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What are the necessary and sufficient conditions for the gauss-seidel method to converge?

Question:

What are the necessary and sufficient conditions for the gauss-seidel method to converge?

Gauss-Seidel Method:

{eq}\\ {/eq}

It is basically defined as an iterative way of solving many linear equations with some unknown. Let there be {eq}n {/eq} number of linear equations and {eq}x {/eq} be the unknown for which the equation is to be solved, then {eq}Ax=b {/eq} can be defined in an iterative way as :-

{eq}L_{*}x^{k+1}=b-Ux^k {/eq}, where {eq}L_{*} {/eq} is the lower triangular component, {eq}U {/eq} is the strictly upper triangular component, {eq}x^k {/eq} refers to the {eq}k^{th} {/eq} iteration approximation and {eq}x^{k+1} {/eq} is the next to {eq}k^{th} {/eq} approximation.

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{eq}\\ {/eq}

Gauss-Seidel method is generally an iterative method to solve the linear system of equations. The covergent properties of the Gauss...

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