBZ = 'V', then if INFO = 0, the first M elements of

IFAIL are zero. If INFO > 0, then IFAIL contains the

indices of the eigenvectors that failed to converge.

If JOBZ = 'N', then IFAIL is not referenced.

*INFO*

INFO is INTEGER

= 0: successful exit.

< 0: if INFO = -i, the i-th argument had an illegal value.

> 0: if INFO = i, then i eigenvectors failed to converge.

Their indices are stored in array IFAIL.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

All details about the 2stage techniques are available in:

Azzam Haidar, Hatem Ltaief, and Jack Dongarra.

Parallel reduction to condensed forms for symmetric eigenvalue problems

using aggregated fine-grained and memory-aware kernels. In Proceedings

of 2011 International Conference for High Performance Computing,

Networking, Storage and Analysis (SC '11), New York, NY, USA,

Article 8 , 11 pages.

http://doi.acm.org/10.1145/2063384.2063394

A. Haidar, J. Kurzak, P. Luszczek, 2013.

An improved parallel singular value algorithm and its implementation

for multicore hardware, In Proceedings of 2013 International Conference

for High Performance Computing, Networking, Storage and Analysis (SC '13).

Denver, Colorado, USA, 2013.

Article 90, 12 pages.

http://doi.acm.org/10.1145/2503210.2503292

A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.

A novel hybrid CPU-GPU generalized eigensolver for electronic structure

calculations based on fine-grained memory aware tasks.

International Journal of High Performance Computing Applications.

Volume 28 Issue 2, Pages 196-209, May 2014.

http://hpc.sagepub.com/content/28/2/196

## subroutine ssbgv (character JOBZ, character UPLO, integer N, integer KA, integer KB, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( ldbb, * ) BB, integer LDBB, real, dimension( * ) W, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer INFO)¶

**SSBGV**

**Purpose:**

SSBGV computes all the eigenvalues, and optionally, the eigenvectors

of a real generalized symmetric-definite banded eigenproblem, of

the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric

and banded, and B is also positive definite.

**Parameters**

*JOBZ*

JOBZ is CHARACTER*1

= 'N': Compute eigenvalues only;

= 'V': Compute eigenvalues and eigenvectors.

*UPLO*

UPLO is CHARACTER*1

= 'U': Upper triangles of A and B are stored;

= 'L': Lower triangles of A and B are stored.

*N*

N is INTEGER

The order of the matrices A and B. N >= 0.

*KA*

KA is INTEGER

The number of superdiagonals of the matrix A if UPLO = 'U',

or the number of subdiagonals if UPLO = 'L'. KA >= 0.

*KB*

KB is INTEGER

The number of superdiagonals of the matrix B if UPLO = 'U',

or the number of subdiagonals if UPLO = 'L'. KB >= 0.

*AB*

AB is REAL array, dimension (LDAB, N)

On entry, the upper or lower triangle of the symmetric band

matrix A, stored in the first ka+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(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;

if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka).

On exit, the contents of AB are destroyed.

*LDAB*

LDAB is INTEGER

The leading dimension of the array AB. LDAB >= KA+1.

*BB*

BB is REAL array, dimension (LDBB, N)

On entry, the upper or lower triangle of the symmetric band

matrix B, stored in the first kb+1 rows of the array. The

j-th column of B is stored in the j-th column of the array BB

as follows:

if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;

if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb).

On exit, the factor S from the split Cholesky factorization

B = S**T*S, as returned by SPBSTF.

*LDBB*

LDBB is INTEGER

The leading dimension of the array BB. LDBB >= KB+1.

*W*

W is REAL array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

*Z*

Z is REAL array, dimension (LDZ, N)

If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of

eigenvectors, with the i-th column of Z holding the

eigenvector associated with W(i). The eigenvectors are

normalized so that Z**T*B*Z = I.

If JOBZ = 'N', then Z is not referenced.

*LDZ*

LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if

JOBZ = 'V', LDZ >= N.

*WORK*

WORK is REAL array, dimension (3*N)

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, and i is:

<= N: the algorithm failed to converge:

i off-diagonal elements of an intermediate

tridiagonal form did not converge to zero;

> N: if INFO = N + i, for 1 <= i <= N, then SPBSTF

returned INFO = i: B is not positive definite.

The factorization of B could not be completed and

no eigenvalues or eigenvectors were computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine ssbgvd (character JOBZ, character UPLO, integer N, integer KA, integer KB, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( ldbb, * ) BB, integer LDBB, real, dimension( * ) W, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)¶

**SSBGVD**

**Purpose:**

SSBGVD computes all the eigenvalues, and optionally, the eigenvectors

of a real generalized symmetric-definite banded eigenproblem, of the

form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric and

banded, and B is also positive definite. If eigenvectors are

desired, it uses a divide and conquer algorithm.

The divide and conquer algorithm makes very mild assumptions about

floating point arithmetic. It will work on machines with a guard

digit in add/subtract, or on those binary machines without guard

digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or

Cray-2. It could conceivably fail on hexadecimal or decimal machines

without guard digits, but we know of none.

**Parameters**

*JOBZ*

JOBZ is CHARACTER*1

= 'N': Compute eigenvalues only;

= 'V': Compute eigenvalues and eigenvectors.

*UPLO*

UPLO is CHARACTER*1

= 'U': Upper triangles of A and B are stored;

= 'L': Lower triangles of A and B are stored.

*N*

N is INTEGER

The order of the matrices A and B. N >= 0.

*KA*

KA is INTEGER

The number of superdiagonals of the matrix A if UPLO = 'U',

or the number of subdiagonals if UPLO = 'L'. KA >= 0.

*KB*

KB is INTEGER

The number of superdiagonals of the matrix B if UPLO = 'U',

or the number of subdiagonals if UPLO = 'L'. KB >= 0.

*AB*

AB is REAL array, dimension (LDAB, N)

On entry, the upper or lower triangle of the symmetric band

matrix A, stored in the first ka+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(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;

if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka).

On exit, the contents of AB are destroyed.

*LDAB*

LDAB is INTEGER

The leading dimension of the array AB. LDAB >= KA+1.

*BB*

BB is REAL array, dimension (LDBB, N)

On entry, the upper or lower triangle of the symmetric band

matrix B, stored in the first kb+1 rows of the array. The

j-th column of B is stored in the j-th column of the array BB

as follows:

if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;

if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb).

