nag_dggev (f08wac) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dggev (f08wac)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dggev (f08wac) computes for a pair of n by n real nonsymmetric matrices A,B the generalized eigenvalues and, optionally, the left and/or right generalized eigenvectors using the QZ algorithm.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_dggev (Nag_OrderType order, Nag_LeftVecsType jobvl, Nag_RightVecsType jobvr, Integer n, double a[], Integer pda, double b[], Integer pdb, double alphar[], double alphai[], double beta[], double vl[], Integer pdvl, double vr[], Integer pdvr, NagError *fail)

3  Description

A generalized eigenvalue for a pair of matrices A,B is a scalar λ or a ratio α/β=λ, such that A-λB is singular. It is usually represented as the pair α,β, as there is a reasonable interpretation for β=0, and even for both being zero.
The right eigenvector vj corresponding to the eigenvalue λj of A,B satisfies
A vj = λj B vj .
The left eigenvector uj corresponding to the eigenvalue λj of A,B satisfies
ujH A = λj ujH B ,
where ujH is the conjugate-transpose of uj.
All the eigenvalues and, if required, all the eigenvectors of the generalized eigenproblem Ax=λBx, where A and B are real, square matrices, are determined using the QZ algorithm. The QZ algorithm consists of four stages:
  1. A is reduced to upper Hessenberg form and at the same time B is reduced to upper triangular form.
  2. A is further reduced to quasi-triangular form while the triangular form of B is maintained. This is the real generalized Schur form of the pair A,B .
  3. The quasi-triangular form of A is reduced to triangular form and the eigenvalues extracted. This function does not actually produce the eigenvalues λj, but instead returns αj and βj such that
    λj=αj/βj,  j=1,2,,n.
    The division by βj becomes your responsibility, since βj may be zero, indicating an infinite eigenvalue. Pairs of complex eigenvalues occur with αj/βj and αj+1/βj+1 complex conjugates, even though αj and αj+1 are not conjugate.
  4. If the eigenvectors are required they are obtained from the triangular matrices and then transformed back into the original coordinate system.

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 http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Wilkinson J H (1979) Kronecker's canonical form and the QZ algorithm Linear Algebra Appl. 28 285–303

