Problem using Integral2 for functions only depending on one variable

I want to compute a lot of integrals over 2D ansatz functions and products of their derivatives using integral2.

The problem is that some integrands will be 0, constant or only depending on 1 variable.

Because of this integral2 will not work. Error Message below

But i have to use integral2 otherwise i would have to do handle way too many exceptions.

I have reduced and simplified my code to the following:

n=10;                                       %number of ansatz functions
ansatzf=cell(n,1);                          %my cell array of ansatz functions
syms x y;          

for i = 1:n
    ansatzf{i}=  @(x,y) x.^i.*y.^i;         %defining the ansatz functions
end

test = matlabFunction(diff(ansatzf{1},x,1)) %test is now only dependent on y
integral2(test,0,1,0,1)                     %this integral2 does not work because test is only depending on y  
>> TestIntegralTwo 

%This is the Error message i get:
test =

  function_handle with value:

    @(y)y

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

Error in integral2Calc>integral2t/tensor (line 228)
        Z = FUN(X,Y);  NFE = NFE + 1;

Error in integral2Calc>integral2t (line 55)
[Qsub,esub] = tensor(thetaL,thetaR,phiB,phiT);

Error in integral2Calc (line 9)
    [q,errbnd] = integral2t(fun,xmin,xmax,ymin,ymax,optionstruct);

Error in integral2 (line 105)
    Q = integral2Calc(fun,xmin,xmax,yminfun,ymaxfun,opstruct);

Error in TestIntegralTwo (line 10)
integral2(test,0,1,0,1)

How can i change my code so that integral2 can work?

Thank you for any help

Answer 1

[Steven Lord]

You can tell matlabFunction exactly what variables the generated function should accept as input arguments.

syms x y
ansatzf = x.*y.^2;
d = diff(ansatzf, x)

d = 

M_justY = matlabFunction(d)

M_justY = function_handle with value:
    @(y)y.^2

M_both = matlabFunction(d, 'Vars', [x y])

M_both = function_handle with value:
    @(x,y)y.^2

M_justY(1:5)

ans = 1×5

1 4 9 16 25

M_both(NaN, 1:5) % M_both doesn't use x so it can be whatever you want

ans = 1×5

1 4 9 16 25

[Lino Marschall]

Hey Steven,

Thank you for your answer. This helped solve most of the problem. But there is still something that i don't know how to solve.

In your code example this works fine if you derive ansatzf only once by x.

The problem which still remains for me however is that i have to compute integrals (with integral2) over derivatives, where i know the derivative is 0 anyways.

I have to do this as it would be much more complicated in my case to find all the integrals that are 0.

In short: I want to integrate functions that are 0 everywhere with integral2

Is there a solution to this problem?

[Bruno Luong]

"I want to integrate functions that are 0 everywhere with integral2"

What's a problem?:

integral2(@(x,y) zeros(size(x)), 0,1,0,1)

ans = 0

It doesn' matter the integrant expression depends on x, y, (x,y) or none. You need to declare it as two-variables function.

[Torsten]

For integral2, define

M_both = matlabFunction(d, 'Vars', [x y]);
M_both = @(x,y)eye(size(x))*M_both(x,y);

[Lino Marschall]

I was too vague sorry.

I meant how can i integrate something like this: (using integral2)

syms x y
exampleFct= @(x,y) x.*y^2;
test = matlabFunction(diff(exampleFct,x,2),'Vars',[x,y]);
integral2(test,0,1,0,1)   

[Torsten]

@Bruno Luong

The OP doesn't know in advance how matlabFunction (returned from a symbolic calculation) is built.

[Torsten]

@Lino Marschall

syms x y
exampleFct=x.*y^2;
ddexampleFct=diff(exampleFct,x,2)
ddexampleFct=matlabFunction(ddexampleFct,'Vars',[x,y]);
ddexampleFct=@(x,y)eye(size(x))*ddexampleFct(x,y);
integral2(ddexampleFct,0,1,0,1) 

[Bruno Luong]

We just tell you

syms x y
exampleFct= @(x,y) x.*y^2;
test = matlabFunction(diff(exampleFct,x,2),'Vars',[x,y]);
integral2(@(x,y) zeros(size(x))+test(x,y),0,1,0,1) 

[Bruno Luong]

@Torsten Do you meant ".*" rather than "*"?

