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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Fall 2003 • Question 5

Question

Math 6610 • Analysis of Numerical Methods I Fall 2003 • Question 5

The Jacobi Method Let \(A\) be a strictly diagonally dominant \(n\times n\) matrix, i.e., \[ \lvert a_{ii}\rvert >\sum _{j\ne i}\lvert a_{ij}\rvert , \qquad i=1,\ldots ,n. \tag {2} \] Describe the Jacobi method applied to the linear system \[ Ax=b \tag {3} \] and prove that it converges for all initial approximations of the solution.

Raw LaTeX

{The Jacobi Method}
Let \(A\) be a strictly diagonally dominant \(n\times n\) matrix, i.e.,
\[
  \lvert a_{ii}\rvert>\sum_{j\ne i}\lvert a_{ij}\rvert,
  \qquad i=1,\ldots,n. \tag{2}
\]
Describe the Jacobi method applied to the linear system
\[
  Ax=b \tag{3}
\]
and prove that it converges for all initial approximations of the solution.

Topic Tags

Iterative Methods for Linear Systems Linear Programming and Optimization Low-Rank Approximation