naginterfaces.library.sparseig.real_​proc

naginterfaces.library.sparseig.real_proc(sigmar, sigmai, resid, v, comm, io_manager=None)

real_proc is a post-processing function in a suite of functions consisting of real_init(), real_option() and real_monit(). It must be called following a final exit from real_iter().

For full information please refer to the NAG Library document for f12ac

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f12/f12acf.html

Parameters

sigmarfloat

If one of the ‘Shifted Inverse Real’ modes have been selected then $sigmar$ contains the real part of the shift used; otherwise $sigmar$ is not referenced.

sigmaifloat

If one of the ‘Shifted Inverse Real’ modes have been selected then $sigmai$ contains the imaginary part of the shift used; otherwise $sigmai$ is not referenced.

residfloat, array-like, shape $\left(\text{n}\right)$

Must not be modified following a call to real_iter() since it contains data required by real_proc.

vfloat, array-like, shape $(\text{n},\text{ncv})$

The $\text{ncv}$ columns of $v$ contain the Arnoldi basis vectors for $op$ as constructed by real_iter().

commdict, communication object, modified in place

Communication structure.

This argument must have been initialized by a prior call to real_init().

io_managerFileObjManager, optional

Manager for I/O in this routine.

Returns

nconvint

The number of converged eigenvalues as found by real_iter().

drfloat, ndarray, shape $(\text{nev}+1)$

The first $nconv$ locations of the array $dr$ contain the real parts of the converged approximate eigenvalues.

difloat, ndarray, shape $(\text{nev}+1)$

The first $nconv$ locations of the array $di$ contain the imaginary parts of the converged approximate eigenvalues.

zfloat, ndarray, shape $(\text{n},:)$

If the default option $\text{‘Vectors'}=\text{'RITZ'}$ (see real_option()) has been selected then $z$ contains the final set of eigenvectors corresponding to the eigenvalues held in $dr$ and $di$. The complex eigenvector associated with the eigenvalue with positive imaginary part is stored in two consecutive columns. The first column holds the real part of the eigenvector and the second column holds the imaginary part. The eigenvector associated with the eigenvalue with negative imaginary part is simply the complex conjugate of the eigenvector associated with the positive imaginary part.

vfloat, ndarray, shape $(\text{n},\text{ncv})$

If the option $\text{‘Vectors'}=\text{'SCHUR'}$ has been set, or the option $\text{‘Vectors'}=\text{'RITZ'}$ has been set and a separate array $z$ has been passed (i.e., $z$ does not equal $v$), then the first $nconv$ columns of $v$ will contain approximate Schur vectors that span the desired invariant subspace.

Raises

NagValueError

(errno $3$)

The number of eigenvalues found to sufficient accuracy, as communicated through the argument $comm$[‘icomm’], is zero. See the function document for further details.

(errno $4$)

Got a different count of the number of converged Ritz values than the value passed to it through the argument $comm$[‘icomm’]: number counted $=⟨value⟩$, number expected $=⟨value⟩$.

(errno $5$)

During calculation of a real Schur form, there was a failure to compute $⟨value⟩$ eigenvalues in a total of $⟨value⟩$ iterations.

(errno $6$)

The computed Schur form could not be reordered by an internal call. This function returned with $\text{errno}=⟨value⟩$. Please contact NAG.

(errno $7$)

In calculating eigenvectors, an internal call returned with an error. Please contact NAG.

(errno $8$)

Either the solver function has not been called prior to the call of this function or a communication array has become corrupted.

Warns

NagAlgorithmicWarning

(errno $2$)

On entry, $\text{‘Vectors'}=\text{'SELECT'}$, but this is not yet implemented.

Notes

The suite of functions is designed to calculate some of the eigenvalues, $\lambda$, (and optionally the corresponding eigenvectors, $x$) of a standard eigenvalue problem $Ax=\lambda x$, or of a generalized eigenvalue problem $Ax=\lambda Bx$ of order $n$, where $n$ is large and the coefficient matrices $A$ and $B$ are sparse, real and nonsymmetric. The suite can also be used to find selected eigenvalues/eigenvectors of smaller scale dense, real and nonsymmetric problems.

Following a call to real_iter(), real_proc returns the converged approximations to eigenvalues and (optionally) the corresponding approximate eigenvectors and/or an orthonormal basis for the associated approximate invariant subspace. The eigenvalues (and eigenvectors) are selected from those of a standard or generalized eigenvalue problem defined by real nonsymmetric matrices. There is negligible additional cost to obtain eigenvectors; an orthonormal basis is always computed, but there is an additional storage cost if both are requested.

real_proc is based on the function dneupd from the ARPACK package, which uses the Implicitly Restarted Arnoldi iteration method. The method is described in Lehoucq and Sorensen (1996) and Lehoucq (2001) while its use within the ARPACK software is described in great detail in Lehoucq et al. (1998). An evaluation of software for computing eigenvalues of sparse nonsymmetric matrices is provided in Lehoucq and Scott (1996). This suite of functions offers the same functionality as the ARPACK software for real nonsymmetric problems, but the interface design is quite different in order to make the option setting clearer and to simplify some of the interfaces.

real_proc is a post-processing function that must be called following a successful final exit from real_iter(). real_proc uses data returned from real_iter() and options, set either by default or explicitly by calling real_option(), to return the converged approximations to selected eigenvalues and (optionally):

• the corresponding approximate eigenvectors;

• an orthonormal basis for the associated approximate invariant subspace;

• both.

References

Lehoucq, R B, 2001, Implicitly restarted Arnoldi methods and subspace iteration, SIAM Journal on Matrix Analysis and Applications (23), 551–562

Lehoucq, R B and Scott, J A, 1996, An evaluation of software for computing eigenvalues of sparse nonsymmetric matrices, Preprint MCS-P547-1195, Argonne National Laboratory

Lehoucq, R B and Sorensen, D C, 1996, Deflation techniques for an implicitly restarted Arnoldi iteration, SIAM Journal on Matrix Analysis and Applications (17), 789–821

Lehoucq, R B, Sorensen, D C and Yang, C, 1998, ARPACK Users’ Guide: Solution of Large-scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, SIAM, Philadelphia