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  "body": "\n## Definition\n\nLet $f \\colon \\R \\times \\R^n \\to \\R^n$, and consider the [[initial value problem|0FXDW992DS]] $y' = f(t, y)$, $y(t_0) = y_0$.\nLet $k \\ge 1$ be an integer, let $h \u003e 0$, let $t_j = t_0 + j h$, and write $f_j = f(t_j, y_j)$.\nLet $\\alpha_0, \\ldots, \\alpha_k$ and $\\beta_0, \\ldots, \\beta_k$ be real numbers with $\\alpha_k = 1$ and $(\\alpha_0, \\beta_0) \\ne (0, 0)$.\nThe *linear $k$-step method* with these coefficients determines $y_{j+k}$ from $y_j, \\ldots, y_{j+k-1}$ by\n$$\\sum_{i=0}^{k} \\alpha_i \\, y_{j+i} = h \\sum_{i=0}^{k} \\beta_i \\, f_{j+i}, \\qquad j = 0, 1, 2, \\ldots$$\nThe method is *explicit* if $\\beta_k = 0$ and *implicit* otherwise.\n\n## Remarks\n\nThe characteristic polynomials of the method are $\\rho(\\zeta) = \\sum_i \\alpha_i \\zeta^i$ and $\\sigma(\\zeta) = \\sum_i \\beta_i \\zeta^i$.\nThe method is consistent if $\\rho(1) = 0$ and $\\rho'(1) = \\sigma(1)$, and zero-stable if every root of $\\rho$ satisfies $\\abs{\\zeta} \\le 1$ and the roots with $\\abs{\\zeta} = 1$ are simple.\nDahlquist's equivalence theorem says that a consistent method converges for all smooth problems with consistent starting values if and only if it is zero-stable.\nExample: the two-step Adams–Bashforth method $y_{j+2} - y_{j+1} = h \\left( \\frac32 f_{j+1} - \\frac12 f_j \\right)$ has $\\rho(\\zeta) = \\zeta^2 - \\zeta$ and $\\sigma(\\zeta) = \\frac32 \\zeta - \\frac12$, so $\\rho(1) = 0$ and $\\rho'(1) = 1 = \\sigma(1)$; it is explicit, consistent and zero-stable.\nThe [[Euler method|0JARTXX4XP]] is the case $k = 1$, $\\alpha_0 = -1$, $\\beta_0 = 1$, $\\beta_1 = 0$.\nNon-example: the classical fourth-order [[Runge–Kutta method|7DBPJKYC6C]] is not a linear multistep method, since its update uses values of $f$ at intermediate points that are not of the form $f_{j+i}$.\nThe explicit method $y_{j+2} + 4 y_{j+1} - 5 y_j = h (4 f_{j+1} + 2 f_j)$ is consistent of order $3$, but $\\rho(\\zeta) = (\\zeta - 1)(\\zeta + 5)$ has the root $-5$, so it is not zero-stable and its errors grow without bound.\n",
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