cgebd2.f man page

cgebd2.f —

Synopsis

Functions/Subroutines

subroutine cgebd2 (M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO)
CGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.

Function/Subroutine Documentation

subroutine cgebd2 (integerM, integerN, complex, dimension( lda, * )A, integerLDA, real, dimension( * )D, real, dimension( * )E, complex, dimension( * )TAUQ, complex, dimension( * )TAUP, complex, dimension( * )WORK, integerINFO)

CGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.

Purpose:

CGEBD2 reduces a complex general m by n matrix A to upper or lower
real 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.

Parameters:

M

M is INTEGER
The number of rows in the matrix A.  M >= 0.

N

N is INTEGER
The number of columns in the matrix A.  N >= 0.

A

A is 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

LDA is INTEGER
The leading dimension of the array A.  LDA >= max(1,M).

D

D is REAL array, dimension (min(M,N))
The diagonal elements of the bidiagonal matrix B:
D(i) = A(i,i).

E

E is 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

TAUQ is COMPLEX array dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the unitary matrix Q. See Further Details.

TAUP

TAUP is COMPLEX array, dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the unitary matrix P. See Further Details.

WORK

WORK is COMPLEX array, dimension (max(M,N))

INFO

INFO is INTEGER
= 0: successful exit 
< 0: if INFO = -i, the i-th argument had an illegal value.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

September 2012

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).

Definition at line 191 of file cgebd2.f.

Author

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Referenced By

cgebd2(3) is an alias of cgebd2.f(3).

Sat Nov 16 2013 Version 3.4.2 LAPACK