Algorithms¶

Univariate polynomial evaluation¶

The evaluation of 1-D polynomials uses Horner’s algorithm.
The evaluation of 1-D Chebyshev and Legendre polynomials uses Clenshaw’s algorithm.

Multivariate polynomial evaluation¶

Multivariate Polynomials are evaluated following the algorithm in [1] . The algorithm uses the following notation:
- multiindex is a tuple of non-negative integers for which the length is defined in the following way:
  
  $\alpha = (\alpha1, \alpha2, \alpha3), |\alpha| = \alpha1+\alpha2+\alpha3$
- inverse lexical order is the ordering of monomials in such a way that ${x^a < x^b}$ if and only if there exists ${1 \le i \le n}$ such that ${a_n = b_n, \dots, a_{i+1} = b_{i+1}, a_i < b_i}$ .
  
  In this ordering $y^2 > x^2*y$ and $x*y > y$
- Multivariate Horner scheme uses d+1 variables $r_0, ...,r_d$ to store intermediate results, where d denotes the number of variables.
  
  Algorithm:
  1. Set di to the max number of variables (2 for a 2-D polynomials).
  2. Set $r_0$ to $c_{\alpha(0)}$ , where c is a list of coeeficients for each multiindex in inverse lexical order.
  3. For each monomial, n, in the polynomial:
    - determine $k = max \{1 \leq j \leq di: \alpha(n)_j \neq \alpha(n-1)_j\}$
    - Set $r_k := l_k(x)* (r_0 + r_1 + \dots + r_k)$
    - Set $r_0 = c_{\alpha(n)}, r_1 = \dots r_{k-1} = 0.$
  4. return $r_0 + \dots + r_{di}$
The evaluation of multivariate Chebyshev and Legendre polynomials uses a variation of the above Horner’s scheme, in which every Legendre or Chebyshev function is considered a separate variable. In this case the length of the $\alpha$ indices tuple is equal to the number of functions in x plus the number of functions in y. In addition the Chebyshev and Legendre functions are cached for efficiency.