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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zggev (f08wn)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zggev (f08wn) computes for a pair of n by n complex nonsymmetric matrices A,B the generalized eigenvalues and, optionally, the left and/or right generalized eigenvectors using the QZ algorithm.

Syntax

[a, b, alpha, beta, vl, vr, info] = f08wn(jobvl, jobvr, a, b, 'n', n)
[a, b, alpha, beta, vl, vr, info] = nag_lapack_zggev(jobvl, jobvr, a, b, 'n', n)

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 generalized eigenvector vj corresponding to the generalized eigenvalue λj of A,B satisfies
A vj = λj B vj .  
The left generalized eigenvector uj corresponding to the generalized 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 complex generalized eigenproblem Ax=λBx, where A and B are complex, square matrices, are determined using the QZ algorithm. The complex QZ algorithm consists of three stages:
1. A is reduced to upper Hessenberg form (with real, non-negative subdiagonal elements) and at the same time B is reduced to upper triangular form.
2. A is further reduced to triangular form while the triangular form of B is maintained and the diagonal elements of B are made real and non-negative. This is the generalized Schur form of the pair A,B .
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.
3. If the eigenvectors are required they are obtained from the triangular matrices and then transformed back into the original coordinate system.

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

Parameters

Compulsory Input Parameters

1:     jobvl – string (length ≥ 1)
If jobvl='N', do not compute the left generalized eigenvectors.
If jobvl='V', compute the left generalized eigenvectors.
Constraint: jobvl='N' or 'V'.
2:     jobvr – string (length ≥ 1)
If jobvr='N', do not compute the right generalized eigenvectors.
If jobvr='V', compute the right generalized eigenvectors.
Constraint: jobvr='N' or 'V'.
3:     alda: – complex 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 matrix A in the pair A,B.
4:     bldb: – complex 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 matrix B in the pair A,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: – complex array
The first dimension of the array a will be max1,n.
The second dimension of the array a will be max1,n.
2:     bldb: – complex array
The first dimension of the array b will be max1,n.
The second dimension of the array b will be max1,n.
b has been overwritten.
3:     alphan – complex array
See the description of beta.
4:     betan – complex array
alphaj/betaj, for j=1,2,,n, will be the generalized eigenvalues.
Note:  the quotients alphaj/betaj may easily overflow or underflow, and betaj 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.
5:     vlldvl: – complex array
The first dimension, ldvl, of the array vl will be
  • if jobvl='V', ldvl= max1,n ;
  • otherwise ldvl=1.
The second dimension of the array vl will be max1,n if jobvl='V' and 1 otherwise.
If jobvl='V', the left generalized eigenvectors uj are stored one after another in the columns of vl, in the same order as the corresponding eigenvalues. Each eigenvector will be scaled so the largest component will have real part+imag. part=1.
If jobvl='N', vl is not referenced.
6:     vrldvr: – complex array
The first dimension, ldvr, of the array vr will be
  • if jobvr='V', ldvr= max1,n ;
  • otherwise ldvr=1.
The second dimension of the array vr will be max1,n if jobvr='V' and 1 otherwise.
If jobvr='V', the right generalized eigenvectors vj are stored one after another in the columns of vr, in the same order as the corresponding eigenvalues. Each eigenvector will be scaled so the largest component will have real part+imag. part=1.
If jobvr='N', vr is not referenced.
7:     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: jobvl, 2: jobvr, 3: n, 4: a, 5: lda, 6: b, 7: ldb, 8: alpha, 9: beta, 10: vl, 11: ldvl, 12: vr, 13: ldvr, 14: work, 15: lwork, 16: rwork, 17: 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. No eigenvectors have been calculated, but alphaj and betaj should be correct for j=info+1,,n.
   info=n+1
Unexpected error returned from nag_lapack_zhgeqz (f08xs).
   info=n+2
Error returned from nag_lapack_ztgevc (f08yx).

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.

