prof. muhammad saeed ( interpolation and curve fitting )
TRANSCRIPT
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Prof. Muhammad Saeed
( Interpolation and Curve Fitting )
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2M.Sc. Physics
1.1.InterpoInterpolationlationa) Newton-Gregory Forward
Formula
Evenly Spaced Data
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M.Sc. Physics 3
a) Lagrange Polynomials (Cubic)
Unevenly Spaced Data
b)Divided Difference
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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]
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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
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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
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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
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M.Sc. Physics 8
b) Line Regression
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M.Sc. Physics 9
c) Polynomial Regression
‘m’ is the degree of polynomial
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M.Sc. Physics 10