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CGEBD2

Section: LAPACK routine (version 3.3.1) (1)
Updated: April 2011
Index

NAME

LAPACK-3 - reduces a complex general m by n matrix A to upper or lower real bidiagonal form B by a unitary transformation

SYNOPSIS

SUBROUTINE CGEBD2(: M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )
: INTEGERINFO, LDA, M, N
: REALD( * ), E( * )
: COMPLEXA( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * )

PURPOSE

CGEBD2 reduces a complex general m by n matrix A to upper or lowerreal 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 bidiagonal 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 unitary 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) REAL array, dimension (min(M,N)): The diagonal elements of the bidiagonal matrix B:
D(i) = A(i,i).
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.
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, dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the unitary matrix P. See Further Details.
WORK (workspace) COMPLEX array, dimension (max(M,N))
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 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**H  and G(i) = I - taup * u * u**H
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**H  and G(i) = I - taup * u * u**H
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).

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