On exit, the factor S from the split Cholesky factorization

B = S**T*S, as returned by SPBSTF.

*LDBB*

LDBB is INTEGER

The leading dimension of the array BB. LDBB >= KB+1.

*W*

W is REAL array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

*Z*

Z is REAL array, dimension (LDZ, N)

If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of

eigenvectors, with the i-th column of Z holding the

eigenvector associated with W(i). The eigenvectors are

normalized so Z**T*B*Z = I.

If JOBZ = 'N', then Z is not referenced.

*LDZ*

LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if

JOBZ = 'V', LDZ >= max(1,N).

*WORK*

WORK is REAL array, dimension (MAX(1,LWORK))

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK is INTEGER

The dimension of the array WORK.

If N <= 1, LWORK >= 1.

If JOBZ = 'N' and N > 1, LWORK >= 3*N.

If JOBZ = 'V' and N > 1, LWORK >= 1 + 5*N + 2*N**2.

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the optimal sizes of the WORK and IWORK

arrays, returns these values as the first entries of the WORK

and IWORK arrays, and no error message related to LWORK or

LIWORK is issued by XERBLA.

*IWORK*

IWORK is INTEGER array, dimension (MAX(1,LIWORK))

On exit, if LIWORK > 0, IWORK(1) returns the optimal LIWORK.

*LIWORK*

LIWORK is INTEGER

The dimension of the array IWORK.

If JOBZ = 'N' or N <= 1, LIWORK >= 1.

If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N.

If LIWORK = -1, then a workspace query is assumed; the

routine only calculates the optimal sizes of the WORK and

IWORK arrays, returns these values as the first entries of

the WORK and IWORK arrays, and no error message related to

LWORK or LIWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, and i is:

<= N: the algorithm failed to converge:

i off-diagonal elements of an intermediate

tridiagonal form did not converge to zero;

> N: if INFO = N + i, for 1 <= i <= N, then SPBSTF

returned INFO = i: B is not positive definite.

The factorization of B could not be completed and

no eigenvalues or eigenvectors were computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

## subroutine ssbgvx (character JOBZ, character RANGE, character UPLO, integer N, integer KA, integer KB, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( ldbb, * ) BB, integer LDBB, real, dimension( ldq, * ) Q, integer LDQ, real VL, real VU, integer IL, integer IU, real ABSTOL, integer M, real, dimension( * ) W, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)¶

**SSBGVX**

**Purpose:**

SSBGVX computes selected eigenvalues, and optionally, eigenvectors

of a real generalized symmetric-definite banded eigenproblem, of

the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric

and banded, and B is also positive definite. Eigenvalues and

eigenvectors can be selected by specifying either all eigenvalues,

a range of values or a range of indices for the desired eigenvalues.

**Parameters**

*JOBZ*

JOBZ is CHARACTER*1

= 'N': Compute eigenvalues only;

= 'V': Compute eigenvalues and eigenvectors.

*RANGE*

RANGE is CHARACTER*1

= 'A': all eigenvalues will be found.

= 'V': all eigenvalues in the half-open interval (VL,VU]

will be found.

= 'I': the IL-th through IU-th eigenvalues will be found.

*UPLO*

UPLO is CHARACTER*1

= 'U': Upper triangles of A and B are stored;

= 'L': Lower triangles of A and B are stored.

*N*

N is INTEGER

The order of the matrices A and B. N >= 0.

*KA*

KA is INTEGER

The number of superdiagonals of the matrix A if UPLO = 'U',

or the number of subdiagonals if UPLO = 'L'. KA >= 0.

*KB*

KB is INTEGER

The number of superdiagonals of the matrix B if UPLO = 'U',

or the number of subdiagonals if UPLO = 'L'. KB >= 0.

*AB*

AB is REAL array, dimension (LDAB, N)

On entry, the upper or lower triangle of the symmetric band

matrix A, stored in the first ka+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(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;

if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka).

On exit, the contents of AB are destroyed.

*LDAB*

LDAB is INTEGER

The leading dimension of the array AB. LDAB >= KA+1.

*BB*

BB is REAL array, dimension (LDBB, N)

On entry, the upper or lower triangle of the symmetric band

matrix B, stored in the first kb+1 rows of the array. The

j-th column of B is stored in the j-th column of the array BB

as follows:

if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;

if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb).

On exit, the factor S from the split Cholesky factorization

B = S**T*S, as returned by SPBSTF.

*LDBB*

LDBB is INTEGER

The leading dimension of the array BB. LDBB >= KB+1.

*Q*

Q is REAL array, dimension (LDQ, N)

If JOBZ = 'V', the n-by-n matrix used in the reduction of

A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x,

and consequently C to tridiagonal form.

If JOBZ = 'N', the array Q is not referenced.

*LDQ*

LDQ is INTEGER

The leading dimension of the array Q. If JOBZ = 'N',

LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).

*VL*

VL is REAL

If RANGE='V', the lower bound of the interval to

be searched for eigenvalues. VL < VU.

Not referenced if RANGE = 'A' or 'I'.

*VU*

VU is REAL

If RANGE='V', the upper bound of the interval to

be searched for eigenvalues. VL < VU.

Not referenced if RANGE = 'A' or 'I'.

*IL*

IL is INTEGER

If RANGE='I', the index of the

smallest eigenvalue to be returned.

1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.

Not referenced if RANGE = 'A' or 'V'.

*IU*

IU is INTEGER

If RANGE='I', the index of the

largest eigenvalue to be returned.

1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.

Not referenced if RANGE = 'A' or 'V'.

*ABSTOL*

ABSTOL is REAL

The absolute error tolerance for the eigenvalues.

An approximate eigenvalue is accepted as converged

when it is determined to lie in an interval [a,b]

of width less than or equal to

ABSTOL + EPS * max( |a|,|b| ) ,

where EPS is the machine precision. If ABSTOL is less than

or equal to zero, then EPS*|T| will be used in its place,

where |T| is the 1-norm of the tridiagonal matrix obtained

by reducing A to tridiagonal form.

Eigenvalues will be computed most accurately when ABSTOL is

set to twice the underflow threshold 2*SLAMCH('S'), not zero.

If this routine returns with INFO>0, indicating that some

eigenvectors did not converge, try setting ABSTOL to

2*SLAMCH('S').

*M*

M is INTEGER

The total number of eigenvalues found. 0 <= M <= N.

