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  "body": "\n## Statement\n\nLet $A$ be a real symmetric [[positive definite|0KX66CZMBB]] $n \\times n$ matrix, and let $b \\in \\R^n$.\nLet $0 \u003c \\alpha \u003c \\beta$ be numbers such that every [[eigenvalue|0BQAA2BK3Q]] of $A$ lies in $[\\alpha, \\beta]$, and let $\\kappa = \\beta / \\alpha$.\nLet $T_k$ be the [[Chebyshev polynomial|M6EAR1JHPP]] of the first kind of degree $k$, and set $\\theta = (\\beta + \\alpha) / (\\beta - \\alpha)$.\nThe *Chebyshev iteration* produces iterates $x^{(k)}$ whose errors $e_k = x^{(k)} - A^{-1} b$ satisfy $e_k = p_k(A) e_0$, where\n$$p_k(t) = \\frac{T_k\\!\\left( \\frac{\\beta + \\alpha - 2t}{\\beta - \\alpha} \\right)}{T_k(\\theta)}.$$\nThe iterates can be computed by a three-term recurrence that uses one product with $A$ per step and no inner products.\nFor every $k \\ge 0$,\n$$\\norm{e_k}_2 \\le \\frac{\\norm{e_0}_2}{T_k(\\theta)} \\le 2 \\left( \\frac{\\sqrt{\\kappa} - 1}{\\sqrt{\\kappa} + 1} \\right)^k \\norm{e_0}_2.$$\n\n## Remarks\n\nSince $A$ is symmetric, $\\norm{p_k(A)}_2 = \\max_\\lambda \\abs{p_k(\\lambda)}$ over its eigenvalues, and on $[\\alpha, \\beta]$ the numerator of $p_k$ is at most $1$ in absolute value.\nThe bound $T_k(\\theta) \\ge \\frac12 \\sigma^{-k}$ with $\\sigma = (\\sqrt{\\kappa} - 1)/(\\sqrt{\\kappa} + 1)$ follows from $T_k(x) = \\frac12 \\big( (x + \\sqrt{x^2 - 1})^k + (x - \\sqrt{x^2 - 1})^k \\big)$ for $x \\ge 1$.\nAmong all polynomials of degree $k$ with value $1$ at $0$, the scaled Chebyshev polynomial $p_k$ has the least maximum on $[\\alpha, \\beta]$.\nSo no method of the form $e_k = q(A) e_0$ with such $q$ can do better in the worst case over matrices with spectrum in $[\\alpha, \\beta]$.\nFor comparison, the optimal [[Richardson iteration|7CESYMHQEA]] contracts by $(\\kappa - 1)/(\\kappa + 1)$ per step.\nExample: for $\\kappa = 100$ the Chebyshev factor is $9/11 \\approx 0.82$, while the Richardson factor is $99/101 \\approx 0.98$.\nIf the interval does not contain the whole spectrum, the bound fails and the method can diverge.\n\n## Explanations\n\n### L1 human\n\nChebyshev iteration is Richardson's method with a cleverly varying step size.\nThe steps are chosen so that the error is damped evenly over the whole range of eigenvalues.\nThe price is that one must know bounds for the smallest and largest eigenvalues in advance.\n\n### L2 human\n\nThe error factor depends on $\\sqrt{\\kappa}$ instead of $\\kappa$, like the conjugate gradient method, where $\\kappa$ is the [[condition number|AB3S6MHJ8X]].\nUnlike conjugate gradients it needs no inner products, which helps on parallel machines where global sums are costly.\nIts iterates lie in the [[Krylov subspace|QB45PCGQPV]] shifted by $x^{(0)}$, and the conjugate gradient method finds an iterate there that is at least as good in the energy norm.\nChebyshev polynomials are also used as smoothers in multigrid and as polynomial preconditioners.\n",
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