NAG CL Interface
f08ysc (ztgsja)

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1 Purpose

f08ysc computes the generalized singular value decomposition (GSVD) of two complex upper trapezoidal matrices A and B, where A is an m×n matrix and B is a p×n matrix.
A and B are assumed to be in the form returned by f08vuc.

2 Specification

#include <nag.h>
void  f08ysc (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)
The function may be called by the names: f08ysc, nag_lapackeig_ztgsja or nag_ztgsja.

3 Description

f08ysc computes the GSVD of the matrices A and B which are assumed to have the form as returned by f08vuc
A= { n-k-lklk0A12A13l00A23m-k-l000() ,   if ​ m-k-l 0; n-k-lklk0A12A13m-k00A23() ,   if ​ m-k-l < 0 ; B= n-k-lkll00B13p-l000() ,  
where the k×k matrix A12 and the 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.
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-l)0 then D1, D2 and R have the form
D1= klkI0l0Cm-k-l00() ,  
D2= kll0Sp-l00() ,  
R = klkR11R12l0R22() ,  
where C=diag(αk+1,,,,,,αk+l),  S=diag(βk+1,,,,,,βk+l).
If (m-k-l)<0 then D1, D2 and R have the form
D1= km-kk+l-mkI00m-k0C0() ,  
D2= km-kk+l-mm-k0S0k+l-m00Ip-l000() ,  
R = km-kk+l-mkR11R12R13m-k0R22R23k+l-m00R33() ,  
where C=diag(αk+1,,,,,,αm),  S=diag(βk+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 https://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: 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, 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_AllU, Nag_InitU or Nag_NotU.
3: jobv Nag_ComputeVType Input
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: jobq Nag_ComputeQType Input
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: 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: k Integer Input
9: l Integer Input
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 f08ysc.
10: 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.
where A(i,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×n matrix A.
On exit: if m-k-l0, A(1:k+l,n-k-l+1:n) contains the (k+l)×(k+l) upper triangular matrix R.
If m-k-l<0, A(1:m,n-k-l+1:n) contains the first m rows of the (k+l)×(k+l) upper triangular matrix R, and the submatrix R33 is returned in B(m-k+1:l,n+m-k-l+1:n) .
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] 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.
where B(i,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×n matrix B.
On exit: if m-k-l<0 , B(m-k+1:l,n+m-k-l+1:n) contains the submatrix R33 of R.
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: tola double Input
15: tolb double Input
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 f08vuc, say
tola=max(m,n)Aε, tolb=max(p,n)Bε,  
where ε is the machine precision.
16: alpha[n] double Output
On exit: see the description of beta.
17: beta[n] double Output
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] Complex Input/Output
Note: the dimension, dim, of the array u must be at least
  • max(1,pdu×m) when jobu=Nag_AllU or Nag_InitU;
  • 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 entry: if jobu=Nag_AllU, u must contain an m×m matrix U1 (usually the unitary matrix returned by f08vuc).
On exit: if jobu=Nag_AllU, u contains the product U1U.
If jobu=Nag_InitU, u contains the unitary matrix U.
If jobu=Nag_NotU, u is not referenced.
19: 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 or Nag_InitU, pdu max(1,m) ;
  • otherwise pdu1.
20: v[dim] Complex Input/Output
Note: the dimension, dim, of the array v must be at least
  • max(1,pdv×p) when jobv=Nag_ComputeV or Nag_InitV;
  • 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 entry: if jobv=Nag_ComputeV, v must contain an p×p matrix V1 (usually the unitary matrix returned by f08vuc).
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: 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 or Nag_InitV, pdv max(1,p) ;
  • otherwise pdv1.
22: q[dim] Complex Input/Output
Note: the dimension, dim, of the array q must be at least
  • max(1,pdq×n) when jobq=Nag_ComputeQ or Nag_InitQ;
  • 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 entry: if jobq=Nag_ComputeQ, q must contain an n×n matrix Q1 (usually the unitary matrix returned by f08vuc).
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: 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 or Nag_InitQ, pdq max(1,n) ;
  • otherwise pdq1.
24: ncycle Integer * Output
On exit: the number of cycles required for convergence.
25: 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 procedure does not converge after 40 cycles.
NE_ENUM_INT_2
On entry, jobq=value, pdq=value and n=value.
Constraint: if jobq=Nag_ComputeQ or Nag_InitQ, pdq max(1,n) ;
otherwise pdq1.
On entry, jobu=value, pdu=value and m=value.
Constraint: if jobu=Nag_AllU or Nag_InitU, pdu max(1,m) ;
otherwise pdu1.
On entry, jobv=value, pdv=value and p=value.
Constraint: if jobv=Nag_ComputeV or Nag_InitV, 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 = 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

Background information to multithreading can be found in the Multithreading documentation.
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 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

The real analogue of this function is 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)