subroutine sbdsvdx (UPLO, JOBZ, RANGE, N, D, E, VL, VU, IL, IU, NS, S, Z, LDZ, WORK, IWORK, INFO)
SBDSVDX
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 decompositon 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 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 California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
Definition at line 228 of file sbdsvdx.f.
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