Spline Interpolation Method Industrial Engineering Majors Authors Autar

Spline Interpolation Method Industrial Engineering Majors Authors: Autar Kaw, Jai Paul http: //numericalmethods. eng. usf. edu Transforming Numerical Methods Education for STEM Undergraduates http: //numericalmethods. eng. usf. edu 1

Spline Method of Interpolation http: //numericalmethods. eng. usf. edu

What is Interpolation ? Given (x 0, y 0), (x 1, y 1), …… (xn, yn), find the value of ‘y’ at a value of ‘x’ that is not given. 3 lmethods. eng. usf. edu http: //numerica

Interpolants Polynomials are the most common choice of interpolants because they are easy to: Evaluate Differentiate, and Integrate. 4 lmethods. eng. usf. edu http: //numerica

Why Splines ? 5 lmethods. eng. usf. edu http: //numerica

Why Splines ? 6 Figure : Higher order polynomial interpolation is a bad ideahttp: //numerica lmethods. eng. usf. edu

Linear Interpolation 7 lmethods. eng. usf. edu http: //numerica

Linear Interpolation (contd) 8 lmethods. eng. usf. edu http: //numerica

Example A curve needs to be fit through the given points to fabricate the cam. If the cam follows a straight line profile between x = 1. 28 to x = 0. 66, what is the value of y at x=1. 1 ? Find using linear spline interpolation. Point 1 2 3 4 5 6 7 9 x (in. ) y (in. ) 2. 20 0. 00 1. 28 0. 88 0. 66 1. 14 0. 00 1. 20 – 0. 60 1. 04 – 1. 04 0. 60 – 1. 20 0. 00 4 3 5 2 6 Y 7 1 X lmethods. eng. usf. edu http: //numerica

Linear Interpolation 10 lmethods. eng. usf. edu http: //numerica

Linear Interpolation Find the cam profile using linear splines. 11 lmethods. eng. usf. edu http: //numerica

Linear Interpolation 12 lmethods. eng. usf. edu http: //numerica

Quadratic Interpolation 13 lmethods. eng. usf. edu http: //numerica

Quadratic Interpolation (contd) 14 lmethods. eng. usf. edu http: //numerica

Quadratic Splines (contd) 15 lmethods. eng. usf. edu http: //numerica

Quadratic Splines (contd) 16 lmethods. eng. usf. edu http: //numerica

Quadratic Splines (contd) 17 lmethods. eng. usf. edu http: //numerica

Example A curve needs to be fit through the given points to fabricate the cam. If the cam follows a straight line profile between x = 1. 28 to x = 0. 66, what is the value of y at x=1. 1 ? Find using a quadratic spline interpolation. Point 1 2 3 4 5 6 7 18 x (in. ) y (in. ) 2. 20 0. 00 1. 28 0. 88 0. 66 1. 14 0. 00 1. 20 – 0. 60 1. 04 – 1. 04 0. 60 – 1. 20 0. 00 4 3 5 2 6 Y 7 1 X lmethods. eng. usf. edu http: //numerica

Solution 19 lmethods. eng. usf. edu http: //numerica

Solution (contd) 20 lmethods. eng. usf. edu http: //numerica

Solution (contd) 21 lmethods. eng. usf. edu http: //numerica

Solution (contd) 22 lmethods. eng. usf. edu http: //numerica

Solution (contd) 23 lmethods. eng. usf. edu http: //numerica

Solution (contd) 24 lmethods. eng. usf. edu http: //numerica

Comparison 25 lmethods. eng. usf. edu http: //numerica

Additional Resources For all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, Math. Cad and MAPLE, blogs, related physical problems, please visit http: //numericalmethods. eng. usf. edu/topics/spline_met hod. html

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