DBDSDC - compute the singular value decomposition (SVD) of
a real N-by-N (upper or lower) bidiagonal matrix B
SYNOPSIS
SUBROUTINE DBDSDC( UPLO, COMPQ, N, D, E, U, LDU, VT, LDVT,
Q, IQ, WORK, IWORK, INFO )
CHARACTER COMPQ, UPLO
INTEGER INFO, LDU, LDVT, N
INTEGER IQ( * ), IWORK( * )
DOUBLE PRECISION D( * ), E( * ), Q( * ), U(
LDU, * ), VT( LDVT, * ), WORK( * )
PURPOSE
DBDSDC computes the singular value decomposition (SVD) of
a real N-by-N (upper or lower) bidiagonal matrix B: B = U
* S * VT, using a divide and conquer method, where S is a
diagonal matrix with non-negative diagonal elements (the
singular values of B), and U and VT are orthogonal matri
ces of left and right singular vectors, respectively.
DBDSDC can be used to compute all singular values, and
optionally, singular vectors or singular vectors in com
pact form.
This code makes very mild assumptions about floating point
arithmetic. It will work on machines with a guard digit in
add/subtract, or on those binary machines without guard
digits which subtract like the Cray X-MP, Cray Y-MP, Cray
C-90, or Cray-2. It could conceivably fail on hexadecimal
or decimal machines without guard digits, but we know of
none. See DLASD3 for details.
The code currently call DLASDQ if singular values only are
desired. However, it can be slightly modified to compute
singular values using the divide and conquer method.
ARGUMENTS
UPLO (input) CHARACTER*1
= 'U': B is upper bidiagonal.
= 'L': B is lower bidiagonal.
COMPQ (input) CHARACTER*1
Specifies whether singular vectors are to be com
puted as follows:
= 'N': Compute singular values only;
= 'P': Compute singular values and compute singu
lar vectors in compact form; = 'I': Compute sin
gular values and singular vectors.
The order of the matrix B. N >= 0.
D (input/output) DOUBLE PRECISION array, dimension
(N)
On entry, the n diagonal elements of the bidiago
nal matrix B. On exit, if INFO=0, the singular
values of B.
E (input/output) DOUBLE PRECISION array, dimension
(N)
On entry, the elements of E contain the offdiago
nal elements of the bidiagonal matrix whose SVD is
desired. On exit, E has been destroyed.
U (output) DOUBLE PRECISION array, dimension (LDU,N)
If COMPQ = 'I', then: On exit, if INFO = 0, U
contains the left singular vectors of the bidiago
nal matrix. For other values of COMPQ, U is not
referenced.
LDU (input) INTEGER
The leading dimension of the array U. LDU >= 1.
If singular vectors are desired, then LDU >= max(
1, N ).
VT (output) DOUBLE PRECISION array, dimension
(LDVT,N)
If COMPQ = 'I', then: On exit, if INFO = 0, VT'
contains the right singular vectors of the bidiag
onal matrix. For other values of COMPQ, VT is not
referenced.
LDVT (input) INTEGER
The leading dimension of the array VT. LDVT >= 1.
If singular vectors are desired, then LDVT >= max(
1, N ).
Q (output) DOUBLE PRECISION array, dimension (LDQ)
If COMPQ = 'P', then: On exit, if INFO = 0, Q and
IQ contain the left and right singular vectors in
a compact form, requiring O(N log N) space instead
of 2*N**2. In particular, Q contains all the DOU
BLE PRECISION data in LDQ >= N*(11 + 2*SMLSIZ +
8*INT(LOG_2(N/(SMLSIZ+1)))) words of memory, where
SMLSIZ is returned by ILAENV and is equal to the
maximum size of the subproblems at the bottom of
the computation tree (usually about 25). For
other values of COMPQ, Q is not referenced.
IQ (output) INTEGER array, dimension (LDIQ)
If COMPQ = 'P', then: On exit, if INFO = 0, Q and
IQ contain the left and right singular vectors in
a compact form, requiring O(N log N) space instead
data in LDIQ >= N*(3 + 3*INT(LOG_2(N/(SMLSIZ+1))))
words of memory, where SMLSIZ is returned by
ILAENV and is equal to the maximum size of the
subproblems at the bottom of the computation tree
(usually about 25). For other values of COMPQ, IQ
is not referenced.
WORK (workspace) DOUBLE PRECISION array, dimension
(LWORK)
If COMPQ = 'N' then LWORK >= (4 * N). If COMPQ =
'P' then LWORK >= (6 * N). If COMPQ = 'I' then
LWORK >= (3 * N**2 + 4 * N).
IWORK (workspace) INTEGER array, dimension (8*N)
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an ille
gal value.
> 0: The algorithm failed to compute an singular
value. The update process of divide and conquer
failed.
FURTHER DETAILS
Based on contributions by
Ming Gu and Huan Ren, Computer Science Division, Uni
versity of
California at Berkeley, USA
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