bdsvdx(3) | Library Functions Manual | bdsvdx(3) |

# NAME

bdsvdx - bdsvdx: bidiagonal SVD, bisection

# SYNOPSIS

## Functions

subroutine **dbdsvdx** (uplo, jobz, range, n, d, e, vl, vu, il,
iu, ns, s, z, ldz, work, iwork, info)

**DBDSVDX** subroutine **sbdsvdx** (uplo, jobz, range, n, d, e, vl, vu,
il, iu, ns, s, z, ldz, work, iwork, info)

**SBDSVDX**

# Detailed Description

# Function Documentation

## subroutine dbdsvdx (character uplo, character jobz, character range, integer n, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision vl, double precision vu, integer il, integer iu, integer ns, double precision, dimension( * ) s, double precision, dimension( ldz, * ) z, integer ldz, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

**DBDSVDX**

**Purpose:**

DBDSVDX computes the singular value decomposition (SVD) of a real N-by-N (upper or lower) bidiagonal matrix B, B = U * S * VT, 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. Given an upper bidiagonal B with diagonal D = [ d_1 d_2 ... d_N ] and superdiagonal E = [ e_1 e_2 ... e_N-1 ], DBDSVDX computes the singular value decomposition of B through the eigenvalues and eigenvectors of the N*2-by-N*2 tridiagonal matrix | 0 d_1 | | d_1 0 e_1 | TGK = | e_1 0 d_2 | | d_2 . . | | . . . | If (s,u,v) is a singular triplet of B with ||u|| = ||v|| = 1, then (+/-s,q), ||q|| = 1, are eigenpairs of TGK, with q = P * ( u' +/-v' ) / sqrt(2) = ( v_1 u_1 v_2 u_2 ... v_n u_n ) / sqrt(2), and P = [ e_{n+1} e_{1} e_{n+2} e_{2} ... ]. Given a TGK matrix, one can either a) compute -s,-v and change signs so that the singular values (and corresponding vectors) are already in descending order (as in DGESVD/DGESDD) or b) compute s,v and reorder the values (and corresponding vectors). DBDSVDX implements a) by calling DSTEVX (bisection plus inverse iteration, to be replaced with a version of the Multiple Relative Robust Representation algorithm. (See P. Willems and B. Lang, A framework for the MR^3 algorithm: theory and implementation, SIAM J. Sci. Comput., 35:740-766, 2013.)

**Parameters**

*UPLO*

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

*JOBZ*

JOBZ is CHARACTER*1 = 'N': Compute singular values only; = 'V': Compute singular values and singular vectors.

*RANGE*

RANGE is CHARACTER*1 = 'A': all singular values will be found. = 'V': all singular values in the half-open interval [VL,VU) will be found. = 'I': the IL-th through IU-th singular values will be found.

*N*

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

*D*

D is DOUBLE PRECISION array, dimension (N) The n diagonal elements of the bidiagonal matrix B.

*E*

E is DOUBLE PRECISION array, dimension (max(1,N-1)) The (n-1) superdiagonal elements of the bidiagonal matrix B in elements 1 to N-1.

*VL*

VL is DOUBLE PRECISION If RANGE='V', the lower bound of the interval to be searched for singular values. VU > VL. Not referenced if RANGE = 'A' or 'I'.

*VU*

VU is DOUBLE PRECISION If RANGE='V', the upper bound of the interval to be searched for singular values. VU > VL. Not referenced if RANGE = 'A' or 'I'.

*IL*

IL is INTEGER If RANGE='I', the index of the smallest singular value to be returned. 1 <= IL <= IU <= min(M,N), if min(M,N) > 0. Not referenced if RANGE = 'A' or 'V'.

*IU*

IU is INTEGER If RANGE='I', the index of the largest singular value to be returned. 1 <= IL <= IU <= min(M,N), if min(M,N) > 0. Not referenced if RANGE = 'A' or 'V'.

*NS*

NS is INTEGER The total number of singular values found. 0 <= NS <= N. If RANGE = 'A', NS = N, and if RANGE = 'I', NS = IU-IL+1.

*S*

S is DOUBLE PRECISION array, dimension (N) The first NS elements contain the selected singular values in ascending order.

*Z*

Z is DOUBLE PRECISION array, dimension (2*N,K) If JOBZ = 'V', then if INFO = 0 the first NS columns of Z contain the singular vectors of the matrix B corresponding to the selected singular values, with U in rows 1 to N and V in rows N+1 to N*2, i.e. Z = [ U ] [ V ] If JOBZ = 'N', then Z is not referenced. Note: The user must ensure that at least K = NS+1 columns are supplied in the array Z; if RANGE = 'V', the exact value of NS is not known in advance and an upper bound must be used.

*LDZ*

LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(2,N*2).

*WORK*

WORK is DOUBLE PRECISION array, dimension (14*N)

*IWORK*

IWORK is INTEGER array, dimension (12*N) If JOBZ = 'V', then if INFO = 0, the first NS elements of IWORK are zero. If INFO > 0, then IWORK contains the indices of the eigenvectors that failed to converge in DSTEVX.