If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

*W*

W is REAL array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

*Z*

Z is REAL array, dimension (LDZ, N)

If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of

eigenvectors, with the i-th column of Z holding the

eigenvector associated with W(i). The eigenvectors are

normalized so Z**T*B*Z = I.

If JOBZ = 'N', then Z is not referenced.

*LDZ*

LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if

JOBZ = 'V', LDZ >= max(1,N).

*WORK*

WORK is REAL array, dimension (7*N)

*IWORK*

IWORK is INTEGER array, dimension (5*N)

*IFAIL*

IFAIL is INTEGER array, dimension (M)

If JOBZ = 'V', then if INFO = 0, the first M elements of

IFAIL are zero. If INFO > 0, then IFAIL contains the

indices of the eigenvalues that failed to converge.

If JOBZ = 'N', then IFAIL is not referenced.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

<= N: if INFO = i, then i eigenvectors failed to converge.

Their indices are stored in IFAIL.

> N: SPBSTF returned an error code; i.e.,

if INFO = N + i, for 1 <= i <= N, then the leading

minor of order i of B is not positive definite.

The factorization of B could not be completed and

no eigenvalues or eigenvectors were computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

## subroutine sspev (character JOBZ, character UPLO, integer N, real, dimension( * ) AP, real, dimension( * ) W, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer INFO)¶

** SSPEV computes the eigenvalues and, optionally, the left
and/or right eigenvectors for OTHER matrices**

**Purpose:**

SSPEV computes all the eigenvalues and, optionally, eigenvectors of a

real symmetric matrix A in packed storage.

**Parameters**

*JOBZ*

JOBZ is CHARACTER*1

= 'N': Compute eigenvalues only;

= 'V': Compute eigenvalues and eigenvectors.

*UPLO*

UPLO is CHARACTER*1

= 'U': Upper triangle of A is stored;

= 'L': Lower triangle of A is stored.

*N*

N is INTEGER

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

*AP*

AP is REAL 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)*(2*n-j)/2) = A(i,j) for j<=i<=n.

On exit, AP is overwritten by values generated during the

reduction to tridiagonal form. If UPLO = 'U', the diagonal

and first superdiagonal of the tridiagonal matrix T overwrite

the corresponding elements of A, and if UPLO = 'L', the

diagonal and first subdiagonal of T overwrite the

corresponding elements of A.

*W*

W is REAL array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

*Z*

Z is REAL array, dimension (LDZ, N)

If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal

eigenvectors of the matrix A, with the i-th column of Z

holding the eigenvector associated with W(i).

If JOBZ = 'N', then Z is not referenced.

*LDZ*

LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if

JOBZ = 'V', LDZ >= max(1,N).

*WORK*

WORK is REAL array, dimension (3*N)

*INFO*

INFO is INTEGER

= 0: successful exit.

< 0: if INFO = -i, the i-th argument had an illegal value.

> 0: if INFO = i, the algorithm failed to converge; i

off-diagonal elements of an intermediate tridiagonal

form did not converge to zero.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine sspevd (character JOBZ, character UPLO, integer N, real, dimension( * ) AP, real, dimension( * ) W, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)¶

** SSPEVD computes the eigenvalues and, optionally, the left
and/or right eigenvectors for OTHER matrices**

**Purpose:**

SSPEVD computes all the eigenvalues and, optionally, eigenvectors

of a real symmetric matrix A in packed storage. If eigenvectors are

desired, it uses a divide and conquer algorithm.

The divide and conquer algorithm makes very mild assumptions about

floating point arithmetic. It will work on machines with a guard

digit in add/subtract, or on those binary machines without guard

digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or

Cray-2. It could conceivably fail on hexadecimal or decimal machines

without guard digits, but we know of none.

**Parameters**

*JOBZ*

JOBZ is CHARACTER*1

= 'N': Compute eigenvalues only;

= 'V': Compute eigenvalues and eigenvectors.

*UPLO*

UPLO is CHARACTER*1

= 'U': Upper triangle of A is stored;

= 'L': Lower triangle of A is stored.

*N*

N is INTEGER

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

*AP*

AP is REAL 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)*(2*n-j)/2) = A(i,j) for j<=i<=n.

On exit, AP is overwritten by values generated during the

reduction to tridiagonal form. If UPLO = 'U', the diagonal

and first superdiagonal of the tridiagonal matrix T overwrite

the corresponding elements of A, and if UPLO = 'L', the

diagonal and first subdiagonal of T overwrite the

corresponding elements of A.

*W*

W is REAL array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

*Z*

Z is REAL array, dimension (LDZ, N)

If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal

eigenvectors of the matrix A, with the i-th column of Z

holding the eigenvector associated with W(i).

If JOBZ = 'N', then Z is not referenced.

*LDZ*

LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if

JOBZ = 'V', LDZ >= max(1,N).

*WORK*

WORK is REAL array, dimension (MAX(1,LWORK))

On exit, if INFO = 0, WORK(1) returns the required LWORK.

*LWORK*

LWORK is INTEGER

The dimension of the array WORK.

If N <= 1, LWORK must be at least 1.

If JOBZ = 'N' and N > 1, LWORK must be at least 2*N.

If JOBZ = 'V' and N > 1, LWORK must be at least

1 + 6*N + N**2.

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the required sizes of the WORK and IWORK

arrays, returns these values as the first entries of the WORK

and IWORK arrays, and no error message related to LWORK or

LIWORK is issued by XERBLA.

*IWORK*

IWORK is INTEGER array, dimension (MAX(1,LIWORK))

On exit, if INFO = 0, IWORK(1) returns the required LIWORK.

*LIWORK*

LIWORK is INTEGER

The dimension of the array IWORK.

If JOBZ = 'N' or N <= 1, LIWORK must be at least 1.

If JOBZ = 'V' and N > 1, LIWORK must be at least 3 + 5*N.

If LIWORK = -1, then a workspace query is assumed; the

routine only calculates the required sizes of the WORK and

IWORK arrays, returns these values as the first entries of

the WORK and IWORK arrays, and no error message related to

LWORK or LIWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value.

