Go to main content
Oracle Developer Studio 12.5 Man Pages

Exit Print View

Updated: June 2017
 
 

zstegr (3p)

Name

zstegr - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T is a relatively robust representation

Synopsis

SUBROUTINE ZSTEGR(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W,
Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO)

CHARACTER*1 JOBZ, RANGE
DOUBLE COMPLEX Z(LDZ,*)
INTEGER N, IL, IU, M, LDZ, LWORK, LIWORK, INFO
INTEGER ISUPPZ(*), IWORK(*)
DOUBLE PRECISION VL, VU, ABSTOL
DOUBLE PRECISION D(*), E(*), W(*), WORK(*)

SUBROUTINE ZSTEGR_64(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M,
W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO)

CHARACTER*1 JOBZ, RANGE
DOUBLE COMPLEX Z(LDZ,*)
INTEGER*8 N, IL, IU, M, LDZ, LWORK, LIWORK, INFO
INTEGER*8 ISUPPZ(*), IWORK(*)
DOUBLE PRECISION VL, VU, ABSTOL
DOUBLE PRECISION D(*), E(*), W(*), WORK(*)




F95 INTERFACE
SUBROUTINE STEGR(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M,
W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO)

CHARACTER(LEN=1) :: JOBZ, RANGE
COMPLEX(8), DIMENSION(:,:) :: Z
INTEGER :: N, IL, IU, M, LDZ, LWORK, LIWORK, INFO
INTEGER, DIMENSION(:) :: ISUPPZ, IWORK
REAL(8) :: VL, VU, ABSTOL
REAL(8), DIMENSION(:) :: D, E, W, WORK

SUBROUTINE STEGR_64(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL,
M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO)

CHARACTER(LEN=1) :: JOBZ, RANGE
COMPLEX(8), DIMENSION(:,:) :: Z
INTEGER(8) :: N, IL, IU, M, LDZ, LWORK, LIWORK, INFO
INTEGER(8), DIMENSION(:) :: ISUPPZ, IWORK
REAL(8) :: VL, VU, ABSTOL
REAL(8), DIMENSION(:) :: D, E, W, WORK




C INTERFACE
#include <sunperf.h>

void zstegr(char jobz, char range, int n, double *d, double *e,  double
vl,  double vu, int il, int iu, double abstol, int *m, double
*w, doublecomplex *z, int ldz, int *isuppz, int *info);

void zstegr_64(char jobz, char range, long n,  double  *d,  double  *e,
double  vl,  double vu, long il, long iu, double abstol, long
*m, double *w, doublecomplex *z, long ldz, long *isuppz, long
*info);

Description

Oracle Solaris Studio Performance Library                           zstegr(3P)



NAME
       zstegr - Compute T-sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T  is
       a relatively robust representation


SYNOPSIS
       SUBROUTINE ZSTEGR(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W,
             Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO)

       CHARACTER*1 JOBZ, RANGE
       DOUBLE COMPLEX Z(LDZ,*)
       INTEGER N, IL, IU, M, LDZ, LWORK, LIWORK, INFO
       INTEGER ISUPPZ(*), IWORK(*)
       DOUBLE PRECISION VL, VU, ABSTOL
       DOUBLE PRECISION D(*), E(*), W(*), WORK(*)

       SUBROUTINE ZSTEGR_64(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M,
             W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO)

       CHARACTER*1 JOBZ, RANGE
       DOUBLE COMPLEX Z(LDZ,*)
       INTEGER*8 N, IL, IU, M, LDZ, LWORK, LIWORK, INFO
       INTEGER*8 ISUPPZ(*), IWORK(*)
       DOUBLE PRECISION VL, VU, ABSTOL
       DOUBLE PRECISION D(*), E(*), W(*), WORK(*)




   F95 INTERFACE
       SUBROUTINE STEGR(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M,
              W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO)

       CHARACTER(LEN=1) :: JOBZ, RANGE
       COMPLEX(8), DIMENSION(:,:) :: Z
       INTEGER :: N, IL, IU, M, LDZ, LWORK, LIWORK, INFO
       INTEGER, DIMENSION(:) :: ISUPPZ, IWORK
       REAL(8) :: VL, VU, ABSTOL
       REAL(8), DIMENSION(:) :: D, E, W, WORK

       SUBROUTINE STEGR_64(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL,
              M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO)

       CHARACTER(LEN=1) :: JOBZ, RANGE
       COMPLEX(8), DIMENSION(:,:) :: Z
       INTEGER(8) :: N, IL, IU, M, LDZ, LWORK, LIWORK, INFO
       INTEGER(8), DIMENSION(:) :: ISUPPZ, IWORK
       REAL(8) :: VL, VU, ABSTOL
       REAL(8), DIMENSION(:) :: D, E, W, WORK




   C INTERFACE
       #include <sunperf.h>

       void zstegr(char jobz, char range, int n, double *d, double *e,  double
                 vl,  double vu, int il, int iu, double abstol, int *m, double
                 *w, doublecomplex *z, int ldz, int *isuppz, int *info);

       void zstegr_64(char jobz, char range, long n,  double  *d,  double  *e,
                 double  vl,  double vu, long il, long iu, double abstol, long
                 *m, double *w, doublecomplex *z, long ldz, long *isuppz, long
                 *info);



PURPOSE
       ZSTEGR computes selected eigenvalues and, optionally, eigenvectors of a
       real symmetric tridiagonal matrix T.  Eigenvalues and eigenvectors  can
       be  selected  by  specifying  either  a  range  of values or a range of
       indices for the desired eigenvalues. The eigenvalues  are  computed  by
       the  dqds  algorithm,  while  orthogonal eigenvectors are computed from
       various ``good'' L D L^T  representations  (also  known  as  Relatively
       Robust  Representations).  Gram-Schmidt orthogonalization is avoided as
       far as possible. More specifically, the various steps of the  algorithm
       are as follows. For the i-th unreduced block of T,
          (a)  Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T is
       a relatively robust representation,
          (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high rel-
       ative accuracy by the dqds algorithm,
          (c)  If  there  is  a cluster of close eigenvalues, "choose" sigma_i
       close to the cluster, and go to step (a),
          (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T, com-
       pute  the corresponding eigenvector by forming a rank-revealing twisted
       factorization.
       The desired accuracy of the output can be specified by the input param-
       eter ABSTOL.

