## table of contents

complex16OTHEReigen(3) | LAPACK | complex16OTHEReigen(3) |

# NAME¶

complex16OTHEReigen

# SYNOPSIS¶

## Functions¶

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

**ZGGGLM** subroutine **zhbev** (JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ,
WORK, RWORK, INFO)

** ZHBEV computes the eigenvalues and, optionally, the left and/or right
eigenvectors for OTHER matrices** subroutine **zhbev_2stage** (JOBZ,
UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, LWORK, RWORK, INFO)

** ZHBEV_2STAGE computes the eigenvalues and, optionally, the left and/or
right eigenvectors for OTHER matrices** subroutine **zhbevd** (JOBZ,
UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK,
INFO)

** ZHBEVD computes the eigenvalues and, optionally, the left and/or right
eigenvectors for OTHER matrices** subroutine **zhbevd_2stage** (JOBZ,
UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK,
INFO)

** ZHBEVD_2STAGE computes the eigenvalues and, optionally, the left and/or
right eigenvectors for OTHER matrices** subroutine **zhbevx** (JOBZ,
RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ,
WORK, RWORK, IWORK, IFAIL, INFO)

** ZHBEVX computes the eigenvalues and, optionally, the left and/or right
eigenvectors for OTHER matrices** subroutine **zhbevx_2stage** (JOBZ,
RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ,
WORK, LWORK, RWORK, IWORK, IFAIL, INFO)

** ZHBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or
right eigenvectors for OTHER matrices** subroutine **zhbgv** (JOBZ,
UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z, LDZ, WORK, RWORK, INFO)

**ZHBGV** subroutine **zhbgvd** (JOBZ, UPLO, N, KA, KB, AB, LDAB, BB,
LDBB, W, Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO)

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

**ZHBGVX** subroutine **zhpev** (JOBZ, UPLO, N, AP, W, Z, LDZ, WORK,
RWORK, INFO)

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

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

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

**ZHPGV** subroutine **zhpgvd** (ITYPE, JOBZ, UPLO, N, AP, BP, W, Z,
LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO)

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

**ZHPGVX**

# Detailed Description¶

This is the group of complex16 Other Eigenvalue routines

# Function Documentation¶

## subroutine zggglm (integer N, integer M, integer P, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( * ) D, complex*16, dimension( * ) X, complex*16, dimension( * ) Y, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)¶

**ZGGGLM**

**Purpose:**

ZGGGLM 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 COMPLEX*16 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 COMPLEX*16 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 COMPLEX*16 array, dimension (N)

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

On exit, D is destroyed.

*X*

X is COMPLEX*16 array, dimension (M)

*Y*

Y is COMPLEX*16 array, dimension (P)

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

*WORK*

WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))

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

*LWORK*

LWORK is INTEGER

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

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

where NB is an upper bound for the optimal blocksizes for

ZGEQRF, ZGERQF, ZUNMQR and ZUNMRQ.

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 zhbev (character JOBZ, character UPLO, integer N, integer KD, complex*16, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) W, complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer INFO)¶

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

**Purpose:**

ZHBEV computes all the eigenvalues and, optionally, eigenvectors of

a complex Hermitian 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 COMPLEX*16 array, dimension (LDAB, N)

On entry, the upper or lower triangle of the Hermitian 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 DOUBLE PRECISION array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

*Z*

Z is COMPLEX*16 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 COMPLEX*16 array, dimension (N)

*RWORK*

RWORK is DOUBLE PRECISION 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 zhbev_2stage (character JOBZ, character UPLO, integer N, integer KD, complex*16, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) W, complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, integer INFO)¶

** ZHBEV_2STAGE computes the eigenvalues and, optionally, the
left and/or right eigenvectors for OTHER matrices**

**Purpose:**

ZHBEV_2STAGE computes all the eigenvalues and, optionally, eigenvectors of

a complex Hermitian 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 COMPLEX*16 array, dimension (LDAB, N)

On entry, the upper or lower triangle of the Hermitian 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 DOUBLE PRECISION array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

*Z*

Z is COMPLEX*16 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 COMPLEX*16 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

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, RWORK and

IWORK arrays, returns these values as the first entries of

the WORK, RWORK and IWORK arrays, and no error message

related to LWORK or LRWORK or LIWORK is issued by XERBLA.

