De Boor's algorithm: Difference between revisions

Revision as of 10:34, 4 August 2017

Revision as of 10:34, 4 August 2017

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== The algorithm ==

We can compute the above <math>\mathbf{s}(x)</math> by defining some <math> x \in [\bar u_{\ell},\bar u_{\ell+1}) </math>, setting <math> \mathbf{d}_i^{[0]} = \mathbf{d}_i</math> for <math>i = \ell-n, \dots, \ell</math>, and with these, computing:

:<math> \mathbf{d}_i^{[k]} = (1-\alpha_{k,i}) \mathbf{d}_{i-1}^{[k-1]} + \alpha_{k,i} \mathbf{d}_i^{[k-1]}; \qquad k=1,\dots,n; \quad i=\ell-n+k,\dots,\ell </math>

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Where the ratio <math>\alpha</math> is described by:

:<math> \alpha_{k,i} = \frac{x-\bar u_i}{\bar u_{i+n+1-k}-\bar u_i}</math>

Doing so gives us <math>\mathbf{s}(x) = \mathbf{d}_{\ell}^{[n]} </math>