dgebrd(3P) Sun Performance Library dgebrd(3P)NAMEdgebrd - reduce a general real M-by-N matrix A to upper or lower bidi‐
agonal form B by an orthogonal transformation
SYNOPSIS
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);
PURPOSEdgebrd reduces a general real M-by-N matrix A to upper or lower bidiag‐
onal 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.
ARGUMENTS
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 over‐
written with the upper bidiagonal 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 ele‐
mentary reflectors, and the elements above the diagonal, with
the array TAUP, represent the orthogonal matrix P as a prod‐
uct 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 repre‐
sent the orthogonal matrix Q. See Further Details.
TAUP (output)
The scalar factors of the elementary reflectors which repre‐
sent 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 opti‐
mal 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
The matrices Q and P are represented as products of elementary reflec‐
tors:
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).
6 Mar 2009 dgebrd(3P)
