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# Notes on Newton-Raph - Notes on Newton-Raphson A root-solving technique 1 Introduction to Newton-Raphson(Updated(Dr Shoane Based on geometric

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Notes on Newton-Raphson – A root-solving technique (Updated 11/3/2010) (Dr. Shoane) 1. Introduction to Newton-Raphson Based on geometric interpretation or Taylor series expansion of the function, f(x) at x i . From the geometry of the situation seen in the graph, we have 1 0 ) ( ) ( + - - = i i i i x x x f x f Hence ) ( ) ( 1 i i i i x f x f x x - = + The Newton-Raphson algorithm attempts to minimize the difference between x i and x i+1 by iteratively updating the next x i value. This continues until x i and x i+1 are very close to each other. The critical point occurs when the function, f, crosses the zero line, which is where the equation f(x)=0 (e.g., x 3 +2x 2 +1=0) is satisfied. It can be shown that at this critical point, x i x i+1. Importantly, the value of x at this point is a root (or solution) of the equation f(x)=0. 1

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2. Examples Example 1 ( 29 ( 29 ( 29 7 10 3 3 7 5 7 10 3 ) ( 0 3 7 5 1 1 3 ) ( 2 2 3 1 2 2 3 + - - + - - = + - = = - + - = - - - = + i i i i i i i x x x x x x x x x x f x x x x x x x f Example 2 dy y x df y x f x x xy x y x f i i / ) , ( ) , ( 0 10 ) , ( 1 2 - = = - + = + 3. Video http://www.youtube.com/watch?v=lFYzdOemDj8 4. Code for function “newton” http://math.fullerton.edu/mathews/n2003/newtonsmethod/New ton%27sMethodProg/Links/Newton%27sMethodProg_lnk_3.html 2
Modified Program Demonstrating Newton-Raphson Technique (Dr. Shoane debugged the above program, and it should run “as is”, after cleaning up any comments that were moved out of position.) ******** Copy the entire code starting from here to the end of the program ********** function newton_test() % From http://math.fullerton.edu/mathews/n2003/newtonsmethod/Newton %27sMethodProg/Links/Newton%27sMethodProg_lnk_3.html % 1) Modified by Prof. George Shoane 11/2/2010: positions of functions, plus comments at each pause. % 2) The test functions have been put at the end of the program. % 3) It is suggested that the students learn how the function "newton" % works, and then write their own functions to solve any given problem. % This is probably more efficient than trying to input a complicated % nonlinear system equation with two variables to fit the format of the function "newton". % 4) If the nonlinear function consists of 2 variables where one of them % is an indpendent variable, then one technique is to update the % independent variable and set it as a constant at each iteration. Then % the "newton" function can be used directly. The output after the function call % provides the estimated value of the dependent variable for that particular % independent variable value. (Updated 11/3/2010) % 5) Hint: Use symbolic representation to define the function and obtain the % derivative of the function. But do not use the symbolic variables in the

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## This note was uploaded on 02/12/2011 for the course 125 305 taught by Professor Madabhushi during the Fall '08 term at Rutgers.

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Notes on Newton-Raph - Notes on Newton-Raphson A root-solving technique 1 Introduction to Newton-Raphson(Updated(Dr Shoane Based on geometric

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