Re: [Maxima-discuss] Bernstein's polynom

[Barton W. <wi...@un...>]

FYI:

42.1 Functions and Variables for Bernstein

Function: bernstein_poly (k, n, x)

Provided k is not a negative integer, the Bernstein polynomials are defined by bernstein_poly(k,n,x) = binomial(n,k) x^k (1-x)^(n-k); for a negative integer k, the Bernstein polynomial bernstein_poly(k,n,x) vanishes. When either k or n are non integers, the option variable bernstein_explicit controls the expansion of the Bernstein polynomials into its explicit form; example:

(%i1) load("bernstein")$

(%i2) bernstein_poly(k,n,x);
(%o2)                bernstein_poly(k, n, x)
(%i3) bernstein_poly(k,n,x), bernstein_explicit : true;
                                       n - k  k
(%o3)            binomial(n, k) (1 - x)      x


The Bernstein polynomials have both a gradef property and an integrate property:

See also  bernstein_approx, bernstein_expand, and bernstein_explicit




--Barton
________________________________
From: Richard Fateman <fa...@gm...>
Sent: Wednesday, March 20, 2024 18:03
To: Pierre CAMPET <pie...@ac...>
Cc: <max...@li...> <max...@li...>
Subject: Re: [Maxima-discuss] Bernstein's polynom

Caution: Non-NU Email

 B[n](fun):=ratsimp(sum(binomial(n,k)*fun(k/n)*x^k*(1-x)^(n-k),k,0,n))$
f(x):=x$
B[2](f);

returns x.
Note that the call B[2](f(x))  is incorrect, since it is immediately changed to B[2](x), and
doesn't define "f" within your function.





On Wed, Mar 20, 2024 at 3:02 PM Pierre CAMPET <pie...@ac...<mailto:pie...@ac...>> wrote:

Hello,

I was expecting x when I evaluate B[2](f(x))

(%i1)    B[n](f):=sum(binomial(n,k)*f(k/n)*x^k*(1-x)^(n-k),k,0,n);
(%o1)    B[n](f):=sum(binomial(n,k)*f(k/n)*x^k*(1-x)^(n-k),k,0,n)
(%i2)    f(x):=x;
(%o2)    f(x):=x
(%i3)    B[2](f(x));
(%o3)    x^2+(1-x)*x


Any idea ?


Pierre

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