in real 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 real generalized Schur form if

S

is block upper triangular with

1

1

and

2

2

diagonal blocks and

T

is upper triangular as returned, for example, by nag_lapack_dgges (f08xa) or nag_lapack_dggesx (f08xb), or nag_lapack_dhgeqz (f08xe) with

job ='S'

. The diagonal elements, or blocks, 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^{T}, B = Q T Z^{T}

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^{T} A x|}^{2} + {|y^{T} 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.

The definition of the reciprocal of the estimated condition number of the right eigenvector

x

and the left eigenvector

y

corresponding to the simple eigenvalue

λ

depends upon whether

λ

is a real eigenvalue, or one of a complex conjugate pair.

If the eigenvalue

λ

is real and

U

and

V

are orthogonal transformations such that

U^{T} (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.

λ

is part of a complex conjugate pair and

U

and

V

are orthogonal transformations such that

U^{T} (A, B) V = (S, T) = (\begin{matrix} S_{11} & * \\ 0 & S_{22} \end{matrix}) (\begin{matrix} T_{11} & * \\ 0 & T_{22} \end{matrix}),

where

S_{11}

and

T_{11}

are two by two matrices,

S_{22}

and

T_{22}

are

(n - 2)

(n - 2)

matrices, and

(S_{11}, T_{11})

corresponds to the complex conjugate eigenvalue pair

λ

\bar{λ}

, then there exist unitary matrices

U_{1}

and

V_{1}

such that

U_{1}^{H} S_{11} V_{1} = (\begin{matrix} s_{11} & s_{12} \\ 0 & s_{22} \end{matrix}) and U_{1}^{H} T_{11} V_{1} = (\begin{matrix} t_{11} & t_{12} \\ 0 & t_{22} \end{matrix}) .

The eigenvalues are given by

λ = s_{11} / t_{11}

and

\bar{λ} = s_{22} / t_{22}

. Then the Frobenius norm-based, estimated reciprocal condition number is bounded by

Dif (x) \equiv Dif (y) \leq \min (d_{1}, \max (1, |Re (s_{11}) / Re (s_{22})|), d_{2})

where

Re (z)

denotes the real part of

z

d_{1} = Dif ((s_{11}, t_{11}), (s_{22}, t_{22})) = σ_{\min} (Z_{1})

Z_{1}

is the complex two by two matrix

Z_{1} = (\begin{matrix} s_{11} & - s_{22} \\ t_{11} & - t_{22} \end{matrix}),

and

d_{2}

is an upper bound on

Dif ((S_{11}, T_{11}), (S_{22}, T_{22}))

; i.e., an upper bound on

σ_{\min} (Z_{2})

, where

Z_{2}

is the

(2 n - 2)

(2 n - 2)

matrix

Z_{2} = (\begin{matrix} S_{11}^{T} \otimes I & - I \otimes S_{22} \\ T_{11}^{T} \otimes I & - I \otimes T_{22} \end{matrix}) .

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'

'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'

'S'

3: $select (:)$ – logical array

The dimension of the array select must be at least

\max (1, n)

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 real eigenvalue

λ_{j}

select (j)

must be set true. To select condition numbers corresponding to a complex conjugate pair of eigenvalues

λ_{j}

and

λ_{j + 1}

select (j)

and/or

select (j + 1)

must be set to true.

howmny ='A'

, select is not referenced.

4: $a (lda :)$ – double array

The first dimension of the array a must be at least

\max (1, n)

The second dimension of the array a must be at least

\max (1, n)

The upper quasi-triangular matrix

S

5: $b (ldb :)$ – double array

The first dimension of the array b must be at least

\max (1, n)

The second dimension of the array b must be at least

\max (1, n)

The upper triangular matrix

T

6: $vl (ldvl :)$ – double array

The first dimension,

ldvl

, of the array vl must satisfy

if $job ='E'$ or $'B'$ , $ldvl \geq \max (1, n)$ ;
otherwise $ldvl \geq 1$ .

The second dimension of the array vl must be at least

\max (1, mm)

job ='E'

'B'

, and at least

1

otherwise.

job ='E'

'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_dggev (f08wa) or nag_lapack_dtgevc (f08yk).

job ='V'

, vl is not referenced.

