NAG CL Interface
f08vcc (dggsvd3)

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1 Purpose

f08vcc computes the generalized singular value decomposition (GSVD) of an m×n real matrix A and a p×n real matrix B.

2 Specification

#include <nag.h>
void  f08vcc (Nag_OrderType order, Nag_ComputeUType jobu, Nag_ComputeVType jobv, Nag_ComputeQType jobq, Integer m, Integer n, Integer p, Integer *k, Integer *l, double a[], Integer pda, double b[], Integer pdb, double alpha[], double beta[], double u[], Integer pdu, double v[], Integer pdv, double q[], Integer pdq, Integer iwork[], NagError *fail)
The function may be called by the names: f08vcc, nag_lapackeig_dggsvd3 or nag_dggsvd3.

3 Description

Given an m×n real matrix A and a p×n real matrix B, the generalized singular value decomposition is given by
UT A Q = D1 ( 0 R ) ,   VT B Q = D2 ( 0 R ) ,  
where U, V and Q are orthogonal matrices. Let l be the effective numerical rank of B and (k+l) be the effective numerical rank of the matrix ( A B ) , then the first k generalized singular values are infinite and the remaining l are finite. R is a (k+l)×(k+l) nonsingular upper triangular matrix, D1 and D2 are m×(k+l) and p×(k+l) ‘diagonal’ matrices structured as follows:
if m-k-l0,
D1= klkI0l0Cm-k-l00()  
D2= kll0Sp-l00()  
( 0R ) = n-k-lklk0R11R12l00R22()  
where
C = diag(αk+1,,αk+l) ,  
S = diag(βk+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-mkI00m-k0C0()  
D2= km-kk+l-mm-k0S0k+l-m00Ip-l000()  
( 0R ) = n-k-lkm-kk+l-mk0R11R12R13m-k00R22R23k+l-m000R33()  
where
C = diag(αk+1,,αm) ,  
S = diag(βk+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 (R33)ij stored as Bm-k+i,n+m-k-l+j.
The function computes C, S, R and, optionally, the orthogonal transformation matrices U, V and Q.
In particular, if B is an n×n nonsingular matrix, then the GSVD of A and B implicitly gives the SVD of AB-1:
A B-1 = U (D1D2−1) VT .  
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:
AT Ax=λ BT Bx .  
In some literature, the GSVD of A and B is presented in the form
UT A X = ( 0D1 ) ,   VT 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 setting
X = Q ( I 0 0 R-1 ) .  
A two stage process is used to compute the GSVD of the matrix pair (A,B). The pair is first reduced to upper triangular form by orthogonal transformations using f08vgc. The GSVD of the resulting upper triangular matrix pair is then performed by f08yec which uses a variant of the Kogbetliantz algorithm (a cyclic Jacobi method).

