NAG CL Interface
f08kmc (dgesvdx)

1 Purpose

f08kmc computes the singular value decomposition (SVD) of a real m by 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

#include <nag.h>
void  f08kmc (Nag_OrderType order, Nag_ComputeSingularVecsType jobu, Nag_ComputeSingularVecsType jobvt, Nag_RangeType range, Integer m, Integer n, double a[], Integer pda, double vl, double vu, Integer il, Integer iu, Integer *ns, double s[], double u[], Integer pdu, double vt[], Integer pdvt, double work[], Integer jfail[], NagError *fail)
The function may be called by the names: f08kmc, nag_lapackeig_dgesvdx or nag_dgesvdx.

3 Description

The SVD is written as
A = UΣVT ,  
where Σ is an m by n matrix which is zero except for its minm,n diagonal elements, U is an m by m orthogonal matrix, and V is an n by 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 minm,n columns of U and V are the left and right singular vectors of A, respectively.
Note that the function 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, σvl,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: order Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: jobu Nag_ComputeSingularVecsType Input
On entry: specifies options for computing all or part of the matrix U.
jobu=Nag_SingularVecs
The ns columns of U, as specified by range, are returned in array u.
jobu=Nag_NotSingularVecs
No columns of U (no left singular vectors) are computed.
Constraint: jobu=Nag_SingularVecs or Nag_NotSingularVecs.
3: jobvt Nag_ComputeSingularVecsType Input
On entry: specifies options for computing all or part of the matrix VT.
jobvt=Nag_SingularVecs
The ns rows of VT, as specified by range, are returned in the array vt.
jobvt=Nag_NotSingularVecs
No rows of VT (no right singular vectors) are computed.
Constraint: jobvt=Nag_SingularVecs or Nag_NotSingularVecs.
4: range Nag_RangeType Input
On entry: indicates which singular values should be returned.
range=Nag_AllValues
All singular values will be found.
range=Nag_Interval
All singular values in the half-open interval vl,vu will be found.
range=Nag_Indices
The ilth through iuth singular values will be found.
Constraint: range=Nag_AllValues, Nag_Interval or Nag_Indices.
5: m Integer Input
On entry: m, the number of rows of the matrix A.
Constraint: m0.
6: n Integer Input
On entry: n, the number of columns of the matrix A.
Constraint: n0.
7: a[dim] double Input/Output
Note: the dimension, dim, of the array a must be at least
  • max1,pda×n when order=Nag_ColMajor;
  • max1,m×pda when order=Nag_RowMajor.
The i,jth element of the matrix A is stored in
  • a[j-1×pda+i-1] when order=Nag_ColMajor;
  • a[i-1×pda+j-1] when order=Nag_RowMajor.
On entry: the m by n matrix A.
On exit: if jobuNag_NotSingularVecs and jobvtNag_NotSingularVecs, the contents of a are destroyed.
8: pda Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
  • if order=Nag_ColMajor, pdamax1,m;
  • if order=Nag_RowMajor, pdamax1,n.
9: vl double Input
On entry: if range=Nag_Interval, the lower bound of the interval to be searched for singular values.
If range=Nag_AllValues or Nag_Indices, vl is not referenced.
Constraint: if range=Nag_Interval, 0.0vl.
10: vu double Input
On entry: if range=Nag_Interval, the upper bound of the interval to be searched for singular values.
If range=Nag_AllValues or Nag_Indices, vu is not referenced.
Constraint: if range=Nag_Interval, vl<vu.
11: il Integer Input
12: iu Integer Input
On entry: if range=Nag_Indices, il and iu specify the indices (in ascending order) of the smallest and largest singular values to be returned, respectively.
If range=Nag_AllValues or Nag_Interval, il and iu are not referenced.
Constraints:
  • if range=Nag_Indices and minm,n=0, il=1 and iu=0;
  • if range=Nag_Indices and minm,n>0, 1 il iu minm,n .
13: ns Integer * Output
On exit: the total number of singular values found. 0nsminm,n.
If range=Nag_AllValues, ns=minm,n.
If range=Nag_Indices, ns=iu-il+1.
If range=Nag_Interval 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., minm,n.
14: s[minm,n] double Output
On exit: the singular values of A, sorted so that s[i-1]s[i].
15: u[dim] double Output
Note: the dimension, dim, of the array u must be at least
