How do I integrate an erfc (complementary error function)?

I'm trying to calculate following in Matlab:

U = 1/pi * integral from [0, 1/Tao] {erfc(gamma*f(x)/sqrt(Tao - f(x))} dx where f(x) = 3 * sin(x) - x * cos(x)/(sin(x))^3 gamma, Tao = variables

I've tried many different ways, but none works. The last attempt looks like this:

gamma = 0.7993;

TAO = [0,0.6127,1.2255,1.8382,2.4509,3.0636,3.6764,4.2891,4.9018,5.5145,6.1273];

f0TAO = zeros();

for i=2:length(TAO) f0TAO(i) = (3*((sin(TAO(i))-(TAO(i)*cos(TAO(i))))/(sin(TAO(i)))^3));

end

f0Tao = 1./f0TAO;

Q = zeros();

for i=2:length(Tao);

    c = TAO(i);
    f = erfc(gamma.*((3.*((sin(x)-(x.*cos(x)))./(sin(x))^3))./sqrt(c-(3.*((sin(x)-(x.*cos(x)))./(sin(x))^3)))));

    Q(i) = 1/pi*int(f,0,f0Tao(i));
end

This resulted in an error like this:

The following error occurred converting from sym to double: Error using mupadmex Error in MuPAD command: DOUBLE cannot convert the input expression into a double array.

If the input expression contains a symbolic variable, use the VPA function instead.

What should I do? Can anyone give me some brilliant clues?

Thanks in advance!

Answer 1

[Torsten]

Q(i)=1/pi*vpa(int(f,0,f0Tao(i)),5);

But I guess the integration will be difficult because you divide by sin(x)^3 which is zero at x=0.

Additionally, the sqrt will become complex-valued.

Best wishes

Torsten.

[Svante Monie]

Still no difference...

Error message:

The following error occurred converting from sym to double: Error using mupadmex Error in MuPAD command: DOUBLE cannot convert the input expression into a double array.

If the input expression contains a symbolic variable, use the VPA function instead.

[Svante Monie]

This is a version where the integral works alright, but I can't find a way to implement the erfc.

gamma = 0.7993;

TAO = [0,0.6127,1.2255,1.8382,2.4509,3.0636,3.6764,4.2891,4.9018,5.5145,6.1273];

f0TAO = zeros();

for i=2:length(TAO)

    f0TAO(i) = (3*((sin(TAO(i))-(TAO(i)*cos(TAO(i))))/(sin(TAO(i)))^3));

end

f0Tao = 1./f0TAO;

Q = zeros();

for i=2:length(Tao);

    c = TAO(i);

    fun = @(x,c) gamma.*((3.*((sin(x)-(x.*cos(x)))./(sin(x))^3))./sqrt(c-(3.*((sin(x)-(x.*cos(x)))./(sin(x))^3))));

    Q(i) = 1/pi*integral(@(x)fun(x,c),0,f0Tao(i),'ArrayValued',true);

end

format long

Q

[Torsten]

Then directly use MATLAB's "integral":

f = @(x) erfc(gamma.*((3.*((sin(x)-(x.*cos(x)))./(sin(x))^3))./sqrt(c-(3.*((sin(x)-(x.*cos(x)))./(sin(x))^3)))));
Q(i) = 1/pi*integral(f,0,f0Tao(i));

Best wishes

Torsten.

[Svante Monie]

Aaw

Tried it, but generated this:

Error using erfc Input must be real and full.

Error in @(x)erfc(gamma.*((3.*((sin(x)-(x.*cos(x)))./(sin(x)).^3))./sqrt(c-(3.*((sin(x)-(x.*cos(x)))./(sin(x)).^3)))))

Error in integralCalc/iterateScalarValued (line 315) 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 88) Q = integralCalc(fun,a,b,opstruct);

Error in integral_test2 (line 18) Q(i) = 1/pi*integral(f,0,f0Tao(i));

Really glad for your input, though!

[Torsten]

What happens if you try to evaluate f for a certain value of x (e.g. x=0, x=1, ...) ?

Best wishes

Torsten.

[Svante Monie]

No difference...

Error using erfc Input must be real and full.

Error in @(x)erfc(gamma*((3.*((sin(x)-(x.*cos(x)))./(sin(x)).^3))./sqrt(c-(3.*((sin(x)-(x.*cos(x)))./(sin(x)).^3)))))

Error in integralCalc/iterateScalarValued (line 315) 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 88) Q = integralCalc(fun,a,b,opstruct);

[Torsten]

If

f=@(x)erfc(gamma.*((3.*((sin(x)-(x.*cos(x)))./(sin(x))^3))./sqrt(c-(3.*((sin(x)-(x.*cos(x)))./(sin(x))^3)))));
f(2);

still gives an error, check the value of the argument to the erfc function and whether gamma and c are real and not symbolic.

Best wishes

Torsten.

[Svante Monie]

Now I have found the cause of the error. In the denominator I have sqrt(c-(3.*((sin..., this gives complex values and thats what MatLab don't like. It needs _real_ input, as stated. When I put abs() around the expression under the sqrt, erfc delievers! Question is then of course, if the integral is valid for my intentions...