NAG C Library Function Document

nag_dggsvp3 (f08vgc)

1
Purpose

nag_dggsvp3 (f08vgc) uses orthogonal transformations to simultaneously reduce the m by n matrix A and the p by n matrix B to upper triangular form. This factorization is usually used as a preprocessing step for computing the generalized singular value decomposition (GSVD). For sufficiently large problems, a blocked algorithm is used to make best use of level 3 BLAS.

2
Specification

#include <nag.h>
#include <nagf08.h>
void  nag_dggsvp3 (Nag_OrderType order, Nag_ComputeUType jobu, Nag_ComputeVType jobv, Nag_ComputeQType jobq, Integer m, Integer p, Integer n, double a[], Integer pda, double b[], Integer pdb, double tola, double tolb, Integer *k, Integer *l, double u[], Integer pdu, double v[], Integer pdv, double q[], Integer pdq, NagError *fail)

3
Description

nag_dggsvp3 (f08vgc) computes orthogonal matrices U, V and Q such that
UTAQ= n-k-lklk0A12A13l00A23m-k-l000() , if ​m-k-l0; n-k-lklk0A12A13m-k00A23() , if ​m-k-l<0; VTBQ= n-k-lkll00B13p-l000()  
where the k by k matrix A12 and 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. k+l is the effective numerical rank of the m+p by n matrix ATBTT.
This decomposition is usually used as the preprocessing step for computing the Generalized Singular Value Decomposition (GSVD), see function nag_dtgsja (f08yec); the two steps are combined in nag_dggsvd3 (f08vcc).

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 (2012) Matrix Computations (4th Edition) Johns Hopkins University Press, Baltimore

5
Arguments

1:     order Nag_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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     jobu Nag_ComputeUTypeInput
On entry: if jobu=Nag_AllU, the orthogonal matrix U is computed.
If jobu=Nag_NotU, U is not computed.
Constraint: jobu=Nag_AllU or Nag_NotU.
3:     jobv Nag_ComputeVTypeInput
On entry: if jobv=Nag_ComputeV, the orthogonal matrix V is computed.
If jobv=Nag_NotV, V is not computed.
Constraint: jobv=Nag_ComputeV or Nag_NotV.
4:     jobq Nag_ComputeQTypeInput
On entry: if jobq=Nag_ComputeQ, the orthogonal matrix Q is computed.
If jobq=Nag_NotQ, Q is not computed.
Constraint: jobq=Nag_ComputeQ or Nag_NotQ.
5:     m IntegerInput
On entry: m, the number of rows of the matrix A.
Constraint: m0.
6:     p IntegerInput
On entry: p, the number of rows of the matrix B.
Constraint: p0.
7:     n IntegerInput
On entry: n, the number of columns of the matrices A and B.
Constraint: n0.
8:     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: contains the triangular (or trapezoidal) matrix described in Section 3.
9:     pda IntegerInput
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.
10:   b[dim] doubleInput/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 described in Section 3.
11:   pdb IntegerInput
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.
12:   tola doubleInput
13:   tolb doubleInput
On entry: tola and tolb are the thresholds to determine the effective numerical rank of matrix B and a subblock of A. Generally, they are set to
tola=maxm,nAε, tolb=maxp,nBε,  
where ε  is the machine precision.
The size of tola and tolb may affect the size of backward errors of the decomposition.
14:   k Integer *Output
15:   l Integer *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 aTbTT.
16:   u[dim] doubleOutput
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 orthogonal matrix U.
If jobu=Nag_NotU, u is not referenced.
17:   pdu IntegerInput
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] doubleOutput
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 orthogonal matrix V.
If jobv=Nag_NotV, v is not referenced.
19:   pdv IntegerInput
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] doubleOutput
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 orthogonal matrix Q.
If jobq=Nag_NotQ, q is not referenced.
21:   pdq IntegerInput
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:   fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_ENUM_INT_2
On entry, jobq=value, pdq=value and n=value.
Constraint: if jobq=Nag_ComputeQ, pdq max1,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.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

7
Accuracy

The computed factorization is nearly the exact factorization for nearby matrices A+E and B+F, where
E2 = OεA2   and   F2= OεB2,  
and ε is the machine precision.

8
Parallelism and Performance

nag_dggsvp3 (f08vgc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dggsvp3 (f08vgc) 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

This function replaces the deprecated function nag_dggsvp (f08vec) which used an unblocked algorithm and therefore did not make best use of level 3 BLAS functions.
The complex analogue of this function is nag_zggsvp3 (f08vuc).

10
Example

This example finds the generalized factorization
A = UΣ1 0 S QT ,   B= VΣ2 0 T QT ,  
of the matrix pair AB, where
A = 123 321 456 788   and   B= -2-33 465 .  

10.1
Program Text

Program Text (f08vgce.c)

10.2
Program Data

Program Data (f08vgce.d)

10.3
Program Results

Program Results (f08vgce.r)