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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_ztgsna (f08yy)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_ztgsna (f08yy) estimates condition numbers for specified eigenvalues and/or eigenvectors of a complex matrix pair in generalized Schur form.

Syntax

[s, dif, m, info] = f08yy(job, howmny, select, a, b, vl, vr, mm, 'n', n)
[s, dif, m, info] = nag_lapack_ztgsna(job, howmny, select, a, b, vl, vr, mm, 'n', n)

Description

nag_lapack_ztgsna (f08yy) estimates condition numbers for specified eigenvalues and/or right eigenvectors of an n by n matrix pair S,T in generalized Schur form. The function actually returns estimates of the reciprocals of the condition numbers in order to avoid possible overflow.
The pair S,T are in generalized Schur form if S and T are upper triangular as returned, for example, by nag_lapack_zgges (f08xn) or nag_lapack_zggesx (f08xp), or nag_lapack_zhgeqz (f08xs) with job='S'. The diagonal elements define the generalized eigenvalues αi,βi, for i=1,2,,n, of the pair S,T and the eigenvalues are given by
λi = αi / βi ,  
so that
βi S xi = αi T xi   or   S xi = λi T xi ,  
where xi is the corresponding (right) eigenvector.
If S and T are the result of a generalized Schur factorization of a matrix pair A,B 
A = QSZH ,   B = QTZH  
then the eigenvalues and condition numbers of the pair S,T are the same as those of the pair A,B.
Let α,β0,0 be a simple generalized eigenvalue of A,B. Then the reciprocal of the condition number of the eigenvalue λ=α/β is defined as
sλ= yHAx 2 + yHBx 2 1/2 x2 y2 ,  
where x and y are the right and left eigenvectors of A,B corresponding to λ. If both α and β are zero, then A,B is singular and sλ=-1 is returned.
If U and V are unitary transformations such that
UH A,B V= S,T = α * 0 S22 β * 0 T22 ,  
where S22 and T22 are n-1 by n-1 matrices, then the reciprocal condition number is given by
Difx Dify = Difα,β,S22,T22 = σmin Z ,  
where σminZ denotes the smallest singular value of the 2n-1 by 2n-1 matrix
Z = αI -1S22 βI -1T22  
and  is the Kronecker product.
See Sections 2.4.8 and 4.11 of Anderson et al. (1999) and Kågström and Poromaa (1996) for further details and 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
Kågström B and Poromaa P (1996) LAPACK-style algorithms and software for solving the generalized Sylvester equation and estimating the separation between regular matrix pairs ACM Trans. Math. Software 22 78–103

Parameters

Compulsory Input Parameters

1:     job – string (length ≥ 1)
Indicates whether condition numbers are required for eigenvalues and/or eigenvectors.
job='E'
Condition numbers for eigenvalues only are computed.
job='V'
Condition numbers for eigenvectors only are computed.
job='B'
Condition numbers for both eigenvalues and eigenvectors are computed.
Constraint: job='E', 'V' or 'B'.
2:     howmny – string (length ≥ 1)
Indicates how many condition numbers are to be computed.
howmny='A'
Condition numbers for all eigenpairs are computed.
howmny='S'
Condition numbers for selected eigenpairs (as specified by select) are computed.
Constraint: howmny='A' or 'S'.
3:     select: – logical array
The dimension of the array select must be at least max1,n if howmny='S', and at least 1 otherwise
Specifies the eigenpairs for which condition numbers are to be computed if howmny='S'. To select condition numbers for the eigenpair corresponding to the eigenvalue λj, selectj must be set to true.
If howmny='A', select is not referenced.
4:     alda: – complex array
The first dimension of the array a must be at least n.
The second dimension of the array a must be at least n.
The upper triangular matrix S.
5:     bldb: – complex array
The first dimension of the array b must be at least n.
The second dimension of the array b must be at least n.
The upper triangular matrix T.
6:     vlldvl: – complex array
The first dimension, ldvl, of the array vl must satisfy
  • if job='E' or 'B', ldvl max1,n ;
  • otherwise ldvl1.
The second dimension of the array vl must be at least max1,mm if job='E' or 'B', and at least 1 otherwise.
If job='E' or 'B', vl must contain left eigenvectors of S,T, corresponding to the eigenpairs specified by howmny and select. The eigenvectors must be stored in consecutive columns of vl, as returned by nag_lapack_zggev (f08wn) or nag_lapack_ztgevc (f08yx).
If job='V', vl is not referenced.
7:     vrldvr: – complex array
The first dimension, ldvr, of the array vr must satisfy
  • if job='E' or 'B', ldvr max1,n ;
  • otherwise ldvr1.
The second dimension of the array vr must be at least max1,mm if job='E' or 'B', and at least 1 otherwise.
If job='E' or 'B', vr must contain right eigenvectors of S,T, corresponding to the eigenpairs specified by howmny and select. The eigenvectors must be stored in consecutive columns of vr, as returned by nag_lapack_zggev (f08wn) or nag_lapack_ztgevc (f08yx).
If job='V', vr is not referenced.
8:     mm int64int32nag_int scalar
The number of elements in the arrays s and dif.
Constraints:
  • if howmny='A', mmn;
  • otherwise mm​ the number of selected eigenvalues.

