naginterfaces.library.sparseig.real_​symm_​band_​init

naginterfaces.library.sparseig.real_symm_band_init(n, nev, ncv)

real_symm_band_init is a setup function for real_symm_band_solve() which can be used to find some eigenvalues (and optionally the corresponding eigenvectors) of a standard or generalized eigenvalue problem defined by real, banded, symmetric matrices. The banded matrix must be stored using the LAPACK storage format for real banded nonsymmetric matrices.

For full information please refer to the NAG Library document for f12ff

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f12/f12fff.html

Parameters

nint

The order of the matrix $A$ (and the order of the matrix $B$ for the generalized problem) that defines the eigenvalue problem.

nevint

The number of eigenvalues to be computed.

ncvint

The number of Lanczos basis vectors to use during the computation.

At present there is no a priori analysis to guide the selection of $ncv$ relative to $nev$.

However, it is recommended that $ncv\ge 2\times nev+1$.

If many problems of the same type are to be solved, you should experiment with increasing $ncv$ while keeping $nev$ fixed for a given test problem.

This will usually decrease the required number of matrix-vector operations but it also increases the work and storage required to maintain the orthogonal basis vectors.

The optimal ‘cross-over’ with respect to CPU time is problem dependent and must be determined empirically.

Returns

commdict, communication object

Communication structure.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n>0$.

(errno $2$)

On entry, $nev=⟨value⟩$.

Constraint: $nev>0$.

(errno $3$)

On entry, $ncv=⟨value⟩$, $nev=⟨value⟩$ and $n=⟨value⟩$.

Constraint: $ncv>nev+1$ and $ncv\le n$.

Notes

The pair of functions real_symm_band_init and real_symm_band_solve() together with the option setting function real_symm_option() are 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 banded real and symmetric.

real_symm_band_init is a setup function which must be called before the option setting function real_symm_option() and the solver function real_symm_band_solve(). Internally, real_symm_band_solve() makes calls to real_symm_iter() and real_symm_proc(); the function documents for real_symm_iter() and real_symm_proc() should be consulted for details of the algorithm used.

This setup function initializes the communication arrays, sets (to their default values) all options that can be set by you via the option setting function real_symm_option(), and checks that the lengths of the communication arrays as passed by you are of sufficient length. For details of the options available and how to set them, see Other Parameters for real_symm_option.

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