*RWORK*

RWORK is DOUBLE PRECISION 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**

**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 zhbevd (character JOBZ, character UPLO, integer N, integer KD, complex*16, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) W, complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, integer LRWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)¶

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

**Purpose:**

ZHBEVD computes all the eigenvalues and, optionally, eigenvectors of

a complex Hermitian 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 COMPLEX*16 array, dimension (LDAB, N)

On entry, the upper or lower triangle of the Hermitian 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 DOUBLE PRECISION array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

*Z*

Z is COMPLEX*16 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 COMPLEX*16 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 must be at least 1.

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

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

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

only calculates the optimal sizes of the WORK, RWORK and

IWORK arrays, returns these values as the first entries of

the WORK, RWORK and IWORK arrays, and no error message

related to LWORK or LRWORK or LIWORK is issued by XERBLA.

*RWORK*

RWORK is DOUBLE PRECISION array,

dimension (LRWORK)

On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.

*LRWORK*

LRWORK is INTEGER

The dimension of array RWORK.

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

If JOBZ = 'N' and N > 1, LRWORK must be at least N.

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

1 + 5*N + 2*N**2.

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

routine only calculates the optimal sizes of the WORK, RWORK

and IWORK arrays, returns these values as the first entries

of the WORK, RWORK and IWORK arrays, and no error message

related to LWORK or LRWORK 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 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 optimal sizes of the WORK, RWORK

and IWORK arrays, returns these values as the first entries

of the WORK, RWORK and IWORK arrays, and no error message

related to LWORK or LRWORK 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 zhbevd_2stage (character JOBZ, character UPLO, integer N, integer KD, complex*16, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) W, complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, integer LRWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)¶

** ZHBEVD_2STAGE computes the eigenvalues and, optionally, the
left and/or right eigenvectors for OTHER matrices**

**Purpose:**

ZHBEVD_2STAGE computes all the eigenvalues and, optionally, eigenvectors of

a complex Hermitian band matrix A using the 2stage technique for

the reduction to tridiagonal. If eigenvectors are desired, it

uses a divide and conquer algorithm.

The divide and conquer algorithm makes very mild assumptions about

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

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

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

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

without guard digits, but we know of none.

**Parameters**

*JOBZ*

JOBZ is CHARACTER*1

= 'N': Compute eigenvalues only;

= 'V': Compute eigenvalues and eigenvectors.

Not available in this release.

*UPLO*

UPLO is CHARACTER*1

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

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

*N*

N is INTEGER

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

*KD*

KD is INTEGER

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

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

*AB*

AB is COMPLEX*16 array, dimension (LDAB, N)

On entry, the upper or lower triangle of the Hermitian 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 DOUBLE PRECISION array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

*Z*

Z is COMPLEX*16 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 COMPLEX*16 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 = (2KD+1)*N + KD*NTHREADS

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, RWORK and

IWORK arrays, returns these values as the first entries of

the WORK, RWORK and IWORK arrays, and no error message

related to LWORK or LRWORK or LIWORK is issued by XERBLA.

*RWORK*

RWORK is DOUBLE PRECISION array,

dimension (LRWORK)

On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.

*LRWORK*

LRWORK is INTEGER

The dimension of array RWORK.

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

If JOBZ = 'N' and N > 1, LRWORK must be at least N.

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

1 + 5*N + 2*N**2.

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

routine only calculates the optimal sizes of the WORK, RWORK

and IWORK arrays, returns these values as the first entries

of the WORK, RWORK and IWORK arrays, and no error message

related to LWORK or LRWORK 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 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 optimal sizes of the WORK, RWORK

and IWORK arrays, returns these values as the first entries

of the WORK, RWORK and IWORK arrays, and no error message

related to LWORK or LRWORK or LIWORK is issued by XERBLA.

*INFO*

INFO is INTEGER

= 0: successful exit.

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

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

off-diagonal elements of an intermediate tridiagonal

form did not converge to zero.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

All details about the 2stage techniques are available in:

Azzam Haidar, Hatem Ltaief, and Jack Dongarra.

Parallel reduction to condensed forms for symmetric eigenvalue problems

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

of 2011 International Conference for High Performance Computing,

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

Article 8 , 11 pages.

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

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

An improved parallel singular value algorithm and its implementation

for multicore hardware, In Proceedings of 2013 International Conference

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

Denver, Colorado, USA, 2013.

Article 90, 12 pages.

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

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

A novel hybrid CPU-GPU generalized eigensolver for electronic structure

calculations based on fine-grained memory aware tasks.

