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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dgees (f08pa)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_lapack_dgees (f08pa) computes the eigenvalues, the real Schur form T, and, optionally, the matrix of Schur vectors Z for an n by n real nonsymmetric matrix A.


[a, sdim, wr, wi, vs, info] = f08pa(jobvs, sort, select, a, 'n', n)
[a, sdim, wr, wi, vs, info] = nag_lapack_dgees(jobvs, sort, select, a, 'n', n)


The real Schur factorization of A is given by
A = Z T ZT ,  
where Z, the matrix of Schur vectors, is orthogonal and T is the real Schur form. A matrix is in real Schur form if it is upper quasi-triangular with 1 by 1 and 2 by 2 blocks. 2 by 2 blocks will be standardized in the form
a b c a  
where bc<0. The eigenvalues of such a block are a±bc.
Optionally, nag_lapack_dgees (f08pa) also orders the eigenvalues on the diagonal of the real Schur form so that selected eigenvalues are at the top left. The leading columns of Z form an orthonormal basis for the invariant subspace corresponding to the selected eigenvalues.


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


Compulsory Input Parameters

1:     jobvs – string (length ≥ 1)
If jobvs='N', Schur vectors are not computed.
If jobvs='V', Schur vectors are computed.
Constraint: jobvs='N' or 'V'.
2:     sort – string (length ≥ 1)
Specifies whether or not to order the eigenvalues on the diagonal of the Schur form.
Eigenvalues are not ordered.
Eigenvalues are ordered (see select).
Constraint: sort='N' or 'S'.
3:     select – function handle or string containing name of m-file
If sort='S', select is used to select eigenvalues to sort to the top left of the Schur form.
If sort='N', select is not referenced and nag_lapack_dgees (f08pa) may be called with the string 'f08paz'.
An eigenvalue wrj+-1×wij is selected if selectwrj,wij is true. If either one of a complex conjugate pair of eigenvalues is selected, then both are. Note that a selected complex eigenvalue may no longer satisfy selectwrj,wij=true after ordering, since ordering may change the value of complex eigenvalues (especially if the eigenvalue is ill-conditioned); in this case info is set to n+2 (see info below).
[result] = select(wr, wi)

Input Parameters

1:     wr – double scalar
2:     wi – double scalar
The real and imaginary parts of the eigenvalue.

Output Parameters

1:     result – logical scalar
result=true for selected eigenvalues.
4:     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 n by n matrix A.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the array a and the second dimension of the array a. (An error is raised if these dimensions are not equal.)
n, the order of the matrix A.
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 real Schur form T.
2:     sdim int64int32nag_int scalar
If sort='N', sdim=0.
If sort='S', sdim= number of eigenvalues (after sorting) for which select is true. (Complex conjugate pairs for which select is true for either eigenvalue count as 2.)
3:     wr: – double array
The dimension of the array wr will be max1,n
See the description of wi.
4:     wi: – double array
The dimension of the array wi will be max1,n
wr and wi contain the real and imaginary parts, respectively, of the computed eigenvalues in the same order that they appear on the diagonal of the output Schur form T. Complex conjugate pairs of eigenvalues will appear consecutively with the eigenvalue having the positive imaginary part first.
5:     vsldvs: – double array
The first dimension, ldvs, of the array vs will be
  • if jobvs='V', ldvs= max1,n ;
  • otherwise ldvs=1.
The second dimension of the array vs will be max1,n if jobvs='V' and 1 otherwise.
If jobvs='V', vs contains the orthogonal matrix Z of Schur vectors.
If jobvs='N', vs is not referenced.
6:     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.

If info=-i, parameter i had an illegal value on entry. The parameters are numbered as follows:
1: jobvs, 2: sort, 3: select, 4: n, 5: a, 6: lda, 7: sdim, 8: wr, 9: wi, 10: vs, 11: ldvs, 12: work, 13: lwork, 14: bwork, 15: 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.
If info=i and in, the QR algorithm failed to compute all the eigenvalues.
W  info=n+1
The eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned).
W  info=n+2
After reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Schur form no longer satisfy select=true. This could also be caused by underflow due to scaling.


The computed Schur factorization satisfies
A+E=ZT ZT ,  
E2 = Oε A2 ,  
and ε is the machine precision. See Section 4.8 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_zgees (f08pn).


This example finds the Schur factorization of the matrix
A = 0.35 0.45 -0.14 -0.17 0.09 0.07 -0.54 0.35 -0.44 -0.33 -0.03 0.17 0.25 -0.32 -0.13 0.11 ,  
such that the real positive eigenvalues of A are the top left diagonal elements of the Schur form, T.
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 f08pa_example

fprintf('f08pa example results\n\n');

% Matrix A
a = [ 0.35,  0.45, -0.14, -0.17;
      0.09,  0.07, -0.54,  0.35;
     -0.44, -0.33, -0.03,  0.17;
      0.25, -0.32, -0.13,  0.11];

% Schur vectors of A, selecting real positive eigenvalues
jobvs = 'Vectors (Schur)';
sortp = 'Sort';
select = @(wr, wi) (wr > 0 && wi == 0);
[~, sdim, wr, wi, vs, info] = ...
f08pa( ...
       jobvs, sortp, select, a);

fprintf('Number of eigenvalues for which SELECT is true = %3d\n',sdim);
fprintf(' (dimension of invariant subspace)\n\n');
disp('Selected eigenvalues');
disp(wr(1:sdim) + i*wi(1:sdim));

f08pa example results

Number of eigenvalues for which SELECT is true =   1
 (dimension of invariant subspace)

Selected eigenvalues

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Chapter Introduction
NAG Toolbox

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