nag_ztgsja (f08ysc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_ztgsja (f08ysc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_ztgsja (f08ysc) computes the generalized singular value decomposition (GSVD) of two complex upper trapezoidal matrices A and B, where A is an m by n matrix and B is a p by n matrix.
A and B are assumed to be in the form returned by nag_zggsvp (f08vsc).

2  Specification

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

3  Description

nag_ztgsja (f08ysc) computes the GSVD of the matrices A and B which are assumed to have the form as returned by nag_zggsvp (f08vsc)
A= n-k-lklk(0A12A13) l 0 0 A23 m-k-l 0 0 0 ,   if ​ m-k-l 0; n-k-lklk(0A12A13) m-k 0 0 A23 ,   if ​ m-k-l < 0 ; B= n-k-lkll(00B13) p-l 0 0 0 ,
where the k by k matrix A12 and the l by l matrix B13 are nonsingular upper triangular, A23 is l by l upper triangular if m-k-l0 and is m-k by l upper trapezoidal otherwise.
nag_ztgsja (f08ysc) computes unitary matrices Q, U and V, diagonal matrices D1 and D2, and an upper triangular matrix R such that
UHAQ = D1 0 R ,   VHBQ = D2 0 R .
Optionally Q, U and V may or may not be computed, or they may be premultiplied by matrices Q1, U1 and V1 respectively.
If m-k-l0 then D1, D2 and R have the form
D1= klk(I0) l 0 C m-k-l 0 0 ,
D2= kll(0S) p-l 0 0 ,
R = klk(R11R12) l 0 R22 ,
where C=diagαk+1,,,,,,αk+l,  S=diagβk+1,,,,,,βk+l.
If m-k-l<0 then D1, D2 and R have the form
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 ,
R = km-kk+l-mk(R11R12R13) m-k 0 R22 R23 k+l-m 0 0 R33 ,
where C=diagαk+1,,,,,,αm,  S=diagβk+1,,,,,,βm.
In both cases the diagonal matrix C has real non-negative diagonal elements, the diagonal matrix S has real positive diagonal elements, so that S is nonsingular, and C2+S2=1. See Section 2.3.5.3 of Anderson et al. (1999) for further information.

