naginterfaces.library.eigen.real_​gen_​eigsys

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

real_gen_eigsys computes selected eigenvalues and eigenvectors of a real general matrix.

For full information please refer to the NAG Library document for f02ec

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f02/f02ecf.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: $w_l\le Re\left({\lambda }_i\right)\le w_u$.

afloat, 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

afloat, 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.

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

The first $m$ elements of $wr$ and $wi$ hold the real and imaginary parts, respectively, of the selected eigenvalues; elements $m+1$ to $\text{n}$ 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 Further Comments.

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

The first $m$ elements of $wr$ and $wi$ hold the real and imaginary parts, respectively, of the selected eigenvalues; elements $m+1$ to $\text{n}$ 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 Further Comments.

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

Contains the real parts of the selected eigenvectors, with the $i$th column holding the real part of the eigenvector associated with the eigenvalue ${\lambda }_i$ (stored in $wr[i-1]$ and $wi[i-1]$).

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

Contains the imaginary parts of the selected eigenvectors, with the $i$th column holding the imaginary part of the eigenvector associated with the eigenvalue ${\lambda }_i$ (stored in $wr[i-1]$ and $wi[i-1]$).

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 $vr$ and $vi=mest\ge m$.

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 $vr$ and $vi$ 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.

real_gen_eigsys computes selected eigenvalues and the corresponding right eigenvectors of a real 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:

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

Note that even though $A$ is real, ${\lambda }_i$ and $x_i$ may be complex. If $x_i$ is an eigenvector corresponding to a complex eigenvalue ${\lambda }_i$, then the complex conjugate vector ${\overline{x}}_i$ is the eigenvector corresponding to the complex conjugate eigenvalue ${\overline{\lambda }}_i$. The eigenvalues in a complex conjugate pair ${\lambda }_i$ and ${\overline{\lambda }}_i$ are either both selected or both not selected.

References

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