Приложение 2.Подпрограммаgeteps.M определения погрешности при заданном параметре сходимости.
function [eps] = geteps(mu)
M = 10; N = 10;
a = 1; b = 1;
h1 = a/M; h2 = b/N; q = h2/h1;
u = ones(M+1, N+1);
x = zeros(M+1,1);
y = zeros(N+1,1);
x = 0:h1:1;
y = 0:h2:1;
for i = 1:(M+1),
u(i,1) = 0;
u(i,N+1) = sin(pi*(i-1)*h1/2);
end
for j = 1:(N+1),
u(1,j) = 0;
u(M+1,j) = (j-1)*h2;
end
k = 0;
MAX = 15;
A = 100;
a1 = 1/(2*(1+q^2));
a2 = q^2;
a3 = h2^2*a1;
for k = 1:MAX,
A = 0;
for i=2:M,
for j=2:N,
X = a1*(u(i,j-1)+u(i,j+1)+a2*u(i-1,j)+a2*u(i+1,j))-a3;
X = mu*X + (1-mu)*u(i,j);
R = abs(X - u(i,j));
if (R > A) A = R; end
u(i,j) = X;
end
end
end
eps = A;
Приложение 3. Программа scanmu.M построения зависимости достигнутой погрешности вычислений от параметра сходимости.
left = input('Input left border of mu-range: ');
right = input('Input right border of mu-range: ');
epsmu = input('Input the quality of mu-parameter definition: ');
mu = left:epsmu:right;
eps = zeros(size(mu));
N = (right-left)/epsmu;
for i=1:(N+1),
eps(i) = geteps(mu(i));
end
eps(N+1) = eps(N);
plot(mu, eps);
Приложение
4. Результат
вычислений программы tikhiy.m
при значениях
N=M=10,
.
|
i \ j |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
1 |
0 |
0.0010 |
0.0057 |
0.0127 |
0.0218 |
0.0333 |
0.0510 |
0.0686 |
0.0915 |
0.1199 |
0.1564 |
|
2 |
0 |
0.0053 |
0.0158 |
0.0305 |
0.0494 |
0.0759 |
0.1068 |
0.1430 |
0.1873 |
0.2416 |
0.3090 |
|
3 |
0 |
0.0114 |
0.0290 |
0.0522 |
0.0840 |
0.1216 |
0.1666 |
0.2195 |
0.2831 |
0.3601 |
0.4540 |
|
4 |
0 |
0.0187 |
0.0445 |
0.0793 |
0.1209 |
0.1700 |
0.2278 |
0.2955 |
0.3758 |
0.4719 |
0.5878 |
|
5 |
0 |
0.0272 |
0.0642 |
0.1084 |
0.1599 |
0.2198 |
0.2891 |
0.3691 |
0.4629 |
0.5740 |
0.7071 |
|
6 |
0 |
0.0389 |
0.0853 |
0.1394 |
0.2011 |
0.2708 |
0.3496 |
0.4392 |
0.5427 |
0.6642 |
0.8090 |
|
7 |
0 |
0.0497 |
0.1077 |
0.1726 |
0.2441 |
0.3226 |
0.4092 |
0.5055 |
0.6146 |
0.7410 |
0.8910 |
|
8 |
0 |
0.0632 |
0.1334 |
0.2093 |
0.2904 |
0.3767 |
0.4692 |
0.5691 |
0.6793 |
0.8042 |
0.9511 |
|
9 |
0 |
0.0791 |
0.1634 |
0.2510 |
0.3416 |
0.4350 |
0.5317 |
0.6325 |
0.7393 |
0.8553 |
0.9877 |
|
10 |
0 |
0.1000 |
0.2000 |
0.3000 |
0.4000 |
0.5000 |
0.6000 |
0.7000 |
0.8000 |
0.9000 |
1.0000 |