## table of contents

realSYeigen(3) | LAPACK | realSYeigen(3) |

# NAME¶

realSYeigen - real

# SYNOPSIS¶

## Functions¶

subroutine **ssyev** (JOBZ, UPLO, N, A, LDA, W, WORK, LWORK,
INFO)

** SSYEV computes the eigenvalues and, optionally, the left and/or right
eigenvectors for SY matrices** subroutine **ssyev_2stage** (JOBZ, UPLO,
N, A, LDA, W, WORK, LWORK, INFO)

** SSYEV_2STAGE computes the eigenvalues and, optionally, the left and/or
right eigenvectors for SY matrices** subroutine **ssyevd** (JOBZ, UPLO,
N, A, LDA, W, WORK, LWORK, IWORK, LIWORK, INFO)

** SSYEVD computes the eigenvalues and, optionally, the left and/or right
eigenvectors for SY matrices** subroutine **ssyevd_2stage** (JOBZ,
UPLO, N, A, LDA, W, WORK, LWORK, IWORK, LIWORK, INFO)

** SSYEVD_2STAGE computes the eigenvalues and, optionally, the left and/or
right eigenvectors for SY matrices** subroutine **ssyevr** (JOBZ,
RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK,
LWORK, IWORK, LIWORK, INFO)

** SSYEVR computes the eigenvalues and, optionally, the left and/or right
eigenvectors for SY matrices** subroutine **ssyevr_2stage** (JOBZ,
RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK,
LWORK, IWORK, LIWORK, INFO)

** SSYEVR_2STAGE computes the eigenvalues and, optionally, the left and/or
right eigenvectors for SY matrices** subroutine **ssyevx** (JOBZ,
RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK,
IWORK, IFAIL, INFO)

** SSYEVX computes the eigenvalues and, optionally, the left and/or right
eigenvectors for SY matrices** subroutine **ssyevx_2stage** (JOBZ,
RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK,
IWORK, IFAIL, INFO)

** SSYEVX_2STAGE computes the eigenvalues and, optionally, the left and/or
right eigenvectors for SY matrices** subroutine **ssygv** (ITYPE, JOBZ,
UPLO, N, A, LDA, B, LDB, W, WORK, LWORK, INFO)

**SSYGV** subroutine **ssygv_2stage** (ITYPE, JOBZ, UPLO, N, A, LDA, B,
LDB, W, WORK, LWORK, INFO)

**SSYGV_2STAGE** subroutine **ssygvd** (ITYPE, JOBZ, UPLO, N, A, LDA, B,
LDB, W, WORK, LWORK, IWORK, LIWORK, INFO)

**SSYGVD** subroutine **ssygvx** (ITYPE, JOBZ, RANGE, UPLO, N, A, LDA,
B, LDB, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL,
INFO)

**SSYGVX**

# Detailed Description¶

This is the group of real eigenvalue driver functions for SY matrices

# Function Documentation¶

## subroutine ssyev (character JOBZ, character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) W, real, dimension( * ) WORK, integer LWORK, integer INFO)¶

** SSYEV computes the eigenvalues and, optionally, the left
and/or right eigenvectors for SY matrices**

**Purpose:**

SSYEV computes all eigenvalues and, optionally, eigenvectors of a

real symmetric 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.

*A*

A is REAL array, dimension (LDA, N)

On entry, the symmetric matrix A. If UPLO = 'U', the

leading N-by-N upper triangular part of A contains the

upper triangular part of the matrix A. If UPLO = 'L',

the leading N-by-N lower triangular part of A contains

the lower triangular part of the matrix A.

On exit, if JOBZ = 'V', then if INFO = 0, A contains the

orthonormal eigenvectors of the matrix A.

If JOBZ = 'N', then on exit the lower triangle (if UPLO='L')

or the upper triangle (if UPLO='U') of A, including the

diagonal, is destroyed.

*LDA*

LDA is INTEGER

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

*W*

W is REAL array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

*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 length of the array WORK. LWORK >= max(1,3*N-1).

For optimal efficiency, LWORK >= (NB+2)*N,

where NB is the blocksize for SSYTRD returned by ILAENV.

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.

