hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zgees (f08pn)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zgees (f08pn) computes the eigenvalues, the Schur form T, and, optionally, the matrix of Schur vectors Z for an n by n complex nonsymmetric matrix A.

Syntax

[a, sdim, w, vs, info] = f08pn(jobvs, sort, select, a, 'n', n)
[a, sdim, w, vs, info] = nag_lapack_zgees(jobvs, sort, select, a, 'n', n)

Description

The Schur factorization of A is given by
A = Z T ZH ,  
where Z, the matrix of Schur vectors, is unitary and T is the Schur form. A complex matrix is in Schur form if it is upper triangular.
Optionally, nag_lapack_zgees (f08pn) also orders the eigenvalues on the diagonal of the 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.

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:     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.
sort='N'
Eigenvalues are not ordered.
sort='S'
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_zgees (f08pn) may be called with the string 'f08pnz'.
An eigenvalue wj is selected if selectwj is true.
[result] = select(w)

Input Parameters

1:     w – complex scalar
The real and imaginary parts of the eigenvalue.

Output Parameters

1:     result – logical scalar
result=true for selected eigenvalues.
4:     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 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: – complex 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 Schur form T.
2:     sdim int64int32nag_int scalar
If sort='N', sdim=0.
If sort='S', sdim= number of eigenvalues for which select is true.
3:     w: – complex array
The dimension of the array w will be max1,n
Contains the computed eigenvalues, in the same order that they appear on the diagonal of the output Schur form T.
4:     vsldvs: – complex 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 unitary matrix Z of Schur vectors.
If jobvs='N', vs is not referenced.
5:     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: jobvs, 2: sort, 3: select, 4: n, 5: a, 6: lda, 7: sdim, 8: w, 9: vs, 10: ldvs, 11: work, 12: lwork, 13: rwork, 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.
   info=1ton
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.

Accuracy

The computed Schur factorization satisfies
A+E=ZT ZH ,  
where
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 real analogue of this function is nag_lapack_dgees (f08pa).

Example

This example finds the Schur factorization of the matrix
A = -3.97-5.04i -4.11+3.70i -0.34+1.01i 1.29-0.86i 0.34-1.50i 1.52-0.43i 1.88-5.38i 3.36+0.65i 3.31-3.85i 2.50+3.45i 0.88-1.08i 0.64-1.48i -1.10+0.82i 1.81-1.59i 3.25+1.33i 1.57-3.44i .  
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 f08pn_example


fprintf('f08pn example results\n\n');

% Complex matrix A
a = [-3.97 - 5.04i, -4.11 + 3.70i, -0.34 + 1.01i,  1.29 - 0.86i;
      0.34 - 1.50i,  1.52 - 0.43i,  1.88 - 5.38i,  3.36 + 0.65i;
      3.31 - 3.85i,  2.50 + 3.45i,  0.88 - 1.08i,  0.64 - 1.48i;
     -1.10 + 0.82i,  1.81 - 1.59i,  3.25 + 1.33i,  1.57 - 3.44i];

% Schur vectors of A, selecting all eigenvalues
jobvs = 'Vectors (Schur)';
sortp = 'No sort';
[~, sdim, w, ~, info] = f08pn( ...
                               jobvs, sortp, @select, a);

disp('Eigenvalues');
disp(w);


f08pn example results

Eigenvalues
  -6.0004 - 6.9998i
  -5.0000 + 2.0060i
   7.9982 - 0.9964i
   3.0023 - 3.9998i


PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2015