nag_zggsvd (f08vnc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zggsvd (f08vnc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zggsvd (f08vnc) computes the generalized singular value decomposition (GSVD) of an m by n complex matrix A and a p by n complex matrix B.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_zggsvd (Nag_OrderType order, Nag_ComputeUType jobu, Nag_ComputeVType jobv, Nag_ComputeQType jobq, Integer m, Integer n, Integer p, Integer *k, Integer *l, Complex a[], Integer pda, Complex b[], Integer pdb, double alpha[], double beta[], Complex u[], Integer pdu, Complex v[], Integer pdv, Complex q[], Integer pdq, Integer iwork[], NagError *fail)

3  Description

The generalized singular value decomposition is given by
UH A Q = D1 0 R ,   VH B Q = D2 0 R ,
where U, V and Q are unitary matrices. Let k+l be the effective numerical rank of the matrix A B , then R is a k+l by k+l nonsingular upper triangular matrix, D1 and D2 are m by k+l and p by k+l ‘diagonal’ matrices structured as follows:
if m-k-l0,
D1= klk(I0) l 0 C m-k-l 0 0
D2= kll(0S) p-l 0 0
0R = n-k-lklk(0R11R12) l 0 0 R22
where
C = diagαk+1,,αk+l ,
S = diagβk+1,,βk+l ,
and
C2 + S2 = I .
R is stored as a submatrix of A with elements Rij stored as Ai,n-k-l+j on exit.
If m-k-l<0 ,
D1= km-kk+l-mk(I00) m-k 0 C 0
D2= km-kk+l-mm-k(0S0) k+l-m 0 0 I p-l 0 0 0
0R = n-k-lkm-kk+l-mk(0R11R12R13) m-k 0 0 R22 R23 k+l-m 0 0 0 R33
where
C = diagαk+1,,αm ,
S = diagβk+1,,βm ,
and
C2 + S2 = I .
R11 R12 R13 0 R22 R23  is stored as a submatrix of A with Rij stored as Ai,n-k-l+j, and R33  is stored as a submatrix of B with R33ij stored as Bm-k+i,n+m-k-l+j.
The function computes C, S, R and, optionally, the unitary transformation matrices U, V and Q.
In particular, if B is an n by n nonsingular matrix, then the GSVD of A and B implicitly gives the SVD of A×B-1:
A B-1 = U D1 D2-1 VH .
If A B  has orthonormal columns, then the GSVD of A and B is also equal to the CS decomposition of A and B. Furthermore, the GSVD can be used to derive the solution of the eigenvalue problem:
AH Ax=λ BH Bx .
In some literature, the GSVD of A and B is presented in the form
UH A X = 0D1 ,   VH B X = 0D2 ,
where U and V are orthogonal and X is nonsingular, and D1 and D2 are ‘diagonal’. The former GSVD form can be converted to the latter form by taking the nonsingular matrix X as
X = Q I 0 0 R-1 .

