NAG CL Interface
f08ykc (dtgevc)

Settings help

CL Name Style:


1 Purpose

f08ykc computes some or all of the right and/or left generalized eigenvectors of a pair of real matrices (A,B) which are in generalized real Schur form.

2 Specification

#include <nag.h>
void  f08ykc (Nag_OrderType order, Nag_SideType side, Nag_HowManyType how_many, const Nag_Boolean select[], Integer n, const double a[], Integer pda, const double b[], Integer pdb, double vl[], Integer pdvl, double vr[], Integer pdvr, Integer mm, Integer *m, NagError *fail)
The function may be called by the names: f08ykc, nag_lapackeig_dtgevc or nag_dtgevc.

3 Description

f08ykc computes some or all of the right and/or left generalized eigenvectors of the matrix pair (A,B) which is assumed to be in generalized upper Schur form. If the matrix pair (A,B) is not in the generalized upper Schur form, then f08xec should be called before invoking f08ykc.
The right generalized eigenvector x and the left generalized eigenvector y of (A,B) corresponding to a generalized eigenvalue λ are defined by
(A-λB)x=0  
and
yH (A-λB)=0.  
If a generalized eigenvalue is determined as 0/0, which is due to zero diagonal elements at the same locations in both A and B, a unit vector is returned as the corresponding eigenvector.
Note that the generalized eigenvalues are computed using f08xec but f08ykc does not explicitly require the generalized eigenvalues to compute eigenvectors. The ordering of the eigenvectors is based on the ordering of the eigenvalues as computed by f08ykc.
If all eigenvectors are requested, the function may either return the matrices X and/or Y of right or left eigenvectors of (A,B), or the products ZX and/or QY, where Z and Q are two matrices supplied by you. Usually, Q and Z are chosen as the orthogonal matrices returned by f08xec. Equivalently, Q and Z are the left and right Schur vectors of the matrix pair supplied to f08xec. In that case, QY and ZX are the left and right generalized eigenvectors, respectively, of the matrix pair supplied to f08xec.
A must be block upper triangular; with 1×1 and 2×2 diagonal blocks. Corresponding to each 2×2 diagonal block is a complex conjugate pair of eigenvalues and eigenvectors; only one eigenvector of the pair is computed, namely the one corresponding to the eigenvalue with positive imaginary part. Each 1×1 block gives a real generalized eigenvalue and a corresponding eigenvector.

