function u = solveLaplace(L, boundaryIds, boundaryVal, tol, maxit)

% Computes the solution u of the Laplace equation L*u = 0

% with boundary conditions u(boundaryIds) = boundaryVal.

%

% u = solveLaplace(L, boundaryIds, boundaryVal, tol, maxit)

%

% Inputs:

%   L: Laplacian matrix computed with cotmatrix() of gptoolbox [numPoints x numPoints]

%   boundaryIds: point IDs for Dirichlet boundary conditions [numBoundaryPoints x 1]

%   boundaryVal: values for Dirichlet boundary conditions [numBoundaryPoints x 1]

%   tol: tolerance for bicgstab

%   maxit: maximum number of iterations for bicgstab

%

% Outputs:

%   u: Laplace solution [numPoints x 1]

%

% Written by Steffen Schuler, Institute of Biomedical Engineering, KIT

if nargin < 5

    maxit = 500;

end

if nargin < 4

    tol = [];

end

if numel(boundaryIds) ~= numel(unique(boundaryIds))

    error('Duplicate entries found in boundaryIds.');

end

N = length(L);

K = numel(boundaryIds);

boundaryIds = double(boundaryIds);

% set up right-hand side

b = -L(:,boundaryIds)*boundaryVal(:);

b(boundaryIds) = boundaryVal;

% add boundary conditions to coefficient matrix

L(boundaryIds,:) = sparse(1:K, boundaryIds, ones(K,1), K, N);

L(:,boundaryIds) = sparse(boundaryIds, 1:K, ones(K,1), N, K);

% apply reverse Cuthill-McKee reordering

p = symrcm(L);

L = L(p,p);

b = b(p);

% compute incomplete LU factorization

[lowTri, upTri] = ilu(L);

% solve linear system

[x, flag, relres, iter] = bicgstab(L, b, tol, maxit, lowTri, upTri);

if flag

    warning('bicgstab failed at iteration %i with flag %i and relative residual %.1e.', iter, flag, relres);

end

u = NaN(N,1);

u(p) = x;

end