St. John's University of Tanzania

MAT210 NUMERICAL ANALYSIS2013/14 Semester II

INTERPOLATIONSplines

Kaw, Chapter 5.05

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● 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

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How do we avoid this?

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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

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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

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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?

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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

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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

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Linear Splines

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Linear Splines

Slope between points

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v(16) … Again

The linear case is unchanged

No surprise there

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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?

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2n from continuity

Each curve must pass through both endpoints

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n-1 from smoothness

a1x2+b1 x+c1⇒2 a1 x+b1

a2 x2+b2 x+c2⇒2 a2 x+b2

Must match at n-1 interior points

2 a1 xi+b1=2 a2 xi+b2∀ i ∈ [1 ,n−1]

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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

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Revisiting the Rocket

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The continuous derivatives

The draftsman is bending his spline!

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The Final Matrix

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The Solution

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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?