Integer type:** int32**** int64**** nag_int** show int32 show int32 show int64 show int64 show nag_int show nag_int

nag_sparseig_complex_init (f12an) is a setup function in a suite of functions consisting of nag_sparseig_complex_init (f12an), nag_sparseig_complex_iter (f12ap), nag_sparseig_complex_proc (f12aq), nag_sparseig_complex_option (f12ar) and nag_sparseig_complex_monit (f12as). It is used to find some of the eigenvalues (and optionally the corresponding eigenvectors) of a standard or generalized eigenvalue problem defined by complex nonsymmetric matrices.

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

The suite of functions is designed to calculate some of the eigenvalues,
λ
$\lambda $, (and optionally the corresponding eigenvectors,
x
$x$) of a standard complex eigenvalue problem
Ax
=
λx
$Ax=\lambda x$, or of a generalized complex eigenvalue problem
Ax
=
λBx
$Ax=\lambda Bx$ of order
n
$n$, where
n
$n$ is large and the coefficient matrices
A
$A$ and
B
$B$ are sparse, complex and nonsymmetric. The suite can also be used to find selected eigenvalues/eigenvectors of smaller scale dense, complex and nonsymmetric problems.

nag_sparseig_complex_init (f12an) is a setup function which must be called before nag_sparseig_complex_iter (f12ap), the reverse communication iterative solver, and before nag_sparseig_complex_option (f12ar), the options setting function. nag_sparseig_complex_proc (f12aq) is a post-processing function that must be called following a successful final exit from nag_sparseig_complex_iter (f12ap), while nag_sparseig_complex_monit (f12as) 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_sparseig_complex_option (f12ar), 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 [Description of the Optional s] in (f12ar).

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

- 1: n – int64int32nag_int scalar
- The order of the matrix A$A$ (and the order of the matrix B$B$ for the generalized problem) that defines the eigenvalue problem.
- 2: nev – int64int32nag_int scalar
- The number of eigenvalues to be computed.
- 3: ncv – int64int32nag_int scalar
- The number of Arnoldi 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 ≥ 2 × nev + 1${\mathbf{ncv}}\ge 2\times {\mathbf{nev}}+1$. 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.

None.

- licomm lcomm

- 1: icomm(max (1,licomm)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,\mathit{licomm})$) – int64int32nag_int array
- Contains data to be communicated to the other functions in the suite.
- 2: comm(max (1,lcomm)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,\mathit{lcomm})$) – complex array
- Contains data to be communicated to the other functions in the suite.
- 3: ifail – int64int32nag_int scalar
- ifail = 0${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

Errors or warnings detected by the function:

- On entry, n ≤ 0${\mathbf{n}}\le 0$.

- On entry, nev ≤ 0${\mathbf{nev}}\le 0$.

- On entry, licomm < 140$\mathit{licomm}<140$ and licomm ≠ − 1$\mathit{licomm}\ne -1$.

Not applicable.

None.

