NAG FL Interface
f08kmf (dgesvdx)

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1 Purpose

f08kmf computes the singular value decomposition (SVD) of a real m×n matrix A, optionally computing the left and/or right singular vectors. All singular values or a selected set of singular values may be computed.

2 Specification

Fortran Interface
Subroutine f08kmf ( jobu, jobvt, range, m, n, a, lda, vl, vu, il, iu, ns, s, u, ldu, vt, ldvt, work, lwork, iwork, info)
Integer, Intent (In) :: m, n, lda, il, iu, ldu, ldvt, lwork
Integer, Intent (Out) :: ns, iwork(12*min(n,m)), info
Real (Kind=nag_wp), Intent (In) :: vl, vu
Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), u(ldu,*), vt(ldvt,*)
Real (Kind=nag_wp), Intent (Out) :: s(min(m,n)), work(max(1,lwork))
Character (1), Intent (In) :: jobu, jobvt, range
C Header Interface
#include <nag.h>
void  f08kmf_ (const char *jobu, const char *jobvt, const char *range, const Integer *m, const Integer *n, double a[], const Integer *lda, const double *vl, const double *vu, const Integer *il, const Integer *iu, Integer *ns, double s[], double u[], const Integer *ldu, double vt[], const Integer *ldvt, double work[], const Integer *lwork, Integer iwork[], Integer *info, const Charlen length_jobu, const Charlen length_jobvt, const Charlen length_range)
The routine may be called by the names f08kmf, nagf_lapackeig_dgesvdx or its LAPACK name dgesvdx.

3 Description

The SVD is written as
A = UΣVT ,  
where Σ is an m×n matrix which is zero except for its min(m,n) diagonal elements, U is an m×m orthogonal matrix, and V is an n×n orthogonal matrix. The diagonal elements of Σ are the singular values of A; they are real and non-negative, and are returned in descending order. The first min(m,n) columns of U and V are the left and right singular vectors of A, respectively.
Note that the routine returns VT, not V.
Alternative to computing all singular values of A, a selected set can be computed. The set is either those singular values lying in a given interval, σ(vl,vu], or those whose index (counting from largest to smallest in magnitude) lies in a given range 1il,,iun. In these cases, the corresponding left and right singular vectors can optionally be computed.

