# NAG Library Function Document

## nag_real_symm_banded_sparse_eigensystem_sol (f12fgc)

Note: this function uses optional parameters to define choices in the problem specification. If you wish to use default settings for all of the optional parameters, then the option setting function nag_real_symm_sparse_eigensystem_option (f12fdc) need not be called. If, however, you wish to reset some or all of the settings please refer to Section 11 in nag_real_symm_sparse_eigensystem_option (f12fdc) for a detailed description of the specification of the optional parameters.

## 1Purpose

nag_real_symm_banded_sparse_eigensystem_sol (f12fgc) is the main solver function in a suite of functions which includes nag_real_symm_sparse_eigensystem_option (f12fdc) and nag_real_symm_banded_sparse_eigensystem_init (f12ffc). nag_real_symm_banded_sparse_eigensystem_sol (f12fgc) must be called following an initial call to nag_real_symm_banded_sparse_eigensystem_init (f12ffc) and following any calls to nag_real_symm_sparse_eigensystem_option (f12fdc).
nag_real_symm_banded_sparse_eigensystem_sol (f12fgc) returns approximations to selected 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.

## 2Specification

 #include #include
 void nag_real_symm_banded_sparse_eigensystem_sol (Integer kl, Integer ku, const double ab[], const double mb[], double sigma, Integer *nconv, double d[], double z[], double resid[], double v[], double comm[], Integer icomm[], 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 banded, real and symmetric.
Following a call to the initialization function nag_real_symm_banded_sparse_eigensystem_init (f12ffc), nag_real_symm_banded_sparse_eigensystem_sol (f12fgc) 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 banded symmetric matrices. There is negligible additional computational cost to obtain eigenvectors; an orthonormal basis is always computed, but there is an additional storage cost if both are requested.
The banded matrices $A$ and $B$ must be stored using the LAPACK storage format for banded nonsymmetric matrices; please refer to Section 3.3.2 in the f07 Chapter Introduction for details on this storage format.
nag_real_symm_banded_sparse_eigensystem_sol (f12fgc) is based on the banded driver functions dsbdr1 to dsbdr6 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). This suite of functions offers the same functionality as the ARPACK banded driver software for real symmetric problems, but the interface design is quite different in order to make the option setting clearer and to combine the different drivers into a general purpose function.
nag_real_symm_banded_sparse_eigensystem_sol (f12fgc), is a general purpose direct communication function that must be called following initialization by nag_real_symm_banded_sparse_eigensystem_init (f12ffc). nag_real_symm_banded_sparse_eigensystem_sol (f12fgc) uses options, set either by default or explicitly by calling nag_real_symm_sparse_eigensystem_option (f12fdc), to return the converged approximations to selected eigenvalues and (optionally):
 – the corresponding approximate eigenvectors; – an orthonormal basis for the associated approximate invariant subspace; – both.

