dbdsdc(3)

NAME

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 ma

trix with non-negative diagonal elements (the singular values of

B), and U and VT are orthogonal matrices of left and right singu

lar vectors, respectively. DBDSDC can be used to compute all sin

gular values, and optionally, singular vectors or singular vec

tors in compact 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 ma

chines 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 singu

lar 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 singular values and

singular vectors.

N (input) INTEGER

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 bidiagonal 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 (LD

VT,N)

If COMPQ = 'I', then: On exit, if INFO = 0, VT'

contains the right singular vectors of the bidiagonal 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 DOUBLE 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 sub

problems 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 of 2*N**2. In particular, IQ

contains all INTEGER data in LDIQ >= N*(3 + 3*INT(LOG_2(N/(SML

SIZ+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

LAPACK version 3.0 15 June 2000

DBDSDC(3)