nag_zgesvd (f08kpc) (PDF version)
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f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zgesvd (f08kpc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zgesvd (f08kpc) computes the singular value decomposition (SVD) of a complex m by n matrix A, optionally computing the left and/or right singular vectors.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_zgesvd (Nag_OrderType order, Nag_ComputeUType jobu, Nag_ComputeVTType jobvt, Integer m, Integer n, Complex a[], Integer pda, double s[], Complex u[], Integer pdu, Complex vt[], Integer pdvt, double rwork[], NagError *fail)

3  Description

The SVD is written as
A = UΣVH ,
where Σ is an m by n matrix which is zero except for its minm,n diagonal elements, U is an m by m unitary matrix, and V is an n by n unitary 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 VH, 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:     jobuNag_ComputeUTypeInput
On entry: specifies options for computing all or part of the matrix U.
jobu=Nag_AllU
All m columns of U are returned in array u.
jobu=Nag_SingularVecsU
The first minm,n columns of U (the left singular vectors) are returned in the array u.
jobu=Nag_Overwrite
The first minm,n columns of U (the left singular vectors) are overwritten on the array a.
jobu=Nag_NotU
No columns of U (no left singular vectors) are computed.
Constraint: jobu=Nag_AllU, Nag_SingularVecsU, Nag_Overwrite or Nag_NotU.
3:     jobvtNag_ComputeVTTypeInput
On entry: specifies options for computing all or part of the matrix VH.
jobvt=Nag_AllVT
All n rows of VH are returned in the array vt.
jobvt=Nag_SingularVecsVT
The first minm,n rows of VH (the right singular vectors) are returned in the array vt.
jobvt=Nag_OverwriteVT
The first minm,n rows of VH (the right singular vectors) are overwritten on the array a.
jobvt=Nag_NotVT
No rows of VH (no right singular vectors) are computed.
Constraints:
  • jobvt=Nag_AllVT, Nag_SingularVecsVT, Nag_OverwriteVT or Nag_NotVT;
  • If jobu=Nag_Overwrite, jobvt cannot be Nag_OverwriteVT.
4:     mIntegerInput
On entry: m, the number of rows of the matrix A.
Constraint: m0.
5:     nIntegerInput
On entry: n, the number of columns of the matrix A.
Constraint: n0.
6:     a[dim]ComplexInput/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 jobu=Nag_Overwrite, a is overwritten with the first minm,n columns of U (the left singular vectors, stored column-wise).
If jobvt=Nag_OverwriteVT, a is overwritten with the first minm,n rows of VH (the right singular vectors, stored row-wise).
If jobuNag_Overwrite and jobvtNag_OverwriteVT, the contents of a are destroyed.
7:     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.
8:     s[dim]doubleOutput
Note: the dimension, dim, of the array s must be at least max1,minm,n .
On exit: the singular values of A, sorted so that s[i-1]s[i].
9:     u[dim]ComplexOutput
Note: the dimension, dim, of the array u must be at least
  • max1,pdu×m when jobu=Nag_AllU;
  • max1,pdu×minm,n when jobu=Nag_SingularVecsU and order=Nag_ColMajor;
  • max1,m×pdu when jobu=Nag_SingularVecsU 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 jobu=Nag_AllU, u contains the m by m matrix U.
If jobu=Nag_SingularVecsU, u contains the first minm,n columns of U (the left singular vectors, stored column-wise).
If jobu=Nag_NotU or Nag_Overwrite, u is not referenced.
10:   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 jobu=Nag_AllU, pdu max1,m ;
    • if jobu=Nag_SingularVecsU, pdu max1,m ;
    • otherwise pdu1;
  • if order=Nag_RowMajor,
    • if jobu=Nag_AllU, pdumax1,m;
    • if jobu=Nag_SingularVecsU, pdumax1,minm,n;
    • otherwise pdu1.
11:   vt[dim]ComplexOutput
Note: the dimension, dim, of the array vt must be at least
  • max1,pdvt×n when jobvt=Nag_AllVT;
  • max1,pdvt×n when jobvt=Nag_SingularVecsVT and order=Nag_ColMajor;
  • max1,minm,n×pdvt when jobvt=Nag_SingularVecsVT 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 jobvt=Nag_AllVT, vt contains the n by n unitary matrix VH.
If jobvt=Nag_SingularVecsVT, vt contains the first minm,n rows of VH (the right singular vectors, stored row-wise).
If jobvt=Nag_NotVT or Nag_OverwriteVT, vt is not referenced.
12:   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 jobvt=Nag_AllVT, pdvt max1,n ;
    • if jobvt=Nag_SingularVecsVT, pdvt max1,minm,n ;
    • otherwise pdvt1;
  • if order=Nag_RowMajor,
    • if jobvt=Nag_AllVT, pdvtmax1,n;
    • if jobvt=Nag_SingularVecsVT, pdvtmax1,n;
    • otherwise pdvt1.
13:   rwork[minm,n]doubleOutput
On exit: if fail.code= NE_CONVERGENCE, RWORK1:minm,n-1 (using the notation described in Section 3.2.1.4 in the Essential Introduction) contains the unconverged superdiagonal elements of an upper bidiagonal matrix B whose diagonal is in S (not necessarily sorted). B satisfies A=UBVH, so it has the same singular values as A, and singular vectors related by U and VH.
14:   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
If nag_zgesvd (f08kpc) did not converge, fail.errnum specifies how many superdiagonals of an intermediate bidiagonal form did not converge to zero.
NE_ENUM_INT_2
On entry, jobu=value, pdu=value and m=value.
Constraint: if jobu=Nag_AllU, pdu max1,m ;
if jobu=Nag_SingularVecsU, pdu max1,m ;
otherwise pdu1.
NE_ENUM_INT_3
On entry, jobu=value, pdu=value, m=value and n=value.
Constraint: if jobu=Nag_AllU, pdumax1,m;
if jobu=Nag_SingularVecsU, pdumax1,minm,n;
otherwise pdu1.
On entry, jobvt=value, pdvt=value, m=value and n=value.
Constraint: if jobvt=Nag_AllVT, pdvt max1,n ;
if jobvt=Nag_SingularVecsVT, pdvt max1,minm,n ;
otherwise pdvt1.
On entry, jobvt=value, pdvt=value, m=value and n=value.
Constraint: if jobvt=Nag_AllVT, pdvtmax1,n;
if jobvt=Nag_SingularVecsVT, 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 real analogue of this function is nag_dgesvd (f08kbc).

9  Example

This example finds the singular values and left and right singular vectors of the 6 by 4 matrix
A = 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.05+0.41i -0.98+1.98i -1.20+0.19i -0.66+0.42i -0.81+0.56i 0.62-0.46i 1.01+0.02i 0.63-0.17i -1.11+0.60i -0.37+0.38i 0.19-0.54i -0.98-0.36i 0.22-0.20i 0.83+0.51i 0.20+0.01i -0.17-0.46i 1.47+1.59i 1.08-0.28i 0.20-0.12i -0.07+1.23i 0.26+0.26i ,
together with approximate error bounds for the computed singular values and vectors.
The example program for nag_zgesdd (f08krc) illustrates finding a singular value decomposition for the case mn.

9.1  Program Text

Program Text (f08kpce.c)

9.2  Program Data

Program Data (f08kpce.d)

9.3  Program Results

Program Results (f08kpce.r)


nag_zgesvd (f08kpc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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