symbolic integral inside a numerical one. How to program it?

Hi guys,

I need to compute an integral

tau(T)=integral(a(u)*exp(-int(b(s),s,0,u),0,T);

so, I need to declare two functions a(u) and b(s) first and then compute the integral for specified T.

Somehow I always encounter problems when using this method. Please, help me!

Daniel.

Answer 1

[Shashank Prasanna]

If you are deriving a(u) and b(s) symbolically then you will need to convert them into matlab functions before you use them for numerical computations:

Here's an example:

syms x
y = x^2
yFun = matlabFunction(y)
integral(yFun,0,1)

Here's the documentation for matlabFunction:

http://www.mathworks.com/help/symbolic/matlabfunction.html

[Daniel]

a1 = 1; b2 = 2; s = .16; u = .07; T=1;

syms u; syms s;

matlabfirst=@ (u) first(u, a1)*exp(-int(matlabsecond,s,0,u)); matlabsecond=@ (s) second(s, b1);

tau=integral(matlabfirst, 0, T);

function y=first(u,a1) y=a1*u;

function z=second(s,b1) z=b1*s;

What do you think?

[Shashank Prasanna]

You can convert your symbolic function to matlab function using the function in the link I provided. Please try it first and post here if it didn't work for you. There are examples at the bottom on how to use it.

[Daniel]

function [Put]=Putprice()

a1=1; a2=2; b1=3; b2=2; s=.16; u=.07; T=1;

syms s; syms u; b=b1*s+b2; bfunc=matlabfunction(b); bfunc=@(s)b1.*s+b2;

a=a1*u+a2; afunc=matlabfunction((a1*u+a2)); afunc = @(u)(a1.*u+a2);

Put=integral(afunc*int(bfunc,s,0,u),0,T); disp(Put);

end

Somehow this doesn't work for me :(

[Shashank Prasanna]

MATLAB is case sensitive, matlabfunction is not same as matlabFunction. Lets try once more, this time give more details on what you see on the command line. 'doesn't work for me', is not enough information to help.

Also:

bfunc=matlabfunction(b); 
bfunc=@(s)b1.*s+b2;

Running those to successively means assigning bfunc twice in a row, it doesn't make much sense.

[Daniel]

a1=0;  a2=1;   b1=0;   b2=0.02;   r1=0;   r2=0.05;  q1=0;  q2=0;  syms u;

%%%%%%%%%%%%%%%%%specify r(t) and q(t)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
r=r1*u+r2;
q=q1*u+q2;
rfunc=@(u)r1*u+r2;
qfunc=@(u)q1*u+q2;

%%%%%%%%%%%%%%%%%specify a(t) and b(t) functions %%%%%%%%%%%%%%%%%%%%%
a=a1*u+a2;
b=b1*u+b2;
alpha=r-q+b;

bfunc=@(u)b1*u+b2;
taufunc=@(u)a^2*exp(-2*abs(beta)*int(alpha,u,0,u));
rbfunc=@(u) r1*u+r2+b1*u+b2;
alphafunc=@(u) r1*u+r2+b1*u+b2-q1*u-q2;

res=taufunc(u);
resfunc=matlabFunction(res);

%%%%%%%%%%%%%%%%%compute integral for tau!%%%%%%%%%%%%%%%%%%%%%%%%%%%%
tau=integral(@(u)resfunc(u),0,T);

That's what I wrote. It is working. However it doesn't work for optimization via fmincon. Could you help me simplify the code?

Is it possible to introduce vectors here? Can matlab calculate vector of integrals?

[Daniel]

When I try to use this code for calibration it gives the following error:

Error using symengine>@()1.0e2
Too many input arguments.

Error in Putprice>@(u)resfunc(u) (line 35)
tau=integral(@(u)resfunc(u),0,T);

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 Putprice (line 35)
tau=integral(@(u)resfunc(u),0,T);

Error in diff_sum (line 18)
       Diff(n)= (mktprice(n)-Putprice(S,K(n),T,x(1),x(2))).^2;

Error in nlconst (line 744)
                f = feval(funfcn{3},x,varargin{:});

Error in fmincon (line 837)
    [X,FVAL,LAMBDA,EXITFLAG,OUTPUT,GRAD,HESSIAN]=...

Error in calibration_main (line 29)
x = fmincon(@diff_sum,x0,[],[],[],[],lb,ub);