NAG FL Interface
f08mdf (dbdsdc)

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1 Purpose

f08mdf computes the singular values and, optionally, the left and right singular vectors of a real n×n (upper or lower) bidiagonal matrix B.

2 Specification

Fortran Interface
Subroutine f08mdf ( uplo, compq, n, d, e, u, ldu, vt, ldvt, q, iq, work, iwork, info)
Integer, Intent (In) :: n, ldu, ldvt
Integer, Intent (Inout) :: iq(*)
Integer, Intent (Out) :: iwork(8*n), info
Real (Kind=nag_wp), Intent (Inout) :: d(*), e(*), u(ldu,*), vt(ldvt,*), q(*), work(*)
Character (1), Intent (In) :: uplo, compq
C Header Interface
#include <nag.h>
void  f08mdf_ (const char *uplo, const char *compq, const Integer *n, double d[], double e[], double u[], const Integer *ldu, double vt[], const Integer *ldvt, double q[], Integer iq[], double work[], Integer iwork[], Integer *info, const Charlen length_uplo, const Charlen length_compq)
The routine may be called by the names f08mdf, nagf_lapackeig_dbdsdc or its LAPACK name dbdsdc.

3 Description

f08mdf computes the singular value decomposition (SVD) of the (upper or lower) bidiagonal matrix B as
B = USVT ,  
where S is a diagonal matrix with non-negative diagonal elements sii=si, such that
s1 s2 sn 0 ,  
and U and V are orthogonal matrices. The diagonal elements of S are the singular values of B and the columns of U and V are respectively the corresponding left and right singular vectors of B.
When only singular values are required the routine uses the QR algorithm, but when singular vectors are required a divide and conquer method is used. The singular values can optionally be returned in compact form, although currently no routine is available to apply U or V when stored in compact form.

4 References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Arguments

1: uplo Character(1) Input
On entry: indicates whether B is upper or lower bidiagonal.
uplo='U'
B is upper bidiagonal.
uplo='L'
B is lower bidiagonal.
Constraint: uplo='U' or 'L'.
2: compq Character(1) Input
On entry: specifies whether singular vectors are to be computed.
compq='N'
Compute singular values only.
compq='P'
Compute singular values and compute singular vectors in compact form.
compq='I'
Compute singular values and singular vectors.
Constraint: compq='N', 'P' or 'I'.
3: n Integer Input
On entry: n, the order of the matrix B.
Constraint: n0.
4: d(*) Real (Kind=nag_wp) array Input/Output
Note: the dimension of the array d must be at least max(1,n).
On entry: the n diagonal elements of the bidiagonal matrix B.
On exit: if info=0, the singular values of B.
5: e(*) Real (Kind=nag_wp) array Input/Output
Note: the dimension of the array e must be at least max(1,n-1).
On entry: the (n-1) off-diagonal elements of the bidiagonal matrix B.
On exit: the contents of e are destroyed.
6: u(ldu,*) Real (Kind=nag_wp) array Output
Note: the second dimension of the array u must be at least max(1,n) if compq='I'.
On exit: if compq='I', then if info=0, u contains the left singular vectors of the bidiagonal matrix B.
If compq'I', u is not referenced.
7: ldu Integer Input
On entry: the first dimension of the array u as declared in the (sub)program from which f08mdf is called.
Constraints:
  • if compq='I', ldu max(1,n) ;
  • otherwise ldu1.
8: vt(ldvt,*) Real (Kind=nag_wp) array Output
Note: the second dimension of the array vt must be at least max(1,n) if compq='I'.
On exit: if compq='I', then if info=0, the rows of vt contain the right singular vectors of the bidiagonal matrix B.
If compq'I', vt is not referenced.
9: ldvt Integer Input
On entry: the first dimension of the array vt as declared in the (sub)program from which f08mdf is called.
Constraints:
  • if compq='I', ldvt max(1,n) ;
  • otherwise ldvt1.
10: q(*) Real (Kind=nag_wp) array Output
Note: the dimension of the array q must be at least max(1,n2+5n,ldq).
On exit: if compq='P', then if info=0, q and iq contain the left and right singular vectors in a compact form, requiring O(nlog2n) space instead of 2×n2. In particular, q contains all the real data in the first ldq =n× (11+2×smlsiz+8×int(log2(n/(smlsiz+1)))) elements of q, where smlsiz is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about 25).
If compq'P', q is not referenced.
11: iq(*) Integer array Output
Note: the dimension of the array iq must be at least max(1,ldiq).
On exit: if compq='P', then if info=0, q and iq contain the left and right singular vectors in a compact form, requiring O(nlog2n) space instead of 2×n2. In particular, iq contains all integer data in the first ldiq =n× (3+3×int(log2(n/(smlsiz+1)))) elements of iq, where smlsiz is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about 25).
If compq'P', iq is not referenced.
12: work(*) Real (Kind=nag_wp) array Workspace
Note: the dimension of the array work must be at least max(1,6×n-2) if compq='N', max(1,6×n) if compq='P', max(1,3×n2+4×n) if compq='I', and at least 1 otherwise.
13: iwork(8×n) Integer array Workspace
14: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
info>0
The algorithm failed to compute a singular value. The update process of divide-and-conquer failed.

7 Accuracy

Each computed singular value of B is accurate to nearly full relative precision, no matter how tiny the singular value. The ith computed singular value, s^i, satisfies the bound
|s^i-si| p(n)εsi  
where ε is the machine precision and p(n) is a modest function of n.
For bounds on the computed singular values, see Section 4.9.1 of Anderson et al. (1999). See also f08flf.

8 Parallelism and Performance

f08mdf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08mdf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

If only singular values are required, the total number of floating-point operations is approximately proportional to n2. When singular vectors are required the number of operations is bounded above by approximately the same number of operations as f08mef, but for large matrices f08mdf is usually much faster.
There is no complex analogue of f08mdf.

10 Example

This example computes the singular value decomposition of the upper bidiagonal matrix
B = ( 3.62 1.26 0.00 0.00 0.00 -2.41 -1.53 0.00 0.00 0.00 1.92 1.19 0.00 0.00 0.00 -1.43 ) .  

10.1 Program Text

Program Text (f08mdfe.f90)

10.2 Program Data

Program Data (f08mdfe.d)

10.3 Program Results

Program Results (f08mdfe.r)