void	nag_real_eigensystem_sel (Nag_Select_Eigenvalues crit, Integer n, double a[], Integer tda, double wl, double wu, Integer mest, Integer m, Complex w[], Complex v[], Integer tdv, NagError fail)

3

Description

nag_real_eigensystem_sel (f02ecc) computes selected eigenvalues and the corresponding right eigenvectors of a real general matrix

A

{A x}_{i} = λ_{i} x_{i} .

Eigenvalues

λ_{i}

may be selected either by modulus, satisfying:

w_{l} \leq |λ_{i}| \leq w_{u},

or by real part, satisfying:

w_{l} \leq Re (λ_{i}) \leq w_{u} .

Note that even though

A

is real,

λ_{i}

and

x_{i}

may be complex. If

x_{i}

is an eigenvector corresponding to a complex eigenvalue

λ_{i}

, then the complex conjugate vector

{\bar{x}}_{i}

is the eigenvector corresponding to the complex conjugate eigenvalue

{\bar{λ}}_{i}

. The eigenvalues in a complex conjugate pair

λ_{i}

and

{\bar{λ}}_{i}

are either both selected or both not selected.

4

References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5

Arguments

1: $crit$ – Nag_Select_EigenvaluesInput

On entry: indicates the criterion for selecting eigenvalues:

if $crit = Nag_Select_Modulus$ , then eigenvalues are selected according to their moduli: $w_{l} \leq |λ_{i}| \leq w_{u}$ .
if $crit = Nag_Select_RealPart$ , then eigenvalues are selected according to their real parts: $w_{l} \leq Re (λ_{i}) \leq w_{u}$ .

Constraint:

crit = Nag_Select_Modulus

Nag_Select_RealPart

2: $n$ – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

3: $a [n \times tda]$ – doubleInput/Output

Note: the

(i, j)

th element of the matrix

A

is stored in

a [(i - 1) \times tda + j - 1]

On entry: the

n

n

general matrix

A

On exit: a contains the Hessenberg form of the balanced input matrix

A^{'}

(see Section 9).

4: $tda$ – IntegerInput

On entry: the stride separating matrix column elements in the array a.

Constraint:

tda \geq \max (1, n)

5: $wl$ – doubleInput

6: $wu$ – doubleInput

On entry:

w_{l}

and

w_{u}

, the lower and upper bounds on the criterion for the selected eigenvalues.

Constraint:

wu > wl

7: $mest$ – IntegerInput

On entry: mest must be an upper bound on

m

, the number of eigenvalues and eigenvectors selected. No eigenvectors are computed if

mest < m

Constraint:

mest \geq \max (1, m)

8: $m$ – Integer *Output

On exit:

m

, the number of eigenvalues actually selected.

9: $w [\max (1, n)]$ – ComplexOutput

On exit: the first m elements of w hold the values of the selected eigenvalues; elements from the index m to

n - 1

contain the other eigenvalues. Complex conjugate pairs of eigenvalues are stored in consecutive elements of the array, with the eigenvalue having the positive imaginary part first.

10: $v [n \times tdv]$ – ComplexOutput

Note: the

(i, j)

th element of the matrix

V

is stored in

v [(i - 1) \times tdv + j - 1]

On exit: v contains the selected eigenvectors, with the

i

th column holding the eigenvector associated with the eigenvalue

λ_{i}

(stored in

w [i - 1]

11: $tdv$ – IntegerInput

On entry: the stride separating matrix column elements in the array v.

Constraint:

tdv \geq mest

12: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_2_REAL_ARG_LE: On entry, $wu = 〈value〉$ while $wl = 〈value〉$ . These arguments must satisfy $wu > wl$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument crit had an illegal value.
NE_EIGVEC: Inverse iteration failed to compute all the specified eigenvectors. If an eigenvector failed to converge, the corresponding column of v is set to zero.
NE_INT_2: On entry, $tda = 〈value〉$ while $n = 〈value〉$ .
Constraint: $tda \geq \max (1, n)$ .

On entry, $tdv = 〈value〉$ while $mest = 〈value〉$ .
Constraint: $tdv \geq \max (1, mest)$ .
NE_INT_ARG_LT: On entry, $mest = 〈value〉$ .
Constraint: $mest \geq 1$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .
NE_QR_FAIL: The QR algorithm failed to compute all the eigenvalues. No eigenvectors have been computed.
NE_REQD_EIGVAL: There are more than mest eigenvalues in the specified range. The actual number of eigenvalues in the range is returned in m. No eigenvectors have been computed.
Rerun with the second dimension of $v = mest \geq m$ .

7

Accuracy

λ_{i}

is an exact eigenvalue, and

{\tilde{λ}}_{i}

is the corresponding computed value, then

|{\tilde{λ}}_{i} - λ_{i}| \leq \frac{c (n) ε {‖A^{'}‖}_{2}}{s_{i}},

where

c (n)

is a modestly increasing function of

n

ε

is the machine precision, and

s_{i}

is the reciprocal condition number of

λ_{i}

;

A^{'}

is the balanced form of the original matrix

A

, and

‖A^{'}‖ \leq ‖A‖

x_{i}

is the corresponding exact eigenvector, and

{\tilde{x}}_{i}

is the corresponding computed eigenvector, then the angle

θ ({\tilde{x}}_{i}, x_{i})

between them is bounded as follows:

θ ({\tilde{x}}_{i}, x_{i}) \leq \frac{c (n) ε {‖A^{'}‖}_{2}}{{s e p}_{i}}

where

{s e p}_{i}

is the reciprocal condition number of

x_{i}

8

Parallelism and Performance

nag_real_eigensystem_sel (f02ecc) is not threaded in any implementation.

9

Further Comments

nag_real_eigensystem_sel (f02ecc) first balances the matrix, using a diagonal similarity transformation to reduce its norm; and then reduces the balanced matrix

A^{'}

to upper Hessenberg form

H

, using an orthogonal similarity transformation:

A^{'} = {Q H Q}^{T}

. The function uses the Hessenberg

Q R

algorithm to compute all the eigenvalues of

H

, which are the same as the eigenvalues of

A

. It computes the eigenvectors of

H

which correspond to the selected eigenvalues, using inverse iteration. It premultiplies the eigenvectors by

Q

to form the eigenvectors of

A^{'}

; and finally transforms the eigenvectors to those of the original matrix

A

Each eigenvector

x

(real or complex) is normalized so that

{‖x‖}_{2} = 1

, and the element of largest absolute value is real and positive.

The inverse iteration function may make a small perturbation to the real parts of close eigenvalues, and this may shift their moduli just outside the specified bounds. If you are relying on eigenvalues being within the bounds, you should test them on return from nag_real_eigensystem_sel (f02ecc).

The time taken by the function is approximately proportional to

n^{3}

The function can be used to compute all eigenvalues and eigenvectors, by setting wl large and negative, and wu large and positive.

10

Example

To compute those eigenvalues of the matrix

A

whose moduli lie in the range

[0.2, 0.5]

, and their corresponding eigenvectors, where

A = (\begin{matrix} 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 \end{matrix})

NAG Library Function Document

nag_real_eigensystem_sel (f02ecc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results