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# Lecture9 - Engineering Analysis ENG 3420 Fall 2009 Dan C...

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1 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 2 Lecture 9 Lecture 9 Last time: Approximations Finding the roots of the equation f(x)=0 Structured programming File creation and file access Relational operators Project1 Today: Bracketing vs Open Methods for finding the roots of the equation f(x) = 0 ; Convergence vs Divergence Open methods for finding the roots of the equation f(x) = 0 Simple Fixed-Point Iteration Newton-Raphson Secant Next Time Optimization
3 Numerical methods Direct methods attempt to solve a numerical problem by a finite sequence of operations. In absence of round off errors deliver an exact solution ; e.g., solving a linear system Ax = b by Gaussian elimination. Iterative methods attempt to solve a numerical problem (for example, finding the root of an equation or system of equations) by finding successive approximations to the solution starting from an initial guess. The stopping criteria: the relative error is smaller than a pre-specified value. When used The only alternative for non-linear systems of equations; Often useful even for linear problems involving a large number of variables where direct methods would be prohibitively expensive or impossible. Convergence of a numerical methods if successive approximations lead to increasingly smaller relative error. Opposite to divergent. ε a = x i + 1 - x i x i + 1 100%

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4 Iterative methods for finding the roots Bracketing methods Open methods : require only a single starting value or two starting values that do not necessarily bracket a root.
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Lecture9 - Engineering Analysis ENG 3420 Fall 2009 Dan C...

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