MATLAB: Solving for simple Integration symbol

integration

When integrate for rho in this equation b/(1-b*rho), I get -log(b*rho-1) which is wrong. It should come out to be -log(1-b*rho).

My code:

syms A B Z a b R T rho

q1 = b/(1-b*rho)

I1 = int(q1,rho)

I1 = – log(b*rho – 1)

Best Answer

HI AC

d/drho (-)*log(b*rho-1) = (-)*1/(b*rho-1)*b = b/(1-b*rho) = q1

so it is correct. But your result is correct as well.

Ignoring the (-) in front for the moment, your result is

log(1-b*rho) = log((-1)*(b*rho-1)) = log(b*rho-1) + log(-1)
= log(b*rho-1) +i*pi

which differs from the 'int' result by a constant of integration i*pi. Whichever result you want to use would usually be decided by keeping the argument of the log function to be positive.

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Integrate with respect to what?

The del operator together with the magnitude operator imply that rho is not a scalar but is instead a function. You are not going to have any success integrating F[rho] without naming a particular rho.

syms rho(x,y) R T alpha beta kappa real
F = rho*R*T*log(rho/(1-beta*rho)) - alpha*rho^2 + kappa/2 * (diff(rho,x).^2 + diff(rho,y).^2)

but integrating F with respect to x or y will not go anywhere without a particular rho.

MATLAB: Accessing function output for different values of variable

rho_grid= -1:0.1:1;
Nrho = length(rho_grid);
for i = Nrho : -1 : 1                        %backwards! For pre-allocation reasons
    rho = rho_grid(i);
    Rq = @(rho) chol(value(Q(rho)),'upper');
    Rp = @(rho) chol(value(P(rho)),'lower');
    pro = @(rho) Rq(rho)*Rp(rho);
    [Uvals(:,:,i), Svals(:,:,i), Vvals(:,:,i)] = svd(pro(rho));
end
U = @(rho) Uvals(:,:,interp1(rho_grid, 1:Nrho, rho, 'nearest'));
S = @(rho) Svals(:,:,interp1(rho_grid, 1:Nrho, rho, 'nearest'));
V = @(rho) Vvals(:,:,interp1(rho_grid, 1:Nrho, rho, 'nearest'));

Instead of using 'nearest' in computing the index, you could go for something more sophisticated such as determining the mixing between the two closest values, and doing a weighted calculation to interpolate at the rho.

Best Answer

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MATLAB: Accessing function output for different values of variable

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