.TH SGEBD2 1 "November 2006" " LAPACK routine (version 3.1) " " LAPACK routine (version 3.1) "
.SH NAME
SGEBD2 - a real general m by n matrix A to upper or lower bidiagonal form B by an orthogonal transformation
.SH SYNOPSIS
.TP 19
SUBROUTINE SGEBD2(
M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )
.TP 19
.ti +4
INTEGER
INFO, LDA, M, N
.TP 19
.ti +4
REAL
A( LDA, * ), D( * ), E( * ), TAUP( * ),
TAUQ( * ), WORK( * )
.SH PURPOSE
SGEBD2 reduces a real general m by n matrix A to upper or lower
bidiagonal form B by an orthogonal transformation: Q\(aq * A * P = B.
If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
.SH ARGUMENTS
.TP 8
M (input) INTEGER
The number of rows in the matrix A. M >= 0.
.TP 8
N (input) INTEGER
The number of columns in the matrix A. N >= 0.
.TP 8
A (input/output) REAL 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
overwritten 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
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) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
.TP 8
D (output) REAL array, dimension (min(M,N))
The diagonal elements of the bidiagonal matrix B:
D(i) = A(i,i).
.TP 8
E (output) REAL 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.
.TP 8
TAUQ (output) REAL array dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the orthogonal matrix Q. See Further Details.
TAUP (output) REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the orthogonal matrix P. See Further Details.
WORK (workspace) REAL array, dimension (max(M,N))
.TP 8
INFO (output) INTEGER
= 0: successful exit.
.br
< 0: if INFO = -i, the i-th argument had an illegal value.
.SH FURTHER DETAILS
The matrices Q and P are represented as products of elementary
reflectors:
.br
If m >= n,
.br
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:
.br
H(i) = I - tauq * v * v\(aq and G(i) = I - taup * u * u\(aq
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).
.br
If m < n,
.br
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:
.br
H(i) = I - tauq * v * v\(aq and G(i) = I - taup * u * u\(aq
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).
.br
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 )
.br
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).
.br