Man page - bdsvdx(3)

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bdsvdx

NAME
SYNOPSIS
Functions
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, doubleprecision, dimension( ldz, * ) z, integer ldz, double precision,dimension( * ) work, integer, dimension( * ) iwork, integer info)
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)
Author

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, doubleprecision, 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 MRˆ3
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.

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 MRˆ3
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.

Author

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