# 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 coefficients 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.

 [1] Pena, Thomas Sauer, “On the Multivariate Horner Scheme”, SIAM Journal on Numerical Analysis, Vol 37, No. 4