SORBDB (3)  Linux Manuals
NAME
sorbdb.f 
SYNOPSIS
Functions/Subroutines
subroutine sorbdb (TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12, X21, LDX21, X22, LDX22, THETA, PHI, TAUP1, TAUP2, TAUQ1, TAUQ2, WORK, LWORK, INFO)
SORBDB
Function/Subroutine Documentation
subroutine sorbdb (characterTRANS, characterSIGNS, integerM, integerP, integerQ, real, dimension( ldx11, * )X11, integerLDX11, real, dimension( ldx12, * )X12, integerLDX12, real, dimension( ldx21, * )X21, integerLDX21, real, dimension( ldx22, * )X22, integerLDX22, real, dimension( * )THETA, real, dimension( * )PHI, real, dimension( * )TAUP1, real, dimension( * )TAUP2, real, dimension( * )TAUQ1, real, dimension( * )TAUQ2, real, dimension( * )WORK, integerLWORK, integerINFO)
SORBDB
Purpose:

SORBDB simultaneously bidiagonalizes the blocks of an MbyM partitioned orthogonal matrix X: [ B11  B12 0 0 ] [ X11  X12 ] [ P1  ] [ 0  0 I 0 ] [ Q1  ]**T X = [] = [] [] [] . [ X21  X22 ] [  P2 ] [ B21  B22 0 0 ] [  Q2 ] [ 0  0 0 I ] X11 is PbyQ. Q must be no larger than P, MP, or MQ. (If this is not the case, then X must be transposed and/or permuted. This can be done in constant time using the TRANS and SIGNS options. See SORCSD for details.) The orthogonal matrices P1, P2, Q1, and Q2 are PbyP, (MP)by (MP), QbyQ, and (MQ)by(MQ), respectively. They are represented implicitly by Householder vectors. B11, B12, B21, and B22 are QbyQ bidiagonal matrices represented implicitly by angles THETA, PHI.
Parameters:

TRANS
TRANS is CHARACTER = 'T': X, U1, U2, V1T, and V2T are stored in rowmajor order; otherwise: X, U1, U2, V1T, and V2T are stored in column major order.
SIGNSSIGNS is CHARACTER = 'O': The lowerleft block is made nonpositive (the "other" convention); otherwise: The upperright block is made nonpositive (the "default" convention).
MM is INTEGER The number of rows and columns in X.
PP is INTEGER The number of rows in X11 and X12. 0 <= P <= M.
QQ is INTEGER The number of columns in X11 and X21. 0 <= Q <= MIN(P,MP,MQ).
X11X11 is REAL array, dimension (LDX11,Q) On entry, the topleft block of the orthogonal matrix to be reduced. On exit, the form depends on TRANS: If TRANS = 'N', then the columns of tril(X11) specify reflectors for P1, the rows of triu(X11,1) specify reflectors for Q1; else TRANS = 'T', and the rows of triu(X11) specify reflectors for P1, the columns of tril(X11,1) specify reflectors for Q1.
LDX11LDX11 is INTEGER The leading dimension of X11. If TRANS = 'N', then LDX11 >= P; else LDX11 >= Q.
X12X12 is REAL array, dimension (LDX12,MQ) On entry, the topright block of the orthogonal matrix to be reduced. On exit, the form depends on TRANS: If TRANS = 'N', then the rows of triu(X12) specify the first P reflectors for Q2; else TRANS = 'T', and the columns of tril(X12) specify the first P reflectors for Q2.
LDX12LDX12 is INTEGER The leading dimension of X12. If TRANS = 'N', then LDX12 >= P; else LDX11 >= MQ.
X21X21 is REAL array, dimension (LDX21,Q) On entry, the bottomleft block of the orthogonal matrix to be reduced. On exit, the form depends on TRANS: If TRANS = 'N', then the columns of tril(X21) specify reflectors for P2; else TRANS = 'T', and the rows of triu(X21) specify reflectors for P2.
LDX21LDX21 is INTEGER The leading dimension of X21. If TRANS = 'N', then LDX21 >= MP; else LDX21 >= Q.
X22X22 is REAL array, dimension (LDX22,MQ) On entry, the bottomright block of the orthogonal matrix to be reduced. On exit, the form depends on TRANS: If TRANS = 'N', then the rows of triu(X22(Q+1:MP,P+1:MQ)) specify the last MPQ reflectors for Q2, else TRANS = 'T', and the columns of tril(X22(P+1:MQ,Q+1:MP)) specify the last MPQ reflectors for P2.
LDX22LDX22 is INTEGER The leading dimension of X22. If TRANS = 'N', then LDX22 >= MP; else LDX22 >= MQ.
THETATHETA is REAL array, dimension (Q) The entries of the bidiagonal blocks B11, B12, B21, B22 can be computed from the angles THETA and PHI. See Further Details.
PHIPHI is REAL array, dimension (Q1) The entries of the bidiagonal blocks B11, B12, B21, B22 can be computed from the angles THETA and PHI. See Further Details.
TAUP1TAUP1 is REAL array, dimension (P) The scalar factors of the elementary reflectors that define P1.
TAUP2TAUP2 is REAL array, dimension (MP) The scalar factors of the elementary reflectors that define P2.
TAUQ1TAUQ1 is REAL array, dimension (Q) The scalar factors of the elementary reflectors that define Q1.
TAUQ2TAUQ2 is REAL array, dimension (MQ) The scalar factors of the elementary reflectors that define Q2.
WORKWORK is REAL array, dimension (LWORK)
LWORKLWORK is INTEGER The dimension of the array WORK. LWORK >= MQ. 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.
INFOINFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith 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 bidiagonal blocks B11, B12, B21, and B22 are represented implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q1). B11 and B21 are upper bidiagonal, while B21 and B22 are lower bidiagonal. Every entry in each bidiagonal band is a product of a sine or cosine of a THETA with a sine or cosine of a PHI. See [1] or SORCSD for details. P1, P2, Q1, and Q2 are represented as products of elementary reflectors. See SORCSD for details on generating P1, P2, Q1, and Q2 using SORGQR and SORGLQ.
References:
 [1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):3365, 2009.
Definition at line 286 of file sorbdb.f.
Author
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