Prof. Muhammad Saeed

( Interpolation and Curve Fitting )

2M.Sc. Physics

1.1.InterpoInterpolationlationa) Newton-Gregory Forward

Formula

Evenly Spaced Data

M.Sc. Physics 3

a) Lagrange Polynomials (Cubic)

Unevenly Spaced Data

b)Divided Difference

M.Sc. Physics 4

c) Cubic Spline

For condition 1 (Natural Spline):

2(h0+h1) h1 0 0 0 0   S1   f[x1,x2] - f[x0,x1]

h1 2(h1+h2) h2 0 0 0   S2   f[x2,x3] - f[x1,x2]

0 h2 2(h2+h3) h3 0 0   S3   f[x3,x4] - f[x2,x3]

0 0 h3 2(h3+h4) h4 0   S4   = 6   f[x4,x5] - f[x3,x4]

0 0 0 .. .. ..   ….   ….

0 0 0 0   ….   …..

0 0 0 0 hn-2 2(hn-2+hn-1)   Sn-1   f[xn-1,xn] - f[xn-2,xn-1]

M.Sc. Physics 5

2.2.Curve Curve FittingFitting

Least-Squares Approximations

Functions to Fit1) y = mx+c2) Polynomial2) y = aebx

3) y = a log(x) + b4) y = axb

5)

6) y = ax2 +bx

baxy

1

y= a

M.Sc. Physics 6

N ∑x ∑x2 ∑x3 …. ∑xn   a0   ∑ Y

∑x ∑x2 ∑x3 ∑x4 …. ∑xn+1   a1   ∑xY

∑x2 ∑x3 ∑x4 ∑x5 ….. ∑xn+2   a2   ∑x2Y

… …. …. …. ….. …   a3   =    ….

…. … … … ….. …   ….   …

… … …. … ….. ….   ….   …

∑xn ∑xn+1 ∑xn+2 ∑xn+3 …. ∑x2n   an   ∑xnY

a) Polynomial Fit

The Best Fit is determined by the minimum value of

M.Sc. Physics 7

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Weight 10.0 12.0 15.0 17.0 20.0 27.0 35.0 41.0 48.0 50.0 51.0 54.0 59.0 66.0 75.0

Height 0.75 0.86 0.95 1.08 1.12 1.26 1.35 1.51 1.55 1.60 1.63 1.67 1.71 1.78 1.85

Use W=aHb as mathematical model

Problem:Weight to Height Ratio of Human Beings

M.Sc. Physics 8

b) Line Regression

M.Sc. Physics 9

c) Polynomial Regression

‘m’ is the degree of polynomial

M.Sc. Physics 10