The FEAST algorithm for Hermitian eigenproblems
FEAST is an algorithm for computing a few eigenpairs of a large sparse eigenvalue problem , where is a Hermitian matrix, and is Hermitian positive definite . FEAST belongs to the class of contour-based eigensolvers which have recently attracted a lot of attention .
Mathematically, FEAST can be interpreted as a subspace iteration applied to a rational matrix function which serves the purpose of separating the wanted eigenvalues from the unwanted (i.e., it acts as a filter function). The RKFUN calculus of the RKToolbox is a convenient tool for working with rational functions and it allows for a simple implementation of the basic FEAST iteration. Of course, the full FEAST implementation comes with many more "bells and whistles," and in this tutorial we only aim to demonstrate the basic idea.
Let us consider an eigenvalue problem , where is a finite-difference discretization of the Laplacian on a disk with homogeneous Dirichlet boundary conditions. Using MATLAB's eigs function, we first compute and visualize the eigenvectors corresponding to the smallest eigenvalues of :
n = 150; G = numgrid('D', n); A = (n+1)^2*delsq(G); N = size(A, 1); B = speye(N); [V, D] = eigs(A, 9, 'SM'); [D, ind] = sort(diag(D)); V = V(:, ind); figure(1) for j = 1:9 subplot(3, 3, j) v = V(:, j); U = NaN*G; U(G>0) = full(v(G(G>0))); contourf(U); prism, axis square ij, axis off title(['\lambda = ' num2str(D(j))]); end
Suppose we want to find these eigenmodes using FEAST, then we need to contruct a rational filter which would separate the corresponding eigenvalues of from the others. To this end, we combine a type step function approximation for the interval with a linear transformation mapping to The result is a near-optimal rational filter for the interval , attaining values very close to on that interval, and oscillating about outside:
lmin = 0; lmax = 220; x = rkfun(); % independent variable x t = 2/(lmax-lmin)*x - (lmin+lmax)/(lmax-lmin); % map [lmin,lmax] -> [-1,1] s = rkfun('step', 5); r = s(t); figure, ezplot(r), hold on
Let's compute the separation ratio between the approximate values for and by finding the real local extrema of , and then finding the smallest/largest modulus of inside/outside at these extrema. Such computations can become quite sensitive, in particular with rational functions of high degree. We therefore recommend performing this using the Advanpix Multipleprecision Toolbox :
r = mp(r); extrema = roots(diff(r), 'real'); values = r(extrema); plot(extrema, values, 'ro') mini = min(abs(r(extrema(lmin < extrema & extrema < lmax)))); maxi = max(abs(r(extrema(lmin > extrema | extrema > lmax)))); plot([lmin, lmax], [mini, mini], 'k--') plot([-1e6, lmin], [maxi, maxi], 'k--') plot([lmax, 1e6], [maxi, maxi], 'k--')
According to , the approximate convergence factor of FEAST for finding the eigenpairs in the interior of is given by the following ratio:
ratio = double(maxi/mini)
ratio = 1.9351e-02
We can easily verify this numerically by implementing a simple subspace iteration applied to . Conveniently, the RKToolbox allows us to simply type for computing , the action of the rational filter on a basis for the search space.
% Change to double precision. r = double(r); % Search space basis V of dimension m. m = 10; V = randn(N, m); for iter = 1:8 % Apply rational filter to V. V = r(A, V); % Compute and sort Ritz pairs. Am = V'*A*V; Bm = V'*B*V; [W, D] = eig(Am, Bm); [D, ind] = sort(diag(D)); W = W(:, ind); % B-normalize W. nrm = sqrt(diag(W'*Bm*W)); W = W/diag(nrm); % Form approximate eigenvectors. V = V*W; % Check residuals and number of eigenpairs inside % search iterval. Di = diag(D(lmin < D & D < lmax)); Vi = V(:, lmin < D & D < lmax); resid(iter) = norm(A*Vi - B*Vi*Di, 'fro'); nrvec(iter) = size(Vi, 2); end
Here is a plot of the residual norms after each FEAST iteration, together with the expected convergence factor, and the number of Ritz values inside the search iterval:
figure semilogy(resid, 'b-o'), hold on semilogy(1e10*ratio.^(1:iter), 'k--') for j = 1:iter hdl = text(j-.1,5*resid(j), num2str(nrvec(j))); set(hdl, 'FontSize', 14, 'Color', 'b') end xlim([.5, iter+.5]), xlabel('iteration') legend('residual norm','expected rate') ylim([1e-8, 1e6])
There is a very good agreement between the predicted and observed convergence. In the first iteration, no Ritz value is in the search interval, but after only iterations we have found Ritz values. Indeed, looking at the $9$th eigenvector in the plot above, we expect by symmetry that there will be a $10$th eigenvalue corresponding to the same eigenvalue .
As mentioned above, our basic FEAST implementation is not at all efficient and serves only the purpose of a demonstration. A main feature of FEAST is its parallelism in evaluating by using a partial fraction expansion of of the form
We can easily compute such an expansion using the residue command of the RKToolbox, again in multiple precision to avoid accuracy loss. Here is a plot of the poles of in the complex plane, relative to the search interval :
[alpha,beta] = residue(mp(r)); figure plot(beta, 'ro'), hold on plot([lmin, lmax], [0, 0], 'k-')
That's it. The following command creates a thumbnail
figure(1), hold on, plot(NaN)
 Advanpix LLC., Multiprecision Computing Toolbox for MATLAB, ver 18.104.22.16813, Tokyo, Japan, 2017. http://www.advanpix.com/.
 S. Güttel, E. Polizzi, P. Tang, and G. Viaud. Zolotarev quadrature rules and load balancing for the FEAST eigensolver, SIAM J. Sci. Comput., 37(4):A2100--A2122, 2015.
 A. Imakura, L. Du, and T. Sakurai. A map of contour integral-based eigensolvers for solving generalized eigenvalue problems, http://arxiv.org/abs/1510.02572, 2015
 E. Polizzi. Density-matrix-based algorithm for solving eigenvalue problems, Phys. Rev. B, 79:115112, 2009.