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, then i eigenvectors failed to converge in DSTEVX. The indices of the eigenvectors (as returned by DSTEVX) are stored in the array IWORK. if INFO = N*2 + 1, an internal error occurred.

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **224** of file **dbdsvdx.f**.

## subroutine sbdsvdx (character uplo, character jobz, character range, integer n, real, dimension( * ) d, real, dimension( * ) e, real vl, real vu, integer il, integer iu, integer ns, real, dimension( * ) s, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

**SBDSVDX**

**Purpose:**

SBDSVDX computes the singular value decomposition (SVD) of a real N-by-N (upper or lower) bidiagonal matrix B, B = U * S * VT, 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. Given an upper bidiagonal B with diagonal D = [ d_1 d_2 ... d_N ] and superdiagonal E = [ e_1 e_2 ... e_N-1 ], SBDSVDX computes the singular value decomposition of B through the eigenvalues and eigenvectors of the N*2-by-N*2 tridiagonal matrix | 0 d_1 | | d_1 0 e_1 | TGK = | e_1 0 d_2 | | d_2 . . | | . . . | If (s,u,v) is a singular triplet of B with ||u|| = ||v|| = 1, then (+/-s,q), ||q|| = 1, are eigenpairs of TGK, with q = P * ( u' +/-v' ) / sqrt(2) = ( v_1 u_1 v_2 u_2 ... v_n u_n ) / sqrt(2), and P = [ e_{n+1} e_{1} e_{n+2} e_{2} ... ]. Given a TGK matrix, one can either a) compute -s,-v and change signs so that the singular values (and corresponding vectors) are already in descending order (as in SGESVD/SGESDD) or b) compute s,v and reorder the values (and corresponding vectors). SBDSVDX implements a) by calling SSTEVX (bisection plus inverse iteration, to be replaced with a version of the Multiple Relative Robust Representation algorithm. (See P. Willems and B. Lang, A framework for the MR^3 algorithm: theory and implementation, SIAM J. Sci. Comput., 35:740-766, 2013.)

**Parameters**

*UPLO*

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

*JOBZ*

JOBZ is CHARACTER*1 = 'N': Compute singular values only; = 'V': Compute singular values and singular vectors.

*RANGE*

RANGE is CHARACTER*1 = 'A': all singular values will be found. = 'V': all singular values in the half-open interval [VL,VU) will be found. = 'I': the IL-th through IU-th singular values will be found.

*N*

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

*D*

D is REAL array, dimension (N) The n diagonal elements of the bidiagonal matrix B.

*E*

E is REAL array, dimension (max(1,N-1)) The (n-1) superdiagonal elements of the bidiagonal matrix B in elements 1 to N-1.

*VL*

VL is REAL If RANGE='V', the lower bound of the interval to be searched for singular values. VU > VL. Not referenced if RANGE = 'A' or 'I'.

*VU*

VU is REAL If RANGE='V', the upper bound of the interval to be searched for singular values. VU > VL. Not referenced if RANGE = 'A' or 'I'.

*IL*

IL is INTEGER If RANGE='I', the index of the smallest singular value to be returned. 1 <= IL <= IU <= min(M,N), if min(M,N) > 0. Not referenced if RANGE = 'A' or 'V'.

*IU*

IU is INTEGER If RANGE='I', the index of the largest singular value to be returned. 1 <= IL <= IU <= min(M,N), if min(M,N) > 0. Not referenced if RANGE = 'A' or 'V'.

*NS*

NS is INTEGER The total number of singular values found. 0 <= NS <= N. If RANGE = 'A', NS = N, and if RANGE = 'I', NS = IU-IL+1.

*S*

S is REAL array, dimension (N) The first NS elements contain the selected singular values in ascending order.

*Z*

Z is REAL array, dimension (2*N,K) If JOBZ = 'V', then if INFO = 0 the first NS columns of Z contain the singular vectors of the matrix B corresponding to the selected singular values, with U in rows 1 to N and V in rows N+1 to N*2, i.e. Z = [ U ] [ V ] If JOBZ = 'N', then Z is not referenced. Note: The user must ensure that at least K = NS+1 columns are supplied in the array Z; if RANGE = 'V', the exact value of NS is not known in advance and an upper bound must be used.

*LDZ*

LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(2,N*2).

*WORK*

WORK is REAL array, dimension (14*N)

*IWORK*

IWORK is INTEGER array, dimension (12*N) If JOBZ = 'V', then if INFO = 0, the first NS elements of IWORK are zero. If INFO > 0, then IWORK contains the indices of the eigenvectors that failed to converge in DSTEVX.

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, then i eigenvectors failed to converge in SSTEVX. The indices of the eigenvectors (as returned by SSTEVX) are stored in the array IWORK. if INFO = N*2 + 1, an internal error occurred.

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **224** of file **sbdsvdx.f**.

# Author

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