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De Boor algorithm In mathematical subfield of numerical analysis the De Boor algorithm is a fast and numerically stable algorithm for evaluating spline curves in B-spline form. It is a generalization of De Casteljau's algorithm for Bézier curves. Introduction The general setting is as follows. We would like to construct a curve passing through a sequence of p points $\backslash vec\_0,\; \backslash vec\_1,\; \backslash dots,\; \backslash vec\_$. The curve can be described as a function $\backslash vec(x)$ of one parameter x. To pass through the sequence of points, the curve must satisfy $\backslash vec(u\_0)=\backslash vec\_0,\; \backslash dots,$ \vec(u_)=\vec_. We assume that u0, ..., up-1 are given to us along with $\backslash vec\_0,\; \backslash vec\_1,\; \backslash dots,\; \backslash vec\_$. This problem is called interpolation. One approach to solving this problem is by splines. A spline is a curve that is piecewise nth degree polynomial. This means that, on any interval [ui , ui+1), the curve must be equal to a polynomial of degree at most n. It may be equal to a different polynomials on different intervals. The polynomials must be synchronized: when the polynomials from intervals [ui-1 , ui) and [ui , ui+1) meet at the point ui, they must have the same value at this point and their derivatives must be equal (to ensure that the curve is smooth). De Boor algorithm is an algorithm which, given u0, ..., up-1 and $\backslash vec\_0,\; \backslash vec\_1,\; \backslash dots,\; \backslash vec\_$, finds the value of spline curve $\backslash vec(x)$ at a point x. It uses O(n2) operations. Notice that the running time of the algorithm depends only on degree n and not on the number of points p. Outline of the algorithmm Suppose we want to evaluate the spline curve for a parameter value $x\; \backslash in\; [u\_,u\_)$. We can express the curve as $\backslash vec(x)\; =\; \backslash sum\_^\; \backslash vec\_i\; N\_i^n(x),$ where Nin(x) are polynomials in x with coefficients depending on u0, ..., up but not $\backslash vec\_i$. Due to the spline locality property, $\backslash vec(x)\; =\; \backslash sum\_^\; \backslash vec\_i\; N\_i^n(x)$ So the value $\backslash vec(x)$ is determined by the controlpoints $\backslash vec\_,\backslash vec\_,\backslash dots,\backslash vec\_$; the other control points $\backslash vec\_i$ have no influence. De Boor's algorithm, described in the next section, is a procedure which efficiently evaluates the expression for $\backslash vec(x)$. The algorithm Suppose $x\; \backslash in\; [u\_,u\_)$ and $\backslash vec\_i^\; =\; \backslash vec\_i$ for i = l-n+k, ..., l. Now calculate $\backslash vec\_i^\; =\; (1-\backslash alpha\_)\; \backslash vec\_^\; +\; \backslash alpha\_\; \backslash vec\_i^;\; \backslash qquad\; k=1,\backslash dots,n;\; \backslash quad\; i=\backslash ell-n+k,\backslash dots,\backslash ell$ with $\backslash alpha\_\; =\; \backslash frac-u\_i\}.$ Then $\backslash vec(x)\; =\; \backslash vec\_^$. Most Popular Articles: Music    Musical Instruments    Guitar    Classical Music    History    Music Definition    Country Music    Rock Music    Pop Music    Hip Hop    Rap    DJ    Video Games    Casino Games    Poker    Blackjack    Casinos    Real Estate    Loan    Auction    eBay    Marketing    Search Engine Optimization    File Sharing    MP3    Music Industry    Kazaa    Internet    Supermodel    Fashion    Fashion Designers    Erotica    Modeling    Clothing    Boots    Shoes    Diamond Engagement Rings    Diamond    Rugs    Gospel Music    Affiliate    E-Commerce    Radio This article is licensed under the GNU Free Documentation License at http://www.gnu.org/copyleft/fdl.html You may copy and modify it as long as the entire work (including additions) remains under this license. You must provide a link to http://www.gnu.org/copyleft/fdl.html To view or edit this article at Wikipedia go to http://www.wikipedia.org/wiki/De_Boor_algorithm