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### 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 matrix with non-negative diagonal elements (the singular values of B), and U and VT are orthogo‐ nal matrices of left and right singular vectors, respectively. DBDSDC can be used to compute all singular values, and optionally, singular vectors or singular vectors 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 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 computed as fol‐ lows: = ’N’: Compute singular values only; = ’P’: Compute singular values and compute singular vectors in compact form; = ’I’: Compute singular values and singular vec‐ tors. 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 bidiagonal 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 offdiagonal 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 val‐ ues 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 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, requir‐ ing O(N log N) space instead of 2*N**2. In particular, Q con‐ tains 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, requir‐ ing O(N log N) space instead of 2*N**2. In particular, IQ con‐ tains 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 bot‐ tom of the computation tree (usually about 25). For other val‐ ues 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 illegal value. > 0: The algorithm failed to compute an singular value. The update process of divide and conquer failed. Based on contributions by Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA