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(*)
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
#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 bidiagonal.
D(i)
= A(i,i).
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.
WORK(1)
returns the optimal LWORK.
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.
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
The matrices Q and P are represented as products of elementary 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).