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
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)