## subroutine ssyev_2stage (character JOBZ, character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) W, real, dimension( * ) WORK, integer LWORK, integer INFO)¶

** SSYEV_2STAGE computes the eigenvalues and, optionally, the
left and/or right eigenvectors for SY matrices**

**Purpose:**

SSYEV_2STAGE computes all eigenvalues and, optionally, eigenvectors of a

real symmetric 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.

*A*

A is REAL array, dimension (LDA, N)

On entry, the symmetric matrix A. If UPLO = 'U', the

leading N-by-N upper triangular part of A contains the

upper triangular part of the matrix A. If UPLO = 'L',

the leading N-by-N lower triangular part of A contains

the lower triangular part of the matrix A.

On exit, if JOBZ = 'V', then if INFO = 0, A contains the

orthonormal eigenvectors of the matrix A.

If JOBZ = 'N', then on exit the lower triangle (if UPLO='L')

or the upper triangle (if UPLO='U') of A, including the

diagonal, is destroyed.

*LDA*

LDA is INTEGER

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

*W*

W is REAL array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

*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 = max(stage1,stage2) + (KD+1)*N + 2*N

= N*KD + N*max(KD+1,FACTOPTNB)

+ max(2*KD*KD, KD*NTHREADS)

+ (KD+1)*N + 2*N

where KD is the blocking size of the reduction,

FACTOPTNB is the blocking used by the QR or LQ

algorithm, usually FACTOPTNB=128 is a good choice

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.

**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 ssyevd (character JOBZ, character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) W, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)¶

** SSYEVD computes the eigenvalues and, optionally, the left
and/or right eigenvectors for SY matrices**

**Purpose:**

SSYEVD computes all eigenvalues and, optionally, eigenvectors of a

real symmetric 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.

Because of large use of BLAS of level 3, SSYEVD needs N**2 more

workspace than SSYEVX.

**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.

*A*

A is REAL array, dimension (LDA, N)

On entry, the symmetric matrix A. If UPLO = 'U', the

leading N-by-N upper triangular part of A contains the

upper triangular part of the matrix A. If UPLO = 'L',

the leading N-by-N lower triangular part of A contains

the lower triangular part of the matrix A.

On exit, if JOBZ = 'V', then if INFO = 0, A contains the

orthonormal eigenvectors of the matrix A.

If JOBZ = 'N', then on exit the lower triangle (if UPLO='L')

or the upper triangle (if UPLO='U') of A, including the

diagonal, is destroyed.

*LDA*

LDA is INTEGER

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

*W*

W is REAL array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

*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 > 1, LWORK must be at least 2*N+1.

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

1 + 6*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 N <= 1, LIWORK must be at least 1.

If JOBZ = 'N' and 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 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 JOBZ = 'N', then the algorithm failed

to converge; i off-diagonal elements of an intermediate

tridiagonal form did not converge to zero;

if INFO = i and JOBZ = 'V', then the algorithm failed

to compute an eigenvalue while working on the submatrix

lying in rows and columns INFO/(N+1) through

mod(INFO,N+1).

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

Modified by Francoise Tisseur, University of Tennessee

Modified description of INFO. Sven, 16 Feb 05.

## subroutine ssyevd_2stage (character JOBZ, character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) W, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)¶

** SSYEVD_2STAGE computes the eigenvalues and, optionally, the
left and/or right eigenvectors for SY matrices**

**Purpose:**

SSYEVD_2STAGE computes all eigenvalues and, optionally, eigenvectors of a

real symmetric 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.

*A*

A is REAL array, dimension (LDA, N)

On entry, the symmetric matrix A. If UPLO = 'U', the

leading N-by-N upper triangular part of A contains the

upper triangular part of the matrix A. If UPLO = 'L',

the leading N-by-N lower triangular part of A contains

the lower triangular part of the matrix A.

On exit, if JOBZ = 'V', then if INFO = 0, A contains the

orthonormal eigenvectors of the matrix A.

If JOBZ = 'N', then on exit the lower triangle (if UPLO='L')

or the upper triangle (if UPLO='U') of A, including the

diagonal, is destroyed.

*LDA*

LDA is INTEGER

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

*W*

W is REAL array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

*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 > 1, LWORK must be queried.

