Plotting a function in matlab involving an integral - function

Question

I'm trying to plot a function that contains a definite integral. My code uses all anonymous functions. When I run the file, it gives me an error. My code is below:
%%% List of Parameters %%%
gamma_sp = 1;
cap_gamma = 15;
gamma_ph = 0;
omega_0 = -750;
d_omega_0 = 400;
omega_inh = 100;
d_omega_inh = 1000;
%%% Formulae %%%
gamma_t = gamma_sp/2 + cap_gamma/2 + gamma_ph;
G = #(x) exp(-(x-omega_inh).^2./(2*d_omega_inh.^2))./(sqrt(2*pi)*d_omega_inh);
F = #(x) exp(-(x-omega_0).^2./(2*d_omega_0.^2))./(sqrt(2*pi)*d_omega_0);
A_integral = #(x,y) G(x)./(y - x + 1i*gamma_t);
Q_integral = #(x,y) F(x)./(y - x + 1i*gamma_t);
A = #(y) integral(#(x)A_integral(x,y),-1000,1000);
Q = #(y) integral(#(x)Q_integral(x,y),-3000,0);
P1 = #(y) -1./(1i.*(gamma_sp + cap_gamma)).*(1./(y + 2.*1i.*gamma_t)*(A(y)-conj(A(0)))-1./y.*(A(y)-A(0))+cap_gamma./gamma_sp.*Q(y).*(A(0)-conj(A(0))));
P2 = #(y) conj(P1(y));
P = #(y) P1(y) - P2(y);
sig = #(y) abs(P(y)).^2;
rng = -2000:0.05:1000;
plot(rng,sig(rng))
It seems to me that when the program runs the plot command, it should put each value of rng into sig(y), and that value will be used as the y value in A_integral and Q_integral. However, matlab throws an error when I try to run the program.
Error using -
Matrix dimensions must agree.
Error in #(x,y)G(x)./(y-x+1i*gamma_t)
Error in #(x)A_integral(x,y)
Error in integralCalc/iterateScalarValued (line 314)
fx = FUN(t);
Error in integralCalc/vadapt (line 133)
[q,errbnd] = iterateScalarValued(u,tinterval,pathlen);
Error in integralCalc (line 76)
[q,errbnd] = vadapt(#AtoBInvTransform,interval);
Error in integral (line 89)
Q = integralCalc(fun,a,b,opstruct);
Error in #(y)integral(#(x)A_integral(x,y),-1000,1000)
Error in
#(y)-1./(1i.*(gamma_sp+cap_gamma)).*(1./(y+2.*1i.*gamma_t)*(A(y)-conj(A(0)))-1. /y.*(A(y)-A(0))+cap_gamma./gamma_sp.*Q(y).*(A(0)-conj(A(0))))
Error in #(y)P1(y)-P2(y)
Error in #(y)abs(P(y)).^2
Error in fwm_spec_diff_paper_eqn (line 26)
plot(rng,sig(rng))
Any ideas about what I'm doing wrong?

Answer 1

You have
>> rng = -2000:0.05:1000;
>> numel(rng)
ans =
60001
all 60001 elements get passed down to
A = #(y) integral(#(x)A_integral(x,y),-1000,1000);
which calls
A_integral = #(x,y) G(x)./(y - x + 1i*gamma_t);
(similar for Q). The thing is, integral is an adaptive quadrature method, meaning (roughly) that the amount of x's it will insert into A_integral varies with how A_integral behaves at certain x.
Therefore, the amount of elements in y will generally be different from the elements in x in the call to A_integral. This is why y-x +1i*gamma_t fails.
Considering the complexity of what you're trying to do, I think it is best to re-define all anonymous functions as proper functions, and integrate a few of them into single functions. Look into the documentation of bsxfun to see if that can help (e.g., bsxfun(#minus, y.', x) instead of y-x could perhaps fix a few of these issues), otherwise, vectorize only in x and loop over y.

Answer 2

Thanks Rody, that made sense to me. I keep trying to use matlab like mathematica and I forget how matlab does things. I modified the code a bit, and it produces the right result. The integrals are evaluated very roughly, but it should be easy to fix that. I've posted my modified code below.
%%% List of Parameters %%%
gamma_sp = 1;
cap_gamma = 15;
gamma_ph = 0;
omega_0 = -750;
d_omega_0 = 400;
omega_inh = 100;
d_omega_inh = 1000;
%%% Formulae %%%
gamma_t = gamma_sp/2 + cap_gamma/2 + gamma_ph;
G = #(x) exp(-(x-omega_inh).^2./(2*d_omega_inh.^2))./(sqrt(2*pi)*d_omega_inh);
F = #(x) exp(-(x-omega_0).^2./(2*d_omega_0.^2))./(sqrt(2*pi)*d_omega_0);
A_integral = #(x,y) G(x)./(y - x + 1i*gamma_t);
Q_integral = #(x,y) F(x)./(y - x + 1i*gamma_t);
w = -2000:0.05:1000;
sigplot = zeros(size(w));
P1plot = zeros(size(w));
P2plot = zeros(size(w));
Pplot = zeros(size(w));
aInt_range = -1000:0.1:1200;
qInt_range = -2000:0.1:100;
A_0 = sum(A_integral(aInt_range,0).*0.1);
for k=1:size(w,2)
P1plot(k) = -1./(1i*(gamma_sp + cap_gamma)).*(1./(w(k)+2.*1i.*gamma_t).*(sum(A_integral(aInt_range,w(k)).*0.1)-conj(A_0))-1./w(k).*(sum(A_integral(aInt_range,w(k)).*0.1)-A_0)+cap_gamma./gamma_sp.*sum(Q_integral(qInt_range,w(k)).*0.1).*(A_0-conj(A_0)));
P2plot(k) = conj(P1plot(k));
Pplot(k) = P1plot(k) - P2plot(k);
sigplot(k) = abs(Pplot(k)).^2;
end
plot(w,sigplot)