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NAME
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( LWORK )
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 bidi-
agonal.
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 bidiag-
onal 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 lead-
ing 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).
E (output) 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) DOUBLE PRECISION array dimension (min(M,N))
The scalar factors of the elementary reflectors
which represent the orthogonal matrix Q. See Further
Details. TAUP (output) DOUBLE PRECISION array,
dimension (min(M,N)) The scalar factors of the ele-
mentary reflectors which represent the orthogonal
matrix P. See Further Details. WORK (workspace)
DOUBLE PRECISION array, dimension (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.
INFO (output) 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 elemen-
tary 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).