NAG CL Interface
f08xec (dhgeqz)

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1 Purpose

f08xec implements the QZ method for finding generalized eigenvalues of the real matrix pair (A,B) of order n, which is in the generalized upper Hessenberg form.

2 Specification

#include <nag.h>
void  f08xec (Nag_OrderType order, Nag_JobType job, Nag_ComputeQType compq, Nag_ComputeZType compz, Integer n, Integer ilo, Integer ihi, double a[], Integer pda, double b[], Integer pdb, double alphar[], double alphai[], double beta[], double q[], Integer pdq, double z[], Integer pdz, NagError *fail)
The function may be called by the names: f08xec, nag_lapackeig_dhgeqz or nag_dhgeqz.

3 Description

f08xec implements a single-double-shift version of the QZ method for finding the generalized eigenvalues of the real matrix pair (A,B) which is in the generalized upper Hessenberg form. If the matrix pair (A,B) is not in the generalized upper Hessenberg form, then the function f08wfc should be called before invoking f08xec.
This problem is mathematically equivalent to solving the equation
det(A-λB)=0.  
Note that, to avoid underflow, overflow and other arithmetic problems, the generalized eigenvalues λj are never computed explicitly by this function but defined as ratios between two computed values, αj and βj:
λj=αj/βj.  
The arguments αj, in general, are finite complex values and βj are finite real non-negative values.
If desired, the matrix pair (A,B) may be reduced to generalized Schur form. That is, the transformed matrix B is upper triangular and the transformed matrix A is block upper triangular, where the diagonal blocks are either 1×1 or 2×2. The 1×1 blocks provide generalized eigenvalues which are real and the 2×2 blocks give complex generalized eigenvalues.
The argument job specifies two options. If job=Nag_Schur then the matrix pair (A,B) is simultaneously reduced to Schur form by applying one orthogonal transformation (usually called Q) on the left and another (usually called Z) on the right. That is,
AQTAZ BQTBZ  
The 2×2 upper-triangular diagonal blocks of B corresponding to 2×2 blocks of a will be reduced to non-negative diagonal matrices. That is, if A(j+1,j) is nonzero, then B(j+1,j)=B(j,j+1)=0 and B(j,j) and B(j+1,j+1) will be non-negative.
If job=Nag_EigVals, then at each iteration the same transformations are computed but they are only applied to those parts of A and B which are needed to compute α and β. This option could be used if generalized eigenvalues are required but not generalized eigenvectors.
If job=Nag_Schur and compq=Nag_AccumulateQ or Nag_InitQ, and compz=Nag_AccumulateZ or Nag_InitZ, then the orthogonal transformations used to reduce the pair (A,B) are accumulated into the input arrays q and z. If generalized eigenvectors are required then job must be set to job=Nag_Schur and if left (right) generalized eigenvectors are to be computed then compq (compz) must be set to compq=Nag_AccumulateQ or Nag_InitQ and not compqNag_NotQ.
If compq=Nag_InitQ, then eigenvectors are accumulated on the identity matrix and on exit the array q contains the left eigenvector matrix Q. However, if compq=Nag_AccumulateQ then the transformations are accumulated on the user-supplied matrix Q0 in array q on entry and thus on exit q contains the matrix product QQ0. A similar convention is used for compz.

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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Moler C B and Stewart G W (1973) An algorithm for generalized matrix eigenproblems SIAM J. Numer. Anal. 10 241–256
Stewart G W and Sun J-G (1990) Matrix Perturbation Theory Academic Press, London

