NAG CL Interface
f08kdc (dgesdd)

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1 Purpose

f08kdc computes the singular value decomposition (SVD) of a real m×n matrix A, optionally computing the left and/or right singular vectors, by using a divide-and-conquer method.

2 Specification

#include <nag.h>
void  f08kdc (Nag_OrderType order, Nag_JobType job, Integer m, Integer n, double a[], Integer pda, double s[], double u[], Integer pdu, double vt[], Integer pdvt, NagError *fail)
The function may be called by the names: f08kdc, nag_lapackeig_dgesdd or nag_dgesdd.

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.
Note that the function returns VT, not V.

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: job Nag_JobType Input
On entry: specifies options for computing all or part of the matrix U.
job=Nag_DoAll
All m columns of U and all n rows of VT are returned in the arrays u and vt.
job=Nag_DoSquare
The first min(m,n) columns of U and the first min(m,n) rows of VT are returned in the arrays u and vt.
job=Nag_DoOverwrite
If mn, the first n columns of U are overwritten on the array a and all rows of VT are returned in the array vt. Otherwise, all columns of U are returned in the array u and the first m rows of VT are overwritten in the array vt.
job=Nag_DoNothing
No columns of U or rows of VT are computed.
Constraint: job=Nag_DoAll, Nag_DoSquare, Nag_DoOverwrite or Nag_DoNothing.
3: m Integer Input
On entry: m, the number of rows of the matrix A.
Constraint: m0.
4: n Integer Input
On entry: n, the number of columns of the matrix A.
Constraint: n0.
5: a[dim] double Input/Output
Note: the dimension, dim, of the array a must be at least
  • max(1,pda×n) when order=Nag_ColMajor;
  • max(1,m×pda) when order=Nag_RowMajor.
The (i,j)th 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×n matrix A.
On exit: if job=Nag_DoOverwrite, a is overwritten with the first n columns of U (the left singular vectors, stored column-wise) if mn; a is overwritten with the first m rows of VT (the right singular vectors, stored row-wise) otherwise.
If jobNag_DoOverwrite, the contents of a are destroyed.
6: 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, pdamax(1,m);
  • if order=Nag_RowMajor, pdamax(1,n).
7: s[min(m,n)] double Output
On exit: the singular values of A, sorted so that s[i-1]s[i].
8: u[dim] double Output
Note: the dimension, dim, of the array u must be at least
  • max(1,pdu×m) when job=Nag_DoAll or (job=Nag_DoOverwrite and m<n);
  • max(1,pdu×min(m,n)) when job=Nag_DoSquare and order=Nag_ColMajor;
  • max(1,m×pdu) when job=Nag_DoSquare and order=Nag_RowMajor;
  • max(1,m) otherwise.
The (i,j)th 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 job=Nag_DoAll or job=Nag_DoOverwrite and m<n, u contains the m×m orthogonal matrix U.
If job=Nag_DoSquare, u contains the first min(m,n) columns of U (the left singular vectors, stored column-wise).
If job=Nag_DoOverwrite and mn, or job=Nag_DoNothing, u is not referenced.
9: 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 job=Nag_DoAll or (job=Nag_DoOverwrite and m<n), pdu max(1,m) ;
    • if job=Nag_DoSquare, pdu max(1,m) ;
    • otherwise pdu1;
  • if order=Nag_RowMajor,
    • if job=Nag_DoAll or (job=Nag_DoOverwrite and m<n), pdumax(1,m);
    • if job=Nag_DoSquare, pdumax(1,min(m,n));
    • otherwise pdu1.
10: vt[dim] double Output
Note: the dimension, dim, of the array vt must be at least
  • max(1,pdvt×n) when job=Nag_DoAll or (job=Nag_DoOverwrite and mn);
  • max(1,pdvt×n) when job=Nag_DoSquare and order=Nag_ColMajor;
  • max(1,min(m,n)×pdvt) when job=Nag_DoSquare and order=Nag_RowMajor;
  • max(1,min(m,n)) otherwise.
The (i,j)th 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 job=Nag_DoAll or job=Nag_DoOverwrite and mn, vt contains the n×n orthogonal matrix VT.
If job=Nag_DoSquare, vt contains the first min(m,n) rows of VT (the right singular vectors, stored row-wise).
If job=Nag_DoOverwrite and m<n, or job=Nag_DoNothing, vt is not referenced.
11: 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 job=Nag_DoAll or (job=Nag_DoOverwrite and mn), pdvt max(1,n) ;
    • if job=Nag_DoSquare, pdvt max(1,min(m,n)) ;
    • otherwise pdvt1;
  • if order=Nag_RowMajor,
    • if job=Nag_DoAll or (job=Nag_DoOverwrite and mn), pdvtmax(1,n);
    • if job=Nag_DoSquare, pdvtmax(1,n);
    • otherwise pdvt1.
12: 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
f08kdc did not converge, the updating process failed.
NE_ENUM_INT_3
On entry, job=value, pdu=value, m=value and n=value.
Constraint: if job=Nag_DoAll or (job=Nag_DoOverwrite and m<n), pdu max(1,m) ;
if job=Nag_DoSquare, pdu max(1,m) ;
otherwise pdu1.
On entry, job=value, pdu=value, m=value and n=value.
Constraint: if job=Nag_DoAll or (job=Nag_DoOverwrite and m<n), pdumax(1,m);
if job=Nag_DoSquare, pdumax(1,min(m,n));
otherwise pdu1.
On entry, job=value, pdvt=value, m=value and n=value.
Constraint: if job=Nag_DoAll or (job=Nag_DoOverwrite and mn), pdvt max(1,n) ;
if job=Nag_DoSquare, pdvt max(1,min(m,n)) ;
otherwise pdvt1.
On entry, job=value, pdvt=value, m=value and n=value.
Constraint: if job=Nag_DoAll or (job=Nag_DoOverwrite and mn), pdvtmax(1,n);
if job=Nag_DoSquare, pdvtmax(1,n);
otherwise pdvt1.
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: pdamax(1,m).
On entry, pda=value and n=value.
Constraint: pdamax(1,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 = 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

f08kdc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08kdc 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 f08krc.

10 Example

This example finds the singular values and left and right singular vectors of the 4×6 matrix
A = ( 2.27 0.28 -0.48 1.07 -2.35 0.62 -1.54 -1.67 -3.09 1.22 2.93 -7.39 1.15 0.94 0.99 0.79 -1.45 1.03 -1.94 -0.78 -0.21 0.63 2.30 -2.57 ) ,  
together with approximate error bounds for the computed singular values and vectors.
The example program for f08kbc illustrates finding a singular value decomposition for the case mn.

10.1 Program Text

Program Text (f08kdce.c)

10.2 Program Data

Program Data (f08kdce.d)

10.3 Program Results

Program Results (f08kdce.r)