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 matrix pair S,T.
Constraint: n0.

Output Parameters

1:     s: – double array
The dimension of the array s will be max1,mm if job='E' or 'B' and 1 otherwise
If job='E' or 'B', the reciprocal condition numbers of the selected eigenvalues, stored in consecutive elements of the array.
If job='V', s is not referenced.
2:     dif: – double array
The dimension of the array dif will be max1,mm if job='V' or 'B' and 1 otherwise
If job='V' or 'B', the estimated reciprocal condition numbers of the selected eigenvectors, stored in consecutive elements of the array. If the eigenvalues cannot be reordered to compute difj, difj is set to 0; this can only occur when the true value would be very small anyway.
If job='E', dif is not referenced.
3:     m int64int32nag_int scalar
The number of elements of the arrays s and dif used to store the specified condition numbers; for each selected eigenvalue one element is used.
If howmny='A', m is set to n.
4:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   info=-i
If info=-i, parameter i had an illegal value on entry. The parameters are numbered as follows:
1: job, 2: howmny, 3: select, 4: n, 5: a, 6: lda, 7: b, 8: ldb, 9: vl, 10: ldvl, 11: vr, 12: ldvr, 13: s, 14: dif, 15: mm, 16: m, 17: work, 18: lwork, 19: iwork, 20: 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.

Accuracy

None.

Further Comments

An approximate asymptotic error bound on the chordal distance between the computed eigenvalue λ~ and the corresponding exact eigenvalue λ is
χλ~,λ εA,BF / Sλ  
where ε is the machine precision.
An approximate asymptotic error bound for the right or left computed eigenvectors x~ or y~ corresponding to the right and left eigenvectors x and y is given by
θz~,z ε A,BF / Dif .  
The real analogue of this function is nag_lapack_dtgsna (f08yl).

Example

This example estimates condition numbers and approximate error estimates for all the eigenvalues and right eigenvectors of the pair S,T given by
S = 4.0+4.0i 1.0+1.0i 1.0+1.0i 2.0-1.0i 0.0i+0.0 2.0+1.0i 1.0+1.0i 1.0+1.0i 0.0i+0.0 0.0i+0.0 2.0-1.0i 1.0+1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 6.0-2.0i  
and
T = 2.0 1.0+1.0i 1.0+1.0i 3.0-1.0i 0.0 1.0i+0.0 2.0+1.0i 1.0+1.0i 0.0 0.0i+0.0 1.0i+0.0 1.0+1.0i 0.0 0.0i+0.0 0.0i+0.0 2.0i+0.0 .  
The eigenvalues and eigenvectors are computed by calling nag_lapack_ztgevc (f08yx).
function f08yy_example


fprintf('f08yy example results\n\n');

% Generalized Schur form pair (S,T)
n = int64(4);
S = [ 4 + 4i,  1 + 1i,  1 + 1i,  2 - 1i;
      0 + 0i,  2 + 1i,  1 + 1i,  1 + 1i;
      0 + 0i,  0 + 0i,  2 - 1i,  1 + 1i;
      0 + 0i,  0 + 0i,  0 + 0i,  6 - 2i];
T = [2,  1 + 1i,  1 + 1i,  3 - 1i;
      0 + 0i,  1 + 0i,  2 + 1i,  1 + 1i;
      0 + 0i,  0 + 0i,  1 + 0i,  1 + 1i;
      0 + 0i,  0 + 0i,  0 + 0i,  2 + 0i];

% Obtain scaled eigenvectors from Schur form
job    = 'Both';
howmny = 'All';
select = [false];
Q = complex(eye(n));
Z = Q;

[VL, VR, m, info] = f08yx( ...
                           job, howmny, select, S, T, Q, Z, n);

% Estimate condition numbers for eigenvalues and right eigenvectors
[rconde, rcondv, m, info] = ...
  f08yy( ...
         job, howmny, select, S, T, VL, VR, n);

disp('Reciprocal condition numbers for eigenvalues of (S,T)');
fprintf('%11.1e',rconde);
fprintf('\n\n');
disp('Reciprocal condition numbers for right eigenvectors of (S,T)');
fprintf('%11.1e',rcondv);
fprintf('\n\n');

% Calculate approximate error estimates
        
snorm = norm(S,1);
tnorm = norm(T,1);
stnorm = sqrt(snorm^2 + tnorm^2);

disp('Approximate error estimates for eigenvalues of (S,T)')
erre = x02aj*stnorm./rconde;
fprintf('%11.1e',erre);
fprintf('\n\n');
disp('Approximate error estimates for right eigenvectors of (S,T)')
errv = x02aj*stnorm./rcondv;
fprintf('%11.1e',errv);
fprintf('\n');


f08yy example results

Reciprocal condition numbers for eigenvalues of (S,T)
    1.0e+00    8.2e-01    7.2e-01    8.2e-01

Reciprocal condition numbers for right eigenvectors of (S,T)
    3.2e-01    3.6e-01    5.5e-01    2.8e-01

Approximate error estimates for eigenvalues of (S,T)
    1.5e-15    1.9e-15    2.1e-15    1.9e-15

Approximate error estimates for right eigenvectors of (S,T)
    4.8e-15    4.3e-15    2.8e-15    5.5e-15

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