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Unformatted text preview: Gauss– Seidel method I ntroduction In numerical linear algebra, the Gauss–Seidel method, also known as the Liebmann method or the method of successive displacement, is an iterative method used to solve a linear system of equations. I t is named after the German mathematicians Carl Friedrich Gauss and Philipp Ludwig von Seidel, and is similar to the Jacobi method. Though it can be applied to any matrix with nonzero elements on the diagonals, convergence is guaranteed if the matrix is either diagonally dominant, or symmetric and positive definite. Description Given a square system of n linear equations with unknown x: Where 1 Then A can be decomposed into a lower t riangular component L*, and a strictly upper t riangular component U: The system of linear equations may be rewritten as: The Gauss–Seidel method is an iterative technique that solves the left hand side of this expression for x, using previous value for x on the right hand side. Analytically, this may be written as : 2 However, by taking advantage of the t riangular form of L*, the elements of x(k+1) can be computed sequentially using forward substitution : Note that the sum inside this computation of xi(k+1) requires each element in x(k) except xi(k) itself.each element in x(k) except xi(k) itself....
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 Spring '10
 chandrasekhar
 Linear Algebra, Algebra, Equations, Gauss–Seidel method, Jacobi method, Philipp Ludwig von Seidel

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