Double Numerical Integration Error

问题

In the script below, in the for loop for calculating the double integrations, I keep receiving an error, and I'm unsure why:

Error using  * 
Inner matrix dimensions must agree.

Error in @(t,p)-sin(t)*(G(:,1)*cos(p)+H(:,1)*sin(p))


Error in @(t,p)B{k}(t,p).*A{k}(t,p).*(V(t)-Veq).*sin(t)


Error in integral2Calc>integral2t/tensor (line 228)
        Z = FUN(X,Y);  NFE = NFE + 1;

Error in integral2Calc>integral2t (line 55)
[Qsub,esub] = tensor(thetaL,thetaR,phiB,phiT);

Error in integral2Calc (line 9)
    [q,errbnd] = integral2t(fun,xmin,xmax,ymin,ymax,optionstruct);

Error in integral2 (line 106)
    Q = integral2Calc(fun,xmin,xmax,yminfun,ymaxfun,opstruct);

This script was formed from another which only had to deal with one variable t, so I'm assuming that I've done something wrong in adapting it to two variable functions.

Thanks!

%%Calculation of dH/dt for mode m=1 for the entire sphere, NH and SH
clear all
%%Radius of photosphere
r = 6.957*(10^5); %In km
R = 1/r; %This will come in handy later

%%Call in spherical harmonic coefficients, change the 535 figure as more
%%data is added to the spreadsheets
G(:,1) = xlsread('G Coefficients.xls', 'D3:D535');
G(:,2) = xlsread('G Coefficients.xls', 'F3:F535');
G(:,3) = xlsread('G Coefficients.xls', 'I3:I535');
G(:,4) = xlsread('G Coefficients.xls', 'M3:M535');
G(:,5) = xlsread('G Coefficients.xls', 'R3:R535');
G(:,6) = xlsread('G Coefficients.xls', 'X3:X535');
G(:,7) = xlsread('G Coefficients.xls', 'AE3:AE535');
G(:,8) = xlsread('G Coefficients.xls', 'AM3:AM535');
G(:,9) = xlsread('G Coefficients.xls', 'AV3:AV535');

H(:,1) = xlsread('H Coefficients.xls', 'D3:D535');
H(:,2) = xlsread('H Coefficients.xls', 'F3:F535');
H(:,3) = xlsread('H Coefficients.xls', 'I3:I535');
H(:,4) = xlsread('H Coefficients.xls', 'M3:M535');
H(:,5) = xlsread('H Coefficients.xls', 'R3:R535');
H(:,6) = xlsread('H Coefficients.xls', 'X3:X535');
H(:,7) = xlsread('H Coefficients.xls', 'AE3:AE535');
H(:,8) = xlsread('H Coefficients.xls', 'AM3:AM535');
H(:,9) = xlsread('H Coefficients.xls', 'AV3:AV535');

%%Set function v which always remains the same
nhztoradperday = 2*pi*86400*(10^(-9));
a = 460.7*nhztoradperday;
b = -62.69*nhztoradperday;
c = -67.13*nhztoradperday;

B{1} = @(t,p) -sin(t)*(G(:,1)*cos(p) + H(:,1)*sin(p));

B{2} = @(t,p) -3*sin(t)*cos(t)*(G(:,2)*cos(p) + H(:,2)*sin(p));

B{3} = @(t,p) -1.5*(5*(cos(t)^2)-1)*sin(t)*(G(:,3)*cos(p) + H(:,3)*sin(p));

B{4} = @(t,p) -2.5*(7*(cos(t)^3)-3*cos(t))*sin(t)*(G(:,4)*cos(p) + H(:,4)*sin(p));

B{5} = @(t,p) -1.875*sin(t)*(21*(cos(t)^4)-14*(cos(t)^2)+1)*(G(:,5)*cos(p) + H(:,5)*sin(p));

B{6} = @(t,p) -2.625*cos(t)*sin(t)*(33*(cos(t)^4)-30*(cos(t)^2)+5)*(G(:,6)*cos(p) + H(:,6)*sin(p));

B{7} = @(t,p) -0.4375*sin(t)*(429*(cos(t)^6)-495*(cos(t)^4)+135*(cos(t)^2)-5)*(G(:,7)*cos(p) + H(:,7)*sin(p));

