= Jacobi preconditioner
{c}
{title2=$M=\operatorname{diag}(a_{11},\ldots,a_{nn})$}
= Jacobi preconditioning
{c}
{synonym}
The <Jacobi preconditioner> keeps only the diagonal entries of the <matrix> $A$. For a <symmetric positive-definite matrix>, these entries are positive, so its auxiliary solve is $z_i=r_i/a_{ii}$. It rescales coordinates with very different diagonal sizes, but cannot remove arbitrary off-diagonal coupling.<Math 221, Matrix Computations, Fall 2023/Notes for Ma221 Lecture 15, Dec 1, 2023>[Math 221, Matrix Computations, Fall 2023; Notes for Ma221 Lecture 15, Dec 1, 2023; PDF p. 1, example (1), diagonal or Jacobi preconditioning]{ref}
This shares a <diagonal matrix> with the <Jacobi method>, but the roles differ. The <Jacobi method> specifies a whole <stationary iterative method for a linear system>; a <Jacobi preconditioner> supplies an auxiliary solve to another <iterative method>, including the <Preconditioned conjugate gradient method>.
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