41. Bernstein

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41.1 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.