       For  more details, see "A new O(n^2) algorithm for the symmetric tridi-
       agonal eigenvalue/eigenvector problem", by Inderjit  Dhillon,  Computer
       Science  Division Technical Report No. UCB/CSD-97-971, UC Berkeley, May
       1997.

       Note 1 : Currently ZSTEGR is only set up to find ALL the n  eigenvalues
       and eigenvectors of T in O(n^2) time
       Note  2  :  Currently  the routine ZSTEIN is called when an appropriate
       sigma_i cannot be chosen in step (c)  above.  ZSTEIN  invokes  modified
       Gram-Schmidt when eigenvalues are close.
       Note  3 : ZSTEGR works only on machines which follow ieee-754 floating-
       point standard in their handling of infinities and NaNs.  Normal execu-
       tion  of  ZSTEGR may create NaNs and infinities and hence may abort due
       to a floating point exception in environments which do not  conform  to
       the ieee standard.


ARGUMENTS
       JOBZ (input)
                 = 'N':  Compute eigenvalues only;
                 = 'V':  Compute eigenvalues and eigenvectors.


       RANGE (input)
                 = 'A': all eigenvalues will be found.
                 = 'V': all eigenvalues in the half-open interval (VL,VU] will
                 be found.  = 'I': the IL-th through IU-th eigenvalues will be
                 found.


       N (input) The order of the matrix.  N >= 0.


       D (input/output)
                 On  entry,  the n diagonal elements of the tridiagonal matrix
                 T. On exit, D is overwritten.


       E (input/output)
                 On entry, the (n-1) subdiagonal elements of  the  tridiagonal
                 matrix T in elements 1 to N-1 of E; E(N) need not be set.  On
                 exit, E is overwritten.


       VL (input)
                 If RANGE='V', the lower and upper bounds of the  interval  to
                 be  searched  for  eigenvalues.  VL  < VU.  Not referenced if
                 RANGE = 'A' or 'I'.


       VU (input)
                 If RANGE='V', the lower and upper bounds of the  interval  to
                 be  searched  for  eigenvalues.  VL  < VU.  Not referenced if
                 RANGE = 'A' or 'I'.


       IL (input)
                 If RANGE='I', the indices (in ascending order) of the  small-
                 est and largest eigenvalues 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 (input)
                 If  RANGE='I', the indices (in ascending order) of the small-
                 est and largest eigenvalues 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 (input)
                 The absolute error tolerance  for  the  eigenvalues/eigenvec-
                 tors.  IF JOBZ = 'V', the eigenvalues and eigenvectors output
                 have residual norms bounded by ABSTOL, and the  dot  products
                 between  different  eigenvectors  are  bounded  by ABSTOL. If
                 ABSTOL is less than N*EPS*|T|, then N*EPS*|T| will be used in
                 its  place, where EPS is the machine precision and |T| is the
                 1-norm of the tridiagonal matrix. The  eigenvalues  are  com-
                 puted  to  an  accuracy of EPS*|T| irrespective of ABSTOL. If
                 high relative accuracy is important, set  ABSTOL  to  DLAMCH(
                 'Safe  minimum' ).  See Barlow and 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 (output)
                 The total number of eigenvalues found.  0  <=  M  <=  N.   If
                 RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.


       W (output)
                 The  first  M  elements  contain  the selected eigenvalues in
                 ascending order.


       Z (output)
                 If JOBZ = 'V', then if INFO = 0, the first  M  columns  of  Z
                 contain  the  orthonormal eigenvectors of the matrix T corre-
                 sponding 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.


       LDZ (input)
                 The  leading dimension of the array Z.  LDZ >= 1, and if JOBZ
                 = 'V', LDZ >= max(1,N).


       ISUPPZ (output)
                 The support of the eigenvectors in Z, i.e., the indices indi-
                 cating  the  nonzero  elements  in Z. The i-th eigenvector is
                 nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ(  2*i
                 ).


       WORK (workspace)
                 On  exit, if INFO = 0, WORK(1) returns the optimal (and mini-
                 mal) LWORK.


       LWORK (input)
                 The dimension of the array WORK.  LWORK >= max(1,18*N)

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


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

                 If  LIWORK  = -1, then a workspace query is assumed; the rou-
                 tine 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 (output)
                 = 0:  successful exit
                 < 0:  if INFO = -i, the i-th argument had an illegal value
                 > 0:  if INFO = 1, internal error in ZLARRE,  if  INFO  =  2,
                 internal error in ZLARRV.

FURTHER DETAILS
       Based on contributions by
          Inderjit Dhillon, IBM Almaden, USA
          Osni Marques, LBNL/NERSC, USA
          Ken Stanley, Computer Science Division, University of
            California at Berkeley, USA




                                  7 Nov 2015                        zstegr(3P)