naginterfaces.library.eigen.complex_​gen_​eigsys

naginterfaces.library.eigen.complex_gen_eigsys(crit, n, a, wl, wu, mest)

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

For full information please refer to the NAG Library document for f02gc

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f02/f02gcf.html

Parameters

critstr, length 1

Indicates the criterion for selecting eigenvalues.

$crit=\text{'M'}$

Eigenvalues are selected according to their moduli: $w_l\le \left|{\lambda }_i\right|\le w_u$.

$crit=\text{'R'}$

Eigenvalues are selected according to their real parts: $w_l\le Re\left({\lambda }_i\right)\le w_u$.

nint

$n$, the order of the matrix $A$.

acomplex, array-like, shape $(n,n)$

The $n\times n$ general matrix $A$.

wlfloat

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

wufloat

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

mestint

$mest$ must be an upper bound on $m$, the number of eigenvalues and eigenvectors selected. No eigenvectors are computed if $mest<m$.

Returns

acomplex, ndarray, shape $(n,n)$

Contains the Hessenberg form of the balanced input matrix $A^{\prime }$ (see Further Comments).

mint

$m$, the number of eigenvalues actually selected.

wcomplex, ndarray, shape $\left(n\right)$

The first $m$ elements of $w$ hold the selected eigenvalues; elements $m+1$ to $n$ contain the other eigenvalues.

vcomplex, ndarray, shape $(n,mest)$

Contains the selected eigenvectors, with the $i$th column holding the eigenvector associated with the eigenvalue ${\lambda }_i$ (stored in $w[i-1]$).

Raises

NagValueError

(errno $1$)

On entry, $mest=⟨value⟩$.

Constraint: $mest\ge 1$.

(errno $1$)

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

Constraint: $wu>wl$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

(errno $1$)

On entry, $crit=⟨value⟩$.

Constraint: $crit=\text{'M'}$ or $\text{'R'}$.

(errno $2$)

The $QR$ algorithm failed to converge: only $⟨value⟩$ eigenvalues have been computed; no eigenvectors have been computed.

(errno $3$)

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 $v=mest\ge m$.

Warns

NagAlgorithmicWarning

(errno $4$)

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.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

complex_gen_eigsys computes selected eigenvalues and the corresponding right eigenvectors of a complex general matrix $A$:

$$A{x}_i={\lambda }_i{x}_i\text{.}$$

Eigenvalues ${\lambda }_i$ may be selected either by modulus, satisfying

$$w_l\le \left|{\lambda }_i\right|\le w_u\text{,}$$

or by real part, satisfying

$$w_l\le Re\left({\lambda }_i\right)\le w_u\text{.}$$

References

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