dorbdb4 - simultaneously bidiagonalize the blocks of a tall and skinny matrix with orthonomal columns
SUBROUTINE DORBDB4(M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, PHANTOM, WORK, LWORK, INFO) INTEGER INFO, LWORK, M, P, Q, LDX11, LDX21 DOUBLE PRECISION PHI(*), THETA(*) DOUBLE PRECISION PHANTOM(*), TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*), X11(LDX11,*), X21(LDX21,*) SUBROUTINE DORBDB4_64(M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, PHANTOM, WORK, LWORK, INFO) INTEGER*8 INFO, LWORK, M, P, Q, LDX11, LDX21 DOUBLE PRECISION PHI(*), THETA(*) DOUBLE PRECISION PHANTOM(*), TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*), X11(LDX11,*), X21(LDX21,*) F95 INTERFACE SUBROUTINE ORBDB4(M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, PHANTOM, WORK, LWORK, INFO) INTEGER :: M, P, Q, LDX11, LDX21, LWORK, INFO REAL(8), DIMENSION(:,:) :: X11, X21 REAL(8), DIMENSION(:) :: THETA, PHI, TAUP1, TAUP2, TAUQ1, PHANTOM, WORK SUBROUTINE ORBDB4_64(M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, PHANTOM, 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, PHANTOM, WORK C INTERFACE #include <sunperf.h> void dorbdb4 (int m, int p, int q, double *x11, int ldx11, double *x21, int ldx21, double *theta, double *phi, double *taup1, double *taup2, double *tauq1, double *phantom, int *info); void dorbdb4_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, double *phantom, long *info);
Oracle Solaris Studio Performance Library dorbdb4(3P) NAME dorbdb4 - simultaneously bidiagonalize the blocks of a tall and skinny matrix with orthonomal columns SYNOPSIS SUBROUTINE DORBDB4(M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, PHANTOM, WORK, LWORK, INFO) INTEGER INFO, LWORK, M, P, Q, LDX11, LDX21 DOUBLE PRECISION PHI(*), THETA(*) DOUBLE PRECISION PHANTOM(*), TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*), X11(LDX11,*), X21(LDX21,*) SUBROUTINE DORBDB4_64(M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, PHANTOM, WORK, LWORK, INFO) INTEGER*8 INFO, LWORK, M, P, Q, LDX11, LDX21 DOUBLE PRECISION PHI(*), THETA(*) DOUBLE PRECISION PHANTOM(*), TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*), X11(LDX11,*), X21(LDX21,*) F95 INTERFACE SUBROUTINE ORBDB4(M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, PHANTOM, WORK, LWORK, INFO) INTEGER :: M, P, Q, LDX11, LDX21, LWORK, INFO REAL(8), DIMENSION(:,:) :: X11, X21 REAL(8), DIMENSION(:) :: THETA, PHI, TAUP1, TAUP2, TAUQ1, PHANTOM, WORK SUBROUTINE ORBDB4_64(M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, PHANTOM, 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, PHANTOM, WORK C INTERFACE #include <sunperf.h> void dorbdb4 (int m, int p, int q, double *x11, int ldx11, double *x21, int ldx21, double *theta, double *phi, double *taup1, double *taup2, double *tauq1, double *phantom, int *info); void dorbdb4_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, double *phantom, long *info); PURPOSE dorbdb4 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-Q must be no larger than P, M- P, or Q. Routines DORBDB1, DORBDB2, and DORBDB3 handle cases in which M-Q 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-Q)-by-(M-Q) 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. Q (input) Q is INTEGER The number of columns in X11 and X21. 0 <= Q <= M and M-Q <= min(P,M-P,Q). 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. PHANTOM (output) PHANTOM is DOUBLE PRECISION array, dimension (M) The routine computes an M-by-1 column vector Y that is orthogonal to the columns of [ X11; X21 ]. PHANTOM(1:P) and PHANTOM(P+1:M) contain Householder vectors for Y(1:P) and Y(P+1:M), respectively. 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 DORCSD2BY1 for details on generating P1, P2, and Q1 using DORGQR and DORGLQ. REFERENCES [1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009. 7 Nov 2015 dorbdb4(3P)