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programming - How to define the Bernstein function like the built-in BernsteinBasis?


We know the Bernstein function defined as below:


$$B_{n,i}(u)=\binom n i u^i(1-u)^{n-i}$$


And we define $B_{n,i}(u)=0$ when $i<0 $ or $i>n$


In addition, the first derivative of $B_{n,i}(u)$ has the below relationship:


$$B'_{n,i}(u)=n\left[B_{n-1,i-1}(u)-B_{n-1,i}(u)\right]$$



So I can utilize this equation to calculate the derivative of order $k$


$$B^{(k)}_{n,i}(u)=n(n-1)\cdots (n-k+1)\\ \left[\{B_{n-k,i-k}(u),\cdots,B_{n-k,i}(u)\} .coefficient\right]$$


Here,$coefficient$ has the style (1,-3,3,1),(1, -4, 6, -4, 1),etc


Implementation


Bernstein[n_, i_, u_] /; i < 0 || i > n := 0
Bernstein[0, 0, u_] := 1
Bernstein[n_, i_, u_?NumericQ] := Binomial[n, i] u^i (1 - u)^(n - i)

The derivative of Bernstein


D[Bernstein[n_, i_, u_], {u_, k_}] ^:=

 Module[{coeff, body},
  coeff = Times @@ Array[n - # &, k, 0];
  body =
   Array[Bernstein[n - k, #, u] &, k + 1, i - k].
    CoefficientList[(1 - u)^k, u];
  coeff* body
]
(*=======================================*)
D[Bernstein[n_, i_, u_], u_] ^:= D[Bernstein[n, i, u], {u, 1}]


However, the Mathematica give me the warining information



enter image description here



Expand the expression


PiecewiseExpand[expr_] ^:=
 expr /. Bernstein[n_, i_, u_Symbol] :> 
  Binomial[n, i] u^i (1 - u)^(n - i)



enter image description here




Tesing


Successful case


Bernstein[3, 2, .4](*0.288*)

D[Bernstein[3, 2, u], u]



3 (Bernstein[2, 1, u] - Bernstein[2, 2, u])

D[Bernstein[3, 2, u], {u, 2}]


6 (Bernstein[1, 0, u] - 2 Bernstein[1, 1, u])

Failture


D[Bernstein[3, 2, u], u] // PiecewiseExpand



no expantion >_<



Question



  • How to fix the warining information about UpSetDelayed?

  • Is it possible to implement the derivative of $B_{n,i}(u)$ by rule-based solution? I have a trial, but failed.



My trial



D[Berns[n_, i_, u_], {u_, k_}] ^:=
 Do[
  Bernstein[n, i, u] /.
   Bernstein[x_, y_, z_] :> 
    x (Bernstein[x - 1, y - 1, u] - Bernstein[x - 1, y, z]), {k}]
 (*failture*)

Built-in function


D[BernsteinBasis[6, 3, u], {u, 3}]



enter image description here



Obviously, the Mathematica utilizes a recursive method by observing the result.




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