| ********** |
| Algorithms |
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| Univariate polynomial evaluation |
| ================================ |
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| * The evaluation of 1-D polynomials uses Horner's algorithm. |
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| * The evaluation of 1-D Chebyshev and Legendre polynomials uses Clenshaw's |
| algorithm. |
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| Multivariate polynomial evaluation |
| ================================== |
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| * Multivariate Polynomials are evaluated following the algorithm in [1]_ . The |
| algorithm uses the following notation: |
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| - **multiindex** is a tuple of non-negative integers for which the length is |
| defined in the following way: |
| |
| .. math:: \alpha = (\alpha1, \alpha2, \alpha3), |\alpha| = \alpha1+\alpha2+\alpha3 |
| |
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| - **inverse lexical order** is the ordering of monomials in such a way that |
| :math:`{x^a < x^b}` if and only if there exists :math:`{1 \le i \le n}` |
| such that :math:`{a_n = b_n, \dots, a_{i+1} = b_{i+1}, a_i < b_i}`. |
| |
| In this ordering :math:`y^2 > x^2*y` and :math:`x*y > y` |
| |
| - **Multivariate Horner scheme** uses d+1 variables :math:`r_0, ...,r_d` to |
| store intermediate results, where *d* denotes the number of variables. |
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| Algorithm: |
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| 1. Set *di* to the max number of variables (2 for a 2-D polynomials). |
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| 2. Set :math:`r_0` to :math:`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: |
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| - determine :math:`k = max \{1 \leq j \leq di: \alpha(n)_j \neq \alpha(n-1)_j\}` |
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| - Set :math:`r_k := l_k(x)* (r_0 + r_1 + \dots + r_k)` |
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| - Set :math:`r_0 = c_{\alpha(n)}, r_1 = \dots r_{k-1} = 0.` |
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| 4. return :math:`r_0 + \dots + r_{di}` |
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| * 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 |
| :math:`\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. |
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| .. [1] J. M. Pena, Thomas Sauer, "On the Multivariate Horner Scheme", SIAM Journal on Numerical Analysis, Vol 37, No. 4 |
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