International Journal of High Performance Computing Applications.

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

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

## subroutine zhbevx (character JOBZ, character RANGE, character UPLO, integer N, integer KD, complex*16, dimension( ldab, * ) AB, integer LDAB, complex*16, dimension( ldq, * ) Q, integer LDQ, double precision VL, double precision VU, integer IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( * ) W, complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)¶

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

**Purpose:**

ZHBEVX computes selected eigenvalues and, optionally, eigenvectors

of a complex Hermitian band matrix A. Eigenvalues and eigenvectors

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

indices for the desired eigenvalues.

**Parameters**

*JOBZ*

JOBZ is CHARACTER*1

= 'N': Compute eigenvalues only;

= 'V': Compute eigenvalues and eigenvectors.

*RANGE*

RANGE is CHARACTER*1

= 'A': all eigenvalues will be found;

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

will be found;

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

*UPLO*

UPLO is CHARACTER*1

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

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

*N*

N is INTEGER

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

*KD*

KD is INTEGER

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

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

*AB*

AB is COMPLEX*16 array, dimension (LDAB, N)

On entry, the upper or lower triangle of the Hermitian 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.

*LDAB*

LDAB is INTEGER

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

*Q*

Q is COMPLEX*16 array, dimension (LDQ, N)

If JOBZ = 'V', the N-by-N unitary 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 DOUBLE PRECISION

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 DOUBLE PRECISION

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 DOUBLE PRECISION

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*DLAMCH('S'), not zero.

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

eigenvectors did not converge, try setting ABSTOL to

2*DLAMCH('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 DOUBLE PRECISION array, dimension (N)

The first M elements contain the selected eigenvalues in

ascending order.

*Z*

Z is COMPLEX*16 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 COMPLEX*16 array, dimension (N)

*RWORK*

RWORK is DOUBLE PRECISION 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 zhbevx_2stage (character JOBZ, character RANGE, character UPLO, integer N, integer KD, complex*16, dimension( ldab, * ) AB, integer LDAB, complex*16, dimension( ldq, * ) Q, integer LDQ, double precision VL, double precision VU, integer IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( * ) W, complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)¶

** ZHBEVX_2STAGE computes the eigenvalues and, optionally, the
left and/or right eigenvectors for OTHER matrices**

**Purpose:**

ZHBEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors

of a complex Hermitian band matrix A using the 2stage technique for

the reduction to tridiagonal. Eigenvalues and eigenvectors

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

indices for the desired eigenvalues.

**Parameters**

*JOBZ*

JOBZ is CHARACTER*1

= 'N': Compute eigenvalues only;

= 'V': Compute eigenvalues and eigenvectors.

Not available in this release.

*RANGE*

RANGE is CHARACTER*1

= 'A': all eigenvalues will be found;

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

will be found;

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

*UPLO*

UPLO is CHARACTER*1

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

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

*N*

N is INTEGER

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

*KD*

KD is INTEGER

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

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

*AB*

AB is COMPLEX*16 array, dimension (LDAB, N)

On entry, the upper or lower triangle of the Hermitian 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.

*LDAB*

LDAB is INTEGER

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

*Q*

Q is COMPLEX*16 array, dimension (LDQ, N)

If JOBZ = 'V', the N-by-N unitary 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 DOUBLE PRECISION

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 DOUBLE PRECISION

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 DOUBLE PRECISION

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*DLAMCH('S'), not zero.

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

eigenvectors did not converge, try setting ABSTOL to

2*DLAMCH('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 DOUBLE PRECISION array, dimension (N)

The first M elements contain the selected eigenvalues in

ascending order.

*Z*

Z is COMPLEX*16 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 COMPLEX*16 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, dimension) where

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

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, RWORK and

IWORK arrays, returns these values as the first entries of

the WORK, RWORK and IWORK arrays, and no error message

related to LWORK or LRWORK or LIWORK is issued by XERBLA.

*RWORK*

RWORK is DOUBLE PRECISION 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**

**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 zhbgv (character JOBZ, character UPLO, integer N, integer KA, integer KB, complex*16, dimension( ldab, * ) AB, integer LDAB, complex*16, dimension( ldbb, * ) BB, integer LDBB, double precision, dimension( * ) W, complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer INFO)¶

**ZHBGV**

**Purpose:**

ZHBGV computes all the eigenvalues, and optionally, the eigenvectors

of a complex generalized Hermitian-definite banded eigenproblem, of

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

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 COMPLEX*16 array, dimension (LDAB, N)

On entry, the upper or lower triangle of the Hermitian 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 COMPLEX*16 array, dimension (LDBB, N)

On entry, the upper or lower triangle of the Hermitian 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**H*S, as returned by ZPBSTF.

