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Numerical-Computing-slides-chapter4

# Numerical-Computing-slides-chapter4 - Numerical Computing...

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Numerical Computing chapter 4- Non-linear Equations Dr. Nguyen V.M. Man Email: [email protected] Faculty of Computer Science and Engineering HCMUT Vietnam April 1, 2010

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Motivation ——————————————————————————— In a specific problem of Identication of Communication channels (used in the GSM telecom standard or in the UMTS third generation standard) we have to determine important quantities by computing parametric normal forms and roots from the system of nonlinear equations α = x 2 1 - x 2 3 - 2 x 1 x 3 - 2 x 2 x 3 , β = - x 2 2 - x 1 x 2 - x 1 x 3 - x 2 x 3 , γ = x 2 1 + x 1 x 2 + x 1 x 3 (source: AAECC 2006, no. 17, 471-485, by Jerome Lebrun, CNRS, France) UMTS: Universal Mobile Telecommunications System one of the third-generation (3G) mobile telecommunications technologies, which is also being developed into a 4G technology
Introduction ——————————————————————————— When a system has n = 1 unknown, we consider numerical methods for finding real solutions to equations of the form f ( x ) = 0 ... (1) Here, x : an unknown variable, and f : a nonlinear function of x . Any real number x * that satisfies f ( x * ) = 0 is a real root of f . For instance, (1) could be sin ( x 3 + 2 x 2 - 1) = 0

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Chapter 4: Non-linear Systems of Equations ——————————————————————————— More generally, we are also interested in solutions to systems of equations f 0 ( x 0 , x 1 , . . . , x n ) = 0 , f 1 ( x 0 , x 1 , . . . , x n ) = 0 , . . . f n - 1 ( x 0 , x 1 , . . . , x n ) = 0 . Again f i ( x 0 , x 1 , . . . , x n ) R , and we assume x k R for all i , k N n := { 0 , 1 , 2 , 3 , . . . , n - 1 } .
Chapter 4: Non-linear Systems ——————————————————————————— Two general features . Numerical methods for solving (1) share two features: they are iterative methods, and they require informed users. Why are these true and necessary when using numerical methods? 1. Iterative method : a method for which, given an initial guess x (0), one can produce a sequence of reals { x ( m ) } = { x (0) , x (1) , x (2) , ... } expect that | x * - x ( m ) | → 0 as m → ∞ · · · ( α ) : x ( m ) converges to x * ; otherwise, it is diverges. Numerical methods to be designed have to make sure that ( α ) happens!

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Introduction —————————————————————————— 2. Analyzing the function f is important, eg., eqn. cos( x 2 ) + π = 0 has no solution. I A few serious problems arising with respect to numerical errors. That occurs when the zeros of f are extremely sensitive to small coefficient/input errors. Eg., notorious ones are the high degree polynomials, like f ( x ) = ( x - 1)( x - 2)( x - 3) · · · ( x - 20) = 20 X i ( x - i ) I How to avoid them?
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Numerical-Computing-slides-chapter4 - Numerical Computing...

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