NAME

ssytd2 - reduce a real symmetric matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation

SYNOPSIS

  SUBROUTINE SSYTD2( UPLO, N, A, LDA, D, E, TAU, INFO)
  CHARACTER * 1 UPLO
  INTEGER N, LDA, INFO
  REAL A(LDA,*), D(*), E(*), TAU(*)

  SUBROUTINE SSYTD2_64( UPLO, N, A, LDA, D, E, TAU, INFO)
  CHARACTER * 1 UPLO
  INTEGER*8 N, LDA, INFO
  REAL A(LDA,*), D(*), E(*), TAU(*)

F95 INTERFACE

  SUBROUTINE SYTD2( UPLO, [N], A, [LDA], D, E, TAU, [INFO])
  CHARACTER(LEN=1) :: UPLO
  INTEGER :: N, LDA, INFO
  REAL, DIMENSION(:) :: D, E, TAU
  REAL, DIMENSION(:,:) :: A

  SUBROUTINE SYTD2_64( UPLO, [N], A, [LDA], D, E, TAU, [INFO])
  CHARACTER(LEN=1) :: UPLO
  INTEGER(8) :: N, LDA, INFO
  REAL, DIMENSION(:) :: D, E, TAU
  REAL, DIMENSION(:,:) :: A

C INTERFACE

#include <sunperf.h>

void ssytd2(char uplo, int n, float *a, int lda, float *d, float *e, float *tau, int *info);

void ssytd2_64(char uplo, long n, float *a, long lda, float *d, float *e, float *tau, long *info);

PURPOSE

ssytd2 reduces a real symmetric matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation: Q' * A * Q = T.

ARGUMENTS

UPLO (input)
Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
```
 = 'U':  Upper triangular
```
```
 = 'L':  Lower triangular
```
N (input)
The order of the matrix A. N > = 0.
A (input/output)
On entry, the symmetric 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 orthogonal 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 orthogonal 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).

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 real scalar, and v is a real 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 real scalar, and v is a real 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).