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: job Nag_JobType Input
On entry: specifies the operations to be performed on (A,B).
job=Nag_EigVals
The matrix pair (A,B) on exit might not be in the generalized Schur form.
job=Nag_Schur
The matrix pair (A,B) on exit will be in the generalized Schur form.
Constraint: job=Nag_EigVals or Nag_Schur.
3: compq Nag_ComputeQType Input
On entry: specifies the operations to be performed on Q:
compq=Nag_NotQ
The array q is unchanged.
compq=Nag_AccumulateQ
The left transformation Q is accumulated on the array q.
compq=Nag_InitQ
The array q is initialized to the identity matrix before the left transformation Q is accumulated in q.
Constraint: compq=Nag_NotQ, Nag_AccumulateQ or Nag_InitQ.
4: compz Nag_ComputeZType Input
On entry: specifies the operations to be performed on Z.
compz=Nag_NotZ
The array z is unchanged.
compz=Nag_AccumulateZ
The right transformation Z is accumulated on the array z.
compz=Nag_InitZ
The array z is initialized to the identity matrix before the right transformation Z is accumulated in z.
Constraint: compz=Nag_NotZ, Nag_AccumulateZ or Nag_InitZ.
5: n Integer Input
On entry: n, the order of the matrices A, B, Q and Z.
Constraint: n0.
6: ilo Integer Input
7: ihi Integer Input
On entry: the indices ilo and ihi, respectively which define the upper triangular parts of A. The submatrices A(1:ilo-1,1:ilo-1) and A(ihi+1:n,ihi+1:n) are then upper triangular. These arguments are provided by f08whc if the matrix pair was previously balanced; otherwise, ilo=1 and ihi=n.
Constraints:
  • if n>0, 1 ilo ihi n ;
  • if n=0, ilo=1 and ihi=0.
8: a[dim] double Input/Output
Note: the dimension, dim, of the array a must be at least max(1,pda×n).
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 n×n upper Hessenberg matrix A. The elements below the first subdiagonal must be set to zero.
On exit: if job=Nag_Schur, the matrix pair (A,B) will be simultaneously reduced to generalized Schur form.
If job=Nag_EigVals, the 1×1 and 2×2 diagonal blocks of the matrix pair (A,B) will give generalized eigenvalues but the remaining elements will be irrelevant.
9: pda Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdamax(1,n).
10: b[dim] double Input/Output
Note: the dimension, dim, of the array b must be at least max(1,pdb×n).
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 n×n upper triangular matrix B. The elements below the diagonal must be zero.
On exit: if job=Nag_Schur, the matrix pair (A,B) will be simultaneously reduced to generalized Schur form.
If job=Nag_EigVals, the 1×1 and 2×2 diagonal blocks of the matrix pair (A,B) will give generalized eigenvalues but the remaining elements will be irrelevant.
11: pdb Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraint: pdbmax(1,n).
12: alphar[n] double Output
On exit: the real parts of αj, for j=1,2,,n.
13: alphai[n] double Output
On exit: the imaginary parts of αj, for j=1,2,,n.
14: beta[n] double Output
On exit: βj, for j=1,2,,n.
15: q[dim] double Input/Output
Note: the dimension, dim, of the array q must be at least
  • max(1,pdq×n) when compq=Nag_AccumulateQ or Nag_InitQ;
  • 1 when compq=Nag_NotQ.
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 compq=Nag_AccumulateQ, the matrix Q0. The matrix Q0 is usually the matrix Q returned by f08wfc.
If compq=Nag_NotQ, q is not referenced.
On exit: if compq=Nag_AccumulateQ, q contains the matrix product QQ0.
If compq=Nag_InitQ, q contains the transformation matrix Q.
16: pdq Integer Input
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 compq=Nag_AccumulateQ or Nag_InitQ, pdqn;
    • if compq=Nag_NotQ, pdq1;
  • if order=Nag_RowMajor,
    • if compq=Nag_AccumulateQ or Nag_InitQ, pdqmax(1,n);
    • if compq=Nag_NotQ, pdq1.
17: z[dim] double Input/Output
Note: the dimension, dim, of the array z must be at least
  • max(1,pdz×n) when compz=Nag_AccumulateZ or Nag_InitZ;
  • 1 when compz=Nag_NotZ.
The (i,j)th element of the matrix Z is stored in
  • z[(j-1)×pdz+i-1] when order=Nag_ColMajor;
  • z[(i-1)×pdz+j-1] when order=Nag_RowMajor.
On entry: if compz=Nag_AccumulateZ, the matrix Z0. The matrix Z0 is usually the matrix Z returned by f08wfc.
If compz=Nag_NotZ, z is not referenced.
On exit: if compz=Nag_AccumulateZ, z contains the matrix product ZZ0.
If compz=Nag_InitZ, z contains the transformation matrix Z.
18: pdz Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array z.
Constraints:
  • if order=Nag_ColMajor,
    • if compz=Nag_AccumulateZ or Nag_InitZ, pdzn;
    • if compz=Nag_NotZ, pdz1;
  • if order=Nag_RowMajor,
    • if compz=Nag_AccumulateZ or Nag_InitZ, pdzmax(1,n);
    • if compz=Nag_NotZ, pdz1.
19: 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, compq=value, pdq=value and n=value.
Constraint: if compq=Nag_AccumulateQ or Nag_InitQ, pdqmax(1,n);
if compq=Nag_NotQ, pdq1.
On entry, compq=value, pdq=value and n=value.
Constraint: if compq=Nag_AccumulateQ or Nag_InitQ, pdqn;
if compq=Nag_NotQ, pdq1.
On entry, compz=value, pdz=value and n=value.
Constraint: if compz=Nag_AccumulateZ or Nag_InitZ, pdzmax(1,n);
if compz=Nag_NotZ, pdz1.
On entry, compz=value, pdz=value and n=value.
Constraint: if compz=Nag_AccumulateZ or Nag_InitZ, pdzn;
if compz=Nag_NotZ, pdz1.
NE_INT
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
On entry, pdb=value.
Constraint: pdb>0.
On entry, pdq=value.
Constraint: pdq>0.
On entry, pdz=value.
Constraint: pdz>0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdamax(1,n).
On entry, pdb=value and n=value.
Constraint: pdbmax(1,n).
NE_INT_3
On entry, n=value, ilo=value and ihi=value.
Constraint: if n>0, 1 ilo ihi n ;
if n=0, ilo=1 and ihi=0.
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.
An unexpected Library error has occurred.
NE_ITERATION_QZ
The QZ iteration did not converge and the matrix pair (A,B) is not in the generalized Schur form. The computed αi and βi should be correct for i=value,,value.
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.
NE_SCHUR
The computation of shifts failed and the matrix pair (A,B) is not in the generalized Schur form. The computed αi and βi should be correct for i=value,,value.

