## 6

## Interpolation

#### 6.1 Introduction

Let the values of a function *y* = *f* (*x*) be given at different *x* values; thus,

*x*_{0} *x*_{1} *x*_{2} ⋯ *x*_{n}

*f*(*x*_{0}) *f*(*x*_{1}) *f*(*x*_{2}) ⋯ *f*(*x*_{n})

Then *interpolation* means to find an approximate value of *f* (*x*) for an *x* between two *x*-values in (*x*_{0}, *x*_{n}). This is done by finding a polynomial called an *interpolating polynomial* which agrees with the function at the nodal points *x*_{i} (*i* = 0, 1, 2, ⋯, *n*). In finding a suitable interpolating polynomial we need to have a formula for errors in polynomial approximation.

#### 6.1.1 Formula for Errors in Polynomial Interpolation

Let *y* = *f* (*x*) be a function defined at the (*n* + 1) points

(*x*_{i}, *y*_{i}) (*i* = 0, 1, 2, ⋯, *n*) (6.1)

Suppose *f* (*x*) is continuous and differentiable ...