# NAG Library Function Document

## 1Purpose

nag_real_symm_sparse_eigensystem_init (f12fac) is a setup function in a suite of functions consisting of nag_real_symm_sparse_eigensystem_init (f12fac), nag_real_symm_sparse_eigensystem_iter (f12fbc), nag_real_symm_sparse_eigensystem_sol (f12fcc), nag_real_symm_sparse_eigensystem_option (f12fdc) and nag_real_symm_sparse_eigensystem_monit (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.

## 2Specification

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

## 3Description

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.
nag_real_symm_sparse_eigensystem_init (f12fac) is a setup function which must be called before nag_real_symm_sparse_eigensystem_iter (f12fbc), the reverse communication iterative solver, and before nag_real_symm_sparse_eigensystem_option (f12fdc), the options setting function. nag_real_symm_sparse_eigensystem_sol (f12fcc), is a post-processing function that must be called following a successful final exit from nag_real_symm_sparse_eigensystem_iter (f12fbc), while nag_real_symm_sparse_eigensystem_monit (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 nag_real_symm_sparse_eigensystem_option (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 nag_real_symm_sparse_eigensystem_option (f12fdc).

## 4References

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, Philidelphia

## 5Arguments

1:    $\mathbf{n}$IntegerInput
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: ${\mathbf{n}}>0$.
2:    $\mathbf{nev}$IntegerInput
On entry: the number of eigenvalues to be computed.
Constraint: $0<{\mathbf{nev}}<{\mathbf{n}}-1$.
3:    $\mathbf{ncv}$IntegerInput
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 ${\mathbf{ncv}}\ge 2×{\mathbf{nev}}+1$. 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: ${\mathbf{nev}}<{\mathbf{ncv}}\le {\mathbf{n}}$.
4:    $\mathbf{icomm}\left[\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{licomm}}\right)\right]$IntegerCommunication Array
On exit: contains data to be communicated to the other functions in the suite.
5:    $\mathbf{licomm}$IntegerInput
On entry: the dimension of the array icomm.
If ${\mathbf{licomm}}=-1$, a workspace query is assumed and the function only calculates the required dimensions of icomm and comm, which it returns in ${\mathbf{icomm}}\left[0\right]$ and ${\mathbf{comm}}\left[0\right]$ respectively.
Constraint: ${\mathbf{licomm}}\ge 140\text{​ or ​}{\mathbf{licomm}}=-1$.
6:    $\mathbf{comm}\left[\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lcomm}}\right)\right]$doubleCommunication Array
On exit: contains data to be communicated to the other functions in the suite.
7:    $\mathbf{lcomm}$IntegerInput
On entry: the dimension of the array comm.
If ${\mathbf{lcomm}}=-1$, a workspace query is assumed and the function only calculates the dimensions of icomm and comm required by nag_real_symm_sparse_eigensystem_iter (f12fbc), which it returns in ${\mathbf{icomm}}\left[0\right]$ and ${\mathbf{comm}}\left[0\right]$ respectively.
Constraint: ${\mathbf{lcomm}}\ge 3×{\mathbf{n}}+{\mathbf{ncv}}×{\mathbf{ncv}}+8×{\mathbf{ncv}}+60\text{​ or ​}{\mathbf{lcomm}}=-1$.
8:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>0$.
On entry, ${\mathbf{nev}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nev}}>0$.
NE_INT_2
The length of the integer array icomm is too small ${\mathbf{licomm}}=〈\mathit{\text{value}}〉$, but must be at least $〈\mathit{\text{value}}〉$.
NE_INT_3
On entry, ${\mathbf{lcomm}}=〈\mathit{\text{value}}〉$, ${\mathbf{n}}=〈\mathit{\text{value}}〉$ and ${\mathbf{ncv}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lcomm}}\ge 3×{\mathbf{n}}+{\mathbf{ncv}}×{\mathbf{ncv}}+8×{\mathbf{ncv}}+60$.
On entry, ${\mathbf{ncv}}=〈\mathit{\text{value}}〉$, ${\mathbf{nev}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ncv}}>{\mathbf{nev}}$ and ${\mathbf{ncv}}\le {\mathbf{n}}$.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

Not applicable.

## 8Parallelism and Performance

nag_real_symm_sparse_eigensystem_init (f12fac) is not threaded in any implementation.

None.

## 10Example

This example solves $Ax=\lambda x$ 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.

### 10.1Program Text

Program Text (f12face.c)

### 10.2Program Data

Program Data (f12face.d)

### 10.3Program Results

Program Results (f12face.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017