ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 21 CURVE FITTING Chapter 18 Function Interpolation and Approximation

Curve Fitting Often we are faced with the problem… what value of y corresponds to x=0.935?

Curve Fitting Question 1 : Is it possible to find a simple and convenient formula that reproduces the points exactly? e.g. Straight Line ? …or smooth line ? …or some other representation? Interpolation

Curve Fitting Question 2 : Is it possible to find a simple and convenient formula that represents data approximately ? e.g. Best Fit ? Approximation

Function Interpolation Linear Interpolation Simplest Form is to connect data points with a straight line (1 st Order Polynomial) x

Function Interpolation STEPS xixi x i+1 Locate Interval x i < x < x i+1 Determine Values f(x i ), f(x i+1 ) f(x i ) f(x i+1 )

Function Interpolation STEPS f(x i ) f(x i+1 ) xixi x i+1 x f 1 (x)

Linear Interpolation Slope of Line 1 st DIVIDED DIFFERENCE f [x i+1,x i ] First order interpolating polynomial

Example what value of y corresponds to x=0.935? Interval: x i = , x i+1 = Values: f(x i )= , f(x i+1 )=

Function Interpolation Quadratic Interpolation Better Accuracy if 2 nd Order Polynomial x

Function Interpolation

How to Determine Coefficients b 0 0

Function Interpolation How to Determine Coefficients b 0

Function Interpolation How to Determine Coefficients b

Function Interpolation

2 nd DIVIDED DIFFERENCE

2 nd Order Interpolating Polynomial

General Form of Newton’s Interpolating Polynomials

EXAMPLE