5  Arguments

1:     orderNag_OrderTypeInput
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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     jobvlNag_LeftVecsTypeInput
On entry: if jobvl=Nag_NotLeftVecs, do not compute the left generalized eigenvectors.
If jobvl=Nag_LeftVecs, compute the left generalized eigenvectors.
Constraint: jobvl=Nag_NotLeftVecs or Nag_LeftVecs.
3:     jobvrNag_RightVecsTypeInput
On entry: if jobvr=Nag_NotRightVecs, do not compute the right generalized eigenvectors.
If jobvr=Nag_RightVecs, compute the right generalized eigenvectors.
Constraint: jobvr=Nag_NotRightVecs or Nag_RightVecs.
4:     nIntegerInput
On entry: n, the order of the matrices A and B.
Constraint: n0.
5:     a[dim]doubleInput/Output
Note: the dimension, dim, of the array a must be at least max1,pda×n.
The i,jth 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 A in the pair A,B.
On exit: a has been overwritten.
6:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdamax1,n.
7:     b[dim]doubleInput/Output
Note: the dimension, dim, of the array b must be at least max1,pdb×n.
The i,jth 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 B in the pair A,B.
On exit: b has been overwritten.
8:     pdbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraint: pdbmax1,n.
9:     alphar[n]doubleOutput
On exit: the element alphar[j-1] contains the real part of αj.
10:   alphai[n]doubleOutput
On exit: the element alphai[j-1] contains the imaginary part of αj.
11:   beta[n]doubleOutput
On exit: alphar[j-1]+alphai[j-1]×i/beta[j-1], for j=1,2,,n, will be the generalized eigenvalues.
If alphai[j-1] is zero, then the jth eigenvalue is real; if positive, then the jth and j+1st eigenvalues are a complex conjugate pair, with alphai[j] negative.
Note:  the quotients alphar[j-1]/beta[j-1] and alphai[j-1]/beta[j-1] may easily overflow or underflow, and beta[j-1] may even be zero. Thus, you should avoid naively computing the ratio αj/βj. However, maxαj will always be less than and usually comparable with a2 in magnitude, and maxβj will always be less than and usually comparable with b2.
12:   vl[dim]doubleOutput
Note: the dimension, dim, of the array vl must be at least
  • max1,pdvl×n when jobvl=Nag_LeftVecs;
  • 1 otherwise.
The ith element of the jth vector is stored in
  • vl[j-1×pdvl+i-1] when order=Nag_ColMajor;
  • vl[i-1×pdvl+j-1] when order=Nag_RowMajor.
On exit: if jobvl=Nag_LeftVecs, the left eigenvectors uj are stored one after another in the columns of vl, in the same order as the corresponding eigenvalues.
If the jth eigenvalue is real, then uj=VL:,j, the jth column of vl.
If the jth and j+1th eigenvalues form a complex conjugate pair, then uj=VL:,j+i×VL:,j+1 and uj+1=VL:,j-i×VL:,j+1. Each eigenvector will be scaled so the largest component has real part+imag. part=1.
If jobvl=Nag_NotLeftVecs, vl is not referenced.
13:   pdvlIntegerInput
On entry: the stride used in the array vl.
Constraints:
  • if jobvl=Nag_LeftVecs, pdvl max1,n ;
  • otherwise pdvl1.
14:   vr[dim]doubleOutput
Note: the dimension, dim, of the array vr must be at least
  • max1,pdvr×n when jobvr=Nag_RightVecs;
  • 1 otherwise.
The ith element of the jth vector is stored in
  • vr[j-1×pdvr+i-1] when order=Nag_ColMajor;
  • vr[i-1×pdvr+j-1] when order=Nag_RowMajor.
On exit: if jobvr=Nag_RightVecs, the right eigenvectors vj are stored one after another in the columns of vr, in the same order as the corresponding eigenvalues.
If the jth eigenvalue is real, then vj=VR:,j, the jth column of VR.
If the jth and j+1th eigenvalues form a complex conjugate pair, then vj=VR:,j+i×VR:,j+1 and vj+1=VR:,j-i×VR:,j+1. Each eigenvector will be scaled so the largest component has real part+imag. part=1.
If jobvr=Nag_NotRightVecs, vr is not referenced.
15:   pdvrIntegerInput
On entry: the stride used in the array vr.
Constraints:
  • if jobvr=Nag_RightVecs, pdvr max1,n ;
  • otherwise pdvr1.
16:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_EIGENVECTORS
A failure occurred in nag_dtgevc (f08ykc) while computing generalized eigenvectors.
NE_ENUM_INT_2
On entry, jobvl=value, pdvl=value and n=value.
Constraint: if jobvl=Nag_LeftVecs, pdvl max1,n ;
otherwise pdvl1.
On entry, jobvr=value, pdvr=value and n=value.
Constraint: if jobvr=Nag_RightVecs, pdvr max1,n ;
otherwise pdvr1.
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: pdamax1,n.
On entry, pdb=value and n=value.
Constraint: pdbmax1,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.
NE_ITERATION_QZ
The QZ iteration failed. No eigenvectors have been calculated but alphar[j], alphai[j], and beta[j] should be correct from element value.
The QZ iteration failed with an unexpected error, please contact NAG.

7  Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrices A+E and B+F, where
E,F F = Oε A,B F ,
and ε is the machine precision. See Section 4.11 of Anderson et al. (1999) for further details.
Note:  interpretation of results obtained with the QZ algorithm often requires a clear understanding of the effects of small changes in the original data. These effects are reviewed in Wilkinson (1979), in relation to the significance of small values of αj and βj. It should be noted that if αj and βj are both small for any j, it may be that no reliance can be placed on any of the computed eigenvalues λi=αi/βi. You are recommended to study Wilkinson (1979) and, if in difficulty, to seek expert advice on determining the sensitivity of the eigenvalues to perturbations in the data.

8  Further Comments

The total number of floating point operations is proportional to n3.
The complex analogue of this function is nag_zggev (f08wnc).

9  Example

This example finds all the eigenvalues and right eigenvectors of the matrix pair A,B, where
A = 3.9 12.5 -34.5 -0.5 4.3 21.5 -47.5 7.5 4.3 21.5 -43.5 3.5 4.4 26.0 -46.0 6.0   and   B= 1.0 2.0 -3.0 1.0 1.0 3.0 -5.0 4.0 1.0 3.0 -4.0 3.0 1.0 3.0 -4.0 4.0 .

9.1  Program Text

Program Text (f08wace.c)

9.2  Program Data

Program Data (f08wace.d)

9.3  Program Results

Program Results (f08wace.r)


nag_dggev (f08wac) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012