7 Accuracy

Please consult Section 4.11 of the LAPACK Users' Guide (see Anderson et al. (1999)) and Chapter 6 of Stewart and Sun (1990), for more information.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08xec 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

f08xec is the fifth step in the solution of the real generalized eigenvalue problem and is called after f08wfc.
The complex analogue of this function is f08xsc.

10 Example

This example computes the α and β arguments, which defines the generalized eigenvalues, of the matrix pair (A,B) given by
A = ( 1.0 1.0 1.0 1.0 1.0 2.0 4.0 8.0 16.0 32.0 3.0 9.0 27.0 81.0 243.0 4.0 16.0 64.0 256.0 1024.0 5.0 25.0 125.0 625.0 3125.0 )  
B = ( 1.0 2.0 3.0 4.0 5.0 1.0 4.0 9.0 16.0 25.0 1.0 8.0 27.0 64.0 125.0 1.0 16.0 81.0 256.0 625.0 1.0 32.0 243.0 1024.0 3125.0 ) .  
This requires calls to five functions: f08whc to balance the matrix, f08aec to perform the QR factorization of B, f08agc to apply Q to A, f08wfc to reduce the matrix pair to the generalized Hessenberg form and f08xec to compute the eigenvalues using the QZ algorithm.

10.1 Program Text

Program Text (f08xece.c)

10.2 Program Data

Program Data (f08xece.d)

10.3 Program Results

Program Results (f08xece.r)