• NAME
• SYNOPSIS
• F95 INTERFACE
• C INTERFACE
• PURPOSE
• ARGUMENTS

NAME

zpteqr - compute all eigenvalues and, optionally, eigenvectors of a symmetric positive definite tridiagonal matrix by first factoring the matrix using SPTTRF and then calling CBDSQR to compute the singular values of the bidiagonal factor

SYNOPSIS

  SUBROUTINE ZPTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO)
  CHARACTER * 1 COMPZ
  DOUBLE COMPLEX Z(LDZ,*)
  INTEGER N, LDZ, INFO
  DOUBLE PRECISION D(*), E(*), WORK(*)

  SUBROUTINE ZPTEQR_64( COMPZ, N, D, E, Z, LDZ, WORK, INFO)
  CHARACTER * 1 COMPZ
  DOUBLE COMPLEX Z(LDZ,*)
  INTEGER*8 N, LDZ, INFO
  DOUBLE PRECISION D(*), E(*), WORK(*)

F95 INTERFACE

  SUBROUTINE PTEQR( COMPZ, [N], D, E, Z, [LDZ], [WORK], [INFO])
  CHARACTER(LEN=1) :: COMPZ
  COMPLEX(8), DIMENSION(:,:) :: Z
  INTEGER :: N, LDZ, INFO
  REAL(8), DIMENSION(:) :: D, E, WORK

  SUBROUTINE PTEQR_64( COMPZ, [N], D, E, Z, [LDZ], [WORK], [INFO])
  CHARACTER(LEN=1) :: COMPZ
  COMPLEX(8), DIMENSION(:,:) :: Z
  INTEGER(8) :: N, LDZ, INFO
  REAL(8), DIMENSION(:) :: D, E, WORK

C INTERFACE

#include <sunperf.h>

void zpteqr(char compz, int n, double *d, double *e, doublecomplex *z, int ldz, int *info);

void zpteqr_64(char compz, long n, double *d, double *e, doublecomplex *z, long ldz, long *info);

PURPOSE

zpteqr computes all eigenvalues and, optionally, eigenvectors of a symmetric positive definite tridiagonal matrix by first factoring the matrix using SPTTRF and then calling CBDSQR to compute the singular values of the bidiagonal factor.

This routine computes the eigenvalues of the positive definite tridiagonal matrix to high relative accuracy. This means that if the eigenvalues range over many orders of magnitude in size, then the small eigenvalues and corresponding eigenvectors will be computed more accurately than, for example, with the standard QR method.

The eigenvectors of a full or band positive definite Hermitian matrix can also be found if CHETRD, CHPTRD, or CHBTRD has been used to reduce this matrix to tridiagonal form. (The reduction to tridiagonal form, however, may preclude the possibility of obtaining high relative accuracy in the small eigenvalues of the original matrix, if these eigenvalues range over many orders of magnitude.)

ARGUMENTS

• COMPZ (input)
  

   = 'N':  Compute eigenvalues only.

   = 'V':  Compute eigenvectors of original Hermitian
  matrix also.  Array Z contains the unitary matrix
  used to reduce the original matrix to tridiagonal
  form.
   = 'I':  Compute eigenvectors of tridiagonal matrix also.

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

• D (input/output)
  On entry, the n diagonal elements of the tridiagonal matrix. On normal exit, D contains the eigenvalues, in descending order.

• E (input/output)
  On entry, the (n-1) subdiagonal elements of the tridiagonal matrix. On exit, E has been destroyed.

• Z (input/output)
  On entry, if COMPZ = 'V', the unitary matrix used in the reduction to tridiagonal form. On exit, if COMPZ = 'V', the orthonormal eigenvectors of the original Hermitian matrix; if COMPZ = 'I', the orthonormal eigenvectors of the tridiagonal matrix. If INFO > 0 on exit, Z contains the eigenvectors associated with only the stored eigenvalues. If COMPZ = 'N', then Z is not referenced.

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

• WORK (workspace)
  dimension(4*N)

• INFO (output)
  

   = 0:  successful exit.

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

   > 0:  if INFO  = i, and i is:
   < = N  the Cholesky factorization of the matrix could
  not be performed because the i-th principal minor
  was not positive definite.
   > N   the SVD algorithm failed to converge;
  if INFO  = N+i, i off-diagonal elements of the
  bidiagonal factor did not converge to zero.