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Glossary and Notation A Glossary and Notation AB: Adams–Bashforth—names of a family of explicit LMMs. AB(2) denotes the two-step, second-order Adams–Bashforth method. AM: Adams–Moulton—names of a family of implicit LMMs. AM(2) denotes the two-step, third-order Adams–Moulton method. BDF: backward differentiation formula. BE: backward Euler (method). Cp+1 : error constant of a pth-order LMM. CF: complementary function—the general solution of a homogeneous linear difference or differential equation. 4E: difference equation. E(·): expected value. fn : the value of f(t, x) at t = tn and x = xn. FE: forward Euler (method). GE: global error—the difference between the exact solution x(tn) at t = tn and the numerical solution xn: en = x(tn) − xn. ∇: gradient operator. ∇F denotes the vector of partial derivatives of F. h: step size—numerical solutions are sought at times tn = t0 + nh for n = 0, 1, 2,... D.F. Griffiths, D.J. Higham, Numerical Methods for Ordinary Differential Equations, Springer Undergraduate Mathematics Series, DOI 10.1007/978-0-85729-148-6, © Springer-Verlag London Limited 2010 244 A. Glossary and Notation bh : λh, where λ is the coefficient in the first-order equation x0(t) = λx(t) or an eigenvalue of the matrix A in the system of ODEs x0(t) = Ax(t). I: identity matrix. =(λ): imaginary part of a complex number λ. IAS: interval of absolute stability. IC: initial condition. IVP: initial value problem—an ODE together with initial condition(s). j = m : n: for integers m < n this is shorthand for the sequence of consecutive integers from m to n. That is, j = m, m + 1, m + 2, . , n. Lh : linear difference operator. LMM: linear multistep method. LTE: local truncation error—generally the remainder term R in a Taylor series expansion. ODE: ordinary differential equation. P(a ≤ X ≤ b): probability that X lies in the interval [a, b]. PS: particular solution—any solution of an inhomogeneous linear difference or differential equation. ρ(r): first characteristic polynomial of a LMM. R: region of absolute stability. R0: interval of absolute stability. <(λ): real part of a complex number λ. RK: Runge–Kutta. RK(p)(p ≤ 4) is a p–stage, pth order RK method. RK(p, q) denotes a pair of RK methods for use in adaptive time–stepping. R(bh): stability function of an RK method. σ(r): second characteristic polynomial of a LMM. SDE: stochastic differential equation. tf : final time at which solution is required. tn: a grid point, generally, tn = t0 + nh, at which the numerical solution is computed. Tn: local truncation error at time t = tn. A. Glossary and Notation 245 Tbn: In Chapter 11 it denotes an approximation to the local truncation error Tn, usually based on the leading term in its Taylor expansion. In Chapter 13 it denotes the local truncation error based on the solution of the modified, rather than the original, IVP. TS: Taylor Series. TS(p)—the Taylor Series method with p + 1 terms (up to, and including, pth derivatives). var(·): variance. x: a scalar-valued quantity while x denotes a vector-valued quantity. x0(t): derivative of x(t) with respect to t. 0 00 0 00 xn, xn, xn,... : approximations to x(tn), x (tn), x (tn),... X ∼ N(µ, σ2): a normally distributed random variable with mean µ and vari- ance σ2. B Taylor Series B.1 Expansions The idea behind Taylor series expansions is to approximate a smooth function g : R → R locally by a polynomial. We suppose here that all derivatives of g are bounded. The approximating polynomial will be chosen to match as many derivatives of g as possible at a certain point, say a. Fixing a, we have, from the definition of a derivative, g(a + h) − g(a) ≈ g0(a), (B.1) h for small h. This rearranges to g(a + h) ≈ g(a) + hg0(a). (B.2) As h varies, the right-hand side in (B.2) defines the tangent line to the function g based on the point x = a. This is a first order polynomial approximation, or Taylor series expansion with two terms—it is close to the function g for sufficiently small h. Differentiating with respect to h and setting h = 0 we see that the polynomial g(a) + hg0(a) in (B.2) matches