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: side Nag_SideType Input
On entry: specifies the required sets of generalized eigenvectors.
side=Nag_RightSide
Only right eigenvectors are computed.
side=Nag_LeftSide
Only left eigenvectors are computed.
side=Nag_BothSides
Both left and right eigenvectors are computed.
Constraint: side=Nag_BothSides, Nag_LeftSide or Nag_RightSide.
3: how_many Nag_HowManyType Input
On entry: specifies further details of the required generalized eigenvectors.
how_many=Nag_ComputeAll
All right and/or left eigenvectors are computed.
how_many=Nag_BackTransform
All right and/or left eigenvectors are computed; they are backtransformed using the input matrices supplied in arrays vr and/or vl.
how_many=Nag_ComputeSelected
Selected right and/or left eigenvectors, defined by the array select, are computed.
Constraint: how_many=Nag_ComputeAll, Nag_BackTransform or Nag_ComputeSelected.
4: select[dim] const Nag_Boolean Input
Note: the dimension, dim, of the array select must be at least
  • n when how_many=Nag_ComputeSelected;
  • otherwise select may be NULL.
On entry: specifies the eigenvectors to be computed if how_many=Nag_ComputeSelected. To select the generalized eigenvector corresponding to the jth generalized eigenvalue, the jth element of select should be set to Nag_TRUE; if the eigenvalue corresponds to a complex conjugate pair, then real and imaginary parts of eigenvectors corresponding to the complex conjugate eigenvalue pair will be computed.
If how_many=Nag_ComputeAll or Nag_BackTransform, select is not referenced and may be NULL.
Constraint: if how_many=Nag_ComputeSelected, select[j]=Nag_TRUE or Nag_FALSE, for j=0,1,,n-1.
5: n Integer Input
On entry: n, the order of the matrices A and B.
Constraint: n0.
6: a[dim] const double Input
Note: the dimension, dim, of the array a must be at least pda×n.
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 matrix pair (A,B) must be in the generalized Schur form. Usually, this is the matrix A returned by f08xec.
7: 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).
8: b[dim] const double Input
Note: the dimension, dim, of the array b must be at least pdb×n.
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 matrix pair (A,B) must be in the generalized Schur form. If A has a 2×2 diagonal block then the corresponding 2×2 block of B must be diagonal with positive elements. Usually, this is the matrix B returned by f08xec.
9: 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).
10: vl[dim] double Input/Output
Note: the dimension, dim, of the array vl must be at least
  • pdvl×mm when side=Nag_LeftSide or Nag_BothSides and order=Nag_ColMajor;
  • n×pdvl when side=Nag_LeftSide or Nag_BothSides and order=Nag_RowMajor;
  • otherwise vl may be NULL.
The (i,j)th element of the matrix is stored in
  • vl[(j-1)×pdvl+i-1] when order=Nag_ColMajor;
  • vl[(i-1)×pdvl+j-1] when order=Nag_RowMajor.
On entry: if how_many=Nag_BackTransform and side=Nag_LeftSide or Nag_BothSides, vl must be initialized to an n×n matrix Q. Usually, this is the orthogonal matrix Q of left Schur vectors returned by f08xec.
On exit: if side=Nag_LeftSide or Nag_BothSides, vl contains:
  • if how_many=Nag_ComputeAll, the matrix Y of left eigenvectors of (A,B);
  • if how_many=Nag_BackTransform, the matrix QY;
  • if how_many=Nag_ComputeSelected, the left eigenvectors of (A,B) specified by select, stored consecutively in the rows or columns (depending on the value of order) of the array vl, in the same order as their corresponding eigenvalues.
A complex eigenvector corresponding to a complex eigenvalue is stored in two consecutive rows or columns, the first holding the real part, and the second the imaginary part.
If side=Nag_RightSide, vl is not referenced and may be NULL.
11: pdvl Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array vl.
Constraints:
  • if order=Nag_ColMajor,
    • if side=Nag_LeftSide or Nag_BothSides, pdvl n ;
    • if side=Nag_RightSide, vl may be NULL;
  • if order=Nag_RowMajor,
    • if side=Nag_LeftSide or Nag_BothSides, pdvlmm;
    • if side=Nag_RightSide, vl may be NULL.
12: vr[dim] double Input/Output
Note: the dimension, dim, of the array vr must be at least
  • pdvr×mm when side=Nag_RightSide or Nag_BothSides and order=Nag_ColMajor;
  • n×pdvr when side=Nag_RightSide or Nag_BothSides and order=Nag_RowMajor;
  • otherwise vr may be NULL.
The (i,j)th element of the matrix is stored in
  • vr[(j-1)×pdvr+i-1] when order=Nag_ColMajor;
  • vr[(i-1)×pdvr+j-1] when order=Nag_RowMajor.
On entry: if how_many=Nag_BackTransform and side=Nag_RightSide or Nag_BothSides, vr must be initialized to an n×n matrix Z. Usually, this is the orthogonal matrix Z of right Schur vectors returned by f08xec.
On exit: if side=Nag_RightSide or Nag_BothSides, vr contains:
  • if how_many=Nag_ComputeAll, the matrix X of right eigenvectors of (A,B);
  • if how_many=Nag_BackTransform, the matrix ZX;
  • if how_many=Nag_ComputeSelected, the right eigenvectors of (A,B) specified by select, stored consecutively in the rows or columns (depending on the value of order) of the array vr, in the same order as their corresponding eigenvalues.
A complex eigenvector corresponding to a complex eigenvalue is stored in two consecutive rows or columns, the first holding the real part, and the second the imaginary part.
If side=Nag_LeftSide, vr is not referenced and may be NULL.
13: pdvr Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array vr.
Constraints:
  • if order=Nag_ColMajor,
    • if side=Nag_RightSide or Nag_BothSides, pdvr n ;
    • if side=Nag_LeftSide, vr may be NULL;
  • if order=Nag_RowMajor,
    • if side=Nag_RightSide or Nag_BothSides, pdvrmm;
    • if side=Nag_LeftSide, vr may be NULL.
14: mm Integer Input
On entry: the number of columns in the arrays vl and/or vr.
Constraints:
  • if how_many=Nag_ComputeAll or Nag_BackTransform, mmn;
  • if how_many=Nag_ComputeSelected, mm must not be less than the number of requested eigenvectors.
15: m Integer * Output
On exit: the number of columns in the arrays vl and/or vr actually used to store the eigenvectors. If how_many=Nag_ComputeAll or Nag_BackTransform, m is set to n. Each selected real eigenvector occupies one row or column and each selected complex eigenvector occupies two rows or columns.
16: 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_CONSTRAINT
Constraint: if how_many=Nag_ComputeSelected, select[j]=Nag_TRUE or Nag_FALSE, for j=0,1,,n-1.
NE_ENUM_INT_2
On entry, how_many=value, n=value and mm=value.
Constraint: if how_many=Nag_ComputeAll or Nag_BackTransform, mmn;
if how_many=Nag_ComputeSelected, mm must not be less than the number of requested eigenvectors.
On entry, side=value, pdvl=value and mm=value.
Constraint: if side=Nag_LeftSide or Nag_BothSides, pdvlmm.
On entry, side=value, pdvl=value and n=value.
Constraint: if side=Nag_LeftSide or Nag_BothSides, pdvl n .
On entry, side=value, pdvr=value and mm=value.
Constraint: if side=Nag_RightSide or Nag_BothSides, pdvrmm.
On entry, side=value, pdvr=value and n=value.
Constraint: if side=Nag_RightSide or Nag_BothSides, pdvr n .
NE_INT
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
On entry, pdb=value.
Constraint: pdb>0.
On entry, pdvl=value.
Constraint: pdvl>0.
On entry, pdvr=value.
Constraint: pdvr>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_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.
NE_NOT_COMPLEX
The 2×2 block (value:value+1) does not have complex eigenvalues.

7 Accuracy

It is beyond the scope of this manual to summarise the accuracy of the solution of the generalized eigenvalue problem. Interested readers should consult Section 4.11 of the LAPACK Users' Guide (see Anderson et al. (1999)) and Chapter 6 of Stewart and Sun (1990).

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08ykc 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

f08ykc is the sixth step in the solution of the real generalized eigenvalue problem and is called after f08xec.
The complex analogue of this function is f08yxc.

10 Example

This example computes the α and β arguments, which defines the generalized eigenvalues and the corresponding left and right eigenvectors, 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 )   and   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 ) .  
To compute generalized eigenvalues, it is required to call 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 via the QZ algorithm.
The computation of generalized eigenvectors is done by calling f08ykc to compute the eigenvectors of the balanced matrix pair. The function f08wjc is called to backward transform the eigenvectors to the user-supplied matrix pair. If both left and right eigenvectors are required then f08wjc must be called twice.

10.1 Program Text

Program Text (f08ykce.c)

10.2 Program Data

Program Data (f08ykce.d)

10.3 Program Results

Program Results (f08ykce.r)