ENGR 391 - Numerical Methods in Engineering ENGR 391 Roots...

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Numerical Methods in Engineering ENGR 391 Faculty of Engineering and Computer Sciences Concordia University Roots of equations Textbook Chapter 3 – Solving Nonlinear Equations
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Newton-Raphson Method •Der iva t ion o f the recurs ive formu la for iteration from Taylor’s series •Roo t f ind ing me thod us secan t l ines KEY POINTS: Summary •Proo f o f quadra t ic convergence for NR method •Unders tand bas a lgor i thm and be able to describe it graphically Secant Method & Method of False Position •Two d i f feren t approaches to der •The add i t cond i t for method of false position Find x r such that F ( x r ) = 0 Bisection Method
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Fixed-point iteration What is a fixed point? •A f i x e d p o i n t f o r a g i v e n f u n c t i o n f ( x) is a number p for which g(p) = p For root finding problem: •s o i n s t e a d o f s o l v i n g f ( x ) = 0 , w e want to do some manipulation of the given function such that we solve x = g(x)
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± x x g x or x x g x or x x g x x x x x f 2 1 ) ( 2 ) ( 2 ) ( 0 2 ) ( 2 2 ± ± ² ² ² Example: Rearrange the function so that x is on the left side of the equation: x x g x f ³ ) ( 0 ) ( Fixed-point iteration Can be rewritten in many different ways!!
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Fixed-point Iteration ... 2, 1, , 0 n , given ) ( ) ( 0 ) ( 1 ± ² o n n x x g x x x g x f •Ma thema t ica l ly , f ixed po in t i tera t ion simply generates again a sequence. •Rearrange the function so that x is on the left side of the equation: Does this method always converge? o NO
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An example ) ( 1 1 3 x g x x ± ² ³ ) ( 1 2 3 / 1 x g x x ± ) ( 3 1 2 1 3 2 3 x g x x x ´ ´ 0 1 3 ± ´ x x
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This note was uploaded on 07/11/2011 for the course ENGR 391 taught by Professor Hoidick during the Winter '09 term at Concordia Canada.

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ENGR 391 - Numerical Methods in Engineering ENGR 391 Roots...

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