*LDBB*

LDBB is INTEGER

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

*W*

W is DOUBLE PRECISION array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

*Z*

Z is COMPLEX*16 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**H*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 COMPLEX*16 array, dimension (N)

*RWORK*

RWORK is DOUBLE PRECISION 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 ZPBSTF

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 zhbgvd (character JOBZ, character UPLO, integer N, integer KA, integer KB, complex*16, dimension( ldab, * ) AB, integer LDAB, complex*16, dimension( ldbb, * ) BB, integer LDBB, double precision, dimension( * ) W, complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, integer LRWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)¶

**ZHBGVD**

**Purpose:**

ZHBGVD computes all the eigenvalues, and optionally, the eigenvectors

of a complex generalized Hermitian-definite banded eigenproblem, of

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

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 COMPLEX*16 array, dimension (LDAB, N)

On entry, the upper or lower triangle of the Hermitian 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 COMPLEX*16 array, dimension (LDBB, N)

On entry, the upper or lower triangle of the Hermitian 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**H*S, as returned by ZPBSTF.

*LDBB*

LDBB is INTEGER

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

*W*

W is DOUBLE PRECISION array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

*Z*

Z is COMPLEX*16 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**H*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 COMPLEX*16 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 >= N.

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

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

only calculates the optimal sizes of the WORK, RWORK and

IWORK arrays, returns these values as the first entries of

the WORK, RWORK and IWORK arrays, and no error message

related to LWORK or LRWORK or LIWORK is issued by XERBLA.

*RWORK*

RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))

On exit, if INFO=0, RWORK(1) returns the optimal LRWORK.

*LRWORK*

LRWORK is INTEGER

The dimension of array RWORK.

If N <= 1, LRWORK >= 1.

If JOBZ = 'N' and N > 1, LRWORK >= N.

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

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

routine only calculates the optimal sizes of the WORK, RWORK

and IWORK arrays, returns these values as the first entries

of the WORK, RWORK and IWORK arrays, and no error message

related to LWORK or LRWORK 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 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, RWORK

and IWORK arrays, returns these values as the first entries

of the WORK, RWORK and IWORK arrays, and no error message

related to LWORK or LRWORK 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 ZPBSTF

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 zhbgvx (character JOBZ, character RANGE, character UPLO, integer N, integer KA, integer KB, complex*16, dimension( ldab, * ) AB, integer LDAB, complex*16, dimension( ldbb, * ) BB, integer LDBB, complex*16, dimension( ldq, * ) Q, integer LDQ, double precision VL, double precision VU, integer IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( * ) W, complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)¶

**ZHBGVX**

**Purpose:**

ZHBGVX computes all the eigenvalues, and optionally, the eigenvectors

of a complex generalized Hermitian-definite banded eigenproblem, of

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

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 COMPLEX*16 array, dimension (LDAB, N)

On entry, the upper or lower triangle of the Hermitian 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 COMPLEX*16 array, dimension (LDBB, N)

On entry, the upper or lower triangle of the Hermitian 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**H*S, as returned by ZPBSTF.

*LDBB*

LDBB is INTEGER

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

*Q*

Q is COMPLEX*16 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 DOUBLE PRECISION

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 DOUBLE PRECISION

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 DOUBLE PRECISION

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*DLAMCH('S'), not zero.

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

eigenvectors did not converge, try setting ABSTOL to

2*DLAMCH('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 DOUBLE PRECISION array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

*Z*

Z is COMPLEX*16 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**H*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 COMPLEX*16 array, dimension (N)

*RWORK*

RWORK is DOUBLE PRECISION 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, and i is:

<= N: then i eigenvectors failed to converge. Their

indices are stored in array IFAIL.