Open in the MATLAB editor: nag_sparseig_complex_init_example

function nag_sparseig_complex_init_examplen = int64(100); nx = int64(10); nev = int64(4); ncv = int64(20); irevcm = int64(0); resid = complex(zeros(100,1)); v = complex(zeros(100,20)); x = complex(zeros(100,1)); mx = complex(zeros(100,1)); sigma = complex(0); % Initialisation Step [icomm, comm, ifail] = nag_sparseig_complex_init(n, nev, ncv); % Solve while (irevcm ~= 5) [irevcm, resid, v, x, mx, nshift, comm, icomm, ifail] = ... nag_sparseig_complex_iter(irevcm, resid, v, x, mx, comm, icomm); if (irevcm == 1 || irevcm == -1) x = f12_av(nx, x); elseif (irevcm == 4) [niter, nconv, ritz, rzest] = nag_sparseig_complex_monit(icomm, comm); fprintf('Iteration %d, No. converged = %d, norm of estimates = %16.8g\n', ... niter, nconv, norm(rzest(1:double(nev)),2)); end end % Post-process to compute eigenvalues/vectors [nconv, d, z, v, comm, icomm, ifail] = ... nag_sparseig_complex_proc(sigma, resid, v, comm, icomm); % sort to avoid different orders showing up as errorfunction [w] = f12_av(nx, v)inx = double(nx); % nx is int64 w = complex(zeros(inx*inx,1)); h2 = double(-(inx+1)^2); w(1:inx) = tv(inx, v(1:inx)); w(1:inx) = h2*v(inx+1:2*inx)+w(1:inx); for j=2:double(inx-1) lo = (j-1)*inx +1; hi = j*inx; w(lo:hi) = tv(inx, v(lo:hi)); w(lo:hi) = h2*v(lo-inx:lo-1)+w(lo:hi); w(lo:hi) = h2*v(hi+1:hi+inx)+w(lo:hi); end lo = (inx-1)*inx +1; hi = inx*inx; w(lo:hi) = tv(inx, v(lo:hi)); w(lo:hi) = h2*v(lo-inx:lo-1)+w(lo:hi);function [y] = tv(inx,x)y = zeros(inx,1); dd = double(4*(inx+1)^2); dl = double(-(inx+1)^2 - 0.5*100*(inx+1)); du = double(-(inx+1)^2 + 0.5*100*(inx+1)); y(1) = dd*x(1) + du*x(2); for j=2:double(inx-1) y(j) = dl*x(j-1) + dd*x(j) + du*x(j+1); end y(inx) = dl*x(inx-1) + dd*x(inx);

Iteration 1, No. converged = 0, norm of estimates = 133.43269 Iteration 2, No. converged = 0, norm of estimates = 99.725711 Iteration 3, No. converged = 0, norm of estimates = 42.48146 Iteration 4, No. converged = 0, norm of estimates = 8.1805323 Iteration 5, No. converged = 0, norm of estimates = 1.7794793 Iteration 6, No. converged = 0, norm of estimates = 0.49794897 Iteration 7, No. converged = 0, norm of estimates = 0.12604717 Iteration 8, No. converged = 0, norm of estimates = 0.026883649 Iteration 9, No. converged = 0, norm of estimates = 0.0070182429 Iteration 10, No. converged = 0, norm of estimates = 0.0014438635 Iteration 11, No. converged = 0, norm of estimates = 0.00040007427 Iteration 12, No. converged = 0, norm of estimates = 0.0001059392 Iteration 13, No. converged = 0, norm of estimates = 2.803229e-05 Iteration 14, No. converged = 0, norm of estimates = 7.7343157e-06 Iteration 15, No. converged = 0, norm of estimates = 1.9579037e-06 Iteration 16, No. converged = 0, norm of estimates = 6.15616e-07 Iteration 17, No. converged = 0, norm of estimates = 1.2591405e-07 Iteration 18, No. converged = 0, norm of estimates = 3.7790713e-08 Iteration 19, No. converged = 2, norm of estimates = 8.3787181e-09 Iteration 20, No. converged = 2, norm of estimates = 1.8827432e-09 Iteration 21, No. converged = 2, norm of estimates = 1.6920778e-10 Iteration 22, No. converged = 2, norm of estimates = 2.299125e-11 Iteration 23, No. converged = 2, norm of estimates = 3.391976e-12 Iteration 24, No. converged = 2, norm of estimates = 5.5954988e-13

