Ch E 310 - Fall 10 - Lecture 14

Ch E 310 - Fall 10 - Lecture 14 - Lecture 14 October 12,...

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Lecture 14 – October 12, 2010 Agenda: Solving systems of non-linear equations Newton-Raphson (multiple equation version) MATLAB built-in root finding methods fzero (any function) roots (any polynomial)
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Systems of Non-linear Equations Often, we’ll need to solve more than one non-linear equation simultaneously: We need to find n unknowns ( x 1 to x n ) that result in all n equations ( f 1 to f n ) equaling zero       1 1 2 2 1 2 12 , ,..., 0 , 0 , 0 n n nn f x x x f x x x f x x x
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Systems of Non-linear Equations Recall that a non-linear system of equations cannot be recast into the linear Ax = b form, thus all of our methods for solving linear system will not work However , non-linear problems can be linearized using Taylor Series approximations taken around specific points For a single non-linear equation: This was set equal to zero and solved for x i +1 :        11 i i i i i f x f x f x x x      1 i ii i fx xx  single-equation Newton-Raphson iteration
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Systems of Non-linear Equations The multiple equation form of Newton-Raphson is similarly derived using Taylor Series estimates For example, consider two functions u and v : Using Taylor Series approximations:         1 1 2 2 1 2 , , 0 , , 0 f x x u x y f x x v x y          1 1 1 1 1 1 i i i i ii i i i i i i x y i i
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This note was uploaded on 09/21/2011 for the course CH E 310 taught by Professor Staff during the Spring '08 term at Iowa State.

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Ch E 310 - Fall 10 - Lecture 14 - Lecture 14 October 12,...

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