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

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 zhpev (character JOBZ, character UPLO, integer N, complex*16, dimension( * ) AP, double precision, dimension( * ) W, complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer INFO)¶

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

**Purpose:**

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

complex Hermitian matrix 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 COMPLEX*16 array, dimension (N*(N+1)/2)

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

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

is stored in the array AP as follows:

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

if UPLO = 'L', AP(i + (j-1)*(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 DOUBLE PRECISION array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

*Z*

Z is COMPLEX*16 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 COMPLEX*16 array, dimension (max(1, 2*N-1))

*RWORK*

RWORK is DOUBLE PRECISION 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 zhpevd (character JOBZ, character UPLO, integer N, complex*16, dimension( * ) AP, double precision, dimension( * ) W, complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, integer LRWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)¶

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

**Purpose:**

ZHPEVD computes all the eigenvalues and, optionally, eigenvectors of

a complex Hermitian 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 COMPLEX*16 array, dimension (N*(N+1)/2)

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

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

is stored in the array AP as follows:

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

if UPLO = 'L', AP(i + (j-1)*(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 DOUBLE PRECISION array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

*Z*

Z is COMPLEX*16 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 COMPLEX*16 array, dimension (MAX(1,LWORK))

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

*LWORK*

LWORK is INTEGER

The dimension of array WORK.

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

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

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

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

only calculates the required sizes of the WORK, RWORK and

IWORK arrays, returns these values as the first entries of

the WORK, RWORK and IWORK arrays, and no error message

related to LWORK or LRWORK or LIWORK is issued by XERBLA.

*RWORK*

RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))

On exit, if INFO = 0, RWORK(1) returns the required LRWORK.

*LRWORK*

LRWORK is INTEGER

The dimension of array RWORK.

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

If JOBZ = 'N' and N > 1, LRWORK must be at least N.

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

1 + 5*N + 2*N**2.

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

routine only calculates the required sizes of the WORK, RWORK

and IWORK arrays, returns these values as the first entries

of the WORK, RWORK and IWORK arrays, and no error message

related to LWORK or LRWORK 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 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, RWORK

and IWORK arrays, returns these values as the first entries

of the WORK, RWORK and IWORK arrays, and no error message

related to LWORK or LRWORK 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 zhpevx (character JOBZ, character RANGE, character UPLO, integer N, complex*16, dimension( * ) AP, double precision VL, double precision VU, integer IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( * ) W, complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)¶

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

**Purpose:**

ZHPEVX computes selected eigenvalues and, optionally, eigenvectors

of a complex Hermitian 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 COMPLEX*16 array, dimension (N*(N+1)/2)

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

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

is stored in the array AP as follows:

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

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

On exit, AP is overwritten by values generated during the

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

and first superdiagonal of the tridiagonal matrix T overwrite

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

diagonal and first subdiagonal of T overwrite the

corresponding elements of A.

*VL*

VL is DOUBLE PRECISION

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 DOUBLE PRECISION

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 DOUBLE PRECISION

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*DLAMCH('S'), not zero.

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

eigenvectors did not converge, try setting ABSTOL to

2*DLAMCH('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 DOUBLE PRECISION array, dimension (N)

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

*Z*

Z is COMPLEX*16 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 COMPLEX*16 array, dimension (2*N)

*RWORK*

RWORK is DOUBLE PRECISION 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 zhpgv (integer ITYPE, character JOBZ, character UPLO, integer N, complex*16, dimension( * ) AP, complex*16, dimension( * ) BP, double precision, dimension( * ) W, complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer INFO)¶

**ZHPGV**

**Purpose:**

ZHPGV computes all the eigenvalues and, optionally, the eigenvectors

of a complex generalized Hermitian-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 Hermitian, stored in packed format,

and B is also positive definite.

**Parameters**

*ITYPE*

ITYPE is INTEGER

Specifies the problem type to be solved:

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

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

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

*JOBZ*

JOBZ is CHARACTER*1

= 'N': Compute eigenvalues only;

= 'V': Compute eigenvalues and eigenvectors.

*UPLO*

UPLO is CHARACTER*1

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

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

*N*

N is INTEGER

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

*AP*

AP is COMPLEX*16 array, dimension (N*(N+1)/2)

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

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

is stored in the array AP as follows:

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

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

On exit, the contents of AP are destroyed.

*BP*

BP is COMPLEX*16 array, dimension (N*(N+1)/2)

On entry, the upper or lower triangle of the Hermitian 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**H*U or B = L*L**H, in the same storage

format as B.

