Lagrange’s, Newton’s and Stirling’s interpolation formulas
and others at use of big number of nodes of interpolation on all segment [
The piece-polynomial function certain on segment [
Let's consider one of the cases most widespread in an expert – interpolation of functions by
Let continuous function
is given on segment [
(6) also we’ll designate , .
1) on each of segments , function is a cubic polynomial;
2) function
, and also its the first and the second derivatives are continuous on segment [
3)
The third condition refers to as interpolation condition. The spline defined by conditions 1) – 3), refers to
Let's consider a way of cubic spline construction [2]. On each of segments , we’ll search for spline-function in the form of the third degree polynomial: (7) where found coefficients. Let’s differentiate (7) three times on х:
hence
From the interpolation condition 3) we recieve: . (8) Besides we’ll consider . From conditions of function continuity follows:
From here in view of (7) we’ll receive:
Having designated and lowering intermediate calculations [2], we’ll finally receive the system of equations for definition of coefficients : (9) By virtue of matrix of coefficients is three-diagonal the system (9) has the unique solution [2]. Having found coefficients , other coefficients we’ll define from the open formulas: (10) Thus, the unique cubic spline, satisfying to conditions 1) – 3) exists and is found. |

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>> Applied Mathematics
>> Numerical Methods
>> Interpolation of Functions
>> The spline interpolation