SYNOPSIS

SUBROUTINE ZSTEDC(

COMPZ, N, D, E, Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO )

    

CHARACTER COMPZ

    

INTEGER INFO, LDZ, LIWORK, LRWORK, LWORK, N

    

INTEGER IWORK( * )

    

DOUBLE PRECISION D( * ), E( * ), RWORK( * )

    

COMPLEX*16 WORK( * ), Z( LDZ, * )

PURPOSE

ZSTEDC computes all eigenvalues and, optionally, eigenvectors of a symmetric tridiagonal matrix using the divide and conquer method. The eigenvectors of a full or band complex Hermitian matrix can also be found if ZHETRD or ZHPTRD or ZHBTRD has been used to reduce this matrix to tridiagonal form.
This code 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. See DLAED3 for details.

ARGUMENTS

COMPZ (input) CHARACTER*1

= 'N': Compute eigenvalues only.
= 'I': Compute eigenvectors of tridiagonal matrix also.
= 'V': Compute eigenvectors of original Hermitian matrix also. On entry, Z contains the unitary matrix used to reduce the original matrix to tridiagonal form.

N (input) INTEGER

The dimension of the symmetric tridiagonal matrix. N >= 0.

D (input/output) DOUBLE PRECISION array, dimension (N)

On entry, the diagonal elements of the tridiagonal matrix. On exit, if INFO = 0, the eigenvalues in ascending order.

E (input/output) DOUBLE PRECISION array, dimension (N-1)

On entry, the subdiagonal elements of the tridiagonal matrix. On exit, E has been destroyed.

Z (input/output) COMPLEX*16 array, dimension (LDZ,N)

On entry, if COMPZ = 'V', then Z contains the unitary matrix used in the reduction to tridiagonal form. On exit, if INFO = 0, then if COMPZ = 'V', Z contains the orthonormal eigenvectors of the original Hermitian matrix, and if COMPZ = 'I', Z contains the orthonormal eigenvectors of the symmetric tridiagonal matrix. If COMPZ = 'N', then Z is not referenced.

LDZ (input) INTEGER

The leading dimension of the array Z. LDZ >= 1. If eigenvectors are desired, then LDZ >= max(1,N).

WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))

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

LWORK (input) INTEGER

The dimension of the array WORK. If COMPZ = 'N' or 'I', or N <= 1, LWORK must be at least 1. If COMPZ = 'V' and N > 1, LWORK must be at least N*N. Note that for COMPZ = 'V', then if N is less than or equal to the minimum divide size, usually 25, then LWORK need only be 1. 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 (workspace/output) DOUBLE PRECISION array,

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

LRWORK (input) INTEGER

The dimension of the array RWORK. If COMPZ = 'N' or N <= 1, LRWORK must be at least 1. If COMPZ = 'V' and N > 1, LRWORK must be at least 1 + 3*N + 2*N*lg N + 3*N**2 , where lg( N ) = smallest integer k such that 2**k >= N. If COMPZ = 'I' and N > 1, LRWORK must be at least 1 + 4*N + 2*N**2 . Note that for COMPZ = 'I' or 'V', then if N is less than or equal to the minimum divide size, usually 25, then LRWORK need only be max(1,2*(N-1)). 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 (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))

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

LIWORK (input) INTEGER

The dimension of the array IWORK. If COMPZ = 'N' or N <= 1, LIWORK must be at least 1. If COMPZ = 'V' or N > 1, LIWORK must be at least 6 + 6*N + 5*N*lg N. If COMPZ = 'I' or N > 1, LIWORK must be at least 3 + 5*N . Note that for COMPZ = 'I' or 'V', then if N is less than or equal to the minimum divide size, usually 25, then LIWORK need only be 1. 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 (output) INTEGER

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

FURTHER DETAILS

Based on contributions by

   Jeff Rutter, Computer Science Division, University of California
   at Berkeley, USA