[Lino Marschall]

This works in my problem.

Thank you so much for your help Bruno, Torsten and Steven.

But i dont quite undestand why this works. Why is the way i tried not working?

[Bruno Luong]

integral2 requires a vectorized user-provided function, x and y input are vectors and it must return the vector result evaluated at (x,y). That why you need to add zeros(size(x)) to expand the matlabFunction output.

[Torsten]

@Bruno Luong

No, I meant usual matrix multiplication.

x and y are passed as nx*nx matrices. So fun(x,y) can always be replaced by eye(size(x))*fun(x,y), I guess.

[Bruno Luong]

@Torsten I don't get your explanation (PS: I see you edit ones to eye)

f=@(x,y) x.*y.^2;
integral2(@(x,y) zeros(size(x))+f(x,y), 0,1,0,1)

ans = 0.1667

integral2(@(x,y) ones(size(x)).*f(x,y), 0,1,0,1)

ans = 0.1667

integral2(@(x,y) ones(size(x))*f(x,y), 0,1,0,1)

Error using *
Incorrect dimensions for matrix multiplication. Check that the number of columns in the first matrix matches the number of rows in the second matrix. To operate on each element of the matrix individually, use TIMES (.*) for
elementwise multiplication.

Error in solution (line 4)
integral2(@(x,y) ones(size(x))*f(x,y), 0,1,0,1)

Error in integral2Calc>integral2t/tensor (line 237)
Z1 = FUN(X(VTSTIDX),Y(VTSTIDX)); NFE = NFE + 1;

Error in integral2Calc>integral2t (line 55)
[Qsub,esub] = tensor(thetaL,thetaR,phiB,phiT);

Error in integral2Calc (line 9)
[q,errbnd] = integral2t(fun,xmin,xmax,ymin,ymax,optionstruct);

Error in integral2 (line 105)
Q = integral2Calc(fun,xmin,xmax,yminfun,ymaxfun,opstruct);

[Torsten]

fun=@(x,y) eye(size(x))*(x.*y.^2);
q = integral2(fun,0,1,0,1)

q = 0.1667

x.*y.^2 is a matrix of the same size as x.

So why should it not be possible to multiply it with eye(size(x)) and get back x.*y.^2 ?

At least in octave, x and y are of the same size. Maybe it's different in MATLAB.

[Bruno Luong]

fun=@(x,y) eye(size(x))*(x.*y.^2);
q = integral2(fun,0,1,0,1)

Error using *
Incorrect dimensions for matrix multiplication. Check that the number of columns in the first matrix matches the number of rows in the second matrix. To operate on each element of the matrix individually, use TIMES (.*) for
elementwise multiplication.

Error in solution (line 1)
fun=@(x,y) eye(size(x))*(x.*y.^2);

Error in integral2Calc>integral2t/tensor (line 237)
Z1 = FUN(X(VTSTIDX),Y(VTSTIDX)); NFE = NFE + 1;

Error in integral2Calc>integral2t (line 55)
[Qsub,esub] = tensor(thetaL,thetaR,phiB,phiT);

Error in integral2Calc (line 9)
[q,errbnd] = integral2t(fun,xmin,xmax,ymin,ymax,optionstruct);

Error in integral2 (line 105)
Q = integral2Calc(fun,xmin,xmax,yminfun,ymaxfun,opstruct);

[Torsten]

Ok, then it's somehow different in MATLAB compared to Octave.

As said, in Octave x and y are quadratic matrices of the same size - thus no reason for this error.

[Bruno Luong]

MATLAB doc only states x and y have the same size, not necessary a square size.

[Torsten]

Ok, this will be the reason for the error.

Then your solution or ones(size(x)).*fun(x,y) will work in MATLAB, too.