Integer, Intent (In)	::	n, lda, mest, ldvr, ldvi, lwork
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	m, iwork(n)
Real (Kind=nag_wp), Intent (In)	::	wl, wu
Real (Kind=nag_wp), Intent (Inout)	::	a(lda,n), vr(ldvr,mest), vi(ldvi,mest)
Real (Kind=nag_wp), Intent (Out)	::	wr(n), wi(n), work(lwork)
Logical, Intent (Out)	::	bwork(n)
Character (1), Intent (In)	::	crit

C Header Interface

#include <nag.h>

void

f02ecf_ (const char *crit, const Integer *n, double a[], const Integer *lda, const double *wl, const double *wu, const Integer *mest, Integer *m, double wr[], double wi[], double vr[], const Integer *ldvr, double vi[], const Integer *ldvi, double work[], const Integer *lwork, Integer iwork[], logical bwork[], Integer *ifail, const Charlen length_crit)

The routine may be called by the names f02ecf or nagf_eigen_real_gen_eigsys.

3 Description

f02ecf 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: $crit$ – Character(1) Input

On entry: indicates the criterion for selecting eigenvalues.

$crit ='M'$: Eigenvalues are selected according to their moduli: $w_{l} \leq | λ_{i} | \leq w_{u}$ .
$crit ='R'$: Eigenvalues are selected according to their real parts: $w_{l} \leq Re (λ_{i}) \leq w_{u}$ .

Constraint:

crit ='M'

'R'

2: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

3: $a (lda, n)$ – Real (Kind=nag_wp) array Input/Output

On entry: the

n \times n

general matrix

A

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

A^{'}

(see Section 9).

4: $lda$ – Integer Input

On entry: the first dimension of the array a as declared in the (sub)program from which f02ecf is called.

Constraint:

lda \geq \max (1, n)

5: $wl$ – Real (Kind=nag_wp) Input

6: $wu$ – Real (Kind=nag_wp) Input

On entry:

w_{l}

and

w_{u}

, the lower and upper bounds on the criterion for the selected eigenvalues (see crit).

Constraint:

wu > wl

7: $mest$ – Integer Input

On entry: the second dimension of the arrays vr and vi as declared in the (sub)program from which f02ecf is called. 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: $wr (n)$ – Real (Kind=nag_wp) array Output

10: $wi (n)$ – Real (Kind=nag_wp) array Output

On exit: the first m elements of wr and wi hold the real and imaginary parts, respectively, of the selected eigenvalues; elements

m + 1

to n contain the other eigenvalues. Complex conjugate pairs of eigenvalues are stored in consecutive elements of the arrays, with the eigenvalue having positive imaginary part first. See also Section 9.

11: $vr (ldvr, mest)$ – Real (Kind=nag_wp) array Output

On exit: contains the real parts of the selected eigenvectors, with the

i

th column holding the real part of the eigenvector associated with the eigenvalue

λ_{i}

(stored in

wr (i)

and

wi (i)

12: $ldvr$ – Integer Input

On entry: the first dimension of the array vr as declared in the (sub)program from which f02ecf is called.

Constraint:

ldvr \geq \max (1, n)

13: $vi (ldvi, mest)$ – Real (Kind=nag_wp) array Output

On exit: contains the imaginary parts of the selected eigenvectors, with the

i

th column holding the imaginary part of the eigenvector associated with the eigenvalue

λ_{i}

(stored in

wr (i)

and

wi (i)

14: $ldvi$ – Integer Input

On entry: the first dimension of the array vi as declared in the (sub)program from which f02ecf is called.

Constraint:

ldvi \geq \max (1, n)

15: $work (lwork)$ – Real (Kind=nag_wp) array Workspace

16: $lwork$ – Integer Input

On entry: the dimension of the array work as declared in the (sub)program from which f02ecf is called.

Constraint:

lwork \geq \max (1, n \times (n + 4))

17: $iwork (n)$ – Integer array Workspace

18: $bwork (n)$ – Logical array Workspace

19: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $crit = ⟨ value ⟩$ .
Constraint: $crit ='M'$ or $'R'$ .

On entry, $lda = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $lda \geq \max (1, n)$ .

On entry, $ldvi = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $ldvi \geq \max (1, n)$ .

On entry, $ldvr = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $ldvr \geq \max (1, n)$ .

On entry, $lwork = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $lwork \geq \max (1, n \times (n + 4))$ .

On entry, $mest = ⟨ value ⟩$ .
Constraint: $mest \geq 1$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

On entry, $wu = ⟨ value ⟩$ and $wl = ⟨ value ⟩$ .
Constraint: $wu > wl$ .

$ifail = 2$: The $Q R$ algorithm failed to converge: only $⟨ value ⟩$ eigenvalues have been computed; no eigenvectors have been computed.

$ifail = 3$: There are more than mest eigenvalues in the specified range. m (number of eigenvalues in range) $= ⟨ value ⟩$ and $mest = ⟨ value ⟩$ . No eigenvectors have been computed. Rerun with second dimension of vr and $vi = mest \geq m$ .

$ifail = 4$: Inverse iteration failed to compute all the specified eigenvectors. The number of eigenvectors which failed to converge is $⟨ value ⟩$ . The corresponding columns of vr and vi are set to zero.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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

(see Section 9), 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}}{{sep}_{i}},

where

{sep}_{i}

is the reciprocal condition number of

x_{i}

The condition numbers

s_{i}

and

{sep}_{i}

may be computed from the Hessenberg form of the balanced matrix

A^{'}

which is returned in the array a. This requires calling f08pef with

job ='S'

to compute the Schur form of

A^{'}

, followed by f08qlf.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

f02ecf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f02ecf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

f02ecf calls routines from LAPACK in Chapter F08. It 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 routine 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 routine 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 f02ecf.

The time taken by the routine is approximately proportional to

n^{3}

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

10 Example

This example computes those eigenvalues of the matrix

A

whose moduli lie in the range

[0.2, 0.5]

, and their corresponding eigenvectors, where

A = (\begin{array}{r} 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{array}) .

f02ec: FL CL CPP AD PY MB

NAG FL Interfacef02ecf (real_​gen_​eigsys)

▸▿ 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

NAG FL Interface
f02ecf (real_gen_eigsys)