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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dggesx (f08xb)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dggesx (f08xb) computes the generalized eigenvalues, the generalized real Schur form S,T  and, optionally, the left and/or right generalized Schur vectors for a pair of n by n real nonsymmetric matrices A,B .
Estimates of condition numbers for selected generalized eigenvalue clusters and Schur vectors are also computed.

Syntax

[a, b, sdim, alphar, alphai, beta, vsl, vsr, rconde, rcondv, info] = f08xb(jobvsl, jobvsr, sort, selctg, sense, a, b, 'n', n)
[a, b, sdim, alphar, alphai, beta, vsl, vsr, rconde, rcondv, info] = nag_lapack_dggesx(jobvsl, jobvsr, sort, selctg, sense, a, b, 'n', n)

Description

The generalized real Schur factorization of A,B  is given by
A = QSZT ,   B = QTZT ,  
where Q and Z are orthogonal, T is upper triangular and S is upper quasi-triangular with 1 by 1 and 2 by 2 diagonal blocks. The generalized eigenvalues, λ , of A,B  are computed from the diagonals of T and S and satisfy
Az = λBz ,  
where z is the corresponding generalized eigenvector. λ  is actually returned as the pair α,β  such that
λ = α/β  
since β , or even both α  and β  can be zero. The columns of Q and Z are the left and right generalized Schur vectors of A,B .
Optionally, nag_lapack_dggesx (f08xb) can order the generalized eigenvalues on the diagonals of S,T  so that selected eigenvalues are at the top left. The leading columns of Q and Z then form an orthonormal basis for the corresponding eigenspaces, the deflating subspaces.
nag_lapack_dggesx (f08xb) computes T to have non-negative diagonal elements, and the 2 by 2 blocks of S correspond to complex conjugate pairs of generalized eigenvalues. The generalized Schur factorization, before reordering, is computed by the QZ algorithm.
The reciprocals of the condition estimates, the reciprocal values of the left and right projection norms, are returned in rconde1  and rconde2  respectively, for the selected generalized eigenvalues, together with reciprocal condition estimates for the corresponding left and right deflating subspaces, in rcondv1  and rcondv2 . See Section 4.11 of Anderson et al. (1999) for further information.

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

Parameters

Compulsory Input Parameters

1:     jobvsl – string (length ≥ 1)
If jobvsl='N', do not compute the left Schur vectors.
If jobvsl='V', compute the left Schur vectors.
Constraint: jobvsl='N' or 'V'.
2:     jobvsr – string (length ≥ 1)
If jobvsr='N', do not compute the right Schur vectors.
If jobvsr='V', compute the right Schur vectors.
Constraint: jobvsr='N' or 'V'.
3:     sort – string (length ≥ 1)
Specifies whether or not to order the eigenvalues on the diagonal of the generalized Schur form.
sort='N'
Eigenvalues are not ordered.
sort='S'
Eigenvalues are ordered (see selctg).
Constraint: sort='N' or 'S'.
4:     selctg – function handle or string containing name of m-file
If sort='S', selctg is used to select generalized eigenvalues to the top left of the generalized Schur form.
If sort='N', selctg is not referenced by nag_lapack_dggesx (f08xb), and may be called with the string 'f08xaz'.
[result] = selctg(ar, ai, b)

Input Parameters

1:     ar – double scalar
2:     ai – double scalar
3:     b – double scalar
An eigenvalue arj + -1 × aij / bj  is selected if selctg arj,aij,bj  is true. If either one of a complex conjugate pair is selected, then both complex generalized eigenvalues are selected.
Note that in the ill-conditioned case, a selected complex generalized eigenvalue may no longer satisfy selctg arj,aij,bj=true  after ordering. info=n+2 in this case.