> 0: if INFO = i, the algorithm failed to converge; i

off-diagonal elements of an intermediate tridiagonal

form did not converge to zero.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine sspevx (character JOBZ, character RANGE, character UPLO, integer N, real, dimension( * ) AP, real VL, real VU, integer IL, integer IU, real ABSTOL, integer M, real, dimension( * ) W, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)¶

** SSPEVX computes the eigenvalues and, optionally, the left
and/or right eigenvectors for OTHER matrices**

**Purpose:**

SSPEVX computes selected eigenvalues and, optionally, eigenvectors

of a real symmetric matrix A in packed storage. Eigenvalues/vectors

can be selected by specifying either a range of values or a range of

indices for the desired eigenvalues.

**Parameters**

*JOBZ*

JOBZ is CHARACTER*1

= 'N': Compute eigenvalues only;

= 'V': Compute eigenvalues and eigenvectors.

*RANGE*

RANGE is CHARACTER*1

= 'A': all eigenvalues will be found;

= 'V': all eigenvalues in the half-open interval (VL,VU]

will be found;

= 'I': the IL-th through IU-th eigenvalues will be found.

*UPLO*

UPLO is CHARACTER*1

= 'U': Upper triangle of A is stored;

= 'L': Lower triangle of A is stored.

*N*

N is INTEGER

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

*AP*

AP is REAL 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)*(2*n-j)/2) = A(i,j) for j<=i<=n.

On exit, AP is overwritten by values generated during the

reduction to tridiagonal form. If UPLO = 'U', the diagonal

and first superdiagonal of the tridiagonal matrix T overwrite

the corresponding elements of A, and if UPLO = 'L', the

diagonal and first subdiagonal of T overwrite the

corresponding elements of A.

*VL*

VL is REAL

If RANGE='V', the lower bound of the interval to

be searched for eigenvalues. VL < VU.

Not referenced if RANGE = 'A' or 'I'.

*VU*

VU is REAL

If RANGE='V', the upper bound of the interval to

be searched for eigenvalues. VL < VU.

Not referenced if RANGE = 'A' or 'I'.

*IL*

IL is INTEGER

If RANGE='I', the index of the

smallest eigenvalue to be returned.

1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.

Not referenced if RANGE = 'A' or 'V'.

*IU*

IU is INTEGER

If RANGE='I', the index of the

largest eigenvalue to be returned.

1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.

Not referenced if RANGE = 'A' or 'V'.

*ABSTOL*

ABSTOL is REAL

The absolute error tolerance for the eigenvalues.

An approximate eigenvalue is accepted as converged

when it is determined to lie in an interval [a,b]

of width less than or equal to

ABSTOL + EPS * max( |a|,|b| ) ,

where EPS is the machine precision. If ABSTOL is less than

or equal to zero, then EPS*|T| will be used in its place,

where |T| is the 1-norm of the tridiagonal matrix obtained

by reducing AP to tridiagonal form.

Eigenvalues will be computed most accurately when ABSTOL is

set to twice the underflow threshold 2*SLAMCH('S'), not zero.

If this routine returns with INFO>0, indicating that some

eigenvectors did not converge, try setting ABSTOL to

2*SLAMCH('S').

See "Computing Small Singular Values of Bidiagonal Matrices

with Guaranteed High Relative Accuracy," by Demmel and

Kahan, LAPACK Working Note #3.

*M*

M is INTEGER

The total number of eigenvalues found. 0 <= M <= N.

If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

*W*

W is REAL array, dimension (N)

If INFO = 0, the selected eigenvalues in ascending order.

*Z*

Z is REAL array, dimension (LDZ, max(1,M))

If JOBZ = 'V', then if INFO = 0, the first M columns of Z

contain the orthonormal eigenvectors of the matrix A

corresponding to the selected eigenvalues, with the i-th

column of Z holding the eigenvector associated with W(i).

If an eigenvector fails to converge, then that column of Z

contains the latest approximation to the eigenvector, and the

index of the eigenvector is returned in IFAIL.

If JOBZ = 'N', then Z is not referenced.

Note: the user must ensure that at least max(1,M) columns are

supplied in the array Z; if RANGE = 'V', the exact value of M

is not known in advance and an upper bound must be used.

*LDZ*

LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if

JOBZ = 'V', LDZ >= max(1,N).

*WORK*

WORK is REAL array, dimension (8*N)

*IWORK*

IWORK is INTEGER array, dimension (5*N)

*IFAIL*

IFAIL is INTEGER array, dimension (N)

If JOBZ = 'V', then if INFO = 0, the first M elements of

IFAIL are zero. If INFO > 0, then IFAIL contains the

indices of the eigenvectors that failed to converge.

If JOBZ = 'N', then IFAIL is not referenced.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, then i eigenvectors failed to converge.

Their indices are stored in array IFAIL.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine sspgv (integer ITYPE, character JOBZ, character UPLO, integer N, real, dimension( * ) AP, real, dimension( * ) BP, real, dimension( * ) W, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer INFO)¶

**SSPGV**

**Purpose:**

SSPGV computes all the eigenvalues and, optionally, the eigenvectors

of a real generalized symmetric-definite eigenproblem, of the form

A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x.

Here A and B are assumed to be symmetric, stored in packed format,

and B is also positive definite.

**Parameters**

*ITYPE*

ITYPE is INTEGER

Specifies the problem type to be solved:

= 1: A*x = (lambda)*B*x

= 2: A*B*x = (lambda)*x

= 3: B*A*x = (lambda)*x

*JOBZ*

JOBZ is CHARACTER*1

= 'N': Compute eigenvalues only;

= 'V': Compute eigenvalues and eigenvectors.

*UPLO*

UPLO is CHARACTER*1

= 'U': Upper triangles of A and B are stored;

= 'L': Lower triangles of A and B are stored.

*N*

N is INTEGER

The order of the matrices A and B. N >= 0.

*AP*

AP is REAL 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)*(2*n-j)/2) = A(i,j) for j<=i<=n.

On exit, the contents of AP are destroyed.

*BP*

BP is REAL array, dimension (N*(N+1)/2)

On entry, the upper or lower triangle of the symmetric matrix

B, packed columnwise in a linear array. The j-th column of B

is stored in the array BP as follows:

if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;

if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.

On exit, the triangular factor U or L from the Cholesky

factorization B = U**T*U or B = L*L**T, in the same storage

format as B.

*W*

W is REAL array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

*Z*

Z is REAL array, dimension (LDZ, N)

If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of

eigenvectors. The eigenvectors are normalized as follows:

if ITYPE = 1 or 2, Z**T*B*Z = I;

if ITYPE = 3, Z**T*inv(B)*Z = I.

If JOBZ = 'N', then Z is not referenced.