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: jobu Character(1) Input
On entry: specifies options for computing all or part of the matrix U.
jobu='V'
The ns columns of U, as specified by range, are returned in array u.
jobu='N'
No columns of U (no left singular vectors) are computed.
Constraint: jobu='V' or 'N'.
2: jobvt Character(1) Input
On entry: specifies options for computing all or part of the matrix VT.
jobvt='V'
The ns rows of VT, as specified by range, are returned in the array vt.
jobvt='N'
No rows of VT (no right singular vectors) are computed.
Constraint: jobvt='V' or 'N'.
3: range Character(1) Input
On entry: indicates which singular values should be returned.
range='A'
All singular values will be found.
range='V'
All singular values in the half-open interval (vl,vu] will be found.
range='I'
The ilth through iuth singular values will be found.
Constraint: range='A', 'V' or 'I'.
4: m Integer Input
On entry: m, the number of rows of the matrix A.
Constraint: m0.
5: n Integer Input
On entry: n, the number of columns of the matrix A.
Constraint: n0.
6: a(lda,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least max(1,n).
On entry: the m×n matrix A.
On exit: if jobu'N' and jobvt'N', the contents of a are destroyed.
7: lda Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08kmf is called.
Constraint: ldamax(1,m).
8: vl Real (Kind=nag_wp) Input
On entry: if range='V', the lower bound of the interval to be searched for singular values.
If range='A' or 'I', vl is not referenced.
Constraint: if range='V', 0.0vl.
9: vu Real (Kind=nag_wp) Input
On entry: if range='V', the upper bound of the interval to be searched for singular values.
If range='A' or 'I', vu is not referenced.
Constraint: if range='V', vl<vu.
10: il Integer Input
11: iu Integer Input
On entry: if range='I', il and iu specify the indices (in ascending order) of the smallest and largest singular values to be returned, respectively.
If range='A' or 'V', il and iu are not referenced.
Constraints:
  • if range='I' and min(m,n)=0, il=1 and iu=0;
  • if range='I' and min(m,n)>0, 1 il iu min(m,n) .
12: ns Integer Output
On exit: the total number of singular values found. 0nsmin(m,n).
If range='A', ns=min(m,n).
If range='I', ns=iu-il+1.
If range='V' then the value of ns is not known in advance and so an upper limit should be used when specifying the dimensions of array u, e.g., min(m,n).
13: s(min(m,n)) Real (Kind=nag_wp) array Output
On exit: the singular values of A, sorted so that s(i)s(i+1).
14: u(ldu,*) Real (Kind=nag_wp) array Output
Note: the second dimension of the array u must be at least max(1,nsmax) if jobu='V', where nsmax is a value larger than the output value ns..
On exit: if jobu='V', u contains the first ns columns of U (the left singular vectors, stored column-wise).
If jobu='N', u is not referenced.
15: ldu Integer Input
On entry: the first dimension of the array u as declared in the (sub)program from which f08kmf is called.
Constraints:
  • if jobu='V', ldu max(1,m) ;
  • otherwise ldu1.
16: vt(ldvt,*) Real (Kind=nag_wp) array Output
Note: the second dimension of the array vt must be at least max(1,n) if jobvt='V'.
On exit: if jobvt='V', vt contains the first ns rows of VT (the right singular vectors, stored row-wise).
If jobvt='N', vt is not referenced.
17: ldvt Integer Input
Note: if jobvt='V' and range='V' then the value of ns is not known in advance and so an upper limit should be used, e.g., min(m,n).
On entry: the first dimension of the array vt as declared in the (sub)program from which f08kmf is called.
Constraints:
  • if jobvt='V', ldvtmax(1,min(m,n));
  • otherwise ldvt1.
18: work(max(1,lwork)) Real (Kind=nag_wp) array Workspace
On exit: if info=0, work(1) returns the optimal lwork.
If info>0, work(2:min(m,n)) contains the unconverged superdiagonal elements of an upper bidiagonal matrix B whose diagonal is in s (not necessarily sorted). B satisfies A=UBVT, so it has the same singular values as A, and left and right singular vectors that are those of A pre-multiplied by UT and VT.
19: lwork Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08kmf is called.
If lwork=−1, a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.
Suggested value: for optimal performance, lwork should generally be larger than the constrained minimum value. Consider increasing lwork beyond the minimum requirement.
Constraint: lwork max(1, min(m,n) × (3×min(m,n)+20) , 4×min(m,n) + max(m,n) ) .
20: iwork(12×min(n,m)) Integer array Workspace
On exit:
  • if info=0, the first ns elements of iwork are zero;
  • if info>0, iwork contains the indices of the eigenvectors that failed to converge in f08jbf and f08mbf, see iwork in f08mbf.
21: 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
If f08kmf did not converge, info specifies how many superdiagonals of an intermediate bidiagonal form did not converge to zero.

7 Accuracy

The computed singular value decomposition is nearly the exact singular value decomposition for a nearby matrix (A+E) , where
E2 = O(ε) A2 ,  
and ε is the machine precision. In addition, the computed singular vectors are nearly orthogonal to working precision. See Section 4.9 of Anderson et al. (1999) for further details.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08kmf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08kmf 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

The total number of floating-point operations is approximately proportional to mn2 when m>n and m2n otherwise.
The singular values are returned in descending order.
The complex analogue of this routine is f08kzf.

10 Example

This example finds the singular values and left and right singular vectors of the 6×4 matrix
A = ( 2.27 -1.54 1.15 -1.94 0.28 -1.67 0.94 -0.78 -0.48 -3.09 0.99 -0.21 1.07 1.22 0.79 0.63 -2.35 2.93 -1.45 2.30 0.62 -7.39 1.03 -2.57 ) ,  
together with approximate error bounds for the computed singular values and vectors.

10.1 Program Text

Program Text (f08kmfe.f90)

10.2 Program Data

Program Data (f08kmfe.d)

10.3 Program Results

Program Results (f08kmfe.r)