f08yy:: Least Squares and Eigenvalue Problems (LAPACK) (NAG Toolbox)

Description

nag_lapack_ztgsna (f08yy) estimates condition numbers for specified eigenvalues and/or right eigenvectors of an

n

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

(α_{i}, β_{i})

, for

i = 1, 2, \dots, n

, of the pair

(S, T)

and the eigenvalues are given by

λ_{i} = α_{i} / β_{i},

so that

β_{i} S x_{i} = α_{i} T x_{i} or S x_{i} = λ_{i} T x_{i},

where

x_{i}

is the corresponding (right) eigenvector.

S

and

T

are the result of a generalized Schur factorization of a matrix pair

(A, B)

A = Q S Z^{H}, B = Q T Z^{H}

then the eigenvalues and condition numbers of the pair

(S, T)

are the same as those of the pair

(A, B)

Let

(α, β) \neq (0, 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

s (λ) = \frac{{({|y^{H} A x|}^{2} + {|y^{H} B x|}^{2})}^{1 / 2}}{({‖x‖}_{2} {‖y‖}_{2})},

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

s (λ) = - 1

is returned.

U

and

V

are unitary transformations such that

U^{H} (A, B) V = (S, T) = (\begin{matrix} α & * \\ 0 & S_{22} \end{matrix}) (\begin{matrix} β & * \\ 0 & T_{22} \end{matrix}),

where

S_{22}

and

T_{22}

are

(n - 1)

(n - 1)

matrices, then the reciprocal condition number is given by

Dif (x) \equiv Dif (y) = Dif ((α, β), (S_{22}, T_{22})) = σ_{\min} (Z),

where

σ_{\min} (Z)

denotes the smallest singular value of the

2 (n - 1)

2 (n - 1)

matrix

Z = (\begin{matrix} α \otimes I & - 1 \otimes S_{22} \\ β \otimes I & - 1 \otimes T_{22} \end{matrix})

and

\otimes

is the Kronecker product.

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

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 = (\begin{array}{r} 4.0 + 4.0 i & 1.0 + 1.0 i & 1.0 + 1.0 i & 2.0 - 1.0 i \\ 0 & 2.0 + 1.0 i & 1.0 + 1.0 i & 1.0 + 1.0 i \\ 0 & 0 & 2.0 - 1.0 i & 1.0 + 1.0 i \\ 0 & 0 & 0 & 6.0 - 2.0 i \end{array})

and

T = (\begin{array}{r} 2.0 & 1.0 + 1.0 i & 1.0 + 1.0 i & 3.0 - 1.0 i \\ 0 & 1.0 & 2.0 + 1.0 i & 1.0 + 1.0 i \\ 0 & 0 & 1.0 & 1.0 + 1.0 i \\ 0 & 0 & 0 & 2.0 \end{array}) .

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

NAG Toolbox: nag_lapack_ztgsna (f08yy)

▸▿ Contents

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

Example