fundamental-engineering-optimization-methods.pdf

Ill conditioned matrices give rise to numerical

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Ill-conditioned matrices give rise to numerical errors in computations. In certain cases, it is possible to improve the condition number by scaling the variables. The convergence property implies that the generated sequence converges to the true solution in the limit. The rate of convergence dictates how quickly the approximate solutions approach the exact solution.
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Download free eBooks at bookboon.com Click on the ad to read more Fundamental Engineering Optimization Methods 31 Mathematical Preliminaries Assume that a sequence of points ሼݔ converges to a solution point ݔ כ and define an error sequence: ݁ ൌ ݔ െ ݔ כ ² Then, we say that the sequence ሼݔ converges to ݔ כ with rate and rate constant ܥ if ௞՜ஶ ԡ௘ ೖశభ ԡ ԡ௘ ԡ ൌ ܥ ² Further, if uniform convergence is assumed, then ԡ݁ ௞ାଵ ԡ ൌ ܥԡ݁ ԡ holds for all ݇ ² Thus, convergence to the limit point is faster if ݎ is larger and ܥ is smaller. Specific cases for different choices of ݎ and ܥ are mentioned below. Linear convergence. For ݎ ൌ ͳ DQG Ͳ ൏ ܥ ൏ ͳ ³ ԡ݁ ௞ାଵ ԡ ൌ ܥԡ݁ ԡ ³ signifying linear convergence. In this case the speed of convergence depends only on ܥ , which can be estimated as ܥ ൎ ௙൫௫ ೖశభ ൯ି௙ሺ௫ כ ௙൫௫ ൯ି௙ሺ௫ כ ² . Quadratic Convergence. For r = 2, the convergence is quadratic, i.e., ԡ݁ ௞ାଵ ԡ ൌ ܥԡ݁ ԡ ² In this case if additionally C = 1, then the number of correct digits in double at every iteration. Superlinear Convergence. For ͳ ൏ ݎ ൏ ʹ , the convergence is superlinear. Superlinear convergence is achieved by numerical algorithms that only use the gradient (first derivative) of the cost function, and thus can qualitatively match quadratic convergence.
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Download free eBooks at bookboon.com Fundamental Engineering Optimization Methods 32 Mathematical Preliminaries 2.12 Conjugate-Gradient Method for Linear Equations The conjugate-gradient method is an iterative method designed to solve a system of linear equations described as: ࡭࢞ ൌ ࢈ǡ where A is assumed normal, i.e., ࡭ ൌ ࡭࡭ ² The method initializes with ൌ ૙ǡ and uses an iterative process to obtain an approximate solution in ݊ iterations. The solution is exact in the case of quadratic functions of the form: ݍሺ࢞ሻ ൌ ࡭࢞ െ ࢈ ² ) For general nonlinear functions, convergence in 2 ݊ iterations is to be expected. The method is named so because ࡭࢞ െ ࢈ represents the gradient of the quadratic function. Solving a linear system of equations thus amounts to solving the minimization problem involving a quadratic function.
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