CGEBRD - reduce a general complex M-by-N matrix A to upper
or lower bidiagonal form B by a unitary transformation
SYNOPSIS
SUBROUTINE CGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK,
LWORK, INFO )
INTEGER INFO, LDA, LWORK, M, N
REAL D( * ), E( * )
COMPLEX A( LDA, * ), TAUP( * ), TAUQ( * ),
WORK( * )
PURPOSE
CGEBRD reduces a general complex M-by-N matrix A to upper
or lower bidiagonal form B by a unitary transformation:
Q**H * 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) COMPLEX 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 unitary matrix
Q as a product of elementary reflectors, and the
elements above the first superdiagonal, with the
array TAUP, represent the unitary 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 unitary matrix Q as a product of
elementary reflectors, and the elements above the
diagonal, with the array TAUP, represent the uni
tary matrix P as a product of elementary reflec
tors. See Further Details. LDA (input) INTE
GER The leading dimension of the array A. LDA >=
max(1,M).
D (output) REAL array, dimension (min(M,N))
The diagonal elements of the bidiagonal matrix B:
D(i) = A(i,i).
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) COMPLEX array dimension (min(M,N))
The scalar factors of the elementary reflectors
which represent the unitary matrix Q. See Further
Details. TAUP (output) COMPLEX array, dimen
sion (min(M,N)) The scalar factors of the elemen
tary reflectors which represent the unitary matrix
P. See Further Details. WORK (workspace/out
put) COMPLEX 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.
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 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)
H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'
where tauq and taup are complex scalars, and 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).
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