St. John's University of Tanzania
MAT210 NUMERICAL ANALYSIS
2013/14 Semester II
INTERPOLATION
Splines
Kaw, Chapter 5.05
MAT210 2013/14 Sem II 2 of 20
● Direct, Newton Divided Difference &
Lagrangian Interpolation
●
Two approaches for finding ...
MAT210 2013/14 Sem II 3 of 20
How do we avoid this?
MAT210 2013/14 Sem II 4 of 20
By observing
● This function has distinct regions
●
The interval from x ≈ -1 to -0.5
● The i...
MAT210 2013/14 Sem II 5 of 20
Piecewise Polynomials
Rather than interpolating n+1 points with a
single polynomial of degre...
MAT210 2013/14 Sem II 6 of 20
Interpretation
● Piecewise linear = connect the dots
●
Piecewise quadratic
= parabolas betwe...
MAT210 2013/14 Sem II 7 of 20
Splines
● In the connect the dots linear case,
the curve is not “smooth”
●
Add “smoothness” ...
MAT210 2013/14 Sem II 8 of 20
Splines
● In the connect the dots linear case,
the curve is not “smooth”
●
Add “smoothness” ...
MAT210 2013/14 Sem II 9 of 20
Linear Splines
MAT210 2013/14 Sem II 10 of 20
Linear Splines
Slope between points
MAT210 2013/14 Sem II 11 of 20
v(16) … Again
The linear case is unchanged
No surprise there
MAT210 2013/14 Sem II 12 of 20
Quadratic Splines
● Now things get interesting
● How to find all the coefficients?
●
3n coe...
MAT210 2013/14 Sem II 13 of 20
2n from continuity
Each curve must pass through both endpoints
MAT210 2013/14 Sem II 14 of 20
n-1 from smoothness
a1
x
2
+b1
x+c1
⇒2a1
x+b1
a2
x2
+b2
x+c2
⇒2a2
x+b2
Must match at n-1 in...
MAT210 2013/14 Sem II 15 of 20
One more assumption
● This is 3n unknowns and 3n -1 equations
●
Need to set one more condit...
MAT210 2013/14 Sem II 16 of 20
Revisiting the Rocket
MAT210 2013/14 Sem II 17 of 20
The continuous derivatives
The draftsman is bending his spline!
MAT210 2013/14 Sem II 18 of 20
The Final Matrix
MAT210 2013/14 Sem II 19 of 20
The Solution
MAT210 2013/14 Sem II 20 of 20
Going Deeper
● The overall curve is smooth and the
accuracy can be quite good
●
Cubic is be...
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SJUT/MAT210/Interpolation/Splines 2013-14S2

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Lecture slides based on Autar Kaw's Numerical Methods text, Chapter 5.05, found at http://nm.mathforcollege.com

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SJUT/MAT210/Interpolation/Splines 2013-14S2

  1. 1. St. John's University of Tanzania MAT210 NUMERICAL ANALYSIS 2013/14 Semester II INTERPOLATION Splines Kaw, Chapter 5.05
  2. 2. MAT210 2013/14 Sem II 2 of 20 ● Direct, Newton Divided Difference & Lagrangian Interpolation ● Two approaches for finding the same nth order polynomial fit for all points in an data set ● Is splines just another way to do the same ● NO! ● It is Piecewise polynomial interpolation ● Each piece can be linear, quadratic or cubic Introduction
  3. 3. MAT210 2013/14 Sem II 3 of 20 How do we avoid this?
  4. 4. MAT210 2013/14 Sem II 4 of 20 By observing ● This function has distinct regions ● The interval from x ≈ -1 to -0.5 ● The interval from x ≈ -0.5 to -0.1 ● The interval from x ≈ -0.1 to +0.1 ● The interval from x ≈ 0.1 to 0.5 ● The interval from x ≈ 0.5 to 1 Though there is some symmetry... ● It would be better to fit different functions to different intervals
  5. 5. MAT210 2013/14 Sem II 5 of 20 Piecewise Polynomials Rather than interpolating n+1 points with a single polynomial of degree n, put different polynomials on each interval S(x)= { s0 (x) , x∈[x0 ,x1) s1 (x) , x∈[x1, x2) ⋮ sn−1 (x) , x∈[xn−1 , xn]}where the sj are polynomials of (usually) small degree
  6. 6. MAT210 2013/14 Sem II 6 of 20 Interpretation ● Piecewise linear = connect the dots ● Piecewise quadratic = parabolas between the dots ● But wait ● Two points uniquely define a line – linear is understandable ● Three points are needed for a parabola – How is the other degree of freedom set?
  7. 7. MAT210 2013/14 Sem II 7 of 20 Splines ● In the connect the dots linear case, the curve is not “smooth” ● Add “smoothness” into the requirement ● Draftsmen achieved this smoothness with splines - a flexible strip of metal or wood
  8. 8. MAT210 2013/14 Sem II 8 of 20 Splines ● In the connect the dots linear case, the curve is not “smooth” ● Add “smoothness” into the requirement ● Draftsmen achieved this smoothness with splines - a flexible strip of metal or wood ● Mathematicians achieve it by matching derivatives at the end points of the intervals
  9. 9. MAT210 2013/14 Sem II 9 of 20 Linear Splines
  10. 10. MAT210 2013/14 Sem II 10 of 20 Linear Splines Slope between points
  11. 11. MAT210 2013/14 Sem II 11 of 20 v(16) … Again The linear case is unchanged No surprise there
  12. 12. MAT210 2013/14 Sem II 12 of 20 Quadratic Splines ● Now things get interesting ● How to find all the coefficients? ● 3n coefficients, n equations, n continuity at end points, whence the other n?
  13. 13. MAT210 2013/14 Sem II 13 of 20 2n from continuity Each curve must pass through both endpoints
  14. 14. MAT210 2013/14 Sem II 14 of 20 n-1 from smoothness a1 x 2 +b1 x+c1 ⇒2a1 x+b1 a2 x2 +b2 x+c2 ⇒2a2 x+b2 Must match at n-1 interior points 2 a1 xi +b1 =2a2 xi +b2 ∀ i ∈ [1 ,n−1]
  15. 15. MAT210 2013/14 Sem II 15 of 20 One more assumption ● This is 3n unknowns and 3n -1 equations ● Need to set one more condition ● Generally set the first spline to be linear ● a1 = 0 ● Now use any technique to solve simultaneous linear equations
  16. 16. MAT210 2013/14 Sem II 16 of 20 Revisiting the Rocket
  17. 17. MAT210 2013/14 Sem II 17 of 20 The continuous derivatives The draftsman is bending his spline!
  18. 18. MAT210 2013/14 Sem II 18 of 20 The Final Matrix
  19. 19. MAT210 2013/14 Sem II 19 of 20 The Solution
  20. 20. MAT210 2013/14 Sem II 20 of 20 Going Deeper ● The overall curve is smooth and the accuracy can be quite good ● Cubic is better, more common – See that next time ● What about finding the distance traveled? ● From 11 to 14s? ● From 11 to 16s? ● From 0 to 30s?

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