NAME

zhetrd - reduce a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation

SYNOPSIS

  SUBROUTINE ZHETRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO)
  CHARACTER * 1 UPLO
  DOUBLE COMPLEX A(LDA,*), TAU(*), WORK(*)
  INTEGER N, LDA, LWORK, INFO
  DOUBLE PRECISION D(*), E(*)

  SUBROUTINE ZHETRD_64( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO)
  CHARACTER * 1 UPLO
  DOUBLE COMPLEX A(LDA,*), TAU(*), WORK(*)
  INTEGER*8 N, LDA, LWORK, INFO
  DOUBLE PRECISION D(*), E(*)

F95 INTERFACE

  SUBROUTINE HETRD( UPLO, [N], A, [LDA], D, E, TAU, [WORK], [LWORK], 
 *       [INFO])
  CHARACTER(LEN=1) :: UPLO
  COMPLEX(8), DIMENSION(:) :: TAU, WORK
  COMPLEX(8), DIMENSION(:,:) :: A
  INTEGER :: N, LDA, LWORK, INFO
  REAL(8), DIMENSION(:) :: D, E

  SUBROUTINE HETRD_64( UPLO, [N], A, [LDA], D, E, TAU, [WORK], [LWORK], 
 *       [INFO])
  CHARACTER(LEN=1) :: UPLO
  COMPLEX(8), DIMENSION(:) :: TAU, WORK
  COMPLEX(8), DIMENSION(:,:) :: A
  INTEGER(8) :: N, LDA, LWORK, INFO
  REAL(8), DIMENSION(:) :: D, E

C INTERFACE

#include <sunperf.h>

void zhetrd(char uplo, int n, doublecomplex *a, int lda, double *d, double *e, doublecomplex *tau, int *info);

void zhetrd_64(char uplo, long n, doublecomplex *a, long lda, double *d, double *e, doublecomplex *tau, long *info);

PURPOSE

zhetrd reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: Q**H * A * Q = T.

ARGUMENTS

UPLO (input)

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

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

N (input)
The order of the matrix A. N > = 0.
A (input/output)
On entry, the Hermitian matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit, if UPLO = 'U', the diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of A are over- written by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors. See Further Details.
LDA (input)
The leading dimension of the array A. LDA > = max(1,N).
D (output)
The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).
E (output)
The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
TAU (output)
The scalar factors of the elementary reflectors (see Further Details).
WORK (workspace)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input)
The dimension of the array WORK. LWORK > = 1. For optimum performance LWORK > = N*NB, where NB is the optimal blocksize.
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 (output)

 = 0:  successful exit

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

FURTHER DETAILS

If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors

   Q  = H(n-1) . . . H(2) H(1).

Each H(i) has the form

   H(i)  = I - tau * v * v'

where tau is a complex scalar, and v is a complex vector with v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in

A(1:i-1,i+1), and tau in TAU(i).

If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors

   Q  = H(1) H(2) . . . H(n-1).

Each H(i) has the form

   H(i)  = I - tau * v * v'

where tau is a complex scalar, and v is a complex vector with v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), and tau in TAU(i).

The contents of A on exit are illustrated by the following examples with n = 5:

if UPLO = 'U': if UPLO = 'L':

  (  d   e   v2  v3  v4 )              (  d                  )
  (      d   e   v3  v4 )              (  e   d              )
  (          d   e   v4 )              (  v1  e   d          )
  (              d   e  )              (  v1  v2  e   d      )
  (                  d  )              (  v1  v2  v3  e   d  )

where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i).