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# 42. Bernstein

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## 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:

      (%i4) diff(bernstein_poly(k,n,x),x);
      (%o4) (bernstein_poly(k - 1, n - 1, x)
                                       - bernstein_poly(k, n - 1, x)) n
      (%i5) integrate(bernstein_poly(k,n,x),x);
      (%o5)
                                                                  k + 1
       hypergeometric([k + 1, k - n], [k + 2], x) binomial(n, k) x
       ----------------------------------------------------------------
                                    k + 1

  For numeric inputs, both real and complex, the Bernstein polynomials evaluate to a numeric result:

      (%i6) bernstein_poly(5,9, 1/2 + %i);
                              39375 %i   39375
      (%o6)                   -------- + -----
                                128       256
      (%i7) bernstein_poly(5,9, 0.5b0 + %i);
      (%o7)           3.076171875b2 %i + 1.5380859375b2

  To use `bernstein_poly`, first `load("bernstein")`.

Variable: bernstein_explicit

  Default value: `false`

  When either `k` or `n` are non integers, the option variable `bernstein_explicit` controls the expansion of `bernstein(k,n,x)` into its explicit form; example:

      (%i1) bernstein_poly(k,n,x);
      (%o1)                bernstein_poly(k, n, x)
      (%i2) bernstein_poly(k,n,x), bernstein_explicit : true;
                                             n - k  k
      (%o2)            binomial(n, k) (1 - x)      x

  When both `k` and `n` are explicitly integers, `bernstein(k,n,x)` always expands to its explicit form.

Function: multibernstein_poly ([k1,k2,…, kp], [n1,n2,…, np], [x1,x2,…, xp])

  The multibernstein polynomial `multibernstein_poly ([k1, k2, ..., kp], [n1, n2, ..., np], [x1, x2, ..., xp])` is the product of bernstein polynomials `bernstein_poly(k1, n1, x1) bernstein_poly(k2, n2, x2) ... bernstein_poly(kp, np, xp)`.

  To use `multibernstein_poly`, first `load("bernstein")`.

Function: bernstein_approx (f, [x1, x1, …, xn], n)

  Return the `n`-th order uniform Bernstein polynomial approximation for the function `(x1, x2, ..., xn) |--> f`. Examples

      (%i1) bernstein_approx(f(x),[x], 2);
                       2       1                          2
      (%o1)      f(1) x  + 2 f(-) (1 - x) x + f(0) (1 - x)
                               2
      (%i2) bernstein_approx(f(x,y),[x,y], 2);
                     2  2       1                2
      (%o2) f(1, 1) x  y  + 2 f(-, 1) (1 - x) x y
                                2
                        2  2          1   2
       + f(0, 1) (1 - x)  y  + 2 f(1, -) x  (1 - y) y
                                      2
             1  1                               1         2
       + 4 f(-, -) (1 - x) x (1 - y) y + 2 f(0, -) (1 - x)  (1 - y) y
             2  2                               2
                  2        2       1                      2
       + f(1, 0) x  (1 - y)  + 2 f(-, 0) (1 - x) x (1 - y)
                                   2
                        2        2
       + f(0, 0) (1 - x)  (1 - y)

  To use `bernstein_approx`, first `load("bernstein")`.

Function: bernstein_expand (e, [x1, x1, …, xn])

  Express the polynomial `e` exactly as a linear combination of multi-variable Bernstein polynomials.

      (%i1) bernstein_expand(x*y+1,[x,y]);
      (%o1)    2 x y + (1 - x) y + x (1 - y) + (1 - x) (1 - y)
      (%i2) expand(%);
      (%o2)                        x y + 1

  Maxima signals an error when the first argument isn't a polynomial.

  To use `bernstein_expand`, first `load("bernstein")`.

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