Contents
dsytd2 - reduce a real symmetric matrix A to symmetric tri-
diagonal form T by an orthogonal similarity transformation
SUBROUTINE DSYTD2(UPLO, N, A, LDA, D, E, TAU, INFO)
CHARACTER * 1 UPLO
INTEGER N, LDA, INFO
DOUBLE PRECISION A(LDA,*), D(*), E(*), TAU(*)
SUBROUTINE DSYTD2_64(UPLO, N, A, LDA, D, E, TAU, INFO)
CHARACTER * 1 UPLO
INTEGER*8 N, LDA, INFO
DOUBLE PRECISION 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(8), DIMENSION(:) :: D, E, TAU
REAL(8), DIMENSION(:,:) :: A
SUBROUTINE SYTD2_64(UPLO, N, A, [LDA], D, E, TAU, [INFO])
CHARACTER(LEN=1) :: UPLO
INTEGER(8) :: N, LDA, INFO
REAL(8), DIMENSION(:) :: D, E, TAU
REAL(8), DIMENSION(:,:) :: A
C INTERFACE
#include <sunperf.h>
void dsytd2(char uplo, int n, double *a, int lda, double *d,
double *e, double *tau, int *info);
void dsytd2_64(char uplo, long n, double *a, long lda, dou-
ble *d, double *e, double *tau, long *info);
dsytd2 reduces a real symmetric matrix A to symmetric tridi-
agonal form T by an orthogonal similarity transformation: Q'
* A * Q = T.
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) On entry, the symmetric matrix A. If UPLO = 'U',
the leading n-by-n upper triangular part of A con-
tains 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 tri-
angular 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 super-
diagonal of A are overwritten by the corresponding
elements of the tridiagonal matrix T, and the ele-
ments 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 tri-
diagonal 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 ille-
gal value.
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).