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August 29, 2015 14:07
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Shifted BiCG method
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function x = shifted_bicg(A, b, sigma, tol, max_iter) | |
%%% initialization | |
if nargin < 4 | |
tol = 1e-6; | |
max_iter = 1000; | |
end | |
if nargin < 5 | |
max_iter = 1000; | |
end | |
shift_convergence_flag = zeros(1, length(sigma)); | |
iters = ones(1, length(sigma)) * max_iter; | |
norm_b = norm(b); | |
r = b; | |
r_shadow = r; | |
x_seed = zeros(length(A), 1); | |
x_shift = zeros(length(A), length(sigma)); | |
p = zeros(length(A), 1); | |
p_shadow = zeros(length(A), 1); | |
p_shift = zeros(length(A), length(sigma)); | |
alpha_old = 1; | |
rho_old = 1; | |
pi_shift_old = ones(1, length(sigma)); | |
pi_shift = ones(1, length(sigma)); | |
for iter = 1:max_iter | |
%%% seed system | |
rho = r_shadow' * r; | |
beta = -rho / rho_old; | |
p = r - beta * p; | |
p_shadow = r_shadow - conj(beta) * p_shadow; | |
q = A * p; | |
alpha = rho / (r_shadow' * q); | |
x_seed = x_seed + alpha * p; | |
%%% shifted system | |
pi_shift_new = (1 + alpha * sigma) .* pi_shift ... | |
+ alpha * beta / alpha_old * (pi_shift_old - pi_shift); | |
beta_shift = power(pi_shift_old ./ pi_shift, 2) * beta; | |
alpha_shift = pi_shift ./ pi_shift_new * alpha; | |
for i = find(shift_convergence_flag == 0) | |
p_shift(:,i) = r / pi_shift(i) - p_shift(:,i) * beta_shift(i); | |
x_shift(:,i) = x_shift(:,i) + p_shift(:,i) * alpha_shift(i); | |
end | |
%%% update residual and scalars | |
r = r - alpha * q; | |
r_shadow = r_shadow - conj(alpha) * A' * p_shadow; | |
alpha_old = alpha; | |
rho_old = rho; | |
pi_shift_old = pi_shift; | |
pi_shift = pi_shift_new; | |
for i = find(shift_convergence_flag == 0) | |
if norm(r/pi_shift(i))/norm_b < tol | |
shift_convergence_flag(i) = 1; | |
iters(i) = iter; | |
end | |
end | |
%%% とりあえずseedが収束したら終わりにする | |
if norm(r) / norm_b < tol | |
break | |
end | |
end | |
x = [x_seed x_shift]; | |
%%% display results | |
disp('results for seed systems...'); | |
disp(strcat('relative residual:', num2str(norm(r)/norm(b)))); | |
disp(strcat('true residual:', num2str(norm(b-A*x_seed)/norm(b)))); | |
disp('results for shifted systems...'); | |
for i = 1:length(sigma) | |
disp(strcat('sigma:', num2str(sigma(i)))); | |
disp(strcat('relative residual:', num2str(norm(r/pi_shift(i))/norm(b)))); | |
disp(strcat('true residual:', num2str(norm(b-(A+sigma(i)*eye(length(A)))*x_shift(:,i))/norm(b)))); | |
end |
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