Newtons Divided Difference Polynomial Method of Interpolation Chemical

Newton’s Divided Difference Polynomial Method of Interpolation Chemical 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

Newton’s Divided Difference 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

Newton’s Divided Difference Method Linear interpolation: Given linear interpolant through the data pass a where 5 lmethods. eng. usf. edu http: //numerica

Example To find how much heat is required to bring a kettle of water to its boiling point, you are asked to calculate the specific heat of water at 61 °C. The specific heat of water is given as a function of time in Table 1. Use Newton’s divided difference method with a first order and then a second order polynomial to determine the value of the specific heat at T = 61°C. Table 1 Specific heat of water as a function of temperature. 6 Temperature, Specific heat, 22 42 52 82 100 4181 4179 4186 4199 4217 Figure 2 Specific heat of water vs. temperature. http: //numerica lmethods. eng. usf. edu

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

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

Quadratic Interpolation 9 lmethods. eng. usf. edu http: //numerica

Quadratic Interpolation (contd) 10 lmethods. eng. usf. edu http: //numerica

Quadratic Interpolation (contd) 11 lmethods. eng. usf. edu http: //numerica

Quadratic Interpolation (contd) 12 lmethods. eng. usf. edu http: //numerica

General Form where Rewriting 13 lmethods. eng. usf. edu http: //numerica

General Form 14 lmethods. eng. usf. edu http: //numerica

General form 15 lmethods. eng. usf. edu http: //numerica

Example To find how much heat is required to bring a kettle of water to its boiling point, you are asked to calculate the specific heat of water at 61 °C. The specific heat of water is given as a function of time in Table 1. Use Newton’s divided difference method with a third order polynomial to determine the value of the specific heat at T = 61°C. Table 1 Specific heat of water as a function of temperature. 16 Temperature, Specific heat, 22 42 52 82 100 4181 4179 4186 4199 4217 Figure 2 Specific heat of water vs. temperature. http: //numerica lmethods. eng. usf. edu

Example 17 lmethods. eng. usf. edu http: //numerica

Example 18 lmethods. eng. usf. edu http: //numerica

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

Comparison Table 20 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/newton_div ided_difference_method. html

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