(maxima.info)Functions and Variables for Bernstein

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42.1 Functions and Variables for Bernstein
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 -- 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")'.