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 (2012) Matrix Computations (4th 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: jobu Nag_ComputeUType Input
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_ComputeVType Input
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_ComputeQType Input
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 Integer Input
On entry: m, the number of rows of the matrix A.
Constraint: m0.
6: n Integer Input
On entry: n, the number of columns of the matrices A and B.
Constraint: n0.
7: p Integer Input
On entry: p, the number of rows of the matrix B.
Constraint: p0.
8: k Integer * Output
9: 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 (AB).
10: a[dim] double 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: contains the triangular matrix R, or part of R. See Section 3 for details.
11: 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).
12: b[dim] double Input/Output
Note: the dimension, dim, of the array b must be at least
  • max(1,pdb×n) when order=Nag_ColMajor;
  • max(1,p×pdb) when order=Nag_RowMajor.
The (i,j)th 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×n matrix B.
On exit: contains the triangular matrix R if m-k-l<0. See Section 3 for details.
13: pdb Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
  • if order=Nag_ColMajor, pdbmax(1,p);
  • if order=Nag_RowMajor, pdbmax(1,n).
14: alpha[n] double Output
On exit: see the description of beta.
15: beta[n] double Output
On exit: alpha and beta contain the generalized singular value pairs of A and B, αi and βi ;
  • ALPHA(1:k) = 1 ,
  • BETA(1:k) = 0 ,
and if m-k-l0 ,
  • ALPHA(k+1:k+l) = C ,
  • BETA(k+1:k+l) = S ,
or if m-k-l<0 ,
  • ALPHA(k+1:m) = C ,
  • ALPHA(m+1:k+l) = 0 ,
  • BETA(k+1:m) = S ,
  • BETA(m+1:k+l) = 1 , and
  • ALPHA(k+l+1:n) = 0 ,
  • BETA(k+l+1:n) = 0 .
The notation ALPHA(k:n) above refers to consecutive elements alpha[i-1], for i=k,,n.
16: u[dim] double Output
Note: the dimension, dim, of the array u must be at least
  • max(1,pdu×m) when jobu=Nag_AllU;
  • 1 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 jobu=Nag_AllU, u contains the m×m orthogonal matrix U.
If jobu=Nag_NotU, u is not referenced.
17: pdu Integer Input
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 max(1,m) ;
  • otherwise pdu1.
18: v[dim] double Output
Note: the dimension, dim, of the array v must be at least
  • max(1,pdv×p) when jobv=Nag_ComputeV;
  • 1 otherwise.
The (i,j)th 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×p orthogonal matrix V.
If jobv=Nag_NotV, v is not referenced.
19: pdv Integer Input
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 max(1,p) ;
  • otherwise pdv1.
20: q[dim] double Output
Note: the dimension, dim, of the array q must be at least
  • max(1,pdq×n) when jobq=Nag_ComputeQ;
  • 1 otherwise.
The (i,j)th 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×n orthogonal matrix Q.
If jobq=Nag_NotQ, q is not referenced.
21: pdq Integer Input
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 max(1,n) ;
  • otherwise pdq1.
22: iwork[n] Integer Output
On exit: stores the sorting information. More precisely, if I is the ordered set of indices of alpha containing C (denote as alpha[I], see beta), then the corresponding elements iwork[I]-1 contain the swap pivots, J, that sorts I such that alpha[I] is in descending numerical order.
The following pseudocode sorts the set I:
for ​iI j=Ji swap ​Ii​ and ​Ij end
23: 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
The Jacobi-type procedure failed to converge.
NE_ENUM_INT_2
On entry, jobq=value, pdq=value and n=value.
Constraint: if jobq=Nag_ComputeQ, pdq max(1,n) ;
otherwise pdq1.
On entry, jobu=value, pdu=value and m=value.
Constraint: if jobu=Nag_AllU, pdu max(1,m) ;
otherwise pdu1.
On entry, jobv=value, pdv=value and p=value.
Constraint: if jobv=Nag_ComputeV, pdv max(1,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: pdamax(1,m).
On entry, pda=value and n=value.
Constraint: pdamax(1,n).
On entry, pdb=value and n=value.
Constraint: pdbmax(1,n).
On entry, pdb=value and p=value.
Constraint: pdbmax(1,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 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 generalized singular value decomposition is nearly the exact generalized singular value decomposition for nearby matrices (A+E) and (B+F) , where
E2 = O(ε) A2 ​ and ​ F2 = O(ε) B2 ,  
and ε is the machine precision. See Section 4.12 of Anderson et al. (1999) for further details.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08vcc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08vcc 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 f08vac which used an unblocked algorithm and, therefore, did not make best use of Level 3 BLAS functions.
The complex analogue of this function is f08vqc.

10 Example

This example finds the generalized singular value decomposition
A = U Σ1 ( 0R ) QT ,   B = V Σ2 ( 0R ) QT ,  
where
A = ( 1 2 3 3 2 1 4 5 6 7 8 8 )   and   B = ( −2 −3 3 4 6 5 ) ,  
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.

10.1 Program Text

Program Text (f08vcce.c)

10.2 Program Data

Program Data (f08vcce.d)

10.3 Program Results

Program Results (f08vcce.r)