Lecture20 - This is called successive substitution...

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Engineering Analysis ENG 3420 Fall 2009 Dan C. Marinescu Office: HEC 439 B Office hours: Tu-Th 11:00-12:00
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2 Lecture 20 Lecture 20 Last time: The inverse of a matrix Iterative methods for solving sytems of linear equations Gauss-Siedel Jacobi Today Relaxation Non-linear systems Random variables, probability distributions, Matlab support for random variables. Next Time Linear regression Linear least squares regression
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Relaxation To enhance convergence, an iterative program can introduce relaxation where the value at a particular iteration is made up of a combination of the old value and the newly calculated value: where λ is a weighting factor that is assigned a value between 0 and 2. 0< λ <1: underrelaxation λ =1: no relaxation 1< λ ≤2: overrelaxation x i new x i new + 1 ( 29 x i o ld
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Nonlinear Systems Nonlinear systems can also be solved using the same strategy as the Gauss-Seidel method - solve each system for one of the unknowns and update each unknown using information from the previous iteration.
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Unformatted text preview: This is called successive substitution . Newton-Raphson Nonlinear systems may also be solved using the Newton-Raphson method for multiple variables. For a two-variable system, the Taylor series approximation and resulting Newton-Raphson equations are: f 1, i + 1 = f 1, i + x 1, i + 1-x 1, i ( 29 ∂ f 1, i x 1 + x 2, i + 1-x 2, i ( 29 f 1, i x 2 x 1, i + 1 = x 1, i-f 1, i f 2, i x 2-f 2, i f 1, i x 2 f 1, i x 1 f 2, i x 2-f 1, i x 2 f 2, i x 1 f 2, i + 1 = f 2, i + x 1, i + 1-x 1, i ( 29 f 2, i x 1 + x 2, i + 1-x 2, i ( 29 f 2, i x 2 x 2, i + 1 = x 2, i-f 2, i f 1, i x 1-f 1, i f 2, i x 1 f 1, i x 1 f 2, i x 2-f 1, i x 2 f 2, i x 1 Probability and statistics concepts See class notes: Probability NASA lecture...
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This note was uploaded on 02/17/2012 for the course EGN 3420 taught by Professor Staff during the Spring '08 term at University of Central Florida.

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Lecture20 - This is called successive substitution...

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