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# Mathematical Algorithm and Computer Procedure for Smooth Interpolation and Numerical Differentiation

IP.com Disclosure Number: IPCOM000083028D
Original Publication Date: 1975-Mar-01
Included in the Prior Art Database: 2005-Feb-28
Document File: 2 page(s) / 36K

IBM

## Related People

Barker, JA: AUTHOR

## Abstract

In the numerical machine tool and process control arts, dynamic interpolation may require an algorithm for smooth interpolation and numerical differentiation in which the interpolating function for x

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Mathematical Algorithm and Computer Procedure for Smooth Interpolation and Numerical Differentiation

In the numerical machine tool and process control arts, dynamic interpolation may require an algorithm for smooth interpolation and numerical differentiation in which the interpolating function for x <x<x(n+1)is determined as a linear combination of "basis" functions, having at x(n) and x(n+1) the desired function values and derivatives, the derivatives at x(n) and x(n+1) being determined from M1 point interpolating functions centered at x(n) and x(n+1), respectively. The resulting interpolating function is continuous and has a certain number of continuous derivatives depending on the value of M.

Most existing processes may be regarded as using either generalizations of Lagrange interpolation or spline interpolation. In Lagrange interpolation, a polynomial function is contracted (or other linear combination of functions) which reproduces exactly a certain number of the function values, and then this polynomial is used to define the function between the tabular points. The drawback of this is that unless the degree of the interpolating polynomial is equal to the total number of points for which the function is defined, then different interpolating polynomials in different ranges must be used. Thus, the derivative of an interpolated function is not continuous at the points where one polynomial changes to another.

Given a set of arguments x(i) and functions values y(...