Further Comments

The total number of floating-point operations is proportional to n3.
The real analogue of this function is nag_lapack_dggev (f08wa).

Example

This example finds all the eigenvalues and right eigenvectors of the matrix pair A,B, where
A = -21.10-22.50i 53.50-50.50i -34.50+127.50i 7.50+00.50i -0.46-07.78i -3.50-37.50i -15.50+058.50i -10.50-01.50i 4.30-05.50i 39.70-17.10i -68.50+012.50i -7.50-03.50i 5.50+04.40i 14.40+43.30i -32.50-046.00i -19.00-32.50i  
and
B = 1.00-5.00i 1.60+1.20i -3.00+0.00i 0.00-1.00i 0.80-0.60i 3.00-5.00i -4.00+3.00i -2.40-3.20i 1.00+0.00i 2.40+1.80i -4.00-5.00i 0.00-3.00i 0.00+1.00i -1.80+2.40i 0.00-4.00i 4.00-5.00i .  
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 f08wn_example


fprintf('f08wn example results\n\n');

% Generalized eigenvalues and right eigenvectors of (A, B):
n = 4;
a = [ -21.10 - 22.50i,  53.5 - 50.5i, -34.5 + 127.5i,    7.5 +  0.5i;
       -0.46 -  7.78i,  -3.5 - 37.5i, -15.5 +  58.5i,  -10.5 -  1.5i;
        4.30 -  5.50i,  39.7 - 17.1i, -68.5 +  12.5i,   -7.5 -  3.5i;
        5.50 +  4.40i,  14.4 + 43.3i, -32.5 -  46.00i, -19.0 - 32.5i];
b = [   1    -  5i,      1.6 +  1.2i,  -3   +   0i,      0   -  1i;
        0.8  -  0.6i,    3   -  5i,    -4   +   3i,     -2.4 -  3.2i;
        1    +  0i,      2.4 +  1.8i,  -4   -   5i,      0   -  3i;
        0    +  1i,     -1.8 +  2.4i,   0   -   4i,      4   -  5i];

jobvl = 'No left vectors';
jobvr = 'Vectors (right)';
[~, ~, alpha, beta, ~, VR, info] = ...
  f08wn( ...
	 jobvl, jobvr, a, b);

small = x02am;

[eigs, pos] = sort(alpha./beta);

for j=1:n
  if (abs(alpha(pos(j)))*small >= abs(beta(pos(j))))
    fprintf('\n%4d: Eigenvalue is numerically infinite or undetermined\n',j);
    fprintf('%4d: alpha = (%11.4e,%11.4e), beta = (%11.4e,%11.4e)\n', ...
	    j, real(alpha(j)), imag(alpha(j)), real(beta(j)), imag(beta(j)));
  else
    fprintf('Eigenvalue  (%d):\n', j);
    disp(eigs(j));
  end

  fprintf('Eigenvector (%d):\n', j);
  disp(VR(:, pos(j))/VR(1, pos(j)));
end


f08wn example results

Eigenvalue  (1):
   3.0000 - 1.0000i

Eigenvector (1):
   1.0000 + 0.0000i
   0.1600 - 0.1200i
   0.1200 - 0.1600i
   0.1600 + 0.1200i

Eigenvalue  (2):
   2.0000 - 5.0000i

Eigenvector (2):
   1.0000 - 0.0000i
   0.0046 - 0.0034i
   0.0629 + 0.0000i
  -0.0000 + 0.0629i

Eigenvalue  (3):
   4.0000 - 5.0000i

Eigenvector (3):
   1.0000 + 0.0000i
   0.0089 - 0.0067i
  -0.0333 + 0.0000i
  -0.0000 + 0.1556i

Eigenvalue  (4):
   3.0000 - 9.0000i

Eigenvector (4):
   1.0000 - 0.0000i
   0.1600 - 0.1200i
   0.1200 + 0.1600i
  -0.1600 + 0.1200i


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Chapter Contents
Chapter Introduction
NAG Toolbox

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