I am new to MATLAB and I am encountering a bit of trouble writing a program to calculate a an integral which itself depends on functions that get integrated each time the integrand is called.
I have already written a function for the integrand, which, when called for a given value of its argument, yields the result that I expect. However, when I try to integrate over this argument, I get multiple errors about disagreeing matrix dimensions. The strange thing is that I can evaluate the integrand at all values for the argument.
My code is as follows:
%DECLARATION OF q1 and Q1
function y = q1(w, tt)
y = exp(-tau*w).*sin(tt.*w)./w;
function y = Q1(tt)
y = integral(@(w) q1(w, tt), 0, Inf, 'AbsTol', 1e-12);
%DECLARATION OF q2 and Q2
function y = q2(w, tt)
if (tt*w) < 1e-5
y = (1/24.0)*exp(-tau.*w).*tt.*tt.*(12.0 - w.*w.*tt.*tt).*coth(0.5*w).*w;
y = exp(-tau.*w).*(1.0 - cos(tt.*w)).*coth(0.5*w)./w;
function y = Q2(tt)
y = integral(@(w) q2(w, tt), 0, Inf, 'AbsTol', 1e-12);
function y = Integrand(tt)
y = cos(Omega*tt + Q1(tt)).*exp(-Q2(tt));
function y = Rate(t)
y = integral(@(tt) Integrand(tt), 1e-3, t);
I appreciate any feedback you can give me.
For efficiency (and it *really* matters for that), MATLAB tries to
vectorize the calls to the integrand function. If you read the doc, it
will tell you that your integrand function must accept an *array* of
inputs and return an array of corresponding outputs. If you have a
function which cannot do that, and most of your functions cannot, you
have a couple of options. The simplest is to regard your problem as a
vector-valued integral with just 1 component. When integrating
vector-valued integrands, INTEGRAL only passes scalars to the integrand
function. To make that work, simply append the option-value pair
to the argument list for each call to INTEGRAL.
The other approach (and almost always a faster one) is to use ARRAYFUN
to vectorize a function that only accepts scalars. So, for example, to
vectorize q2, you might use
q2v = @(w,tt)arrayfun(@q2,w,tt);
instead of q2 in the definition of Q2. I do think ARRAYFUN can get
confusing, so if I were you I would use the 'ArrayValued' option *first*
and then try to remove those one at a time by introducing uses of ARRAYFUN.
I notice that you are only supplying 'AbsTol' below. That is probably a
mistake. INTEGRAL attempts to meet a *mixed* error control condition
involving an estimate of the relative error. Absolute error control is
only relevant when the true answer approaches zero. A good way of
thinking about it is that the 'RelTol' indicates how many significant
digits you want (e.g. 1e-5 means about 5 significant digits, or almost
that), and 'AbsTol' is a kind of "threshold", a "good enough" error when
it becomes difficult to achieve the requested relative tolerance because
cancellation of positive and negative areas lead to a small correct
answer. Technically, only 'AbsTol' only matters if |Q| < AbsTol/RelTol
and only 'RelTol' matters otherwise. The default RelTol is 1e-6, so
when combined with the default RelTol, an AbsTol of 1e-12 only matters
if |Q| < 1e-12/1e-6 = 1e-6.
Finally, you will probably observe that when nesting integrals you will
get more accuracy than you ask for. That sounds like a good thing, but
it is not coming for free. There are a couple of reasons for it, one
being that INTEGRAL has a conservative initial mesh (meaning that there
are quite a few evaluations points at a minimum). The other reason I
suspect is more technical and has to do with truncation error associated
with cross-product terms.
> I am new to MATLAB and I am encountering a bit of trouble writing a
> program to calculate a an integral which itself depends on functions
> that get integrated each time the integrand is called.
> I have already written a function for the integrand, which, when called
> for a given value of its argument, yields the result that I expect.
> However, when I try to integrate over this argument, I get multiple
> errors about disagreeing matrix dimensions. The strange thing is that I
> can evaluate the integrand at all values for the argument.
> My code is as follows:
> %DECLARATION OF q1 and Q1
> function y = q1(w, tt)
> y = exp(-tau*w).*sin(tt.*w)./w;
> function y = Q1(tt)
> y = integral(@(w) q1(w, tt), 0, Inf, 'AbsTol', 1e-12); end
> %DECLARATION OF q2 and Q2
> function y = q2(w, tt)
> if (tt*w) < 1e-5
> y = (1/24.0)*exp(-tau.*w).*tt.*tt.*(12.0 -
> y = exp(-tau.*w).*(1.0 - cos(tt.*w)).*coth(0.5*w)./w;
> function y = Q2(tt)
> y = integral(@(w) q2(w, tt), 0, Inf, 'AbsTol', 1e-12); end
> %RATE INTEGRAL
> function y = Integrand(tt)
> y = cos(Omega*tt + Q1(tt)).*exp(-Q2(tt));
> function y = Rate(t)
> y = integral(@(tt) Integrand(tt), 1e-3, t);
> I appreciate any feedback you can give me.
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