nag_dgesdd (f08kdc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dgesdd (f08kdc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

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

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_dgesdd (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)

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.
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 http://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:     orderNag_OrderTypeInput
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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     jobNag_JobTypeInput
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 minm,n columns of U and the first minm,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:     mIntegerInput
On entry: m, the number of rows of the matrix A.
Constraint: m0.
4:     nIntegerInput
On entry: n, the number of columns of the matrix A.
Constraint: n0.
5:     a[dim]doubleInput/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 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:     pdaIntegerInput
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.
7:     s[minm,n]doubleOutput
On exit: the singular values of A, sorted so that s[i-1]s[i].
8:     u[dim]doubleOutput
Note: the dimension, dim, of the array u must be at least
  • max1,pdu×m when job=Nag_DoAll or job=Nag_DoOverwrite and m<n;
  • max1,pdu×minm,n when job=Nag_DoSquare and order=Nag_ColMajor;
  • max1,m×pdu when job=Nag_DoSquare and order=Nag_RowMajor;
  • 1 otherwise.
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 job=Nag_DoAll or job=Nag_DoOverwrite and m<n, u contains the m by m matrix U.
If job=Nag_DoSquare, u contains the first minm,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:     pduIntegerInput
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 max1,m ;
    • if job=Nag_DoSquare, pdu max1,m ;
    • otherwise pdu1;
  • if order=Nag_RowMajor,
    • if job=Nag_DoAll or job=Nag_DoOverwrite and m<n, pdumax1,m;
    • if job=Nag_DoSquare, pdumax1,minm,n;
    • otherwise pdu1.
10:   vt[dim]doubleOutput
Note: the dimension, dim, of the array vt must be at least
  • max1,pdvt×n when job=Nag_DoAll or job=Nag_DoOverwrite and mn;
  • max1,pdvt×n when job=Nag_DoSquare and order=Nag_ColMajor;
  • max1,minm,n×pdvt when job=Nag_DoSquare and order=Nag_RowMajor;
  • 1 otherwise.
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 job=Nag_DoAll or job=Nag_DoOverwrite and mn, vt contains the n by n orthogonal matrix VT.
If job=Nag_DoSquare, vt contains the first minm,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:   pdvtIntegerInput
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 max1,n ;
    • if job=Nag_DoSquare, pdvt max1,minm,n ;
    • otherwise pdvt1;
  • if order=Nag_RowMajor,
    • if job=Nag_DoAll or job=Nag_DoOverwrite and mn, pdvtmax1,n;
    • if job=Nag_DoSquare, pdvtmax1,n;
    • otherwise pdvt1.
12:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_CONVERGENCE
nag_dgesdd (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 max1,m ;
if job=Nag_DoSquare, pdu max1,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, pdumax1,m;
if job=Nag_DoSquare, pdumax1,minm,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 max1,n ;
if job=Nag_DoSquare, pdvt max1,minm,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, pdvtmax1,n;
if job=Nag_DoSquare, pdvtmax1,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: 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.

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  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 nag_zgesvd (f08kpc).

9  Example

This example finds the singular values and left and right singular vectors of the 4 by 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 nag_dgesvd (f08kbc) illustrates finding a singular value decomposition for the case mn.

9.1  Program Text

Program Text (f08kdce.c)

9.2  Program Data

Program Data (f08kdce.d)

9.3  Program Results

Program Results (f08kdce.r)


nag_dgesdd (f08kdc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012