DGEBRD - reduce a general real M-by-N matrix A to upper or
lower bidiagonal form B by an orthogonal transformation
SYNOPSIS
SUBROUTINE DGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK,
LWORK, INFO )
INTEGER INFO, LDA, LWORK, M, N
DOUBLE PRECISION A( LDA, * ), D( * ), E( * ),
TAUP( * ), TAUQ( * ), WORK( * )
PURPOSE
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.
ARGUMENTS
M (input) INTEGER
The number of rows in the matrix A. M >= 0.
N (input) INTEGER
The number of columns in the matrix A. N >= 0.
A (input/output) 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 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)
INTEGER The leading dimension of the array A. LDA
>= max(1,M).
D (output) DOUBLE PRECISION array, dimension
(min(M,N))
The diagonal elements of the bidiagonal matrix B:
D(i) = A(i,i).
(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) DOUBLE PRECISION array dimension
(min(M,N))
The scalar factors of the elementary reflectors
which represent the orthogonal matrix Q. See Fur
ther Details. TAUP (output) DOUBLE PRECISION
array, dimension (min(M,N)) The scalar factors of
the elementary reflectors which represent the
orthogonal matrix P. See Further Details. WORK
(workspace/output) DOUBLE PRECISION array, dimen
sion (LWORK) On exit, if INFO = 0, WORK(1) returns
the optimal LWORK.
LWORK (input) INTEGER
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) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille
gal value.
FURTHER DETAILS
The matrices Q and P are represented as products of ele
mentary 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,
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).
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