*LDZ*

LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if

JOBZ = 'V', LDZ >= max(1,N).

*WORK*

WORK is REAL array, dimension (3*N)

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: SPPTRF or SSPEV returned an error code:

<= N: if INFO = i, SSPEV failed to converge;

i off-diagonal elements of an intermediate

tridiagonal form did not converge to zero.

> N: if INFO = n + i, for 1 <= i <= n, then the leading

minor of order i of B is not positive definite.

The factorization of B could not be completed and

no eigenvalues or eigenvectors were computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine sspgvd (integer ITYPE, character JOBZ, character UPLO, integer N, real, dimension( * ) AP, real, dimension( * ) BP, real, dimension( * ) W, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)¶

**SSPGVD**

**Purpose:**

SSPGVD computes all the eigenvalues, and optionally, the eigenvectors

of a real generalized symmetric-definite eigenproblem, of the form

A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and

B are assumed to be symmetric, stored in packed format, and B is also

positive definite.

If eigenvectors are desired, it uses a divide and conquer algorithm.

The divide and conquer algorithm makes very mild assumptions about

floating point arithmetic. It will work on machines with a guard

digit in add/subtract, or on those binary machines without guard

digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or

Cray-2. It could conceivably fail on hexadecimal or decimal machines

without guard digits, but we know of none.

**Parameters**

*ITYPE*

ITYPE is INTEGER

Specifies the problem type to be solved:

= 1: A*x = (lambda)*B*x

= 2: A*B*x = (lambda)*x

= 3: B*A*x = (lambda)*x

*JOBZ*

JOBZ is CHARACTER*1

= 'N': Compute eigenvalues only;

= 'V': Compute eigenvalues and eigenvectors.

*UPLO*

UPLO is CHARACTER*1

= 'U': Upper triangles of A and B are stored;

= 'L': Lower triangles of A and B are stored.

*N*

N is INTEGER

The order of the matrices A and B. N >= 0.

*AP*

AP is REAL 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)*(2*n-j)/2) = A(i,j) for j<=i<=n.

On exit, the contents of AP are destroyed.

*BP*

BP is REAL array, dimension (N*(N+1)/2)

On entry, the upper or lower triangle of the symmetric matrix

B, packed columnwise in a linear array. The j-th column of B

is stored in the array BP as follows:

if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;

if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.

On exit, the triangular factor U or L from the Cholesky

factorization B = U**T*U or B = L*L**T, in the same storage

format as B.

*W*

W is REAL array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

*Z*

Z is REAL array, dimension (LDZ, N)

If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of

eigenvectors. The eigenvectors are normalized as follows:

if ITYPE = 1 or 2, Z**T*B*Z = I;

if ITYPE = 3, Z**T*inv(B)*Z = I.

If JOBZ = 'N', then Z is not referenced.

*LDZ*

LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if

JOBZ = 'V', LDZ >= max(1,N).

*WORK*

WORK is REAL array, dimension (MAX(1,LWORK))

On exit, if INFO = 0, WORK(1) returns the required LWORK.

*LWORK*

LWORK is INTEGER

The dimension of the array WORK.

If N <= 1, LWORK >= 1.

If JOBZ = 'N' and N > 1, LWORK >= 2*N.

If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the required sizes of the WORK and IWORK

arrays, returns these values as the first entries of the WORK

and IWORK arrays, and no error message related to LWORK or

LIWORK is issued by XERBLA.

*IWORK*

IWORK is INTEGER array, dimension (MAX(1,LIWORK))

On exit, if INFO = 0, IWORK(1) returns the required LIWORK.

*LIWORK*

LIWORK is INTEGER

The dimension of the array IWORK.

If JOBZ = 'N' or N <= 1, LIWORK >= 1.

If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N.

If LIWORK = -1, then a workspace query is assumed; the

routine only calculates the required sizes of the WORK and

IWORK arrays, returns these values as the first entries of

the WORK and IWORK arrays, and no error message related to

LWORK or LIWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: SPPTRF or SSPEVD returned an error code:

<= N: if INFO = i, SSPEVD failed to converge;

i off-diagonal elements of an intermediate

tridiagonal form did not converge to zero;

> N: if INFO = N + i, for 1 <= i <= N, then the leading

minor of order i of B is not positive definite.

The factorization of B could not be completed and

no eigenvalues or eigenvectors were computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

## subroutine sspgvx (integer ITYPE, character JOBZ, character RANGE, character UPLO, integer N, real, dimension( * ) AP, real, dimension( * ) BP, real VL, real VU, integer IL, integer IU, real ABSTOL, integer M, real, dimension( * ) W, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)¶

**SSPGVX**

**Purpose:**

SSPGVX computes selected eigenvalues, and optionally, eigenvectors

of a real generalized symmetric-definite eigenproblem, of the form

A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A

and B are assumed to be symmetric, stored in packed storage, and B

is also positive definite. Eigenvalues and eigenvectors can be

selected by specifying either a range of values or a range of indices

for the desired eigenvalues.

**Parameters**

*ITYPE*

ITYPE is INTEGER

Specifies the problem type to be solved:

= 1: A*x = (lambda)*B*x

= 2: A*B*x = (lambda)*x

= 3: B*A*x = (lambda)*x

*JOBZ*

JOBZ is CHARACTER*1

= 'N': Compute eigenvalues only;

= 'V': Compute eigenvalues and eigenvectors.

*RANGE*

RANGE is CHARACTER*1

= 'A': all eigenvalues will be found.

= 'V': all eigenvalues in the half-open interval (VL,VU]

will be found.

= 'I': the IL-th through IU-th eigenvalues will be found.

*UPLO*

UPLO is CHARACTER*1

= 'U': Upper triangle of A and B are stored;

= 'L': Lower triangle of A and B are stored.

*N*

N is INTEGER

The order of the matrix pencil (A,B). N >= 0.

*AP*

AP is REAL 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)*(2*n-j)/2) = A(i,j) for j<=i<=n.

On exit, the contents of AP are destroyed.

*BP*

BP is REAL array, dimension (N*(N+1)/2)

On entry, the upper or lower triangle of the symmetric matrix

B, packed columnwise in a linear array. The j-th column of B

is stored in the array BP as follows:

if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;

if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.

On exit, the triangular factor U or L from the Cholesky

factorization B = U**T*U or B = L*L**T, in the same storage

format as B.

