dgebd2 - reduce a general matrix to bidiagonal form using an unblocked algorithm
SUBROUTINE DGEBD2(M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO) INTEGER INFO, LDA, M, N DOUBLE PRECISION A(LDA,*), D(*), E(*), TAUP(*), TAUQ(*), WORK(*) SUBROUTINE DGEBD2_64(M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO) INTEGER*8 INFO, LDA, M, N DOUBLE PRECISION A(LDA,*), D(*), E(*), TAUP(*), TAUQ(*), WORK(*) F95 INTERFACE SUBROUTINE GEBD2(M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO) INTEGER :: M, N, LDA, INFO REAL(8), DIMENSION(:,:) :: A REAL(8), DIMENSION(:) :: D, E, TAUQ, TAUP, WORK SUBROUTINE GEBD2_64(M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO) INTEGER(8) :: M, N, LDA, INFO REAL(8), DIMENSION(:,:) :: A REAL(8), DIMENSION(:) :: D, E, TAUQ, TAUP, WORK C INTERFACE #include <sunperf.h> void dgebd2 (int m, int n, double *a, int lda, double *d, double *e, double *tauq, double *taup, int *info); void dgebd2_64 (long m, long n, double *a, long lda, double *d, double *e, double *tauq, double *taup, long *info);
Oracle Solaris Studio Performance Library dgebd2(3P)
NAME
dgebd2 - reduce a general matrix to bidiagonal form using an unblocked
algorithm
SYNOPSIS
SUBROUTINE DGEBD2(M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO)
INTEGER INFO, LDA, M, N
DOUBLE PRECISION A(LDA,*), D(*), E(*), TAUP(*), TAUQ(*), WORK(*)
SUBROUTINE DGEBD2_64(M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO)
INTEGER*8 INFO, LDA, M, N
DOUBLE PRECISION A(LDA,*), D(*), E(*), TAUP(*), TAUQ(*), WORK(*)
F95 INTERFACE
SUBROUTINE GEBD2(M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO)
INTEGER :: M, N, LDA, INFO
REAL(8), DIMENSION(:,:) :: A
REAL(8), DIMENSION(:) :: D, E, TAUQ, TAUP, WORK
SUBROUTINE GEBD2_64(M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO)
INTEGER(8) :: M, N, LDA, INFO
REAL(8), DIMENSION(:,:) :: A
REAL(8), DIMENSION(:) :: D, E, TAUQ, TAUP, WORK
C INTERFACE
#include <sunperf.h>
void dgebd2 (int m, int n, double *a, int lda, double *d, double *e,
double *tauq, double *taup, int *info);
void dgebd2_64 (long m, long n, double *a, long lda, double *d, double
*e, double *tauq, double *taup, long *info);
PURPOSE
dgebd2 reduces a real general 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)
M is INTEGER
The number of rows in the matrix A. M >= 0.
N (input)
N is INTEGER
The number of columns in the matrix A. N >= 0.
A (input/output)
A is DOUBLE PRECISION array, dimension (LDA,N)
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 over-
written 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 reflec-
tors, and the elements above the diagonal, with the array
TAUP, represent the orthogonal matrix P as a product of ele-
mentary reflectors. See Further Details.
LDA (input)
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
D (output)
D is DOUBLE PRECISION array, dimension (min(M,N))
The diagonal elements of the bidiagonal matrix B:
D(i) = A(i,i).
E (output)
E is DOUBLE PRECISION array, dimension (min(M,N)-1)
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)
TAUQ is DOUBLE PRECISION array dimension (min(M,N))
The scalar factors of the elementary reflectors which repre-
sent the orthogonal matrix Q. See Further Details.
TAUP (output)
TAUP is DOUBLE PRECISION array, dimension (min(M,N))
The scalar factors of the elementary reflectors which repre-
sent the orthogonal matrix P. See Further Details.
WORK (output)
WORK is DOUBLE PRECISION array, dimension (max(M,N))
INFO (output)
INFO is INTEGER
= 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
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**T and G(i) = I - taup * u * u**T
where tauq and taup are complex scalars, and v and u are complex
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**T and G(i) = I - taup * u * u**T
where tauq and taup are complex scalars, v and u are complex 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).
7 Nov 2015 dgebd2(3P)