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 Lecture 9 Lecture 9 ± Last time: ² Approximations ² Finding the roots of the equation f(x)=0 ² Structured programming ² File creation and file access ² Relational operators ² Project1 ± Today: ² Bracketing vs Open Methods for finding the roots of the equation f(x) = 0 ; ² Convergence vs Divergence ² Open methods for finding the roots of the equation f(x) = 0 ± Simple Fixed-Point Iteration ± Newton-Raphson ± Secant ± Next Time ² Optimization
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3 Numerical methods ± Direct methods Æ 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 Æ 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. ² The stopping criteria: the relative error is smaller than a pre-specified value. ² 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 Æ 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 : ² 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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