使用 * 时出错 - 内维矩阵必须一致

Question

我遇到了一个任务问题，我必须比较求解 ODE 的方法：ode45、Euler 和 Gauss-Legendre 方法。在这里，我必须计算不同步骤的误差。

   h=[0.01 0.05 0.1 0.5];
   func = @(t, y) -2*y+t*sin(t);
   opts = odeset('Reltol',1e-13,'AbsTol',1e-14,'Stats','on');
   f1_f2_matrix = [];
   sinus=@(t) t*(sin(t));
   error2nd = zeros(length(h),1);
   errorInf = zeros(length(h),1);
   error2ndEuler = zeros(length(h),1);
   errorInfEuler = zeros(length(h),1);


   for i=1:length(h)
   t=0:h(i):10;
   euler = zeros(length(t),1);
   fun_builtin = zeros(length(t),1);
   t_builtin = zeros(length(t),1);
   results = zeros(length(t),1);


   %matrix for A=-2 and given h=0.01

   matrix=[ 1-1/2*h, (1/2-sqrt(3)/3)*h; 
  (1/2+sqrt(3)/3)*h, 1-1/2*h];


    for n=2:length(t)
    B = [-2*results(n-1) + sinus(t(n-1)+(1/2-sqrt(3)/6)*h);
        -2*results(n-1) + sinus(t(n-1) + (1/2+sqrt(3)/6)*h)];

    %system of equations solution

    f1_f2_matrix = matrix\B; 

    results(n) = (results(n-1) + h*1/2*(f1_f2_matrix(1) + f1_f2_matrix(2))); 

    %euler
    euler(n) = euler(n-1)+h(i)*func(t(n-1),euler(n-1));

   end


   %ODE45 FUNCTION
[t_builtin,fun_builtin] = ode45(func, t, 0, opts);


   %y'=-2y+tsin(t) errors
 error2nd(i)=norm(fun_builtin-results)/norm(fun_builtin); %root mean square error of my function
errorInf(i)=norm((fun_builtin-results), Inf)/ norm((fun_builtin), Inf); %maximum error of my function

error2ndEuler(i)=norm(euler-results)/norm(fun_builtin); %root mean square error of my function (euler)
errorInfEuler(i)=norm((euler-results), Inf)/ norm((fun_builtin), Inf); %maximum error of my function (euler)

   end

   figure semilogy(h, error2nd, h, errorInf,h, error2ndEuler, h, errorInfEuler)
   title([{('Dependence of root mean square and maximum errors')}; {('from h-step and method of solving differential equation')}])
legend('root mean square error: Gauss-Legendre of 4th order','maximum error: Gauss-Legendre of 4th order method','root mean square error: Euler method', 'maximum error: Euler method', 'Location', 'southeast');
xlabel('step');
ylabel('error');

它应该在不同步骤生成不同方法的误差比较图。但是出现这样的错误：

  Error using  * 
  Inner matrix dimensions must agree.

  Error in untitled>@(t)t*(sin(t))

  Error in untitled (line 27)
    B = [-2*results(n-1) + sinus(t(n-1)+(1/2-sqrt(3)/6)*h);

我找不到错误。我试过应用点 ('.') 但它没有用。我哪里做错了？

Answer 1

使用点.*一定能解决问题。在此之后还有第二个问题，这是因为您没有为变量 h 建立索引。一旦你这样做，代码将运行完美。

Answer 2

h=[0.01 0.05 0.1 0.5 1];
func = @(t, y) -2*y+t*sin(t);
opts = odeset('Reltol',1e-13,'AbsTol',1e-14,'Stats','on');
f1_f2_matrix = [];
sinus=@(t) t*(sin(t));
error2nd = zeros(length(h),1);
errorInf = zeros(length(h),1);
error2ndEuler = zeros(length(h),1);
errorInfEuler = zeros(length(h),1);


for i=1:length(h)
t=0:h(i):10;
euler = zeros(length(t),1);
fun_builtin = zeros(length(t),1);
t_builtin = zeros(length(t),1);
results = zeros(length(t),1);


%matrix for A=-2 and given h=0.01
matrix=[ 1-1/2*h(i), (1/2-sqrt(3)/3)*h(i); 
        (1/2+sqrt(3)/3)*h(i), 1-1/2*h(i)];


    for n=2:length(t)
    B = [-2*results(n-1) + sinus(t(n-1)+(1/2-sqrt(3)/6)*h(i));
         -2*results(n-1) + sinus(t(n-1) + (1/2+sqrt(3)/6)*h(i))];

%system of equations solution

    f1_f2_matrix = matrix\B; 

    results(n) = (results(n-1) + h(i)*0.5*(f1_f2_matrix(1) + f1_f2_matrix(2))); 

%euler
    euler(n) = euler(n-1)+h(i)*func(t(n-1),euler(n-1));

   end


%ODE45 FUNCTION
   [t_builtin,fun_builtin] = ode45(func, t, 0, opts);


%y'=-2y+tsin(t) errors
error2nd(i)=norm(fun_builtin-results)/norm(fun_builtin); %root mean square error of my function
errorInf(i)=norm((fun_builtin-results), Inf)/ norm((fun_builtin), Inf); %maximum error of my function

error2ndEuler(i)=norm(euler-results)/norm(fun_builtin); %root mean square error of my function (euler)
errorInfEuler(i)=norm((euler-results), Inf)/ norm((fun_builtin), Inf); %maximum error of my function (euler)

end

figure 
semilogy(h, error2nd, h, errorInf,h, error2ndEuler, h, errorInfEuler)
title([{('Dependence of root mean square and maximum errors')}; {('from h-step and method of solving differential equation')}])
legend('root mean square error: Gauss-Legendre of 4th order','maximum error: Gauss-Legendre of 4th order method','root mean square error: Euler method', 'maximum error: Euler method', 'Location', 'southeast');
xlabel('step');
ylabel('error');

使用 * 时出错 - 内维矩阵必须一致

Error using *- inner dimension matrix must agree

matlab

matrix

ode