A Unified Proof for the Convergence of Jacobi and Gauss–Seidel Methods | Zendy

Roberto Bagnara | Zendy

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A Unified Proof for the Convergence of Jacobi and Gauss–Seidel Methods

Author(s) -

Roberto Bagnara

Publication year - 1995

Publication title -

siam review

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 4.683

H-Index - 120

eISSN - 1095-7200

pISSN - 0036-1445

DOI - 10.1137/1037008

Subject(s) - gauss–seidel method , diagonally dominant matrix , mathematics , irreducibility , convergence (economics) , jacobi method , diagonal , pure mathematics , algebra over a field , mathematical optimization , iterative method , geometry , economics , invertible matrix , economic growth

We present a new unified proof for the convergence of both the Jacobi and the Gauss–Seidel methods for solving systems of linear equations under the criterion of either (a) strict diagonal dominance of the matrix, or (b) diagonal dominance and irreducibility of the matrix. These results are well known. The proof for criterion (a) makes use of Gersgorin’s theorem, while the proof for criterion (b) uses Taussky’s theorem that extends Gersgorin’s work. Hence the topic is interesting for teaching purposes.

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