SelisOnelLectureNotes_SolLinearEqII

SelisOnelLectureNotes_SolLinearEqII - Solution of Systems...

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Solution of Systems of Linear Equations and Applications with MATLAB® : II - Indirect Methods Selis Önel, PhD
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2 SelisÖnel© Solution methods for linear systems A x = y Solution of Linear systems I-Direct Methods - Cramer’s Rule -Elimination Methods - Inverse of a matrix - LU Decomposition II-Indirect Methods Iterative Methods
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3 SelisÖnel© Iterative Solution Good for large systems of equations when Gauss elimination is NOT good, i.e., if n>>m for |A m,n ||x n,1 |= |y m,1 | (# unknowns is very large compared to # equations) Simple programming Applicable to nonlinear coefficients Requires an initial guess to start the iteration The goal is to: Choose a good initial guess x0 for x Substitute x0 in the equations and check if the right hand side of equations is equal to the left hand side or if x-x0< ε Increment/decrement x0 until all equations are satisfied
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4 SelisÖnel© Iterative Solution Popular technique for finding roots of equations Applied to systems of linear equations to produce accurate results (Generalized fixed point iteration ) Jacobi iteration: Carl Jacobi (1804-1851) Gauss-Seidel iteration: Johann Carl Friedrich Gauss (1777-1855) and Philipp Ludwig von Seidel (1821- 1896)
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5 SelisÖnel© Quotations It is true that Fourier had the opinion that the principal aim of mathematics was public utility and explanation of natural phenomena; but a philosopher like him should have known that the sole end of science is the honor of the human mind, and that under this title a question about numbers is worth as much as a question about the system of the world. Quoted in N Rose Mathematical Maxims and Minims (Raleigh N C 1988). Carl Jacobi There are problems to whose solution I would attach an infinitely greater importance than to those of mathematics, for example touching ethics, or our relation to God, or concerning our destiny and our future; but their solution lies wholly beyond us and completely outside the province of science. Quoted in J R Newman, The World of Mathematics (New York 1956). Carl Friedrich Gauss
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6 SelisÖnel© A x = y Solution by Iteration Input an initial guess for iteration to get started Can be any arbitrary vector x0 Ex: null vector x0=zeros(m,1) Good initial guess →fast convergence Consecutive solution of similar problems: Use the solution of previous problem as the initial guess for the next Iteration does not always converge ! 0
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This note was uploaded on 03/14/2012 for the course CHEMICAL E kmu 206 taught by Professor Onel,selis during the Spring '08 term at Hacettepe Üniversitesi.

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SelisOnelLectureNotes_SolLinearEqII - Solution of Systems...

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