## 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:    $\mathbf{kl}$IntegerInput
On entry: the number of subdiagonals of the matrices $A$ and $B$.
Constraint: ${\mathbf{kl}}\ge 0$.
2:    $\mathbf{ku}$IntegerInput
On entry: the number of superdiagonals of the matrices $A$ and $B$. Since $A$ and $B$ are symmetric, the normal case is ${\mathbf{ku}}={\mathbf{kl}}$.
Constraint: ${\mathbf{ku}}\ge 0$.
3:    $\mathbf{ab}\left[\mathit{dim}\right]$const doubleInput
Note: the dimension, dim, of the array ab must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}},×,\left(2×{\mathbf{kl}}+{\mathbf{ku}}+1\right)\right)$ (see nag_real_symm_banded_sparse_eigensystem_init (f12ffc)).
On entry: must contain the matrix $A$ in LAPACK column-ordered banded storage format for nonsymmetric matrices (see Section 3.3.4 in the f07 Chapter Introduction).
4:    $\mathbf{mb}\left[\mathit{dim}\right]$const doubleInput
Note: the dimension, dim, of the array mb must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}},×,\left(2×{\mathbf{kl}}+{\mathbf{ku}}+1\right)\right)$ (see nag_real_symm_banded_sparse_eigensystem_init (f12ffc)).
On entry: must contain the matrix $B$ in LAPACK column-ordered banded storage format for nonsymmetric matrices (see Section 3.3.4 in the f07 Chapter Introduction).
5:    $\mathbf{sigma}$doubleInput
On entry: if one of the ${\mathbf{Shifted Inverse}}$ (see nag_real_symm_sparse_eigensystem_option (f12fdc)) modes has been selected then sigma contains the real shift used; otherwise sigma is not referenced.
6:    $\mathbf{nconv}$Integer *Output
On exit: the number of converged eigenvalues.
7:    $\mathbf{d}\left[\mathit{dim}\right]$doubleOutput
Note: the dimension, dim, of the array d must be at least ${\mathbf{ncv}}$ (see nag_real_symm_banded_sparse_eigensystem_init (f12ffc)).
On exit: the first nconv locations of the array d contain the converged approximate eigenvalues.
8:    $\mathbf{z}\left[{\mathbf{n}}×\left({\mathbf{nev}}+1\right)\right]$doubleOutput
On exit: if the default option ${\mathbf{Vectors}}=\mathrm{RITZ}$ (see nag_real_symm_sparse_eigensystem_option (f12fdc)) has been selected then z contains the final set of eigenvectors corresponding to the eigenvalues held in d. The real eigenvector associated with eigenvalue $\mathit{i}-1$, for $\mathit{i}=1,2,\dots ,{\mathbf{nconv}}$, is stored at locations ${\mathbf{z}}\left[\mathit{i}-1×n+\mathit{j}-1\right]$, for $\mathit{j}=1,2,\dots ,n$.
9:    $\mathbf{resid}\left[\mathit{dim}\right]$doubleInput/Output
Note: the dimension, dim, of the array resid must be at least ${\mathbf{n}}$ (see nag_real_symm_banded_sparse_eigensystem_init (f12ffc)).
On entry: need not be set unless the option ${\mathbf{Initial Residual}}$ has been set in a prior call to nag_real_symm_sparse_eigensystem_option (f12fdc) in which case resid must contain an initial residual vector.
On exit: contains the final residual vector.
10:  $\mathbf{v}\left[{\mathbf{n}}×{\mathbf{ncv}}\right]$doubleOutput
On exit: if the option ${\mathbf{Vectors}}$ (see nag_real_symm_sparse_eigensystem_option (f12fdc)) has been set to Schur or Ritz and z does not equal v then the first nconv sections of v, of length $n$, will contain approximate Schur vectors that span the desired invariant subspace.
The $j$th Schur vector is stored in locations ${\mathbf{v}}\left[{\mathbf{n}}×\left(\mathit{j}-1\right)+\mathit{i}-1\right]$, for $\mathit{j}=1,2,\dots ,{\mathbf{nconv}}$ and $\mathit{i}=1,2,\dots ,n$.
11:  $\mathbf{comm}\left[\mathit{dim}\right]$doubleCommunication Array
Note: the dimension, dim, of the array comm must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lcomm}}\right)$ (see nag_real_symm_banded_sparse_eigensystem_init (f12ffc)).
On initial entry: must remain unchanged from the prior call to nag_real_symm_sparse_eigensystem_option (f12fdc) and nag_real_symm_banded_sparse_eigensystem_init (f12ffc).
On exit: contains no useful information.
12:  $\mathbf{icomm}\left[\mathit{dim}\right]$IntegerCommunication Array
Note: the dimension, dim, of the array icomm must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{licomm}}\right)$ (see nag_real_symm_banded_sparse_eigensystem_init (f12ffc)).
On initial entry: must remain unchanged from the prior call to nag_real_symm_sparse_eigensystem_iter (f12fbc) and nag_real_symm_sparse_eigensystem_option (f12fdc).
On exit: contains no useful information.
13:  $\mathbf{fail}$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.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_BOTH_ENDS_1
Eigenvalues from both ends of the spectrum were requested, but the number of eigenvalues (nev in nag_real_symm_banded_sparse_eigensystem_init (f12ffc)) requested is one.
NE_INITIALIZATION
Either an initial call to the setup function has not been made or the communication arrays have become corrupted.
NE_INT
On entry, ${\mathbf{kl}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{kl}}\ge 0$.
On entry, ${\mathbf{ku}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ku}}\ge 0$.
The maximum number of iterations $\le 0$, the option ${\mathbf{Iteration Limit}}$ has been set to $〈\mathit{\text{value}}〉$.
NE_INT_2
The maximum number of iterations has been reached: there have been $〈\mathit{\text{value}}〉$ iterations. There are $〈\mathit{\text{value}}〉$ converged eigenvalues.
NE_INTERNAL_EIGVAL_FAIL
Error in internal call to compute eigenvalues and corresponding error bounds of the current upper Hessenberg matrix. Please contact NAG.
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_INVALID_OPTION
On entry, ${\mathbf{Vectors}}=\text{Select}$, but this is not yet implemented.
NE_MAX_ITER
During calculation of a tridiagonal form, there was a failure to compute $〈\mathit{\text{value}}〉$ eigenvalues in a total of $〈\mathit{\text{value}}〉$ iterations.
NE_NO_LANCZOS_FAC
Could not build a Lanczos factorization. The size of the current Lanczos factorization $=〈\mathit{\text{value}}〉$.
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.
NE_NO_SHIFTS_APPLIED
No shifts could be applied during a cycle of the implicitly restarted Lanczos iteration.
NE_OPT_INCOMPAT
The options ${\mathbf{Generalized}}$ and ${\mathbf{Regular}}$ are incompatible.
NE_REAL_BAND_FAC
NE_REAL_BAND_SOL
NE_ZERO_EIGS_FOUND
The number of eigenvalues found to sufficient accuracy is zero.
NE_ZERO_INIT_RESID
The option ${\mathbf{Initial Residual}}$ was selected but the starting vector held in resid is zero.

## 7Accuracy

The relative accuracy of a Ritz value, $\lambda$, is considered acceptable if its Ritz estimate $\le {\mathbf{Tolerance}}×\left|\lambda \right|$. The default ${\mathbf{Tolerance}}$ used is the machine precision given by nag_machine_precision (X02AJC).

## 8Parallelism and Performance

nag_real_symm_banded_sparse_eigensystem_sol (f12fgc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_real_symm_banded_sparse_eigensystem_sol (f12fgc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

None.

## 10Example

This example solves $Ax=\lambda x$ in regular mode, where $A$ is obtained from the standard central difference discretization of the two-dimensional convection-diffusion operator $\frac{{d}^{2}u}{d{x}^{2}}+\frac{{d}^{2}u}{d{y}^{2}}=\rho \frac{du}{dx}$ on the unit square with zero Dirichlet boundary conditions. $A$ is stored in LAPACK banded storage format.

### 10.1Program Text

Program Text (f12fgce.c)

### 10.2Program Data

Program Data (f12fgce.d)

### 10.3Program Results

Program Results (f12fgce.r)

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