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  "body": "\n## Definition\n\nLet $X$ be a [[normed vector space|03J5RAKB30]], let $D \\subseteq X$, and let $x^* \\in X$ be a solution of a given problem.\nAn *iterative method* is a sequence of maps $\\Phi_k \\colon D^{k+1} \\to D$ for $k \\ge 0$.\nFrom an initial guess $x_0 \\in D$, it produces the iterates $x_{k+1} = \\Phi_k(x_0, \\dots, x_k)$.\nThe method is *convergent* for $x_0$ if the iterates [[converge|090AA928Y7]] to $x^*$.\nIt is *stationary* if there is one map $\\Phi \\colon D \\to D$ with $\\Phi_k(x_0, \\dots, x_k) = \\Phi(x_k)$ for all $k$.\n\n## Remarks\n\nA stationary method is a [[fixed-point iteration|K6ZF91051B]] for $\\Phi$.\nFor a [[linear system|0GT4WWGWH2]] $A x = b$ with $A$ invertible, a *stationary linear method* has the form $x_{k+1} = G x_k + c$.\nIt is *consistent* if $x^* = A^{-1} b$ satisfies $x^* = G x^* + c$.\nThen the error $e_k = x_k - x^*$ satisfies $e_{k+1} = G e_k$.\nSo the method converges for every $x_0$ if and only if the [[spectral radius|9V7YQA85PC]] of $G$ is less than $1$.\nExample: $x_{k+1} = x_k / 2 + 1$ on $\\R$ is stationary and consistent for $x / 2 = 1$, with $x^* = 2$.\nIts error halves at each step, so it converges for every $x_0$.\nNon-example: $x_{k+1} = 2 x_k - 2$ has the fixed point $2$, but the error doubles at each step.\nIt is not convergent for any $x_0 \\ne 2$.\nKrylov subspace methods are nonstationary: each iterate depends on all earlier ones, or on several of them.\n",
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