Dear Arnold,
Arnold Doray <[email protected]> writes:
> I'm trying to create a set of orthogonal polynomials using the
> Gram-Schmidt process, but I hit a problem:
>
> dot(f,g) == integrate(f*g*x^2,x=-1..1)
> proj(f,g) == dot(f,g)*f/dot(f,f)
>
> p0 := 1
> p1 := x - proj(p0,x)
> p2 := x^2 - proj(p0,x^2) - proj(p1,x^2)
>
> p0 and p1 evaluate correctly, but p2 hits this error:
as a quick workaround (sorry for answering late), use
dot(f,g) == integrate(f*g*x^2,x=-1..1)::EXPR INT
instead. I'm not yet sure how the "missing" coercion really should go,
since I don't think we want Expression OrderedCompletion Integer, but
rather OrderedCompletion Expression Integer...
(Waldek?)
> I've also tried to define p(n) as a recursive function like so:
>
> p(0) == 1
> p(n | n > 0) == x^n - sum(proj(p(k),x^n),k=0..n-1)
This is a common misconception. sum takes as first argument an
expression, in your case proj(p(k),x^n), and then tries to "simplify"
this.
As always in Axiom (FriCAS, OpenAxiom), arguments are evaluated. Thus,
Axiom tries to evaluate
proj(p(k),x^n)
Next step is (remember: arguments are always evaluated...) to evaluate
p(k). So k is certainly not 0 (it's an Expression Integer), so we try
the other rule, i.e.,
p(n | n > 0) == x^n - sum(proj(p(k),x^n),k=0..n-1)
Thus, what you want is to use
p(n | n > 0) == x^n - reduce(+, [proj(p(k),x^n) for k in 0..n-1])
(1) -> proj(f,g) == dot(f,g)*f/dot(f,f)
Type: Void
(2) -> dot(f,g) == integrate(f*g*x^2,x=-1..1)::EXPR INT
Type: Void
(3) -> p(0) == 1
Type: Void
(4) -> p(n | n > 0) == x^n - reduce(+, [proj(p(k),x^n) for k in 0..n-1])
Type: Void
(5) -> p 1
Compiling function dot with type (PositiveInteger,Fraction(
Polynomial(Integer))) -> Expression(Integer)
Compiling function dot with type (PositiveInteger,PositiveInteger)
-> Expression(Integer)
Compiling function proj with type (PositiveInteger,Fraction(
Polynomial(Integer))) -> Expression(Integer)
Compiling function dot with type (Expression(Integer),Fraction(
Polynomial(Integer))) -> Expression(Integer)
Compiling function dot with type (Expression(Integer),Expression(
Integer)) -> Expression(Integer)
Compiling function proj with type (Expression(Integer),Fraction(
Polynomial(Integer))) -> Expression(Integer)
Compiling function p with type Integer -> Expression(Integer)
(5) x
Type: Expression(Integer)
(6) -> p 2
2
5x - 3
(6) -------
5
Type: Expression(Integer)
Hope this helps,
Martin
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