dsytrd(3P) Sun Performance Library dsytrd(3P)NAMEdsytrd - reduce a real symmetric matrix A to real symmetric tridiagonal
form T by an orthogonal similarity transformation
SYNOPSIS
SUBROUTINE DSYTRD(UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO)
CHARACTER * 1 UPLO
INTEGER N, LDA, LWORK, INFO
DOUBLE PRECISION A(LDA,*), D(*), E(*), TAU(*), WORK(*)
SUBROUTINE DSYTRD_64(UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO)
CHARACTER * 1 UPLO
INTEGER*8 N, LDA, LWORK, INFO
DOUBLE PRECISION A(LDA,*), D(*), E(*), TAU(*), WORK(*)
F95 INTERFACE
SUBROUTINE SYTRD(UPLO, N, A, [LDA], D, E, TAU, [WORK], [LWORK], [INFO])
CHARACTER(LEN=1) :: UPLO
INTEGER :: N, LDA, LWORK, INFO
REAL(8), DIMENSION(:) :: D, E, TAU, WORK
REAL(8), DIMENSION(:,:) :: A
SUBROUTINE SYTRD_64(UPLO, N, A, [LDA], D, E, TAU, [WORK], [LWORK],
[INFO])
CHARACTER(LEN=1) :: UPLO
INTEGER(8) :: N, LDA, LWORK, INFO
REAL(8), DIMENSION(:) :: D, E, TAU, WORK
REAL(8), DIMENSION(:,:) :: A
C INTERFACE
#include <sunperf.h>
void dsytrd(char uplo, int n, double *a, int lda, double *d, double *e,
double *tau, int *info);
void dsytrd_64(char uplo, long n, double *a, long lda, double *d, dou‐
ble *e, double *tau, long *info);
PURPOSEdsytrd reduces a real symmetric matrix A to real 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.
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 triangu‐
lar 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 corre‐
sponding elements of the tridiagonal matrix T, and the ele‐
ments above the first superdiagonal, with the array TAU, rep‐
resent 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 subdi‐
agonal, 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).
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 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).
6 Mar 2009 dsytrd(3P)
