AML702 Applied Computational Methods
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Lecture 12
Solution of set of
simultaneous non-linear
equations
System of Non-linear Equations
• Unlike in linear systems, we obtain curves from
simultaneous non-linear equations.
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System of Non-linear Equations
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The system of non-linear equations can be expressed
as we did in the case of single non-linear equation
Newton-Raphson Method for NL equations
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Recall the N-R method we used earlier based on
derivative. Let us first take a 2 equation system to
illustrate the method
Newton-Raphson Method for 2 NL equations
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Recall the N-R method we used earlier based on
derivative. Let us first take a 2 equation system to
illustrate the method
Newton-Raphson Method for NL equations
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Now we can extend this idea of 2 equations to a multi
equation system
Where [J] is the Jacobian matrix that contains the partial
derivatives w.r.t xis
System of Non-linear equations
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The initial values and the final values of the unknown
variable xi are givens as.
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The function values at i are expressed as:
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Newton-Raphson NL Equations
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Generally Newton-Raphson has speedy quadratic
convergence as in the case of single nonlinear
equation
However, if the initial guesses are not close to the
true roots, it may diverge also.
It is not possible to employ graphical methods in
case of multiequation systems
Evaluation of partial derivatives to determine [J] is
necessary
As accurate initial guesses are necessary for
convergence, some slower alternatives to NewtonRaphson method are used.