numerical integration with nonarray function

Hello, guys,

I have a trouble in using numerical integration command. The integrand in my case is det(x*A),x is the variable, and A is a n*n matrix. I noticed that nearly all the numerical integration command has requirement of array function, which means, they have to use .*, ./ when needed? do we have numerical integration command without such requirement?

Thank you very much!

Clair

Answer 1

[Mike Hosea]

It is, indeed, unnecessary to perform numerical integration on this integrand. However, to answer the question in general, The INTEGRAL function has an option called 'ArrayValued' that allows you to integrate array-valued functions. When this option is set to true, the integrator will only call the integrand function with scalar inputs, and it doesn't matter if the "array value" only has one element. So, this does it

integral(@(x)det(A*x),a,b,'ArrayValued',true)

For example

>> rng(0)
>> A = round(100*rand(3))
A =
    81    91    28
    91    63    55
    13    10    96
>> integral(@(x)det(A*x),0,1,'ArrayValued',true)
ans =
    -7.050624999999999e+04
>> det(A)/4
ans =
    -7.050624999999999e+04

[zoe]

Thank you so much, Mike, it is so helpful. I am considering the case where x is a bivariate. ur methods still works! But,it takes sometime to have the result, do we have the similar result for double integral, and it will produce result really fast?. ps: you are right, it is not necessary. det(A*x) is a simplified representation for my problem, and the function i am used is more complicated. Thanks again.

[Mike Hosea]

ARRAYFUN is usually a faster way for scalar-valued problems, and since INTEGRAL2 and INTEGRAL3 do not support an 'ArrayValued' option, you will have to do something like that (or write a wrapper function with a loop). Here's how to use ARRAYFUN. If f(x,y) is a bivariate integrand but that only works with scalar inputs, integrate

g = @(x,y)arrayfun(f,x,y)

Try this technique with the univariate function as well if speed is an issue, i.e. integrate

g = @(x)arrayfun(f,x);

Here I am assuming in both cases that f is defined as an anonymous function. If f is defined in a MATLAB program file, f.m, then of course you need @f instead of just f as the first argument to ARRAYFUN.

Answer 2

[Yao Li]

Assuming

x=[x0,x1,x2];
  B=[det(x0*A) det(x1*A) det(x2*A)];
  trapz(x,B)

Answer 3

[Roger Stafford]

With x used as a vector, your integrand can be written as:

 (x.^n)*det(A)

with the x factored out, which should integrate very nicely. However, why bother to do numerical integration when the indefinite integral is already known from elementary calculus, namely

 x^(n+!)/(n+1)*det(A)

[zoe]

det(A*x) is a simplified form. The really form is det(A-x*B-y*C),A,B,C is matrix of n*n. So it is indeed a double integral