NAG CL Interface
f08vuc (zggsvp3)

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1 Purpose

f08vuc uses unitary transformations to simultaneously reduce the m×n matrix A and the p×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>
void  f08vuc (Nag_OrderType order, Nag_ComputeUType jobu, Nag_ComputeVType jobv, Nag_ComputeQType jobq, Integer m, Integer p, Integer n, Complex a[], Integer pda, Complex b[], Integer pdb, double tola, double tolb, Integer *k, Integer *l, Complex u[], Integer pdu, Complex v[], Integer pdv, Complex q[], Integer pdq, NagError *fail)
The function may be called by the names: f08vuc, nag_lapackeig_zggsvp3 or nag_zggsvp3.

3 Description

f08vuc computes unitary matrices U, V and Q such that
UHAQ= { n-k-lklk0A12A13l00A23m-k-l000() , if ​m-k-l0; n-k-lklk0A12A13m-k00A23() , if ​m-k-l<0;   VHBQ= n-k-lkll00B13p-l000()  
where the k×k matrix A12 and l×l matrix B13 are nonsingular upper triangular; A23 is l×l upper triangular if m-k-l0 and is (m-k)×l upper trapezoidal otherwise. (k+l) is the effective numerical rank of the (m+p)×n matrix (AHBH)H.
This decomposition is usually used as the preprocessing step for computing the Generalized Singular Value Decomposition (GSVD), see function f08ysc; the two steps are combined in f08vqc.

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 unitary 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 unitary 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 unitary 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: p Integer Input
On entry: p, the number of rows of the matrix B.
Constraint: p0.
7: n Integer Input
On entry: n, the number of columns of the matrices A and B.
Constraint: n0.
8: 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: contains the triangular (or trapezoidal) matrix described in Section 3.
9: 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).
10: b[dim] Complex 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 described in Section 3.
11: 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).
12: tola double Input
13: tolb double Input
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=max(m,n)Aε, tolb=max(p,n)Bε,  
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 (aTbT)T.
16: u[dim] Complex 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 unitary 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] Complex 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 unitary 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] Complex 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 unitary 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: 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_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 factorization is nearly the exact factorization for nearby matrices (A+E) and (B+F), where
E2 = O(ε)A2   and   F2= O(ε)B2,  
and ε is the machine precision.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08vuc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08vuc 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 f08vsc which used an unblocked algorithm and, therefore, did not make best use of Level 3 BLAS functions.
The real analogue of this function is f08vgc.

10 Example

This example finds the generalized factorization
A = UΣ1 ( 0 S ) QH ,   B= VΣ2 ( 0 T ) QH ,  
of the matrix pair (AB), 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 = ( 10−10 010−1 ) .  

10.1 Program Text

Program Text (f08vuce.c)

10.2 Program Data

Program Data (f08vuce.d)

10.3 Program Results

Program Results (f08vuce.r)