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Spline Interpolation Method. Major: All Engineering Majors Authors: Autar Kaw, Jai Paul http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates. Spline Method of Interpolation http://numericalmethods.eng.usf.edu. What is Interpolation ?.

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spline interpolation method

Spline Interpolation Method

Major: All Engineering Majors

Authors: Autar Kaw, Jai Paul

http://numericalmethods.eng.usf.edu

Transforming Numerical Methods Education for STEM Undergraduates

http://numericalmethods.eng.usf.edu

spline method of interpolation http numericalmethods eng usf edu
Spline Method of Interpolationhttp://numericalmethods.eng.usf.edu
what is interpolation
What is Interpolation ?

Given (x0,y0), (x1,y1), …… (xn,yn), find the value of ‘y’ at a value of ‘x’ that is not given.

http://numericalmethods.eng.usf.edu

interpolants
Interpolants

Polynomials are the most common choice of interpolants because they are easy to:

  • Evaluate
  • Differentiate, and
  • Integrate.

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why splines
Why Splines ?

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why splines1
Why Splines ?

Figure : Higher order polynomial interpolation is a bad idea

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linear interpolation
Linear Interpolation

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linear interpolation contd
Linear Interpolation (contd)

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example
Example

The upward velocity of a rocket is given as a function of time in Table 1. Find the velocity at t=16 seconds using linear splines.

Table Velocity as a

function of time

Figure. Velocity vs. time data

for the rocket example

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linear interpolation1
Linear Interpolation

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quadratic interpolation
Quadratic Interpolation

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quadratic interpolation contd
Quadratic Interpolation (contd)

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quadratic splines contd
Quadratic Splines (contd)

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quadratic splines contd1
Quadratic Splines (contd)

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quadratic splines contd2
Quadratic Splines (contd)

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quadratic spline example
Quadratic Spline Example

The upward velocity of a rocket is given as a function of time. Using quadratic splines

Find the velocity at t=16 seconds

Find the acceleration at t=16 seconds

Find the distance covered between t=11 and t=16 seconds

Table Velocity as a

function of time

Figure. Velocity vs. time data

for the rocket example

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solution
Solution

Let us set up the equations

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each spline goes through two consecutive data points
Each Spline Goes Through Two Consecutive Data Points

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slide19

Each Spline Goes Through Two Consecutive Data Points

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derivatives are continuous at interior data points
Derivatives are Continuous at Interior Data Points

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slide21

Derivatives are continuous at Interior Data Points

At t=10

At t=15

At t=20

At t=22.5

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last equation
Last Equation

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final set of equations
Final Set of Equations

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coefficients of spline
Coefficients of Spline

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quadratic spline interpolation part 2 of 2 http numericalmethods eng usf edu

Quadratic Spline InterpolationPart 2 of 2http://numericalmethods.eng.usf.edu

http://numericalmethods.eng.usf.edu

final solution
Final Solution

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velocity at a particular point
Velocity at a Particular Point

a) Velocity at t=16

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slide28

Acceleration from Velocity Profile

b) The quadratic spline valid at t=16 is given by

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distance from velocity profile
Distance from Velocity Profile

c) Find the distance covered by the rocket from t=11s to t=16s.

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additional resources
Additional Resources

For all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit

http://numericalmethods.eng.usf.edu/topics/spline_method.html

slide31
THE END

http://numericalmethods.eng.usf.edu