NAME

dsptrd - reduce a real symmetric matrix A stored in packed form to symmetric tridiagonal form T by an orthogonal similarity transformation

SYNOPSIS

  SUBROUTINE DSPTRD( UPLO, N, AP, D, E, TAU, INFO)
  CHARACTER * 1 UPLO
  INTEGER N, INFO
  DOUBLE PRECISION AP(*), D(*), E(*), TAU(*)

  SUBROUTINE DSPTRD_64( UPLO, N, AP, D, E, TAU, INFO)
  CHARACTER * 1 UPLO
  INTEGER*8 N, INFO
  DOUBLE PRECISION AP(*), D(*), E(*), TAU(*)

F95 INTERFACE

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

  SUBROUTINE SPTRD_64( UPLO, N, AP, D, E, TAU, [INFO])
  CHARACTER(LEN=1) :: UPLO
  INTEGER(8) :: N, INFO
  REAL(8), DIMENSION(:) :: AP, D, E, TAU

C INTERFACE

#include <sunperf.h>

void dsptrd(char uplo, int n, double *ap, double *d, double *e, double *tau, int *info);

void dsptrd_64(char uplo, long n, double *ap, double *d, double *e, double *tau, long *info);

PURPOSE

dsptrd reduces a real symmetric matrix A stored in packed form to symmetric tridiagonal form T by an orthogonal similarity transformation: Q**T * 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.
AP (input/output)
On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1 < =i < =j; if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j < =i < =n. 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.
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 AP, overwriting A(1:i-1,i+1), and tau is stored 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 AP, overwriting A(i+2:n,i), and tau is stored in TAU(i).