Open in the MATLAB editor: f12an_example

function f12an_examplen = int64(100); nx = int64(10); nev = int64(4); ncv = int64(20); irevcm = int64(0); resid = complex(zeros(100,1)); v = complex(zeros(100,20)); x = complex(zeros(100,1)); mx = complex(zeros(100,1)); sigma = complex(0); % Initialisation Step [icomm, comm, ifail] = f12an(n, nev, ncv); % Solve while (irevcm ~= 5) [irevcm, resid, v, x, mx, nshift, comm, icomm, ifail] = ... f12ap(irevcm, resid, v, x, mx, comm, icomm); if (irevcm == 1 || irevcm == -1) x = f12_av(nx, x); elseif (irevcm == 4) [niter, nconv, ritz, rzest] = f12as(icomm, comm); fprintf('Iteration %d, No. converged = %d, norm of estimates = %16.8g\n', ... niter, nconv, norm(rzest(1:double(nev)),2)); end end % Post-process to compute eigenvalues/vectors [nconv, d, z, v, comm, icomm, ifail] = f12aq(sigma, resid, v, comm, icomm); % sort to avoid different orders showing up as errorfunction [w] = f12_av(nx, v)inx = double(nx); % nx is int64 w = complex(zeros(inx*inx,1)); h2 = double(-(inx+1)^2); w(1:inx) = tv(inx, v(1:inx)); w(1:inx) = h2*v(inx+1:2*inx)+w(1:inx); for j=2:double(inx-1) lo = (j-1)*inx +1; hi = j*inx; w(lo:hi) = tv(inx, v(lo:hi)); w(lo:hi) = h2*v(lo-inx:lo-1)+w(lo:hi); w(lo:hi) = h2*v(hi+1:hi+inx)+w(lo:hi); end lo = (inx-1)*inx +1; hi = inx*inx; w(lo:hi) = tv(inx, v(lo:hi)); w(lo:hi) = h2*v(lo-inx:lo-1)+w(lo:hi);function [y] = tv(inx,x)y = zeros(inx,1); dd = double(4*(inx+1)^2); dl = double(-(inx+1)^2 - 0.5*100*(inx+1)); du = double(-(inx+1)^2 + 0.5*100*(inx+1)); y(1) = dd*x(1) + du*x(2); for j=2:double(inx-1) y(j) = dl*x(j-1) + dd*x(j) + du*x(j+1); end y(inx) = dl*x(inx-1) + dd*x(inx);

Iteration 1, No. converged = 0, norm of estimates = 133.43269 Iteration 2, No. converged = 0, norm of estimates = 99.725711 Iteration 3, No. converged = 0, norm of estimates = 42.48146 Iteration 4, No. converged = 0, norm of estimates = 8.1805323 Iteration 5, No. converged = 0, norm of estimates = 1.7794793 Iteration 6, No. converged = 0, norm of estimates = 0.49794897 Iteration 7, No. converged = 0, norm of estimates = 0.12604717 Iteration 8, No. converged = 0, norm of estimates = 0.026883649 Iteration 9, No. converged = 0, norm of estimates = 0.0070182429 Iteration 10, No. converged = 0, norm of estimates = 0.0014438635 Iteration 11, No. converged = 0, norm of estimates = 0.00040007427 Iteration 12, No. converged = 0, norm of estimates = 0.0001059392 Iteration 13, No. converged = 0, norm of estimates = 2.803229e-05 Iteration 14, No. converged = 0, norm of estimates = 7.7343157e-06 Iteration 15, No. converged = 0, norm of estimates = 1.9579037e-06 Iteration 16, No. converged = 0, norm of estimates = 6.15616e-07 Iteration 17, No. converged = 0, norm of estimates = 1.2591405e-07 Iteration 18, No. converged = 0, norm of estimates = 3.7790713e-08 Iteration 19, No. converged = 2, norm of estimates = 8.3787181e-09 Iteration 20, No. converged = 2, norm of estimates = 1.8827432e-09 Iteration 21, No. converged = 2, norm of estimates = 1.6920778e-10 Iteration 22, No. converged = 2, norm of estimates = 2.299125e-11 Iteration 23, No. converged = 2, norm of estimates = 3.391976e-12 Iteration 24, No. converged = 2, norm of estimates = 5.5954988e-13

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