7: $vr (ldvr :)$ – double array

The first dimension,

ldvr

, of the array vr must satisfy

if $job ='E'$ or $'B'$ , $ldvr \geq \max (1, n)$ ;
otherwise $ldvr \geq 1$ .

The second dimension of the array vr must be at least

\max (1, mm)

job ='E'

'B'

, and at least

1

otherwise.

job ='E'

'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_dggev (f08wa) or nag_lapack_dtgevc (f08yk).

job ='V'

, vr is not referenced.

8: $mm$ – int64int32nag_int scalar

The number of elements in the arrays s and dif.

Constraints:

if $howmny ='A'$ , $mm \geq n$ ;
otherwise $mm \geq m$ .

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: $n \geq 0$ .

Output Parameters

1: $s (:)$ – double array: The dimension of the array s will be $\max (1, mm)$ 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. For a complex conjugate pair of eigenvalues two consecutive elements of s are set to the same value. Thus $s (j)$ , $dif (j)$ , and the $j$ th columns of $VL$ and $VR$ all correspond to the same eigenpair (but not in general the $j$ th eigenpair, unless all eigenpairs are selected).
If $job ='V'$ , s is not referenced.
2: $dif (:)$ – double array: The dimension of the array dif will be $\max (1, mm)$ 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. For a complex eigenvector two consecutive elements of dif are set to the same value. If the eigenvalues cannot be reordered to compute $dif (j)$ , $dif (j)$ 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 real eigenvalue one element is used, and for each selected complex conjugate pair of eigenvalues, two elements are 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

\tilde{λ}

and the corresponding exact eigenvalue

λ

χ (\tilde{λ}, λ) \leq ε {‖(A, B)‖}_{F} / S (λ)

where

ε

is the machine precision.

An approximate asymptotic error bound for the right or left computed eigenvectors

\tilde{x}

\tilde{y}

corresponding to the right and left eigenvectors

x

and

y

is given by

θ (\tilde{z}, z) \leq ε {‖(A, B)‖}_{F} / Dif .

The complex analogue of this function is nag_lapack_ztgsna (f08yy).

Example

This example estimates condition numbers and approximate error estimates for all the eigenvalues and eigenvalues and right eigenvectors of the pair

(S, T)

given by

S = (\begin{array}{r} 4.0 & 1.0 & 1.0 & 2.0 \\ 0 & 3.0 & - 1.0 & 1.0 \\ 0 & 1.0 & 3.0 & 1.0 \\ 0 & 0 & 0 & 6.0 \end{array}) and T = (\begin{array}{r} 2.0 & 1.0 & 1.0 & 3.0 \\ 0 & 1.0 & 0.0 & 1.0 \\ 0 & 0 & 1.0 & 1.0 \\ 0 & 0 & 0 & 2.0 \end{array}) .

The eigenvalues and eigenvectors are computed by calling nag_lapack_dtgevc (f08yk).

Open in the MATLAB editor: f08yl_example

function f08yl_example


fprintf('f08yl example results\n\n');

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

% Obtain scaled eigenvectors from Schur form
job = 'Both';
howmny = 'All';
select = [false];
Q = eye(n);
Z = Q;
[VL, VR, m, info] = f08yk( ...
                           job, howmny, select, S, T, Q, Z, n);

% Estimate condition numbers for eigenvalues and right eigenvectors
[rconde, rcondv, m, info] = ...
  f08yl ( ...
          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');

f08yl example results

Reciprocal condition numbers for eigenvalues of (S,T)
    1.6e+00    1.7e+00    1.7e+00    1.4e+00

Reciprocal condition numbers for right eigenvectors of (S,T)
    5.4e-01    1.5e-01    1.5e-01    1.2e-01

Approximate error estimates for eigenvalues of (S,T)
    8.7e-16    7.8e-16    7.8e-16    9.9e-16

Approximate error estimates for right eigenvectors of (S,T)
    2.5e-15    9.0e-15    9.0e-15    1.1e-14

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox