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how to define symbolic function on functions?

asked 2021-03-24 01:03:05 +0200

Performant Data gravatar image

I recently started using Sage in earnest, and was trying to define a function on functions. But:

inner_product(f,g) = integral(f(x) * g(x), (x,0,1))
inner_product

returns:

(f, g) |--> 1/3

Not surprisingly, trying to use it, it behaves as described:

e1(x) = 1
inner_product(e1,e1)

returns:

1/3

And curiously:

inner_product(f,g) = integral(f(x) * 1, (x,0,1))
inner_product

returns:

(f, g) |--> 1/2

What's going on here?

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answered 2021-06-10 11:50:57 +0200

rburing gravatar image

The reason for the results is that callable symbolic expressions are defined by interpreting the input variables as symbolic variables, and calling will amount to substitution of values into those variables in the expression on the right-hand side. I agree this limitation is not obvious and often confusing for beginners.

Indeed, when f and g are symbolic variables, and x is a symbolic variable, then f(x) and g(x) will both evaluate to x, and integral(f(x)*g(x), (x,0,1)) hence evaluates to integral(x^2, (x,0,1)) which is indeed 1/3.

As explained in the other answer, what you want is a Python function instead.

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answered 2021-03-24 02:42:35 +0200

cav_rt gravatar image

Your expressions are ill defined. A python function does the job

def inner_product(f,g): return integral(f*g, (x,-pi,pi))

Thus

inner_product(sin(3*x),sin(3*x))

returns $\pi$.

See here for symbolic computation.

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Comments

Thanks, I'll have a look. But what I really want to know is why I got the above results.

Performant Data gravatar imagePerformant Data ( 2021-03-24 03:08:54 +0200 )edit

I looked over your symbolic computation link but don't see anything there relevant to my question.

Performant Data gravatar imagePerformant Data ( 2021-03-24 18:03:09 +0200 )edit

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Asked: 2021-03-24 01:03:05 +0200

Seen: 107 times

Last updated: Jun 10