bdsdc(3) | Library Functions Manual | bdsdc(3) |

# NAME

bdsdc - bdsdc: bidiagonal SVD, divide and conquer

# SYNOPSIS

## Functions

subroutine **dbdsdc** (uplo, compq, n, d, e, u, ldu, vt, ldvt,
q, iq, work, iwork, info)

**DBDSDC** subroutine **sbdsdc** (uplo, compq, n, d, e, u, ldu, vt,
ldvt, q, iq, work, iwork, info)

**SBDSDC**

# Detailed Description

# Function Documentation

## subroutine dbdsdc (character uplo, character compq, integer n, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( ldu, * ) u, integer ldu, double precision, dimension( ldvt, * ) vt, integer ldvt, double precision, dimension( * ) q, integer, dimension( * ) iq, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

**DBDSDC**

**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 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. The code currently calls DLASDQ if singular values only are desired. However, it can be slightly modified to compute singular values using the divide and conquer method.

**Parameters**

*UPLO*

UPLO is CHARACTER*1 = 'U': B is upper bidiagonal. = 'L': B is lower bidiagonal.

*COMPQ*

COMPQ is CHARACTER*1 Specifies whether singular vectors are to be computed as follows: = 'N': Compute singular values only; = 'P': Compute singular values and compute singular vectors in compact form; = 'I': Compute singular values and singular vectors.

*N*

N is INTEGER The order of the matrix B. N >= 0.

*D*

D is 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*

E is DOUBLE PRECISION array, dimension (N-1) 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*

U is 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*

LDU is INTEGER The leading dimension of the array U. LDU >= 1. If singular vectors are desired, then LDU >= max( 1, N ).

*VT*

VT is DOUBLE PRECISION array, dimension (LDVT,N) If COMPQ = 'I', then: On exit, if INFO = 0, VT**T contains the right singular vectors of the bidiagonal matrix. For other values of COMPQ, VT is not referenced.

*LDVT*

LDVT is INTEGER The leading dimension of the array VT. LDVT >= 1. If singular vectors are desired, then LDVT >= max( 1, N ).

*Q*

Q is 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 subproblems at the bottom of the computation tree (usually about 25). For other values of COMPQ, Q is not referenced.

*IQ*

IQ is 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/(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*

WORK is DOUBLE PRECISION array, dimension (MAX(1,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*

IWORK is INTEGER array, dimension (8*N)

*INFO*

INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: The algorithm failed to compute a singular value. The update process of divide and conquer failed.

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

Ming Gu and Huan Ren, Computer Science Division,
University of California at Berkeley, USA

Definition at line **196** of file **dbdsdc.f**.

## subroutine sbdsdc (character uplo, character compq, integer n, real, dimension( * ) d, real, dimension( * ) e, real, dimension( ldu, * ) u, integer ldu, real, dimension( ldvt, * ) vt, integer ldvt, real, dimension( * ) q, integer, dimension( * ) iq, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

**SBDSDC**

**Purpose:**

SBDSDC 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 matrices of left and right singular vectors, respectively. SBDSDC can be used to compute all singular values, and optionally, singular vectors or singular vectors in compact form. The code currently calls SLASDQ if singular values only are desired. However, it can be slightly modified to compute singular values using the divide and conquer method.

**Parameters**

*UPLO*

UPLO is CHARACTER*1 = 'U': B is upper bidiagonal. = 'L': B is lower bidiagonal.

*COMPQ*

COMPQ is CHARACTER*1 Specifies whether singular vectors are to be computed as follows: = 'N': Compute singular values only; = 'P': Compute singular values and compute singular vectors in compact form; = 'I': Compute singular values and singular vectors.

*N*

N is INTEGER The order of the matrix B. N >= 0.

*D*

D is REAL 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*

E is REAL array, dimension (N-1) 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*

U is REAL 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*

LDU is INTEGER The leading dimension of the array U. LDU >= 1. If singular vectors are desired, then LDU >= max( 1, N ).

*VT*

VT is REAL array, dimension (LDVT,N) If COMPQ = 'I', then: On exit, if INFO = 0, VT**T contains the right singular vectors of the bidiagonal matrix. For other values of COMPQ, VT is not referenced.

*LDVT*

LDVT is INTEGER The leading dimension of the array VT. LDVT >= 1. If singular vectors are desired, then LDVT >= max( 1, N ).

*Q*

Q is REAL 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 REAL 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*

IQ is 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/(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*

WORK is REAL array, dimension (MAX(1,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*

IWORK is INTEGER array, dimension (8*N)

*INFO*

INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: The algorithm failed to compute a singular value. The update process of divide and conquer failed.

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

Ming Gu and Huan Ren, Computer Science Division,
University of California at Berkeley, USA

Definition at line **196** of file **sbdsdc.f**.

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