Output Parameters

1:     result – logical scalar
result=true for selected eigenvalues.
5:     sense – string (length ≥ 1)
Determines which reciprocal condition numbers are computed.
sense='N'
None are computed.
sense='E'
Computed for average of selected eigenvalues only.
sense='V'
Computed for selected deflating subspaces only.
sense='B'
Computed for both.
If sense='E', 'V' or 'B', sort='S'.
Constraint: sense='N', 'E', 'V' or 'B'.
6:     alda: – double array
The first dimension of the array a must be at least max1,n.
The second dimension of the array a must be at least max1,n.
The first of the pair of matrices, A.
7:     bldb: – double array
The first dimension of the array b must be at least max1,n.
The second dimension of the array b must be at least max1,n.
The second of the pair of matrices, B.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the arrays a, b and the second dimension of the arrays a, b. (An error is raised if these dimensions are not equal.)
n, the order of the matrices A and B.
Constraint: n0.

Output Parameters

1:     alda: – double array
The first dimension of the array a will be max1,n.
The second dimension of the array a will be max1,n.
a stores its generalized Schur form S.
2:     bldb: – double array
The first dimension of the array b will be max1,n.
The second dimension of the array b will be max1,n.
b stores its generalized Schur form T.
3:     sdim int64int32nag_int scalar
If sort='N', sdim=0.
If sort='S', sdim= number of eigenvalues (after sorting) for which selctg is true. (Complex conjugate pairs for which selctg is true for either eigenvalue count as 2.)
4:     alpharn – double array
See the description of beta.
5:     alphain – double array
See the description of beta.
6:     betan – double array
alpharj+alphaij×i/betaj, for j=1,2,,n, will be the generalized eigenvalues. alpharj+alphaij×i, and betaj, for j=1,2,,n, are the diagonals of the complex Schur form S,T that would result if the 2 by 2 diagonal blocks of the real Schur form of A,B were further reduced to triangular form using 2 by 2 complex unitary transformations.
If alphaij is zero, then the jth eigenvalue is real; if positive, then the jth and j+1st eigenvalues are a complex conjugate pair, with alphaij+1 negative.
Note:  the quotients alpharj/betaj and alphaij/betaj may easily overflow or underflow, and betaj may even be zero. Thus, you should avoid naively computing the ratio α/β. However, alphar and alphai will always be less than and usually comparable with a2 in magnitude, and beta will always be less than and usually comparable with b2.
7:     vslldvsl: – double array
The first dimension, ldvsl, of the array vsl will be
  • if jobvsl='V', ldvsl= max1,n ;
  • otherwise ldvsl=1.
The second dimension of the array vsl will be max1,n if jobvsl='V' and 1 otherwise.
If jobvsl='V', vsl will contain the left Schur vectors, Q.
If jobvsl='N', vsl is not referenced.
8:     vsrldvsr: – double array
The first dimension, ldvsr, of the array vsr will be
  • if jobvsr='V', ldvsr= max1,n ;
  • otherwise ldvsr=1.
The second dimension of the array vsr will be max1,n if jobvsr='V' and 1 otherwise.
If jobvsr='V', vsr will contain the right Schur vectors, Z.
If jobvsr='N', vsr is not referenced.
9:     rconde2 – double array
If sense='E' or 'B', rconde1 and rconde2 contain the reciprocal condition numbers for the average of the selected eigenvalues.
If sense='N' or 'V', rconde is not referenced.
10:   rcondv2 – double array
If sense='V' or 'B', rcondv1 and rcondv2 contain the reciprocal condition numbers for the selected deflating subspaces.
if sense='N' or 'E', rcondv is not referenced.
11:   info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   info=-i
If info=-i, parameter i had an illegal value on entry. The parameters are numbered as follows:
1: jobvsl, 2: jobvsr, 3: sort, 4: selctg, 5: sense, 6: n, 7: a, 8: lda, 9: b, 10: ldb, 11: sdim, 12: alphar, 13: alphai, 14: beta, 15: vsl, 16: ldvsl, 17: vsr, 18: ldvsr, 19: rconde, 20: rcondv, 21: work, 22: lwork, 23: iwork, 24: liwork, 25: bwork, 26: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.
W  info=1ton
The QZ iteration failed. A,B are not in Schur form, but alpharj, alphaij, and betaj should be correct for j=info+1,,n.
   info=n+1
Unexpected error returned from nag_lapack_dhgeqz (f08xe).
W  info=n+2
After reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the generalized Schur form no longer satisfy selctg=true. This could also be caused by underflow due to scaling.
W  info=n+3
The eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned).

