De Boor's algorithm: Difference between revisions

Revision as of 16:11, 29 January 2005

Revision as of 03:18, 25 February 2005

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\vec{s}(u_{p-1})=\vec{d}_{p-1}</math>. We assume that ''u₀, ..., u_p-1'' are given to us along with <math>\vec{d}_0, \vec{d}_1, \dots, \vec{d}_{p-1}</math>. This problem is called [[interpolation]].

One approach to solving this problem is by [[spline (mathematics)|spline]]s. A spline is a curve that is piecewise ''n^th'' degree polynomial. This means that, on any interval ''<nowiki>[</nowiki>u_i , u_i+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 ''<nowiki>[</nowiki>u_i-1 , u_i)'' and ''<nowiki>[</nowiki>u_i , u_i+1)'' meet at the point ''u_i'', 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, ..., u_p-1'' and <math>\vec{d}_0, \vec{d}_1, \dots, \vec{d}_{p-1}</math>, finds the value of spline curve <math>\vec{s}(x)</math> at a point ''x''. It uses [[Big O notation|O]](n²) operations. Notice that the running time of the algorithm depends only on degree ''n'' and not on the number of points ''p''.