LWORK = MAX(1, dimension) where

dimension = max(stage1,stage2) + (KD+1)*N + 2*N+1

= N*KD + N*max(KD+1,FACTOPTNB)

+ max(2*KD*KD, KD*NTHREADS)

+ (KD+1)*N + 2*N+1

where KD is the blocking size of the reduction,

FACTOPTNB is the blocking used by the QR or LQ

algorithm, usually FACTOPTNB=128 is a good choice

NTHREADS is the number of threads used when

openMP compilation is enabled, otherwise =1.

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

1 + 6*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 N <= 1, LIWORK must be at least 1.

If JOBZ = 'N' and 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 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 JOBZ = 'N', then the algorithm failed

to converge; i off-diagonal elements of an intermediate

tridiagonal form did not converge to zero;

if INFO = i and JOBZ = 'V', then the algorithm failed

to compute an eigenvalue while working on the submatrix

lying in rows and columns INFO/(N+1) through

mod(INFO,N+1).

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

Modified by Francoise Tisseur, University of Tennessee

Modified description of INFO. Sven, 16 Feb 05.

**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 ssyevr (character JOBZ, character RANGE, character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, 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)¶

** SSYEVR computes the eigenvalues and, optionally, the left
and/or right eigenvectors for SY matrices**

**Purpose:**

SSYEVR computes selected eigenvalues and, optionally, eigenvectors

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

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

indices for the desired eigenvalues.

SSYEVR first reduces the matrix A to tridiagonal form T with a call

to SSYTRD. Then, whenever possible, SSYEVR 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 each unreduced block (submatrix) of T,

(a) Compute T - sigma I = L D L^T, so that L and D

define all the wanted eigenvalues to high relative accuracy.

This means that small relative changes in the entries of D and L

cause only small relative changes in the eigenvalues and

eigenvectors. The standard (unfactored) representation of the

tridiagonal matrix T does not have this property in general.

(b) Compute the eigenvalues to suitable accuracy.

If the eigenvectors are desired, the algorithm attains full

accuracy of the computed eigenvalues only right before

the corresponding vectors have to be computed, see steps c) and d).

(c) For each cluster of close eigenvalues, select a new

shift close to the cluster, find a new factorization, and refine

the shifted eigenvalues to suitable accuracy.

(d) For each eigenvalue with a large enough relative separation compute

the corresponding eigenvector by forming a rank revealing twisted

factorization. Go back to (c) for any clusters that remain.

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

parameter ABSTOL.

For more details, see SSTEMR's documentation and:

- Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations

to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"

Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.

- Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and

Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,

2004. Also LAPACK Working Note 154.

- Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric

tridiagonal eigenvalue/eigenvector problem",

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

UC Berkeley, May 1997.

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

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

SSYEVR 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

*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.

*A*

A is REAL array, dimension (LDA, N)

On entry, the symmetric matrix A. If UPLO = 'U', the

leading N-by-N upper triangular part of A contains the

upper triangular part of the matrix A. If UPLO = 'L',

the leading N-by-N lower triangular part of A contains

the lower triangular part of the matrix A.

On exit, the lower triangle (if UPLO='L') or the upper

triangle (if UPLO='U') of A, including the diagonal, is

destroyed.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= 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.

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).

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.

Supplying N columns is always safe.

*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 ). This is an output of SSTEMR (tridiagonal

matrix). The support of the eigenvectors of A is typically

1:N because of the orthogonal transformations applied by SORMTR.

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 LWORK.

*LWORK*

LWORK is INTEGER

The dimension of the array WORK. LWORK >= max(1,26*N).

For optimal efficiency, LWORK >= (NB+6)*N,

where NB is the max of the blocksize for SSYTRD and SORMTR

returned by ILAENV.

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 LWORK.

*LIWORK*

LIWORK is INTEGER

The dimension of the array IWORK. LIWORK >= max(1,10*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.

**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 ssyevr_2stage (character JOBZ, character RANGE, character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, 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)¶

** SSYEVR_2STAGE computes the eigenvalues and, optionally, the
left and/or right eigenvectors for SY matrices**

**Purpose:**

SSYEVR_2STAGE computes selected eigenvalues and, optionally, eigenvectors

of a real symmetric 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.

SSYEVR_2STAGE first reduces the matrix A to tridiagonal form T with a call

to SSYTRD. Then, whenever possible, SSYEVR_2STAGE 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 each unreduced block (submatrix) of T,

(a) Compute T - sigma I = L D L^T, so that L and D

define all the wanted eigenvalues to high relative accuracy.

