## 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*

*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*

*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*

*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*

**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*

*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*

*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*

*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*

*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*

*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*

*VU*

*IL*

*IU*

*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*

*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*

*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*

*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*

*VU*

*IL*

*IU*

*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*

*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*

*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*

*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*

*VU*

*IL*

*IU*

*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*

*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: **

## 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*

*VU*

*IL*

*IU*

*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*

*W*

*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*

*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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