naginterfaces.library.sparseig.real_​symm_​proc

naginterfaces.library.sparseig.real_symm_proc(sigma, resid, v, comm, io_manager=None)

real_symm_proc is a post-processing function in a suite of functions which includes real_symm_init(), real_symm_iter(), real_symm_option() and real_symm_monit(). real_symm_proc must be called following a final exit from real_symm_iter().

For full information please refer to the NAG Library document for f12fc

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f12/f12fcf.html

Parameters

sigmafloat

If one of the ‘Shifted Inverse’ (see real_symm_option()) modes has been selected then $sigma$ contains the real shift used; otherwise $sigma$ is not referenced.

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

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

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

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

commdict, communication object, modified in place

Communication structure.

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

io_managerFileObjManager, optional

Manager for I/O in this routine.

Returns

nconvint

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

dfloat, ndarray, shape $\left(\text{ncv}\right)$

The first $nconv$ locations of the array $d$ contain the converged approximate eigenvalues.

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

If the default option $\text{‘Vectors'}=\text{'RITZ'}$ (see real_symm_option()) has been selected then $z$ contains the final set of eigenvectors corresponding to the eigenvalues held in $d$. The real eigenvector associated with an eigenvalue is stored in the corresponding of $z$.

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 $2$)

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

(errno $3$)

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

(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 tridiagonal form, there was a failure to compute $⟨value⟩$ eigenvalues in a total of $⟨value⟩$ iterations.

(errno $8$)

Either the function was called out of sequence (following an initial call to the setup function and following completion of calls to the reverse communication function) or the communication arrays have become corrupted.

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 symmetric. The suite can also be used to find selected eigenvalues/eigenvectors of smaller scale dense, real and symmetric problems.

Following a call to real_symm_iter(), real_symm_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 symmetric 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_symm_proc is based on the function dseupd from the ARPACK package, which uses the Implicitly Restarted Lanczos 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 symmetric matrices is provided in Lehoucq and Scott (1996). This suite of functions offers the same functionality as the ARPACK software for real symmetric problems, but the interface design is quite different in order to make the option setting clearer and to simplify some of the interfaces.

real_symm_proc, is a post-processing function that must be called following a successful final exit from real_symm_iter(). real_symm_proc uses data returned from real_symm_iter() and options, set either by default or explicitly by calling real_symm_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