*W*

W is DOUBLE PRECISION array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

*Z*

Z is COMPLEX*16 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**H*B*Z = I;

if ITYPE = 3, Z**H*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 COMPLEX*16 array, dimension (max(1, 2*N-1))

*RWORK*

RWORK is DOUBLE PRECISION 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: ZPPTRF or ZHPEV returned an error code:

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

i off-diagonal elements of an intermediate

tridiagonal form did not convergeto 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 zhpgvd (integer ITYPE, character JOBZ, character UPLO, integer N, complex*16, dimension( * ) AP, complex*16, dimension( * ) BP, double precision, dimension( * ) W, complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, integer LRWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)¶

**ZHPGVD**

**Purpose:**

ZHPGVD computes all the eigenvalues and, optionally, the eigenvectors

of a complex generalized Hermitian-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 Hermitian, stored in packed format, and B is also

positive definite.

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

The divide and conquer algorithm makes very mild assumptions about

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

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

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

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

without guard digits, but we know of none.

**Parameters**

*ITYPE*

ITYPE is INTEGER

Specifies the problem type to be solved:

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

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

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

*JOBZ*

JOBZ is CHARACTER*1

= 'N': Compute eigenvalues only;

= 'V': Compute eigenvalues and eigenvectors.

*UPLO*

UPLO is CHARACTER*1

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

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

*N*

N is INTEGER

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

*AP*

AP is COMPLEX*16 array, dimension (N*(N+1)/2)

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

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

is stored in the array AP as follows:

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

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

On exit, the contents of AP are destroyed.

*BP*

BP is COMPLEX*16 array, dimension (N*(N+1)/2)

On entry, the upper or lower triangle of the Hermitian 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**H*U or B = L*L**H, in the same storage

format as B.

*W*

W is DOUBLE PRECISION array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

*Z*

Z is COMPLEX*16 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**H*B*Z = I;

if ITYPE = 3, Z**H*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 COMPLEX*16 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 >= N.

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

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

only calculates the required sizes of the WORK, RWORK and

IWORK arrays, returns these values as the first entries of

the WORK, RWORK and IWORK arrays, and no error message

related to LWORK or LRWORK or LIWORK is issued by XERBLA.

*RWORK*

RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))

On exit, if INFO = 0, RWORK(1) returns the required LRWORK.

*LRWORK*

LRWORK is INTEGER

The dimension of array RWORK.

If N <= 1, LRWORK >= 1.

If JOBZ = 'N' and N > 1, LRWORK >= N.

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

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

routine only calculates the required sizes of the WORK, RWORK

and IWORK arrays, returns these values as the first entries

of the WORK, RWORK and IWORK arrays, and no error message

related to LWORK or LRWORK 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 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, RWORK

and IWORK arrays, returns these values as the first entries

of the WORK, RWORK and IWORK arrays, and no error message

related to LWORK or LRWORK 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: ZPPTRF or ZHPEVD returned an error code:

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

i off-diagonal elements of an intermediate

tridiagonal form did not convergeto 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 zhpgvx (integer ITYPE, character JOBZ, character RANGE, character UPLO, integer N, complex*16, dimension( * ) AP, complex*16, dimension( * ) BP, double precision VL, double precision VU, integer IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( * ) W, complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)¶

**ZHPGVX**

**Purpose:**

ZHPGVX computes selected eigenvalues and, optionally, eigenvectors

of a complex generalized Hermitian-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 Hermitian, stored in packed format, 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 triangles of A and B are stored;

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

*N*

N is INTEGER

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

*AP*

AP is COMPLEX*16 array, dimension (N*(N+1)/2)

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

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

is stored in the array AP as follows:

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

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

On exit, the contents of AP are destroyed.

*BP*

BP is COMPLEX*16 array, dimension (N*(N+1)/2)

On entry, the upper or lower triangle of the Hermitian 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**H*U or B = L*L**H, in the same storage

format as B.

*VL*

VL is DOUBLE PRECISION

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 DOUBLE PRECISION

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 DOUBLE PRECISION

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*DLAMCH('S'), not zero.

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

eigenvectors did not converge, try setting ABSTOL to

2*DLAMCH('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 DOUBLE PRECISION array, dimension (N)

On normal exit, the first M elements contain the selected

eigenvalues in ascending order.

*Z*

Z is COMPLEX*16 array, dimension (LDZ, N)

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**H*B*Z = I;

if ITYPE = 3, Z**H*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 COMPLEX*16 array, dimension (2*N)

*RWORK*

RWORK is DOUBLE PRECISION 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: ZPPTRF or ZHPEVX returned an error code:

<= N: if INFO = i, ZHPEVX 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:**

# Author¶

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