  • pdu×nsmax when jobu=Nag_SingularVecs and order=Nag_ColMajor;
  • m×pdu when jobu=Nag_SingularVecs and order=Nag_RowMajor;
  • otherwise u may be NULL;
where nsmax is a value larger than the output value ns.
The i,jth element of the matrix U is stored in
  • u[j-1×pdu+i-1] when order=Nag_ColMajor;
  • u[i-1×pdu+j-1] when order=Nag_RowMajor.
On exit: if jobu=Nag_SingularVecs, u contains the first ns columns of U (the left singular vectors, stored column-wise).
If jobu=Nag_NotSingularVecs, u is not referenced.
16: pdu Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array u.
Constraints:
  • if order=Nag_ColMajor,
    • if jobu=Nag_SingularVecs, pdu m ;
    • otherwise pdu1;
  • if order=Nag_RowMajor,
    • if jobu=Nag_SingularVecs, pdunsmax;
    • otherwise u may be NULL;
    where nsmax is a value larger than the output value ns.
17: vt[dim] double Output
Note: the dimension, dim, of the array vt must be at least
  • pdvt×n when jobvt=Nag_SingularVecs and order=Nag_ColMajor;
  • minm,n×pdvt when jobvt=Nag_SingularVecs and order=Nag_RowMajor;
  • otherwise vt may be NULL.
The i,jth element of the matrix is stored in
  • vt[j-1×pdvt+i-1] when order=Nag_ColMajor;
  • vt[i-1×pdvt+j-1] when order=Nag_RowMajor.
On exit: if jobvt=Nag_SingularVecs, vt contains the first ns rows of VT (the right singular vectors, stored row-wise).
If jobvt=Nag_NotSingularVecs, vt is not referenced.
18: pdvt Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array vt.
Constraints:
  • if order=Nag_ColMajor,
    • if jobvt=Nag_SingularVecs, pdvtminm,n;
    • otherwise pdvt1;
  • if order=Nag_RowMajor,
    • if jobvt=Nag_SingularVecs, pdvtn;
    • otherwise vt may be NULL.
19: work[minm,n] double Output
On exit: if fail.code= NE_CONVERGENCE, WORK2:minm,n (using the notation described in Section 3.1.4 in the Introduction to the NAG Library CL Interface) 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 singular vectors related by U and VT.
20: jfail[2×minm,n] Integer Output
On exit: if fail.code= NE_CONVERGENCE, jfail contains, in its first k nonzero elements, the indices of the k eigenvectors (associated with a left or right singular vector, see f08mbc) that failed to converge.
21: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_CONVERGENCE
If f08kmc did not converge, fail.errnum specifies how many superdiagonals of an intermediate bidiagonal form did not converge to zero.
NE_ENUM_INT
On entry, jobu=value and pdu=value.
Constraint: if jobu=Nag_SingularVecs, pdunsmax.
NE_ENUM_INT_2
On entry, jobu=value, pdu=value and m=value.
Constraint: if jobu=Nag_SingularVecs, pdu m .
On entry, jobvt=value, pdvt=value and n=value.
Constraint: if jobvt=Nag_SingularVecs, pdvtn.
NE_ENUM_INT_3
On entry, jobvt=value, pdvt=value, n=value and m=value.
Constraint: if jobvt=Nag_SingularVecs, pdvtminm,n.
NE_ENUM_INT_4
On entry, range=value, il=value, iu=value, n=value and m=value.
Constraint: if range=Nag_Indices and minm,n=0, il=1 and iu=0;
if range=Nag_Indices and minm,n>0, 1 il iu minm,n .
NE_ENUM_REAL_1
On entry, range=value and vl=value.
Constraint: if range=Nag_Interval, 0.0vl.
NE_ENUM_REAL_2
On entry, range=value, vl=value and vu=value.
Constraint: if range=Nag_Interval, vl<vu.
NE_INT
On entry, m=value.
Constraint: m0.
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
On entry, pdu=value.
Constraint: pdu>0.
On entry, pdvt=value.
Constraint: pdvt>0.
NE_INT_2
On entry, pda=value and m=value.
Constraint: pdamax1,m.
On entry, pda=value and n=value.
Constraint: pdamax1,n.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

The computed singular value decomposition is nearly the exact singular value decomposition for a nearby matrix A+E , where
E2 = Oε 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

f08kmc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08kmc 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 function. 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 function is f08kzc.

10 Example

This example finds the singular values and left and right singular vectors of the 6 by 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 (f08kmce.c)

10.2 Program Data

Program Data (f08kmce.d)

10.3 Program Results

Program Results (f08kmce.r)