*VL*

VL is REAL

If RANGE='V', the lower bound of the interval to

be searched for eigenvalues. VL < VU.

Not referenced if RANGE = 'A' or 'I'.

*VU*

VU is REAL

If RANGE='V', the upper bound of the interval to

be searched for eigenvalues. VL < VU.

Not referenced if RANGE = 'A' or 'I'.

*IL*

IL is INTEGER

If RANGE='I', the index of the

smallest eigenvalue to be returned.

1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.

Not referenced if RANGE = 'A' or 'V'.

*IU*

IU is INTEGER

If RANGE='I', the index of the

largest eigenvalue to be returned.

1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.

Not referenced if RANGE = 'A' or 'V'.

*ABSTOL*

ABSTOL is REAL

The absolute error tolerance for the eigenvalues.

An approximate eigenvalue is accepted as converged

when it is determined to lie in an interval [a,b]

of width less than or equal to

ABSTOL + EPS * max( |a|,|b| ) ,

where EPS is the machine precision. If ABSTOL is less than

or equal to zero, then EPS*|T| will be used in its place,

where |T| is the 1-norm of the tridiagonal matrix obtained

by reducing A to tridiagonal form.

Eigenvalues will be computed most accurately when ABSTOL is

set to twice the underflow threshold 2*SLAMCH('S'), not zero.

If this routine returns with INFO>0, indicating that some

eigenvectors did not converge, try setting ABSTOL to

2*SLAMCH('S').

*M*

M is INTEGER

The total number of eigenvalues found. 0 <= M <= N.

If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

*W*

W is REAL array, dimension (N)

On normal exit, the first M elements contain the selected

eigenvalues in ascending order.

*Z*

Z is REAL array, dimension (LDZ, max(1,M))

If JOBZ = 'N', then Z is not referenced.

If JOBZ = 'V', then if INFO = 0, the first M columns of Z

contain the orthonormal eigenvectors of the matrix A

corresponding to the selected eigenvalues, with the i-th

column of Z holding the eigenvector associated with W(i).

The eigenvectors are normalized as follows:

if ITYPE = 1 or 2, Z**T*B*Z = I;

if ITYPE = 3, Z**T*inv(B)*Z = I.

If an eigenvector fails to converge, then that column of Z

contains the latest approximation to the eigenvector, and the

index of the eigenvector is returned in IFAIL.

Note: the user must ensure that at least max(1,M) columns are

supplied in the array Z; if RANGE = 'V', the exact value of M

is not known in advance and an upper bound must be used.

*LDZ*

LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if

JOBZ = 'V', LDZ >= max(1,N).

*WORK*

WORK is REAL array, dimension (8*N)

*IWORK*

IWORK is INTEGER array, dimension (5*N)

*IFAIL*

IFAIL is INTEGER array, dimension (N)

If JOBZ = 'V', then if INFO = 0, the first M elements of

IFAIL are zero. If INFO > 0, then IFAIL contains the

indices of the eigenvectors that failed to converge.

If JOBZ = 'N', then IFAIL is not referenced.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: SPPTRF or SSPEVX returned an error code:

<= N: if INFO = i, SSPEVX failed to converge;

i eigenvectors failed to converge. Their indices

are stored in array IFAIL.

> N: if INFO = N + i, for 1 <= i <= N, then the leading

minor of order i of B is not positive definite.

The factorization of B could not be completed and

no eigenvalues or eigenvectors were computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

## subroutine sstev (character JOBZ, integer N, real, dimension( * ) D, real, dimension( * ) E, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer INFO)¶

** SSTEV computes the eigenvalues and, optionally, the left
and/or right eigenvectors for OTHER matrices**

**Purpose:**

SSTEV computes all eigenvalues and, optionally, eigenvectors of a

real symmetric tridiagonal matrix A.

**Parameters**

*JOBZ*

JOBZ is CHARACTER*1

= 'N': Compute eigenvalues only;

= 'V': Compute eigenvalues and eigenvectors.

*N*

N is INTEGER

The order of the matrix. N >= 0.

*D*

D is REAL array, dimension (N)

On entry, the n diagonal elements of the tridiagonal matrix

A.

On exit, if INFO = 0, the eigenvalues in ascending order.

*E*

E is REAL array, dimension (N-1)

On entry, the (n-1) subdiagonal elements of the tridiagonal

matrix A, stored in elements 1 to N-1 of E.

On exit, the contents of E are destroyed.

*Z*

Z is REAL array, dimension (LDZ, N)

If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal

eigenvectors of the matrix A, with the i-th column of Z

holding the eigenvector associated with D(i).

If JOBZ = 'N', then Z is not referenced.

*LDZ*

LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if

JOBZ = 'V', LDZ >= max(1,N).

*WORK*

WORK is REAL array, dimension (max(1,2*N-2))

If JOBZ = 'N', WORK is not referenced.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, the algorithm failed to converge; i

off-diagonal elements of E did not converge to zero.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine sstevd (character JOBZ, integer N, real, dimension( * ) D, real, dimension( * ) E, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)¶

** SSTEVD computes the eigenvalues and, optionally, the left
and/or right eigenvectors for OTHER matrices**

**Purpose:**

SSTEVD computes all eigenvalues and, optionally, eigenvectors of a

real symmetric tridiagonal matrix. If eigenvectors are desired, it

uses a divide and conquer algorithm.

The divide and conquer algorithm makes very mild assumptions about

floating point arithmetic. It will work on machines with a guard

digit in add/subtract, or on those binary machines without guard

digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or

Cray-2. It could conceivably fail on hexadecimal or decimal machines

without guard digits, but we know of none.

**Parameters**

*JOBZ*

JOBZ is CHARACTER*1

= 'N': Compute eigenvalues only;

= 'V': Compute eigenvalues and eigenvectors.

*N*

N is INTEGER

The order of the matrix. N >= 0.

*D*

D is REAL array, dimension (N)

On entry, the n diagonal elements of the tridiagonal matrix

A.

On exit, if INFO = 0, the eigenvalues in ascending order.

*E*

E is REAL array, dimension (N-1)

On entry, the (n-1) subdiagonal elements of the tridiagonal

matrix A, stored in elements 1 to N-1 of E.

On exit, the contents of E are destroyed.

*Z*

Z is REAL array, dimension (LDZ, N)

If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal

eigenvectors of the matrix A, with the i-th column of Z

holding the eigenvector associated with D(i).