B{8} = @(t,p) -0.5625*cos(t)*sin(t)*(715*(cos(t)^6)-1001*(cos(t)^4)+385*(cos(t)^2)-35)*(G(:,8)*cos(p) + H(:,8)*sin(p));



A{1} = @(t,p) 0.5*R*cos(t)*(G(:,1)*cos(p) + H(:,1)*sin(p));

A{2} = @(t,p) 0.5*R*cos(2*t)*(G(:,2)*cos(p) + H(:,2)*sin(p));

A{3} = @(t,p) 0.125*R*cos(t)*(15*(cos(t)^2)-11)*(G(:,3)*cos(p) + H(:,3)*sin(p));

A{4} = @(t,p) 0.125*R*(-3*cos(2*t)+7*(cos(t)^4-3*sin(t)^2*cos(t)^2))*(G(:,4)*cos(p) + H(:,4)*sin(p));

A{5} = @(t,p) 0.0625*R*(21*(cos(t)^5)-(cos(t)^3)*(14+84*(sin(t)^2))+cos(t)*(1+28*(sin(t)^2)))*(G(:,5)*cos(p) + H(:,5)*sin(p));

A{6} = @(t,p) 0.0625*R*(33*(cos(t)^6)-(cos(t)^4)*(165*(sin(t)^2)+30)+90*(sin(t)^2)*(cos(t)^2)+5*cos(2*t))*(G(:,6)*cos(p) + H(:,6)*sin(p));

A{7} = @(t,p) 1/128*R*(429*(cos(t)^7)-(cos(t)^5)*(495+2574*(sin(t)^2))+(cos(t)^3)*(135+1980*(sin(t)^2))-cos(t)*(5+270*(sin(t)^2)))*(G(:,7)*cos(p) + H(:,7)*sin(p));

A{8} = @(t,p) 1/128*R*(715*(cos(t)^8)-1001*(cos(t)^6)+385*(cos(t)^4)-35*(cos(t)^2)+(sin(t)^2)*(-5005*(cos(t)^6)+5005*(cos(t)^4)-1155*(cos(t)^2)+35))*(G(:,8)*cos(p) + H(:,8)*sin(p));

V = @(t) a + (b*cos(t)^2) + (c*cos(t)^4);
Veq = V(0);

intNH = zeros(length(G),9);
intSH = zeros(length(G),9);
intSun = zeros(length(G),9);
for k=1:8
   fun{k} = @(t,p) B{k}(t,p).*A{k}(t,p).*(V(t)-Veq).*sin(t);
   intNH(:,k) = integral2(fun{k},0,pi/2,0,2*pi);
   intSH(:,k) = integral2(fun{k},pi/2,pi,0,2*pi);
   intSun(:,k) = integral2(fun{k},0,pi,0,2*pi);
end

for i=1:length(G)
   NH(i) = sum(intNH(i,:));
   SH(i) = sum(intSH(i,:));
   Sun(i) = sum(intSun(i,:));
end

回答1:

Unfortunately what you are trying to attempt will probably not work as-is. Considering that I know the history of the question, I know you're trying to integrate an array-valued function. This worked in 1d, but I'm afraid it won't work in 2d.

If you look at the help of integral2 as it has already been suggested in comments, you'll see this:

All input functions must accept arrays as input and operate elementwise. The function Z = FUN(X,Y) must accept arrays X and Y of the same size and return an array of corresponding values.

This means that the output from the fun entering integral2 can't have more dimensions than the input; in other words integral2 will only integrate scalar functions for you.

After a cursory glance over the options, I don't think the built-in 2d integrators support this. You have two options, and each are inefficient, so you should try both and see which is better for your application.

Option one you already know: loop over each index of your array-valued function, and integrate these scalars using interp2. This will be slow because you need a loop over the elements of your array, and interp2d has to be called for each.

Option two is to perform the double integral as two single integrals. I mean formally doing

integrated_result = integral(@(t)integral(@(p) fun(t,p),p1,p2,'arrayvalued',true),...
                             t1,t2,'arrayvalued',true);

to integrate in p from p1 to p2 and from t1 to t2. This will be slow because for each value of the outer variables you need to call integral.

来源：https://stackoverflow.com/questions/38104305/double-numerical-integration-error

标签

matlab

numerical-integration