NAG FL Interface
f02gcf (complex_​gen_​eigsys)

Settings help

FL Name Style:

FL Specification Language:

1 Purpose

f02gcf computes selected eigenvalues and eigenvectors of a complex general matrix.

2 Specification

Fortran Interface
Subroutine f02gcf ( crit, n, a, lda, wl, wu, mest, m, w, v, ldv, work, lwork, rwork, iwork, bwork, ifail)
Integer, Intent (In) :: n, lda, mest, ldv, lwork
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: m, iwork(n)
Real (Kind=nag_wp), Intent (In) :: wl, wu
Real (Kind=nag_wp), Intent (Out) :: rwork(2*n)
Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*), v(ldv,mest)
Complex (Kind=nag_wp), Intent (Out) :: w(n), work(lwork)
Logical, Intent (Out) :: bwork(n)
Character (1), Intent (In) :: crit
C Header Interface
#include <nag.h>
void  f02gcf_ (const char *crit, const Integer *n, Complex a[], const Integer *lda, const double *wl, const double *wu, const Integer *mest, Integer *m, Complex w[], Complex v[], const Integer *ldv, Complex work[], const Integer *lwork, double rwork[], Integer iwork[], logical bwork[], Integer *ifail, const Charlen length_crit)
The routine may be called by the names f02gcf or nagf_eigen_complex_gen_eigsys.

3 Description

f02gcf computes selected eigenvalues and the corresponding right eigenvectors of a complex general matrix A:
Eigenvalues λi may be selected either by modulus, satisfying
or by real part, satisfying

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.
Eigenvalues are selected according to their moduli: wl|λi|wu.
Eigenvalues are selected according to their real parts: wlRe(λi)wu.
Constraint: crit='M' or 'R'.
2: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
3: a(lda,*) Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least max(1,n).
On entry: the n×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 f02gcf is called.
Constraint: ldamax(1,n).
5: wl Real (Kind=nag_wp) Input
6: wu Real (Kind=nag_wp) Input
On entry: wl and wu, 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 array v as declared in the (sub)program from which f02gcf 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: mestmax(1,m).
8: m Integer Output
On exit: m, the number of eigenvalues actually selected.
9: w(n) Complex (Kind=nag_wp) array Output
On exit: the first m elements of w hold the selected eigenvalues; elements m+1 to n contain the other eigenvalues.
10: v(ldv,mest) Complex (Kind=nag_wp) array Output
On exit: contains the selected eigenvectors, with the ith column holding the eigenvector associated with the eigenvalue λi (stored in w(i)).
11: ldv Integer Input
On entry: the first dimension of the array v as declared in the (sub)program from which f02gcf is called.
Constraint: ldvmax(1,n).
12: work(lwork) Complex (Kind=nag_wp) array Workspace
13: lwork Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f02gcf is called.
Constraint: lworkmax(1,n×(n+2)).
14: rwork(2×n) Real (Kind=nag_wp) array Workspace
15: iwork(n) Integer array Workspace
16: bwork(n) Logical array Workspace
17: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 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 or 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 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, crit=value.
Constraint: crit='M' or 'R'.
On entry, lda=value and n=value.
Constraint: ldamax(1,n).
On entry, ldv=value and n=value.
Constraint: ldvmax(1,n).
On entry, lwork=value and n=value.
Constraint: lworkmax(1,n×(n+2)).
On entry, mest=value.
Constraint: mest1.
On entry, n=value.
Constraint: n0.
On entry, wu=value and wl=value.
Constraint: wu>wl.
The QR algorithm failed to converge: only value eigenvalues have been computed; no eigenvectors have been computed.
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 v=mestm.
Inverse iteration failed to compute all the specified eigenvectors. The number of eigenvectors which failed to converge is value. The corresponding columns of v are set to zero.
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.
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.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

If λi is an exact eigenvalue, and λ~i is the corresponding computed value, then
|λ~i-λi| c(n)εA2si,  
where c(n) is a modestly increasing function of n, ε is the machine precision, and si is the reciprocal condition number of λi; A is the balanced form of the original matrix A (see Section 9), and AA.
If xi is the corresponding exact eigenvector, and x~i is the corresponding computed eigenvector, then the angle θ(x~i,xi) between them is bounded as follows:
where sepi is the reciprocal condition number of xi.
The condition numbers si and sepi may be computed from the Hessenberg form of the balanced matrix A which is returned in the array a. This requires calling f08psf with job='S' to compute the Schur form of A, followed by f08qyf.

8 Parallelism and Performance

f02gcf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f02gcf 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

f02gcf 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 a unitary similarity transformation: A=QHQH. The routine uses the Hessenberg QR 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 is normalized so that x2=1, and the element of largest absolute value is real.
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 f02gcf.
The time taken by the routine is approximately proportional to n3.
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 which lie in the range [-5.5,+5.5] , and their corresponding eigenvectors, where
A = ( -3.97-5.04i -4.11+3.70i -0.34+1.01i 1.29-0.86i 0.34-1.50i 1.52-0.43i 1.88-5.38i 3.36+0.65i 3.31-3.85i 2.50+3.45i 0.88-1.08i 0.64-1.48i -1.10+0.82i 1.81-1.59i 3.25+1.33i 1.57-3.44i ) .  

10.1 Program Text

Program Text (f02gcfe.f90)

10.2 Program Data

Program Data (f02gcfe.d)

10.3 Program Results

Program Results (f02gcfe.r)