If JOBZ = 'N', then Z is not referenced.

*LDZ*

LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if

JOBZ = 'V', LDZ >= max(1,N).

*WORK*

WORK is REAL array,

dimension (LWORK)

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK is INTEGER

The dimension of the array WORK.

If JOBZ = 'N' or N <= 1 then LWORK must be at least 1.

If JOBZ = 'V' and N > 1 then LWORK must be at least

( 1 + 4*N + N**2 ).

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the optimal sizes of the WORK and IWORK

arrays, returns these values as the first entries of the WORK

and IWORK arrays, and no error message related to LWORK or

LIWORK is issued by XERBLA.

*IWORK*

IWORK is INTEGER array, dimension (MAX(1,LIWORK))

On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

*LIWORK*

LIWORK is INTEGER

The dimension of the array IWORK.

If JOBZ = 'N' or N <= 1 then LIWORK must be at least 1.

If JOBZ = 'V' and N > 1 then LIWORK must be at least 3+5*N.

If LIWORK = -1, then a workspace query is assumed; the

routine only calculates the optimal sizes of the WORK and

IWORK arrays, returns these values as the first entries of

the WORK and IWORK arrays, and no error message related to

LWORK or LIWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, the algorithm failed to converge; i

off-diagonal elements of E did not converge to zero.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine sstevr (character JOBZ, character RANGE, integer N, real, dimension( * ) D, real, dimension( * ) E, real VL, real VU, integer IL, integer IU, real ABSTOL, integer M, real, dimension( * ) W, real, dimension( ldz, * ) Z, integer LDZ, integer, dimension( * ) ISUPPZ, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)¶

** SSTEVR computes the eigenvalues and, optionally, the left
and/or right eigenvectors for OTHER matrices**

**Purpose:**

SSTEVR computes selected eigenvalues and, optionally, eigenvectors

of a real symmetric tridiagonal matrix T. Eigenvalues and

eigenvectors can be selected by specifying either a range of values

or a range of indices for the desired eigenvalues.

Whenever possible, SSTEVR calls SSTEMR to compute the

eigenspectrum using Relatively Robust Representations. SSTEMR

computes eigenvalues by the dqds algorithm, while orthogonal

eigenvectors are computed from various "good" L D L^T representations

(also known as Relatively Robust Representations). Gram-Schmidt

orthogonalization is avoided as far as possible. More specifically,

the various steps of the algorithm are as follows. For the i-th

unreduced block of T,

(a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T

is a relatively robust representation,

(b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high

relative accuracy by the dqds algorithm,

(c) If there is a cluster of close eigenvalues, "choose" sigma_i

close to the cluster, and go to step (a),

(d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T,

compute the corresponding eigenvector by forming a

rank-revealing twisted factorization.

The desired accuracy of the output can be specified by the input

parameter ABSTOL.

For more details, see "A new O(n^2) algorithm for the symmetric

tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon,

Computer Science Division Technical Report No. UCB//CSD-97-971,

UC Berkeley, May 1997.

Note 1 : SSTEVR calls SSTEMR when the full spectrum is requested

on machines which conform to the ieee-754 floating point standard.

SSTEVR calls SSTEBZ and SSTEIN on non-ieee machines and

when partial spectrum requests are made.

Normal execution of SSTEMR may create NaNs and infinities and

hence may abort due to a floating point exception in environments

which do not handle NaNs and infinities in the ieee standard default

manner.

**Parameters**

*JOBZ*

JOBZ is CHARACTER*1

= 'N': Compute eigenvalues only;

= 'V': Compute eigenvalues and eigenvectors.

*RANGE*

RANGE is CHARACTER*1

= 'A': all eigenvalues will be found.

= 'V': all eigenvalues in the half-open interval (VL,VU]

will be found.

= 'I': the IL-th through IU-th eigenvalues will be found.

For RANGE = 'V' or 'I' and IU - IL < N - 1, SSTEBZ and

SSTEIN are called

*N*

N is INTEGER

The order of the matrix. N >= 0.

*D*

D is REAL array, dimension (N)

On entry, the n diagonal elements of the tridiagonal matrix

A.

On exit, D may be multiplied by a constant factor chosen

to avoid over/underflow in computing the eigenvalues.

*E*

E is REAL array, dimension (max(1,N-1))

On entry, the (n-1) subdiagonal elements of the tridiagonal

matrix A in elements 1 to N-1 of E.

On exit, E may be multiplied by a constant factor chosen

to avoid over/underflow in computing the eigenvalues.

*VL*

VL is REAL

If RANGE='V', the lower bound of the interval to

be searched for eigenvalues. VL < VU.

Not referenced if RANGE = 'A' or 'I'.

*VU*

VU is REAL

If RANGE='V', the upper bound of the interval to

be searched for eigenvalues. VL < VU.

Not referenced if RANGE = 'A' or 'I'.

*IL*

IL is INTEGER

If RANGE='I', the index of the

smallest eigenvalue to be returned.

1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.

Not referenced if RANGE = 'A' or 'V'.

*IU*

IU is INTEGER

If RANGE='I', the index of the

largest eigenvalue to be returned.

1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.

Not referenced if RANGE = 'A' or 'V'.

*ABSTOL*

ABSTOL is REAL

The absolute error tolerance for the eigenvalues.

An approximate eigenvalue is accepted as converged

when it is determined to lie in an interval [a,b]

of width less than or equal to

ABSTOL + EPS * max( |a|,|b| ) ,

where EPS is the machine precision. If ABSTOL is less than

or equal to zero, then EPS*|T| will be used in its place,

where |T| is the 1-norm of the tridiagonal matrix obtained

by reducing A to tridiagonal form.

See "Computing Small Singular Values of Bidiagonal Matrices

with Guaranteed High Relative Accuracy," by Demmel and

Kahan, LAPACK Working Note #3.

If high relative accuracy is important, set ABSTOL to

SLAMCH( 'Safe minimum' ). Doing so will guarantee that

eigenvalues are computed to high relative accuracy when

possible in future releases. The current code does not

make any guarantees about high relative accuracy, but

future releases will. See J. Barlow and J. Demmel,

"Computing Accurate Eigensystems of Scaled Diagonally

Dominant Matrices", LAPACK Working Note #7, for a discussion

of which matrices define their eigenvalues to high relative

accuracy.

*M*

M is INTEGER

The total number of eigenvalues found. 0 <= M <= N.

