Solve for parameters of an integral (fsolve?)

Dear All, let a,c,d,e be real number parameters. Let x also be a parameter, which I would like to solve the attached equation for (using matlab). f(y) is a density function (for the beginning it can be a normal distribution). Can anyone point towards the general strategy to solve this and some useful commands?

Thanks in advance!

Answer 1

[Matt J]

FZERO would be enough, I expect, since it is a 1D root finding problem. You can use commands like TRAPZ or INTEGRAL to get a numerical approximation of the integral. Also, in the case of a normal distribution, f(y), the integral of the terms (d+x)*f(y) can be evaluated using calls to the ERF command while the term y*f(y) has a closed analytical form.

[Matt J]

roblocks replied

Dear Matt, thanks! I tried to implement, but I am running into a problem using the integral command. In a later version I want to solve for r. For now assume r is given and I would just like to compute the integral. The following code produces an error:

clear all; clc;

epsilon   = 0.5;
r         = 1.08;
D         = 10;
r_s       = 1.01;
mu        = -1;
sigma     = 8;
E_guess   = 12;
r_guess   = 1.08;

% define my function that is to be integrated over epsilon

second_summand = @(epsilon,r,D,r_s, mu, sigma, E_guess, r_guess)
((r*D+E_guess - (r_guess-1)-epsilon)*normpdf(epsilon,mu,sigma));

second_summand(epsilon,r,D,r_s, mu, sigma, E_guess, r_guess)

disp('second_summand function seems to be working.')

% clear epsilon to have it as a variable
clear epsilon

lb = E_guess - (r_guess-1)
ub = E_guess - (r_guess-1)+r*D

% compute the integral
integral( @(epsilon)
second_summand(epsilon,r,D,r_s, mu, sigma, E_guess, r_guess) ,lb   , ub)

It simply copied what the integral reference page is suggesting for syntax if one uses parametric integrals.

[Matt J]

Use .* in your definition of second_summand so that it gives vectorized output.