NAG CL Interface
f08krc (zgesdd)

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1 Purpose

f08krc computes the singular value decomposition (SVD) of a complex 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  f08krc (Nag_OrderType order, Nag_JobType job, Integer m, Integer n, Complex a[], Integer pda, double s[], Complex u[], Integer pdu, Complex vt[], Integer pdvt, NagError *fail)
The function may be called by the names: f08krc, nag_lapackeig_zgesdd or nag_zgesdd.

3 Description

The SVD is written as
A = UΣVH ,  
where Σ is an m×n matrix which is zero except for its min(m,n) diagonal elements, U is an m×m unitary matrix, and V is an n×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 min(m,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 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 VH 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 VH 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 VH are returned in the array vt. Otherwise, all columns of U are returned in the array u and the first m rows of VH are overwritten in the array vt.
job=Nag_DoNothing
No columns of U or rows of VH 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] Complex 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 VH (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] Complex 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 unitary 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] Complex 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 unitary matrix VH.
If job=Nag_DoSquare, vt contains the first min(m,n) rows of VH (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
f08krc 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

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

10 Example

This example finds the singular values and left and right singular vectors of the 4×6 matrix
A = ( 0.96+0.81i -0.98-1.98i 0.62+0.46i -0.37-0.38i 0.83-0.51i 1.08+0.28i -0.03-0.96i -1.20-0.19i 1.01-0.02i 0.19+0.54i 0.20-0.01i 0.20+0.12i -0.91-2.06i -0.66-0.42i 0.63+0.17i -0.98+0.36i -0.17+0.46i -0.07-1.23i -0.05-0.41i -0.81-0.56i -1.11-0.60i 0.22+0.20i 1.47-1.59i 0.26-0.26i ) ,  
together with approximate error bounds for the computed singular values and vectors.
The example program for f08kpc illustrates finding a singular value decomposition for the case mn.

10.1 Program Text

Program Text (f08krce.c)

10.2 Program Data

Program Data (f08krce.d)

10.3 Program Results

Program Results (f08krce.r)