This means that small relative changes in the entries of D and L

cause only small relative changes in the eigenvalues and

eigenvectors. The standard (unfactored) representation of the

tridiagonal matrix T does not have this property in general.

(b) Compute the eigenvalues to suitable accuracy.

If the eigenvectors are desired, the algorithm attains full

accuracy of the computed eigenvalues only right before

the corresponding vectors have to be computed, see steps c) and d).

(c) For each cluster of close eigenvalues, select a new

shift close to the cluster, find a new factorization, and refine

the shifted eigenvalues to suitable accuracy.

(d) For each eigenvalue with a large enough relative separation compute

the corresponding eigenvector by forming a rank revealing twisted

factorization. Go back to (c) for any clusters that remain.

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

parameter ABSTOL.

For more details, see SSTEMR's documentation and:

- Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations

to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"

Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.

- Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and

Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,

2004. Also LAPACK Working Note 154.

- Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric

tridiagonal eigenvalue/eigenvector problem",

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

UC Berkeley, May 1997.

Note 1 : SSYEVR_2STAGE calls SSTEMR when the full spectrum is requested

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

SSYEVR_2STAGE 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.

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.

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

SSTEIN are called

*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.

*A*

A is REAL array, dimension (LDA, N)

On entry, the symmetric matrix A. If UPLO = 'U', the

leading N-by-N upper triangular part of A contains the

upper triangular part of the matrix A. If UPLO = 'L',

the leading N-by-N lower triangular part of A contains

the lower triangular part of the matrix A.

On exit, the lower triangle (if UPLO='L') or the upper

triangle (if UPLO='U') of A, including the diagonal, is

destroyed.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= 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.

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).

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.

Supplying N columns is always safe.

*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 ). This is an output of SSTEMR (tridiagonal

matrix). The support of the eigenvectors of A is typically

1:N because of the orthogonal transformations applied by SORMTR.

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 LWORK.

*LWORK*

LWORK is INTEGER

The dimension of the array WORK.

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

LWORK = MAX(1, 26*N, dimension) where

dimension = max(stage1,stage2) + (KD+1)*N + 5*N

= N*KD + N*max(KD+1,FACTOPTNB)

+ max(2*KD*KD, KD*NTHREADS)

+ (KD+1)*N + 5*N

where KD is the blocking size of the reduction,

FACTOPTNB is the blocking used by the QR or LQ

algorithm, usually FACTOPTNB=128 is a good choice

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 (MAX(1,LIWORK))

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

*LIWORK*

LIWORK is INTEGER

The dimension of the array IWORK. LIWORK >= max(1,10*N).

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

routine only calculates the optimal size of the IWORK array,

returns this value as the first entry of the IWORK array, and

no error message related to 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.

**Contributors:**

Inderjit Dhillon, IBM Almaden, USA 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

**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 ssyevx (character JOBZ, character RANGE, character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, 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)¶

** SSYEVX computes the eigenvalues and, optionally, the left
and/or right eigenvectors for SY matrices**

**Purpose:**

SSYEVX computes selected eigenvalues and, optionally, eigenvectors

of a real symmetric 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.

*A*

A is REAL array, dimension (LDA, N)

On entry, the symmetric matrix A. If UPLO = 'U', the

leading N-by-N upper triangular part of A contains the

upper triangular part of the matrix A. If UPLO = 'L',

the leading N-by-N lower triangular part of A contains

the lower triangular part of the matrix A.

On exit, the lower triangle (if UPLO='L') or the upper

triangle (if UPLO='U') of A, including the diagonal, is

destroyed.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= 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').

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)

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 = '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 (MAX(1,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 8*N.

For optimal efficiency, LWORK >= (NB+3)*N,

where NB is the max of the blocksize for SSYTRD and SORMTR

returned by ILAENV.

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.

## subroutine ssyevx_2stage (character JOBZ, character RANGE, character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, 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)¶

** SSYEVX_2STAGE computes the eigenvalues and, optionally, the
left and/or right eigenvectors for SY matrices**

**Purpose:**

SSYEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors

of a real symmetric 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.