the zeroth and first derivatives of g. We may improve the accuracy of the approximation by adding a second order term of the form h2g00(a)/2; so h2 g(a + h) ≈ g(a) + hg0(a) + g00(a). (B.3) 2 248 B. Taylor Series The right-hand side now matches a further derivative of g at h = 0. Generally, a Taylor series expansion for g about the point a with p + 1 terms has the form 2 p h 00 h g(a + h) ≈ g(a) + hg0(a) + g (a) + ··· + g(p)(a), (B.4) 2 p! where g(p) denotes the pth derivative. This pth-degree polynomial matches derivatives of g at the point a from order zero to p. It is sometimes more convenient to rewrite these expansions with a + h as a single independent variable, say x. The expansions (B.2) and (B.3) then become g(x) ≈ g(a) + (x − a)g0(a) and (x − a)2 g(x) ≈ g(a) + (x − a)g0(a) + g00(a) 2 respectively, and x is required to be close to a for the approximations to be close to g. For the general Taylor expansion (B.4) we then have (x − a)2 (x − a)p g(x) ≈ g(a) + (x − a)g0(a) + g00(a) + ··· + g(p)(a). (B.5) 2 p! 2 Figure B.1 illustrates the case of p = 1, 2, 3 when g(x) = x e1−x and a = 1. The expansions developed in this chapter are extended in Appendix C to functions of two variables. When the point of expansion is the origin, Taylor series are often referred to as Maclaurin series. B.2 Accuracy Increasing the order of our Taylor series expansion by one involves adding a term with an extra power of h. The next term that we would include in the right-hand side of (B.4) to improve the accuracy would be proportional hp+1. Using the asymptotic order notation, as introduced in Section 2.2, we may write h2 hp g(a + h) = g(a) + hg0(a) + g00(a) + ··· + g(p)(a) + O(hp+1), 2 p! as h → 0, to quantify the accuracy in (B.4). For the version in (B.5), this becomes (x − a)2 (x − a)p g(x) = g(a)+(x−a)g0(a)+ g00(a)+···+ g(p)(a)+O((x−a)p+1), 2 p! as x → a. B.2 Accuracy 249 Two terms Three terms Four terms 1.5 1.5 1.5 1 1 1 0.5 0.5 0.5 0 0 0 0 1 2 0 1 2 0 1 2 x x x 2 Fig. B.1 In each picture, the function g(x) = x e1−x is shown as a solid curve. On the left, the Taylor series with two terms (linear) about the point x = 1 is shown as a dashed curve. The middle and right pictures show the Taylor series with three terms (quadratic) and four terms (cubic) respectively The error introduced by the Taylor series can be expressed succinctly, albeit slightly mysteriously. Returning to (B.1), the Mean Value Theorem tells us that the secant approximation (g(a + h) − g(a))/h must match the first derivative of g for at least one point,1 say c, between a and a + h; that is, g(a + h) − g(a) = g0(c). h This shows that g(a + h) = g(a) + hg0(c). (B.6) This idea extends to a general Taylor series expansion. For (B.4), we have h2 hp hp+1 g(a+h) = g(a)+hg0(a)+ g00(a)+···+ g(p)(a)+ g(p+1)(c), (B.7) 2 p! (p + 1)! where c is a point between a and a + h. We emphasize that 1For a concrete example, suppose that a car is driven at an average speed of 100 miles per hour over a stretch of road. This average speed is computed as the ratio of total distance to total time, corresponding to the secant curve on the left hand side of (B.1). It is intuitively reasonable that this car cannot always have been travelling more slowly than 100 miles per hour over this period, and similarly it cannot always have been travelling more quickly than 100 miles per hour. So at some point in time over that period, it must have been travelling at exactly 100 miles per hour. (Of course more than one such time may exist.) 250 B. Taylor Series – although we know that at least one such c exists in that interval, its exact location is unknown; – c may be different for each choice of h. For (B.5), we may write (x − a)2 g(x) = g(a) + (x − a)g0(a) + g00(a)+ 2 (x − a)p (x − a)p+1 ··· + g(p)(a) + g(p+1)(c), (B.8) p! (p + 1)! where c is a point between a and x. We refer to (B.7) and (B.8) as Taylor series expansions with remainder. EXERCISES 2 B.1.? Show that the first four terms in the Taylor series of g(x) = x e1−x about the point a = 1 are 2 5 3 g(x) ≈ 1 − (x − 1) − (x − 1) + 3 (x − 1) .
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