f12fac is a setup function in a suite of functions consisting of f12fac, f12fbc, f12fcc, f12fdc and f12fec. It is used to find some of the eigenvalues (and optionally the corresponding eigenvectors) of a standard or generalized eigenvalue problem defined by real symmetric matrices.

The suite of functions is suitable for the solution of large sparse, standard or generalized, symmetric eigenproblems where only a few eigenvalues from a selected range of the spectrum are required.

2 Specification

#include <nag.h>

void	f12fac (Integer n, Integer nev, Integer ncv, Integer icomm[], Integer licomm, double comm[], Integer lcomm, NagError *fail)

The function may be called by the names: f12fac, nag_sparseig_real_symm_init or nag_real_symm_sparse_eigensystem_init.

3 Description

The suite of functions is designed to calculate some of the eigenvalues,

λ

, (and optionally the corresponding eigenvectors,

x

) of a standard eigenvalue problem

A x = λ x

, or of a generalized eigenvalue problem

A x = λ B x

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.

f12fac is a setup function which must be called before f12fbc, the reverse communication iterative solver, and before f12fdc, the options setting function. f12fcc, is a post-processing function that must be called following a successful final exit from f12fbc, while f12fec can be used to return additional monitoring information during the computation.

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 f12fdc, 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 Section 11.1 in f12fdc.

4 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

5 Arguments

1: $n$ – Integer Input: On entry: the order of the matrix $A$ (and the order of the matrix $B$ for the generalized problem) that defines the eigenvalue problem.

Constraint: $n > 0$ .
2: $nev$ – Integer Input: On entry: the number of eigenvalues to be computed.

Constraint: $0 < nev < n - 1$ .
3: $ncv$ – Integer Input: On entry: 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 $ncv \geq 2 \times nev + 1$ . 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.

Constraint: $nev < ncv \leq n$ .
4: $icomm [\max (1, licomm)]$ – Integer Communication Array: On exit: contains data to be communicated to the other functions in the suite.
5: $licomm$ – Integer Input: On entry: the dimension of the array icomm.
If $licomm = −1$ , a workspace query is assumed and the function only calculates the required dimensions of icomm and comm, which it returns in $icomm [0]$ and $comm [0]$ respectively.

Constraint: $licomm \geq 140$ or $licomm = −1$ .
6: $comm [\max (1, lcomm)]$ – double Communication Array: On exit: contains data to be communicated to the other functions in the suite.
7: $lcomm$ – Integer Input: On entry: the dimension of the array comm.
If $lcomm = −1$ , a workspace query is assumed and the function only calculates the dimensions of icomm and comm required by f12fbc, which it returns in $icomm [0]$ and $comm [0]$ respectively.

Constraint: $lcomm \geq 3 \times n + ncv \times ncv + 8 \times ncv + 60$ or $lcomm = −1$ .
8: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n > 0$ .

On entry, $nev = ⟨ value ⟩$ .
Constraint: $nev > 0$ .
NE_INT_2: The length of the integer array icomm is too small $licomm = ⟨ value ⟩$ , but must be at least $⟨ value ⟩$ .
NE_INT_3: On entry, $lcomm = ⟨ value ⟩$ , $n = ⟨ value ⟩$ and $ncv = ⟨ value ⟩$ .
Constraint: $lcomm \geq 3 \times n + ncv \times ncv + 8 \times ncv + 60$ .

On entry, $ncv = ⟨ value ⟩$ , $nev = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $ncv > nev$ and $ncv \leq n$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

f12fac is not threaded in any implementation.

9 Further Comments

None.

10 Example

This example solves

A x = λ x

in regular mode, where

A

is obtained from the standard central difference discretization of the Laplacian operator

\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}}

on the unit square, with zero Dirichlet boundary conditions. Eigenvalues of smallest magnitude are selected.

f12fa: FL CL CPP AD

NAG CL Interfacef12fac (real_​symm_​init)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
f12fac (real_symm_init)