*A*

A is REAL array, dimension (LDA, N)

On entry, the symmetric matrix A. If UPLO = 'U', the

leading N-by-N upper triangular part of A contains the

upper triangular part of the matrix A. If UPLO = 'L',

the leading N-by-N lower triangular part of A contains

the lower triangular part of the matrix A.

On exit, the lower triangle (if UPLO='L') or the upper

triangle (if UPLO='U') of A, including the diagonal, is

destroyed.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= 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').

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)

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 = '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 (MAX(1,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, 8*N, dimension) where

dimension = max(stage1,stage2) + (KD+1)*N + 3*N

= N*KD + N*max(KD+1,FACTOPTNB)

+ max(2*KD*KD, KD*NTHREADS)

+ (KD+1)*N + 3*N

where KD is the blocking size of the reduction,

FACTOPTNB is the blocking used by the QR or LQ

algorithm, usually FACTOPTNB=128 is a good choice

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.

**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 ssygv (integer ITYPE, character JOBZ, character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( * ) W, real, dimension( * ) WORK, integer LWORK, integer INFO)¶

**SSYGV**

**Purpose:**

SSYGV 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 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.

*A*

A is REAL array, dimension (LDA, N)

On entry, the symmetric matrix A. If UPLO = 'U', the

leading N-by-N upper triangular part of A contains the

upper triangular part of the matrix A. If UPLO = 'L',

the leading N-by-N lower triangular part of A contains

the lower triangular part of the matrix A.

On exit, if JOBZ = 'V', then if INFO = 0, A 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 on exit the upper triangle (if UPLO='U')

or the lower triangle (if UPLO='L') of A, including the

diagonal, is destroyed.

*LDA*

LDA is INTEGER

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

*B*

B is REAL array, dimension (LDB, N)

On entry, the symmetric positive definite matrix B.

If UPLO = 'U', the leading N-by-N upper triangular part of B

contains the upper triangular part of the matrix B.

If UPLO = 'L', the leading N-by-N lower triangular part of B

contains the lower triangular part of the matrix B.

On exit, if INFO <= N, the part of B containing the matrix is

overwritten by the triangular factor U or L from the Cholesky

factorization B = U**T*U or B = L*L**T.

*LDB*

LDB is INTEGER

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

*W*

W is REAL array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

*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 length of the array WORK. LWORK >= max(1,3*N-1).

For optimal efficiency, LWORK >= (NB+2)*N,

where NB is the blocksize for SSYTRD returned by ILAENV.

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: SPOTRF or SSYEV returned an error code:

<= N: if INFO = i, SSYEV 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.

## subroutine ssygv_2stage (integer ITYPE, character JOBZ, character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( * ) W, real, dimension( * ) WORK, integer LWORK, integer INFO)¶

**SSYGV_2STAGE**

**Purpose:**

SSYGV_2STAGE 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 and B is also

positive definite.

This routine use the 2stage technique for the reduction to tridiagonal

which showed higher performance on recent architecture and for large

sizes N>2000.

**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.

Not available in this release.

*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.

*A*

A is REAL array, dimension (LDA, N)

On entry, the symmetric matrix A. If UPLO = 'U', the

leading N-by-N upper triangular part of A contains the

upper triangular part of the matrix A. If UPLO = 'L',

the leading N-by-N lower triangular part of A contains

the lower triangular part of the matrix A.

On exit, if JOBZ = 'V', then if INFO = 0, A 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 on exit the upper triangle (if UPLO='U')

or the lower triangle (if UPLO='L') of A, including the

diagonal, is destroyed.

*LDA*

LDA is INTEGER

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

*B*

B is REAL array, dimension (LDB, N)

On entry, the symmetric positive definite matrix B.

If UPLO = 'U', the leading N-by-N upper triangular part of B

contains the upper triangular part of the matrix B.

If UPLO = 'L', the leading N-by-N lower triangular part of B

contains the lower triangular part of the matrix B.

On exit, if INFO <= N, the part of B containing the matrix is

overwritten by the triangular factor U or L from the Cholesky

factorization B = U**T*U or B = L*L**T.

