CGEBRD (3) Linux Manual Page
cgebrd.f –
Synopsis
Functions/Subroutines
subroutine cgebrd (M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO)CGEBRD
Function/Subroutine Documentation
subroutine cgebrd (integerM, integerN, complex, dimension( lda, * )A, integerLDA, real, dimension( * )D, real, dimension( * )E, complex, dimension( * )TAUQ, complex, dimension( * )TAUP, complex, dimension( * )WORK, integerLWORK, integerINFO)
CGEBRD 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.
Parameters:
- M
M is INTEGER
N
The number of rows in the matrix A. M >= 0.N is INTEGER
A
The number of columns in the matrix A. N >= 0.A is COMPLEX array, dimension (LDA,N)
LDA
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 is INTEGER
D
The leading dimension of the array A. LDA >= max(1,M).D is REAL array, dimension (min(M,N))
E
The diagonal elements of the bidiagonal matrix B:
D(i) = A(i,i).E is REAL array, dimension (min(M,N)-1)
TAUQ
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 is COMPLEX array dimension (min(M,N))
TAUP
The scalar factors of the elementary reflectors which
represent the unitary matrix Q. See Further Details.TAUP is COMPLEX array, dimension (min(M,N))
WORK
The scalar factors of the elementary reflectors which
represent the unitary matrix P. See Further Details.WORK is COMPLEX array, dimension (MAX(1,LWORK))
LWORK
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.LWORK is INTEGER
INFO
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 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:
- November 2011
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, 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).
Definition at line 206 of file cgebrd.f.