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: if jobu=Nag_AllU, the unitary matrix U is computed.
If jobu=Nag_NotU, U is not computed.
Constraint: jobu=Nag_AllU or Nag_NotU.
3:     jobvNag_ComputeVTypeInput
On entry: if jobv=Nag_ComputeV, the unitary matrix V is computed.
If jobv=Nag_NotV, V is not computed.
Constraint: jobv=Nag_ComputeV or Nag_NotV.
4:     jobqNag_ComputeQTypeInput
On entry: if jobq=Nag_ComputeQ, the unitary matrix Q is computed.
If jobq=Nag_NotQ, Q is not computed.
Constraint: jobq=Nag_ComputeQ or Nag_NotQ.
5:     mIntegerInput
On entry: m, the number of rows of the matrix A.
Constraint: m0.
6:     nIntegerInput
On entry: n, the number of columns of the matrices A and B.
Constraint: n0.
7:     pIntegerInput
On entry: p, the number of rows of the matrix B.
Constraint: p0.
8:     kInteger *Output
9:     lInteger *Output
On exit: k and l specify the dimension of the subblocks k and l as described in Section 3; k+l is the effective numerical rank of AB.
10:   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: contains the triangular matrix R, or part of R. See Section 3 for details.
11:   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.
12:   b[dim]ComplexInput/Output
Note: the dimension, dim, of the array b must be at least
  • max1,pdb×n when order=Nag_ColMajor;
  • max1,p×pdb when order=Nag_RowMajor.
The i,jth element of the matrix B is stored in
  • b[j-1×pdb+i-1] when order=Nag_ColMajor;
  • b[i-1×pdb+j-1] when order=Nag_RowMajor.
On entry: the p by n matrix B.
On exit: contains the triangular matrix R if m-k-l<0. See Section 3 for details.
13:   pdbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
  • if order=Nag_ColMajor, pdbmax1,p;
  • if order=Nag_RowMajor, pdbmax1,n.
14:   alpha[n]doubleOutput
On exit: see the description of beta.
15:   beta[n]doubleOutput
On exit: alpha and beta contain the generalized singular value pairs of A and B, αi  and βi ;
  • ALPHA1:k = 1 ,
  • BETA1:k = 0 ,
and if m-k-l0 ,
  • ALPHAk+1:k+l = C ,
  • BETAk+1:k+l = S ,
or if m-k-l<0 ,
  • ALPHAk+1:m = C ,
  • ALPHAm+1:k+l = 0 ,
  • BETAk+1:m = S ,
  • BETAm+1:k+l = 1 , and
  • ALPHAk+l+1:n = 0 ,
  • BETAk+l+1:n = 0 .
The notation ALPHAk:n above refers to consecutive elements alpha[i-1], for i=k,,n.
16:   u[dim]ComplexOutput
Note: the dimension, dim, of the array u must be at least
  • max1,pdu×m when jobu=Nag_AllU;
  • 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 unitary matrix U.
If jobu=Nag_NotU, u is not referenced.
17:   pduIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array u.
Constraints:
  • if jobu=Nag_AllU, pdu max1,m ;
  • otherwise pdu1.
18:   v[dim]ComplexOutput
Note: the dimension, dim, of the array v must be at least
  • max1,pdv×p when jobv=Nag_ComputeV;
  • 1 otherwise.
The i,jth element of the matrix V is stored in
  • v[j-1×pdv+i-1] when order=Nag_ColMajor;
  • v[i-1×pdv+j-1] when order=Nag_RowMajor.
On exit: if jobv=Nag_ComputeV, v contains the p by p unitary matrix V.
If jobv=Nag_NotV, v is not referenced.
19:   pdvIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array v.
Constraints:
  • if jobv=Nag_ComputeV, pdv max1,p ;
  • otherwise pdv1.
20:   q[dim]ComplexOutput
Note: the dimension, dim, of the array q must be at least
  • max1,pdq×n when jobq=Nag_ComputeQ;
  • 1 otherwise.
The i,jth element of the matrix Q is stored in
  • q[j-1×pdq+i-1] when order=Nag_ColMajor;
  • q[i-1×pdq+j-1] when order=Nag_RowMajor.
On exit: if jobq=Nag_ComputeQ, q contains the n by n unitary matrix Q.
If jobq=Nag_NotQ, q is not referenced.
21:   pdqIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array q.
Constraints:
  • if jobq=Nag_ComputeQ, pdq max1,n ;
  • otherwise pdq1.
22:   iwork[n]IntegerOutput
On exit: stores the sorting information. More precisely, the following loop will sort alpha such that alpha[0]alpha[1]alpha[n-1].
23:   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
The Jacobi-type procedure failed to converge.
NE_ENUM_INT_2
On entry, jobq=value, pdq=value, n=value.
Constraint: if jobq=Nag_ComputeQ, pdqmax1,n;
otherwise pdq1.
On entry, jobu=value, pdu=value and m=value.
Constraint: if jobu=Nag_AllU, pdu max1,m ;
otherwise pdu1.
On entry, jobv=value, pdv=value and p=value.
Constraint: if jobv=Nag_ComputeV, pdv max1,p ;
otherwise pdv1.
On entry, pdq=value, jobq=value and n=value.
Constraint: if jobq=Nag_ComputeQ, pdq max1,n ;
otherwise pdq1.
NE_INT
On entry, m=value.
Constraint: m0.
On entry, n=value.
Constraint: n0.
On entry, p=value.
Constraint: p0.
On entry, pda=value.
Constraint: pda>0.
On entry, pdb=value.
Constraint: pdb>0.
On entry, pdq=value.
Constraint: pdq>0.
On entry, pdu=value.
Constraint: pdu>0.
On entry, pdv=value.
Constraint: pdv>0.
NE_INT_2
On entry, pda=value and m=value.
Constraint: pdamax1,m.
On entry, pda=value and n=value.
Constraint: pdamax1,n.
On entry, pdb=value and n=value.
Constraint: pdbmax1,n.
On entry, pdb=value and p=value.
Constraint: pdbmax1,p.
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 generalized singular value decomposition is nearly the exact generalized singular value decomposition for nearby matrices A+E  and B+F , where
E2 = Oε A2 ​ and ​ F2 = Oε B2 ,
and ε  is the machine precision. See Section 4.12 of Anderson et al. (1999) for further details.

8  Further Comments

The diagonal elements of the matrix R are real.
The real analogue of this function is nag_dggsvd (f08vac).

9  Example

This example finds the generalized singular value decomposition
A = U Σ1 0R QH ,   B = V Σ2 0R QH ,
where
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
and
B = 1 0 -1 0 0 1 0 -1 ,
together with estimates for the condition number of R and the error bound for the computed generalized singular values.
The example program assumes that mn, and would need slight modification if this is not the case.

9.1  Program Text

Program Text (f08vnce.c)

9.2  Program Data

Program Data (f08vnce.d)

9.3  Program Results

Program Results (f08vnce.r)


nag_zggsvd (f08vnc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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