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.

Answer and Explanation:

{eq}\\ {/eq}

Gauss-Seidel method is generally an iterative method to solve the linear system of equations. The covergent properties of the Gauss Siedel method generally depend on the nature of matrix i.e. convergence is possible only if the two conditions hold :-

(i) The matrix is symmetric positive-definite.

(ii) The matrix is strictly diagonally dominant.


Learn more about this topic:

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How to Solve Linear Systems Using Gaussian Elimination

from Algebra II Textbook

Chapter 10 / Lesson 6
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