dorbdb3 - simultaneously bidiagonalize the blocks of a tall and skinny matrix with orthonomal columns
SUBROUTINE DORBDB3(M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO) INTEGER INFO, LWORK, M, P, Q, LDX11, LDX21 DOUBLE PRECISION PHI(*), THETA(*) DOUBLE PRECISION TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*), X11(LDX11,*), X21(LDX21,*) SUBROUTINE DORBDB3_64(M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO) INTEGER*8 INFO, LWORK, M, P, Q, LDX11, LDX21 DOUBLE PRECISION PHI(*), THETA(*) DOUBLE PRECISION TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*), X11(LDX11,*), X21(LDX21,*) F95 INTERFACE SUBROUTINE ORBDB3(M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO) INTEGER :: M, P, Q, LDX11, LDX21, LWORK, INFO REAL(8), DIMENSION(:,:) :: X11, X21 REAL(8), DIMENSION(:) :: THETA, PHI, TAUP1, TAUP2, TAUQ1, WORK SUBROUTINE ORBDB3_64(M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO) INTEGER(8) :: M, P, Q, LDX11, LDX21, LWORK, INFO REAL(8), DIMENSION(:,:) :: X11, X21 REAL(8), DIMENSION(:) :: THETA, PHI, TAUP1, TAUP2, TAUQ1, WORK C INTERFACE #include <sunperf.h> void dorbdb3 (int m, int p, int q, double *x11, int ldx11, double *x21, int ldx21, double *theta, double *phi, double *taup1, double *taup2, double *tauq1, int *info); void dorbdb3_64 (long m, long p, long q, double *x11, long ldx11, dou- ble *x21, long ldx21, double *theta, double *phi, double *taup1, double *taup2, double *tauq1, long *info);
Oracle Solaris Studio Performance Library dorbdb3(3P)
NAME
dorbdb3 - simultaneously bidiagonalize the blocks of a tall and skinny
matrix with orthonomal columns
SYNOPSIS
SUBROUTINE DORBDB3(M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1,
TAUP2, TAUQ1, WORK, LWORK, INFO)
INTEGER INFO, LWORK, M, P, Q, LDX11, LDX21
DOUBLE PRECISION PHI(*), THETA(*)
DOUBLE PRECISION TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*), X11(LDX11,*),
X21(LDX21,*)
SUBROUTINE DORBDB3_64(M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI,
TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO)
INTEGER*8 INFO, LWORK, M, P, Q, LDX11, LDX21
DOUBLE PRECISION PHI(*), THETA(*)
DOUBLE PRECISION TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*), X11(LDX11,*),
X21(LDX21,*)
F95 INTERFACE
SUBROUTINE ORBDB3(M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1,
TAUP2, TAUQ1, WORK, LWORK, INFO)
INTEGER :: M, P, Q, LDX11, LDX21, LWORK, INFO
REAL(8), DIMENSION(:,:) :: X11, X21
REAL(8), DIMENSION(:) :: THETA, PHI, TAUP1, TAUP2, TAUQ1, WORK
SUBROUTINE ORBDB3_64(M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI,
TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO)
INTEGER(8) :: M, P, Q, LDX11, LDX21, LWORK, INFO
REAL(8), DIMENSION(:,:) :: X11, X21
REAL(8), DIMENSION(:) :: THETA, PHI, TAUP1, TAUP2, TAUQ1, WORK
C INTERFACE
#include <sunperf.h>
void dorbdb3 (int m, int p, int q, double *x11, int ldx11, double *x21,
int ldx21, double *theta, double *phi, double *taup1, double
*taup2, double *tauq1, int *info);
void dorbdb3_64 (long m, long p, long q, double *x11, long ldx11, dou-
ble *x21, long ldx21, double *theta, double *phi, double
*taup1, double *taup2, double *tauq1, long *info);
PURPOSE
dorbdb3 simultaneously bidiagonalizes the blocks of a tall and skinny
matrix X with orthonomal columns:
[ B11 ]
[ X11 ] [ P1 | ] [ 0 ]
[-----] = [---------] [-----] Q1**T .
[ X21 ] [ | P2 ] [ B21 ]
[ 0 ]
X11 is P-by-Q, and X21 is (M-P)-by-Q. M-P must be no larger than P, Q,
or M-Q. Routines DORBDB1, DORBDB2, and DORBDB4 handle cases in which M-
P is not the minimum dimension.
The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P), and
(M-Q)-by-(M-Q), respectively. They are represented implicitly by House-
holder vectors.
B11 and B12 are (M-P)-by-(M-P) bidiagonal matrices represented implic-
itly by angles THETA, PHI.
ARGUMENTS
M (input)
M is INTEGER
The number of rows X11 plus the number of rows in X21.
P (input)
P is INTEGER
The number of rows in X11. 0 <= P <= M. M-P <= min(P,Q,M-Q).
Q (input)
Q is INTEGER
The number of columns in X11 and X21. 0 <= Q <= M.
X11 (input/output)
X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
On entry, the top block of the matrix X to be reduced.
On exit, the columns of tril(X11) specify reflectors for P1
and the rows of triu(X11,1) specify reflectors for Q1.
LDX11 (input)
LDX11 is INTEGER
The leading dimension of X11. LDX11 >= P.
X21 (input/output)
X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
On entry, the bottom block of the matrix X to be reduced.
On exit, the columns of tril(X21) specify reflectors for P2.
LDX21 (input)
LDX21 is INTEGER
The leading dimension of X21. LDX21 >= M-P.
THETA (output)
THETA is DOUBLE PRECISION array, dimension (Q)
The entries of the bidiagonal blocks B11, B21 are defined by
THETA and PHI. See Further Details.
PHI (output)
PHI is DOUBLE PRECISION array, dimension (Q-1)
The entries of the bidiagonal blocks B11, B21 are defined by
THETA and PHI. See Further Details.
TAUP1 (output)
TAUP1 is DOUBLE PRECISION array, dimension (P)
The scalar factors of the elementary reflectors that define
P1.
TAUP2 (output)
TAUP2 is DOUBLE PRECISION array, dimension (M-P)
The scalar factors of the elementary reflectors that define
P2.
TAUQ1 (output)
TAUQ1 is DOUBLE PRECISION array, dimension (Q)
The scalar factors of the elementary reflectors that define
Q1.
WORK (output)
WORK is DOUBLE PRECISION array, dimension (LWORK)
LWORK (input)
LWORK is INTEGER
The dimension of the array WORK. LWORK >= M-Q.
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)
INFO is INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal value.
FURTHER DETAILS
The upper-bidiagonal blocks B11, B21 are represented implicitly by
angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). 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, and Q1 are represented as products of elementary reflectors.
See SORCSD2BY1 for details on generating P1, P2, and Q1 using SORGQR
and SORGLQ.
REFERENCES
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
Algorithms, 50(1):33-65, 2009.
7 Nov 2015 dorbdb3(3P)