Hermite interpolation - Cornell University

Bindel, Spring 2012 Intro to Scientific Computing (CS 3220)Week 10: Monday, Apr 2 Hermite interpolationFor standard polynomial interpolation problems, we seek to satisfy conditionsof the formp(xj) =yj,whereyjis frequently a sampled function valuef(xj). If all we know isfunction values, this is a reasonable approach. But sometimes we have constructs an interpolant based not onlyon equations for the function values, but also for the example, consider the important special case of finding a cubic poly-nomial that satisfies proscribed conditions on the values and derivatives atthe endpoints of the interval [ 1,1]. That is, we requirep(1) =f(1)p( 1) =f( 1)p (1) =f (1)p ( 1) =f ( 1).As with polynomial interpolation based just on function values, we can ex-press the cubic that satisfies these conditions with respect to several differentbases: monomial, Lagrange, or the monomial basis, we havep(x) =c0+c1x+c2x2+c3x3,which yields the linear system 11111 11 1012301 23 c0c1c2c3 = f(1)f( 1)f (1)f ( 1).

mials that satisfy Hermite interpolation conditions (sometimes referred to by the acronym PCHIP or Piecewise Cubic Hermite Interpolating Polynomials). That is, the function values and derivatives are speci ed at each nodal point. If we don’t actually have derivative values prescribed at the nodal points,

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  Polynomials, Hermite

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