Interpolation Used to estimate values between data points difference from regression - goes through data points no error in data points

Most common method is polynomial interpolation Given n+1 data points, a unique n th order polynomial fits them. Polynomial interpolation determines a’s of this polynomial A number of methods

Newton divided difference interpolating polynomials Start with linear interpolation

x0x0 xx1x1 f(x 0 ) f 1 (x) f(x 1 ) From similar triangles

Can rearrange to get linear interpolation formula Example: Interpolate exp(2) using 1) exp(1) and exp(6) and 2) exp(1.5) and exp (2.5) 1) 2)

Quadratic interpolation - need three points Use parabola This is the same as with

To get b’s 1) set x=x 0 in quadratic 2) use b 0 and x=x 1 in quadratic

3) use b 0 and b 1 and x=x 2 b 0 is a constant (0th order) b 1 gives slope (finite difference) b 2 give curvature (difference of finite differences)

Example: interpolate exp(2) using exp(1), exp(3) and exp(4)

Example: interpolate exp(2) using exp(1), exp(1.5) and exp(2.5)

General form for Newton’s interpolating polynomials Bracketed functions are finite differences

First finite difference Second finite difference The difference of two finite differences

The nth finite difference An interative proceedure 1) make all first order finite differences; save f(x 0 ) for b 0 2) make second order from firsts; save f[x 1,x 0 ] for b 1 3) continue to nth order, saving needed ones

Example: estimate exp(2) using 6 points - 0,1, 3, 4, 5, 6 Do first differences, get b 1

Use firsts to get seconds and save b 2

Use seconds to get thirds and get b 3

Use thirds to get fourths and b 4 Use fourths to get the fifth finite difference and b 5

So

Blow up

Error for Newton’s polynomial - estimate from Thinking of interpolation like a Taylor series

Example: use exp(2.5)=12.18 Calculate

Matlab code Excel code