Accuracy

The computed generalized Schur factorization satisfies
A+E = QS ZT ,   B+F = QT ZT ,  
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.

Further Comments

The total number of floating-point operations is proportional to n3.
The complex analogue of this function is nag_lapack_zggesx (f08xp).

Example

This example finds the generalized Schur factorization 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 ,  
such that the real positive eigenvalues of A,B correspond to the top left diagonal elements of the generalized Schur form, S,T. Estimates of the condition numbers for the selected eigenvalue cluster and corresponding deflating subspaces are also returned.
Note that the block size (NB) of 64 assumed in this example is not realistic for such a small problem, but should be suitable for large problems.
function f08xb_example


fprintf('f08xb example results\n\n');

% Matrix pair (A,B)
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];
B = [1,    2,    -3,    1;
     1,    3,    -5,    4;
     1,    3,    -4,    3;
     1,    3,    -4,    4];


% Generalized Schur form (S,T) of (A,B), generalized eigenvalues
% and Schur vectors Q and Z with sorting and selecting only
% real eigenvalues
jobvsl = 'Vectors (left)';
jobvsr = 'Vectors (right)';
sortp = 'Sort';
selctg = @(ar, ai, b) (ai == 0) ;
sense = 'Both reciprocal condition numbers';

[S, T, sdim, alphar, alphai, beta, VSL, VSR, rconde, rcondv, info] = ...
  f08xb( ...
         jobvsl, jobvsr, sortp, selctg, sense, A, B);

fprintf('Number of selected eigenvalues = %4d\n\n', sdim);
disp('Selected generalized eigenvalues')
eigs = alphar./beta + i*alphai./beta;
disp(eigs(1:sdim));

fprintf('%s\n%s\n%s = %8.1e, %s = %8.1e\n\n', ...
        'Reciprocals of left and right projection norms onto', ...
        'the deflating subspaces for the selected eigenvalues', ...
        'rconde(1)', rconde(1), 'rconde(2)', rconde(2));
fprintf('%s\n%s\n%s = %8.1e, %s = %8.1e\n\n', ...
        'Reciprocals condition numbers for the left and right', ...
        'deflating subspaces', 'rcondv(1)', rcondv(1), ...
        'rcondv(2)', rcondv(2));

anorm = norm(A,2);
bnorm = norm(B,2);
abnorm = sqrt(anorm^2+bnorm^2);
fprintf('%s = %8.1e\n', ...
        'Approximate asymptotic error bound for selected eigenvalues   ', ...
        x02aj*abnorm/rconde(1));
fprintf('%s = %8.1e\n', ...
        'Approximate asymptotic error bound for the deflating subspaces', ...
        x02aj*abnorm/rcondv(2));


f08xb example results

Number of selected eigenvalues =    2

Selected generalized eigenvalues
    2.0000
    4.0000

Reciprocals of left and right projection norms onto
the deflating subspaces for the selected eigenvalues
rconde(1) =  1.9e-01, rconde(2) =  1.8e-02

Reciprocals condition numbers for the left and right
deflating subspaces
rcondv(1) =  5.4e-02, rcondv(2) =  9.0e-02

Approximate asymptotic error bound for selected eigenvalues    =  5.7e-14
Approximate asymptotic error bound for the deflating subspaces =  1.2e-13

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Chapter Contents
Chapter Introduction
NAG Toolbox

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