SORBDB(3) LAPACK routine of NEC Numeric Library Collection SORBDB(3)
NAME
SORBDB
SYNOPSIS
SUBROUTINE SORBDB (TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12, X21,
LDX21, X22, LDX22, THETA, PHI, TAUP1, TAUP2, TAUQ1, TAUQ2, WORK,
LWORK, INFO)
PURPOSE
SORBDB 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 SORCSD
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.
ARGUMENTS
TRANS (input)
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 (input)
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 (input)
M is INTEGER
The number of rows and columns in X.
P (input)
P is INTEGER
The number of rows in X11 and X12. 0 <= P <= M.
Q (input)
Q is INTEGER
The number of columns in X11 and X21. 0 <= Q <=
MIN(P,M-P,M-Q).
X11 (input/output)
X11 is REAL 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 (input)
LDX11 is INTEGER
The leading dimension of X11. If TRANS = 'N', then LDX11 >=
P; else LDX11 >= Q.
X12 (input/output)
X12 is REAL 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 (input)
LDX12 is INTEGER
The leading dimension of X12. If TRANS = 'N', then LDX12 >=
P; else LDX11 >= M-Q.
X21 (input/output)
X21 is REAL 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 (input)
LDX21 is INTEGER
The leading dimension of X21. If TRANS = 'N', then LDX21 >=
M-P; else LDX21 >= Q.
X22 (input/output)
X22 is REAL 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 (input)
LDX22 is INTEGER
The leading dimension of X22. If TRANS = 'N', then LDX22 >=
M-P; else LDX22 >= M-Q.
THETA (output)
THETA 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.
PHI (output)
PHI is REAL 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 (output)
TAUP1 is REAL array, dimension (P)
The scalar factors of the elementary reflectors that define
P1.
TAUP2 (output)
TAUP2 is REAL array, dimension (M-P)
The scalar factors of the elementary reflectors that define
P2.
TAUQ1 (output)
TAUQ1 is REAL array, dimension (Q)
The scalar factors of the elementary reflectors that define
Q1.
TAUQ2 (output)
TAUQ2 is REAL array, dimension (M-Q)
The scalar factors of the elementary reflectors that define
Q2.
WORK (output)
WORK is REAL 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 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
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.
LAPACK routine 31 October 2017 SORBDB(3)