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, u must contain a unitary matrix U1 on entry, and the product U1U is returned.
If jobu=Nag_InitU, u is initialized to the unit matrix, and the unitary matrix U is returned.
If jobu=Nag_NotU, U is not computed.
Constraint: jobu=Nag_InitU, Nag_AllU or Nag_NotU.
3:     jobvNag_ComputeVTypeInput
On entry: if jobv=Nag_ComputeV, v must contain a unitary matrix V1 on entry, and the product V1V is returned.
If jobv=Nag_InitV, v is initialized to the unit matrix, and the unitary matrix V is returned.
If jobv=Nag_NotV, V is not computed.
Constraint: jobv=Nag_ComputeV, Nag_InitV or Nag_NotV.
4:     jobqNag_ComputeQTypeInput
On entry: if jobq=Nag_ComputeQ, q must contain a unitary matrix Q1 on entry, and the product Q1Q is returned.
If jobq=Nag_InitQ, q is initialized to the unit matrix, and the unitary matrix Q is returned.
If jobq=Nag_NotQ, Q is not computed.
Constraint: jobq=Nag_ComputeQ, Nag_InitQ or Nag_NotQ.
5:     mIntegerInput
On entry: m, the number of rows of the matrix A.
Constraint: m0.
6:     pIntegerInput
On entry: p, the number of rows of the matrix B.
Constraint: p0.
7:     nIntegerInput
On entry: n, the number of columns of the matrices A and B.
Constraint: n0.
8:     kIntegerInput
9:     lIntegerInput
On entry: k and l specify the sizes, k and l, of the subblocks of A and B, whose GSVD is to be computed by nag_ztgsja (f08ysc).
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.
Where Ai,j appears in this document, it refers to the array element
  • 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 m-k-l0, A1:k+l,n-k-l+1:n  contains the k+l by k+l upper triangular matrix R.
If m-k-l<0, A1:m,n-k-l+1:n  contains the first m rows of the k+l by k+l upper triangular matrix R, and the submatrix R33 is returned in Bm-k+1:l,n+m-k-l+1:n .
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.
Where Bi,j appears in this document, it refers to the array element
  • 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: if m-k-l<0 , Bm-k+1:l,n+m-k-l+1:n  contains the submatrix R33 of R.
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:   toladoubleInput
15:   tolbdoubleInput
On entry: tola and tolb are the convergence criteria for the Jacobi–Kogbetliantz iteration procedure. Generally, they should be the same as used in the preprocessing step performed by nag_zggsvp (f08vsc), say
tola=maxm,nAε, tolb=maxp,nBε,
where ε  is the machine precision.
16:   alpha[n]doubleOutput
On exit: see the description of beta.
17:   beta[n]doubleOutput
On exit: alpha and beta contain the generalized singular value pairs of A and B;
  • alpha[i]=1 , beta[i]=0 , for i=0,1,,k-1, and
  • if m-k-l0 , alpha[i]=αi , beta[i]=βi , for i=k,,k+l-1, or
  • if m-k-l<0 , alpha[i]=αi , beta[i]=βi , for i=k,,m-1 and alpha[i]=0 , beta[i]=1 , for i=m,,k+l-1.
Furthermore, if k+l<n, alpha[i]= beta[i]=0 , for i=k+l,,n-1.
18:   u[dim]ComplexInput/Output
Note: the dimension, dim, of the array u must be at least
  • max1,pdu×m when jobu=Nag_AllU or Nag_InitU and order=Nag_ColMajor;
  • max1,×pdu when jobu=Nag_AllU or Nag_InitU 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 entry: if jobu=Nag_AllU, u must contain an m by m matrix U1 (usually the unitary matrix returned by nag_zggsvp (f08vsc)).
On exit: if jobu=Nag_InitU, u contains the unitary matrix U.
If jobu=Nag_AllU, u contains the product U1U.
If jobu=Nag_NotU, u is not referenced.
19:   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 jobuNag_NotU, pdu max1,m ;
    • otherwise pdu1;
  • if order=Nag_RowMajor,
    • if jobu=Nag_AllU or Nag_InitU, pdumax1,m;
    • otherwise pdu1.
20:   v[dim]ComplexInput/Output
Note: the dimension, dim, of the array v must be at least
  • max1,pdv×p when jobv=Nag_ComputeV or Nag_InitV and order=Nag_ColMajor;
  • max1,×pdv when jobv=Nag_ComputeV or Nag_InitV and order=Nag_RowMajor;
  • 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 entry: if jobv=Nag_ComputeV, v must contain an p by p matrix V1 (usually the unitary matrix returned by nag_zggsvp (f08vsc)).
On exit: if jobv=Nag_InitV, v contains the unitary matrix V.
If jobv=Nag_ComputeV, v contains the product V1V.
If jobv=Nag_NotV, v is not referenced.
21:   pdvIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array v.
Constraints:
  • if order=Nag_ColMajor,
    • if jobvNag_NotV, pdv max1,p ;
    • otherwise pdv1;
  • if order=Nag_RowMajor,
    • if jobv=Nag_ComputeV or Nag_InitV, pdvmax1,p;
    • otherwise pdv1.
22:   q[dim]ComplexInput/Output
Note: the dimension, dim, of the array q must be at least
  • max1,pdq×n when jobq=Nag_ComputeQ or Nag_InitQ and order=Nag_ColMajor;
  • max1,×pdq when jobq=Nag_ComputeQ or Nag_InitQ and order=Nag_RowMajor;
  • 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 entry: if jobq=Nag_ComputeQ, q must contain an n by n matrix Q1 (usually the unitary matrix returned by nag_zggsvp (f08vsc)).
On exit: if jobq=Nag_InitQ, q contains the unitary matrix Q.
If jobq=Nag_ComputeQ, q contains the product Q1Q.
If jobq=Nag_NotQ, q is not referenced.
23:   pdqIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array q.
Constraints:
  • if order=Nag_ColMajor,
    • if jobqNag_NotQ, pdq max1,n ;
    • otherwise pdq1;
  • if order=Nag_RowMajor,
    • if jobq=Nag_ComputeQ or Nag_InitQ, pdqmax1,n;
    • otherwise pdq1.
24:   ncycleInteger *Output
On exit: the number of cycles required for convergence.
25:   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 procedure does not converge after 40 cycles.
NE_ENUM_INT_2
On entry, jobq=value, pdq=value, n=value.
Constraint: if jobq=Nag_ComputeQ or Nag_InitQ, pdqmax1,n;
otherwise pdq1.
On entry, jobu=value, pdu=value, m=value.
Constraint: if jobu=Nag_AllU or Nag_InitU, pdumax1,m;
otherwise pdu1.
On entry, jobu=value, pdu=value and m=value.
Constraint: if jobuNag_NotU, pdu max1,m ;
otherwise pdu1.
On entry, jobv=value, pdv=value, p=value.
Constraint: if jobv=Nag_ComputeV or Nag_InitV, pdvmax1,p;
otherwise pdv1.
On entry, jobv=value, pdv=value and p=value.
Constraint: if jobvNag_NotV, pdv max1,p ;
otherwise pdv1.
On entry, pdq=value, jobq=value and n=value.
Constraint: if jobqNag_NotQ, 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  Parallelism and Performance

nag_ztgsja (f08ysc) is not threaded by NAG in any implementation.
nag_ztgsja (f08ysc) 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 Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

The real analogue of this function is nag_dtgsja (f08yec).

10  Example

This example finds the generalized singular value decomposition
A = UΣ1 0 R QH ,   B= VΣ2 0 R QH ,
of the matrix pair A,B, 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 .

10.1  Program Text

Program Text (f08ysce.c)

10.2  Program Data

Program Data (f08ysce.d)

10.3  Program Results

Program Results (f08ysce.r)


nag_ztgsja (f08ysc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014