*LDB*

LDB is INTEGER

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

*W*

W is REAL array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

*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 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 = max(stage1,stage2) + (KD+1)*N + 2*N

= N*KD + N*max(KD+1,FACTOPTNB)

+ max(2*KD*KD, KD*NTHREADS)

+ (KD+1)*N + 2*N

where KD is the blocking size of the reduction,

FACTOPTNB is the blocking used by the QR or LQ

algorithm, usually FACTOPTNB=128 is a good choice

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: SPOTRF or SSYEV returned an error code:

<= N: if INFO = i, SSYEV 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.

**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 ssygvd (integer ITYPE, character JOBZ, character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( * ) W, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)¶

**SSYGVD**

**Purpose:**

SSYGVD 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 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.

*A*

A is REAL array, dimension (LDA, N)

On entry, the symmetric matrix A. If UPLO = 'U', the

leading N-by-N upper triangular part of A contains the

upper triangular part of the matrix A. If UPLO = 'L',

the leading N-by-N lower triangular part of A contains

the lower triangular part of the matrix A.

On exit, if JOBZ = 'V', then if INFO = 0, A 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 on exit the upper triangle (if UPLO='U')

or the lower triangle (if UPLO='L') of A, including the

diagonal, is destroyed.

*LDA*

LDA is INTEGER

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

*B*

B is REAL array, dimension (LDB, N)

On entry, the symmetric matrix B. If UPLO = 'U', the

leading N-by-N upper triangular part of B contains the

upper triangular part of the matrix B. If UPLO = 'L',

the leading N-by-N lower triangular part of B contains

the lower triangular part of the matrix B.

On exit, if INFO <= N, the part of B containing the matrix is

overwritten by the triangular factor U or L from the Cholesky

factorization B = U**T*U or B = L*L**T.

*LDB*

LDB is INTEGER

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

*W*

W is REAL array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

*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 >= 2*N+1.

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 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 N <= 1, LIWORK >= 1.

If JOBZ = 'N' and 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: SPOTRF or SSYEVD returned an error code:

<= N: if INFO = i and JOBZ = 'N', then the algorithm

failed to converge; i off-diagonal elements of an

intermediate tridiagonal form did not converge to

zero;

if INFO = i and JOBZ = 'V', then the algorithm

failed to compute an eigenvalue while working on

the submatrix lying in rows and columns INFO/(N+1)

through mod(INFO,N+1);

> 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.

**Further Details:**

Modified so that no backsubstitution is performed if SSYEVD fails to

converge (NEIG in old code could be greater than N causing out of

bounds reference to A - reported by Ralf Meyer). Also corrected the

description of INFO and the test on ITYPE. Sven, 16 Feb 05.

**Contributors:**

## subroutine ssygvx (integer ITYPE, character JOBZ, character RANGE, character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, 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)¶

**SSYGVX**

**Purpose:**

SSYGVX 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 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.

*A*

A is REAL array, dimension (LDA, N)

On entry, the symmetric matrix A. If UPLO = 'U', the

leading N-by-N upper triangular part of A contains the

upper triangular part of the matrix A. If UPLO = 'L',

the leading N-by-N lower triangular part of A contains

the lower triangular part of the matrix A.

On exit, the lower triangle (if UPLO='L') or the upper

triangle (if UPLO='U') of A, including the diagonal, is

destroyed.

*LDA*

LDA is INTEGER

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

*B*

B is REAL array, dimension (LDB, N)

On entry, the symmetric matrix B. If UPLO = 'U', the

leading N-by-N upper triangular part of B contains the

upper triangular part of the matrix B. If UPLO = 'L',

the leading N-by-N lower triangular part of B contains

the lower triangular part of the matrix B.

On exit, if INFO <= N, the part of B containing the matrix is

overwritten by the triangular factor U or L from the Cholesky

factorization B = U**T*U or B = L*L**T.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= 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 C to tridiagonal form, where C is the symmetric

matrix of the standard symmetric problem to which the

generalized problem is transformed.

Eigenvalues will be computed most accurately when ABSTOL is

set to twice the underflow threshold 2*DLAMCH('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 (MAX(1,LWORK))

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

*LWORK*

LWORK is INTEGER

The length of the array WORK. LWORK >= max(1,8*N).

For optimal efficiency, LWORK >= (NB+3)*N,

where NB is the blocksize for SSYTRD returned by ILAENV.

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: SPOTRF or SSYEVX returned an error code:

<= N: if INFO = i, SSYEVX 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.

**Contributors:**

# Author¶

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