Contents
dgebrd - reduce a general real M-by-N matrix A to upper or
lower bidiagonal form B by an orthogonal transformation
SUBROUTINE DGEBRD(M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO)
INTEGER M, N, LDA, LWORK, INFO
DOUBLE PRECISION A(LDA,*), D(*), E(*), TAUQ(*), TAUP(*),
WORK(*)
SUBROUTINE DGEBRD_64(M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK,
INFO)
INTEGER*8 M, N, LDA, LWORK, INFO
DOUBLE PRECISION A(LDA,*), D(*), E(*), TAUQ(*), TAUP(*),
WORK(*)
F95 INTERFACE
SUBROUTINE GEBRD([M], [N], A, [LDA], D, E, TAUQ, TAUP, [WORK], [LWORK],
[INFO])
INTEGER :: M, N, LDA, LWORK, INFO
REAL(8), DIMENSION(:) :: D, E, TAUQ, TAUP, WORK
REAL(8), DIMENSION(:,:) :: A
SUBROUTINE GEBRD_64([M], [N], A, [LDA], D, E, TAUQ, TAUP, [WORK],
[LWORK], [INFO])
INTEGER(8) :: M, N, LDA, LWORK, INFO
REAL(8), DIMENSION(:) :: D, E, TAUQ, TAUP, WORK
REAL(8), DIMENSION(:,:) :: A
C INTERFACE
#include <sunperf.h>
void dgebrd(int m, int n, double *a, int lda, double *d,
double *e, double *tauq, double *taup, int *info);
void dgebrd_64(long m, long n, double *a, long lda, double
*d, double *e, double *tauq, double *taup, long
*info);
dgebrd reduces a general real M-by-N matrix A to upper or
lower bidiagonal form B by an orthogonal transformation:
Q**T * A * P = B.
If m >= n, B is upper bidiagonal; if m < n, B is lower bidi-
agonal.
M (input) The number of rows in the matrix A. M >= 0.
N (input) The number of columns in the matrix A. N >= 0.
A (input/output)
On entry, the M-by-N general matrix to be reduced.
On exit, if m >= n, the diagonal and the first
superdiagonal are overwritten with the upper bidi-
agonal matrix B; the elements below the diagonal,
with the array TAUQ, represent the orthogonal
matrix Q as a product of elementary reflectors,
and the elements above the first superdiagonal,
with the array TAUP, represent the orthogonal
matrix P as a product of elementary reflectors; if
m < n, the diagonal and the first subdiagonal are
overwritten with the lower bidiagonal matrix B;
the elements below the first subdiagonal, with the
array TAUQ, represent the orthogonal matrix Q as a
product of elementary reflectors, and the elements
above the diagonal, with the array TAUP, represent
the orthogonal matrix P as a product of elementary
reflectors. See Further Details.
LDA (input)
The leading dimension of the array A. LDA >=
max(1,M).
D (output)
The diagonal elements of the bidiagonal matrix B:
D(i) = A(i,i).
E (output)
The off-diagonal elements of the bidiagonal matrix
B: if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-
1; if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
TAUQ (output)
The scalar factors of the elementary reflectors
which represent the orthogonal matrix Q. See
Further Details.
TAUP (output)
The scalar factors of the elementary reflectors
which represent the orthogonal matrix P. See
Further Details.
WORK (workspace)
On exit, if INFO = 0, WORK(1) returns the optimal
LWORK.
LWORK (input)
The length of the array WORK. LWORK >=
max(1,M,N). For optimum performance LWORK >=
(M+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 ille-
gal value.
The matrices Q and P are represented as products of elemen-
tary reflectors:
If m >= n,
Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
Each H(i) and G(i) has the form:
H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'
where tauq and taup are real scalars, and v and u are real
vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on
exit in A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is
stored on exit in A(i,i+2:n); tauq is stored in TAUQ(i) and
taup in TAUP(i).
If m < n,
Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
Each H(i) and G(i) has the form:
H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'
where tauq and taup are real scalars, and v and u are real
vectors; v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on
exit in A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is
stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and
taup in TAUP(i).
The contents of A on exit are illustrated by the following
examples:
m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
( d e u1 u1 u1 ) ( d u1 u1 u1 u1
u1 )
( v1 d e u2 u2 ) ( e d u2 u2 u2
u2 )
( v1 v2 d e u3 ) ( v1 e d u3 u3
u3 )
( v1 v2 v3 d e ) ( v1 v2 e d u4
u4 )
( v1 v2 v3 v4 d ) ( v1 v2 v3 e d
u5 )
( v1 v2 v3 v4 v5 )
where d and e denote diagonal and off-diagonal elements of
B, vi denotes an element of the vector defining H(i), and ui
an element of the vector defining G(i).