Gauss sedial method

# Gauss sedial method - Gauss– Seidel method I ntroduction...

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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 non-zero 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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Gauss sedial method - Gauss– Seidel method I ntroduction...

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