subroutine dorbdb (TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12, X21, LDX21, X22, LDX22, THETA, PHI, TAUP1, TAUP2, TAUQ1, TAUQ2, WORK, LWORK, INFO)
DORBDB
DORBDB
Purpose:
DORBDB simultaneously bidiagonalizes the blocks of an M-by-M 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 P-by-Q. Q must be no larger than P, M-P, or M-Q. (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 DORCSD for details.) The orthogonal matrices P1, P2, Q1, and Q2 are P-by-P, (M-P)-by- (M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. They are represented implicitly by Householder vectors. B11, B12, B21, and B22 are Q-by-Q bidiagonal matrices represented implicitly by angles THETA, PHI.
Parameters:
TRANS is CHARACTER = 'T': X, U1, U2, V1T, and V2T are stored in row-major order; otherwise: X, U1, U2, V1T, and V2T are stored in column- major order.
SIGNS
SIGNS is CHARACTER = 'O': The lower-left block is made nonpositive (the "other" convention); otherwise: The upper-right block is made nonpositive (the "default" convention).
M
M is INTEGER The number of rows and columns in X.
P
P is INTEGER The number of rows in X11 and X12. 0 <= P <= M.
Q
Q is INTEGER The number of columns in X11 and X21. 0 <= Q <= MIN(P,M-P,M-Q).
X11
X11 is DOUBLE PRECISION array, dimension (LDX11,Q) On entry, the top-left 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.
LDX11
LDX11 is INTEGER The leading dimension of X11. If TRANS = 'N', then LDX11 >= P; else LDX11 >= Q.
X12
X12 is DOUBLE PRECISION array, dimension (LDX12,M-Q) On entry, the top-right 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.
LDX12
LDX12 is INTEGER The leading dimension of X12. If TRANS = 'N', then LDX12 >= P; else LDX11 >= M-Q.
X21
X21 is DOUBLE PRECISION array, dimension (LDX21,Q) On entry, the bottom-left 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.
LDX21
LDX21 is INTEGER The leading dimension of X21. If TRANS = 'N', then LDX21 >= M-P; else LDX21 >= Q.
X22
X22 is DOUBLE PRECISION array, dimension (LDX22,M-Q) On entry, the bottom-right 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:M-P,P+1:M-Q)) specify the last M-P-Q reflectors for Q2, else TRANS = 'T', and the columns of tril(X22(P+1:M-Q,Q+1:M-P)) specify the last M-P-Q reflectors for P2.
LDX22
LDX22 is INTEGER The leading dimension of X22. If TRANS = 'N', then LDX22 >= M-P; else LDX22 >= M-Q.
THETA
THETA is DOUBLE PRECISION 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.
PHI
PHI is DOUBLE PRECISION array, dimension (Q-1) The entries of the bidiagonal blocks B11, B12, B21, B22 can be computed from the angles THETA and PHI. See Further Details.
TAUP1
TAUP1 is DOUBLE PRECISION array, dimension (P) The scalar factors of the elementary reflectors that define P1.
TAUP2
TAUP2 is DOUBLE PRECISION array, dimension (M-P) The scalar factors of the elementary reflectors that define P2.
TAUQ1
TAUQ1 is DOUBLE PRECISION array, dimension (Q) The scalar factors of the elementary reflectors that define Q1.
TAUQ2
TAUQ2 is DOUBLE PRECISION array, dimension (M-Q) The scalar factors of the elementary reflectors that define Q2.
WORK
WORK is DOUBLE PRECISION array, dimension (LWORK)
LWORK
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
INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
Further Details:
The bidiagonal blocks B11, B12, B21, and B22 are represented implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). 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 DORCSD for details. P1, P2, Q1, and Q2 are represented as products of elementary reflectors. See DORCSD for details on generating P1, P2, Q1, and Q2 using DORGQR and DORGLQ.
References:
Definition at line 289 of file dorbdb.f.
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