If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

*W*

W is REAL array, dimension (N)

The first M elements contain the selected eigenvalues in

ascending order.

*Z*

Z is REAL array, dimension (LDZ, max(1,M) )

If JOBZ = 'V', then if INFO = 0, the first M columns of Z

contain the orthonormal eigenvectors of the matrix A

corresponding to the selected 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; if RANGE = 'V', the exact value of M

is not known in advance and an upper bound must be used.

*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 ).

Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1

*WORK*

WORK is REAL array, dimension (MAX(1,LWORK))

On exit, if INFO = 0, WORK(1) returns the optimal (and

minimal) LWORK.

*LWORK*

LWORK is INTEGER

The dimension of the array WORK. LWORK >= 20*N.

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the optimal sizes of the WORK and IWORK

arrays, returns these values as the first entries of the WORK

and IWORK arrays, and no error message related to LWORK or

LIWORK is issued by XERBLA.

*IWORK*

IWORK is INTEGER array, dimension (MAX(1,LIWORK))

On exit, if INFO = 0, IWORK(1) returns the optimal (and

minimal) LIWORK.

*LIWORK*

LIWORK is INTEGER

The dimension of the array IWORK. LIWORK >= 10*N.

If LIWORK = -1, then a workspace query is assumed; the

routine only calculates the optimal sizes of the WORK and

IWORK arrays, returns these values as the first entries of

the WORK and IWORK arrays, and no error message related to

LWORK or LIWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: Internal error

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

Osni Marques, LBNL/NERSC, USA

Ken Stanley, Computer Science Division, University of California at Berkeley, USA

Jason Riedy, Computer Science Division, University of California at Berkeley, USA

## subroutine sstevx (character JOBZ, character RANGE, integer N, real, dimension( * ) D, real, dimension( * ) E, real VL, real VU, integer IL, integer IU, real ABSTOL, integer M, real, dimension( * ) W, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)¶

** SSTEVX computes the eigenvalues and, optionally, the left
and/or right eigenvectors for OTHER matrices**

**Purpose:**

SSTEVX computes selected eigenvalues and, optionally, eigenvectors

of a real symmetric tridiagonal matrix A. Eigenvalues and

eigenvectors can be selected by specifying either a range of values

or a range of indices for the desired eigenvalues.

**Parameters**

*JOBZ*

JOBZ is CHARACTER*1

= 'N': Compute eigenvalues only;

= 'V': Compute eigenvalues and eigenvectors.

*RANGE*

RANGE is CHARACTER*1

= 'A': all eigenvalues will be found.

= 'V': all eigenvalues in the half-open interval (VL,VU]

will be found.

= 'I': the IL-th through IU-th eigenvalues will be found.

*N*

N is INTEGER

The order of the matrix. N >= 0.

*D*

D is REAL array, dimension (N)

On entry, the n diagonal elements of the tridiagonal matrix

A.

On exit, D may be multiplied by a constant factor chosen

to avoid over/underflow in computing the eigenvalues.

*E*

E is REAL array, dimension (max(1,N-1))

On entry, the (n-1) subdiagonal elements of the tridiagonal

matrix A in elements 1 to N-1 of E.

On exit, E may be multiplied by a constant factor chosen

to avoid over/underflow in computing the eigenvalues.

*VL*

VL is REAL

If RANGE='V', the lower bound of the interval to

be searched for eigenvalues. VL < VU.

Not referenced if RANGE = 'A' or 'I'.

*VU*

VU is REAL

If RANGE='V', the upper bound of the interval to

be searched for eigenvalues. VL < VU.

Not referenced if RANGE = 'A' or 'I'.

*IL*

IL is INTEGER

If RANGE='I', the index of the

smallest eigenvalue to be returned.

1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.

Not referenced if RANGE = 'A' or 'V'.

*IU*

IU is INTEGER

If RANGE='I', the index of the

largest eigenvalue to be returned.

1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.

Not referenced if RANGE = 'A' or 'V'.

*ABSTOL*

ABSTOL is REAL

The absolute error tolerance for the eigenvalues.

An approximate eigenvalue is accepted as converged

when it is determined to lie in an interval [a,b]

of width less than or equal to

ABSTOL + EPS * max( |a|,|b| ) ,

where EPS is the machine precision. If ABSTOL is less

than or equal to zero, then EPS*|T| will be used in

its place, where |T| is the 1-norm of the tridiagonal

matrix.

Eigenvalues will be computed most accurately when ABSTOL is

set to twice the underflow threshold 2*SLAMCH('S'), not zero.

If this routine returns with INFO>0, indicating that some

eigenvectors did not converge, try setting ABSTOL to

2*SLAMCH('S').

See "Computing Small Singular Values of Bidiagonal Matrices

with Guaranteed High Relative Accuracy," by Demmel and

Kahan, LAPACK Working Note #3.

*M*

M is INTEGER

The total number of eigenvalues found. 0 <= M <= N.

If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

*W*

W is REAL array, dimension (N)

The first M elements contain the selected eigenvalues in

ascending order.

*Z*

Z is REAL array, dimension (LDZ, max(1,M) )

If JOBZ = 'V', then if INFO = 0, the first M columns of Z

contain the orthonormal eigenvectors of the matrix A

corresponding to the selected eigenvalues, with the i-th

column of Z holding the eigenvector associated with W(i).

If an eigenvector fails to converge (INFO > 0), then that

column of Z contains the latest approximation to the

eigenvector, and the index of the eigenvector is returned

in IFAIL. If JOBZ = 'N', then Z is not referenced.

Note: the user must ensure that at least max(1,M) columns are

supplied in the array Z; if RANGE = 'V', the exact value of M

is not known in advance and an upper bound must be used.

*LDZ*

LDZ is INTEGER

The leading dimension of the array Z. LDZ >= 1, and if

JOBZ = 'V', LDZ >= max(1,N).

*WORK*

WORK is REAL array, dimension (5*N)

*IWORK*

IWORK is INTEGER array, dimension (5*N)

*IFAIL*

IFAIL is INTEGER array, dimension (N)

If JOBZ = 'V', then if INFO = 0, the first M elements of

IFAIL are zero. If INFO > 0, then IFAIL contains the

indices of the eigenvectors that failed to converge.

If JOBZ = 'N', then IFAIL is not referenced.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, then i eigenvectors failed to converge.

Their indices are stored in array IFAIL.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

# Author¶

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