Griffiths D.F., Higham D.J. Numerical Methods for Ordinary Differential Equations: Initial Value Problems

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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)

≈ g

(a), (B.1)

for small h. This rearranges to

g(a + h) ≈ g(a) + hg

(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 clos e to the function g for

suﬃciently small h. Diﬀerentiating with respect to h and setting h = 0 we

see that the polynomial g(a) + hg

(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 h

(a)/2; so

g(a + h) ≈ g(a) + hg

(a) +

(a). (B.3)

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

g(a + h) ≈ g(a) + hg

(a) +

(a) + ··· +

(p)

(a), (B.4)

where g

(p)

denotes the pth derivative. This pth-degree polynomial matches

derivatives of g at the point a from order z ero 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)g

(a)

and

g(x) ≈ g(a) + (x − a)g

(a) +

(x − a)

(a)

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

g(x) ≈ g(a) + (x − a)g

(a) +

(x − a)

(a) + ··· +

(x − a)

(p)

(a). (B.5)

Figure B.1 illustrates the case of p = 1, 2, 3 when g(x) = x e

1−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 h

p+1

Using the asymptotic order notation, as introduced in Section 2.2, we may

write

g(a + h) = g(a) + hg

(a) +

(a) + ··· +

(p)

(a) + O(h

p+1

as h → 0, to quantify the accuracy in (B.4). For the version in (B.5), this

becomes

g(x) = g(a)+(x−a)g

(a)+

(x − a)

(a)+···+

(x − a)

(p)

(a)+O((x−a)

p+1

as x → a.

B.2 Accuracy 249

0 1 2

0.5

1.5

Two terms

0 1 2

0.5

1.5

Th ree term s

0 1 2

0.5

1.5

Four terms

Fig. B.1 In each picture, the function g(x) = x e

1−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,

say c, between a and a + h; that is,

g(a + h) − g(a)

= g

(c).

This shows that

g(a + h) = g(a) + hg

(c). (B.6)

This idea extends to a general Taylor series expansion. For (B.4), we have

g(a+ h) = g(a)+ hg

(a)+

(a)+ ···+

(p)

(a)+

p+1

(p + 1)!

(p+1)

(c), (B.7)

where c is a point between a and a + h. We emphasize that

For 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 pe r 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

g(x) = g(a) + (x − a)g

(a) +

(x − a)

(a)+

··· +

(x − a)

(p)

(a) +

(x − a)

p+1

(p + 1)!

(p+1)

(c), (B.8)

where c is a point between a and x.

We refer to (B.7) and (B.8) as Taylor series expansions with remainder.

EXERCISES

B.1.

Show that the ﬁrst four terms in the Taylor series of g(x) = x e

1−x

about the point a = 1 are

g(x) ≈ 1 − (x − 1) − (x − 1)

(x − 1)

B.2.

Determine the ﬁrst three terms in the Maclaurin expansion of e

−x

Hence, or otherwise, show that the ﬁrst three non-zero terms in the

Maclaurin expansion of g(x) = x e

1−x

are given by

g(x) ≈ e



x − x



Jacobians and Variational Equations

The Taylor series expansion of a function of more than one variable was used in

the process of linearization in Chapter 12 and, in Section 15.1, it was applied to

discrete m aps with two components. To see how such expansions can be derived

from the scalar version developed in Appendix B, we consider a general map

g : R

→ R

, which we write more explicitly as

g(x) =



, x

)

, x

)



Extension to the general case of g : R

→ R

is straightforward. Now, if we

make a small perturbation to the argument x, what happens to the value of

g? Introducing a small quantity ε ∈ R

with components ε

and ε

, we may

expand in the ﬁrst argument—using a two-term Taylor series as described in

Appendix B—to obtain

+ ε

, x

+ ε

) ≈ g

, x

+ ε

) +

∂g

∂x

, x

+ ε

)ε

Then expanding g

, x

+ ε

) in the second argument gives

+ ε

, x

+ ε

) ≈ g

, x

) +

∂g

∂x

, x

)ε

∂g

∂x

, x

+ ε

)ε

The ﬁnal term on the right-hand side could be expanded as ∂g

/∂x

, x

)ε

plus second order terms in ε

and ε

. So we arrive at the expansion

+ ε

, x

+ ε

) ≈ g

, x

) +

∂g

∂x

, x

)ε

∂g

∂x

, x

)ε

. (C.1)

252 C. Jacobians and Variational Equations

Analogously,

+ ε

, x

+ ε

) ≈ g

, x

) +

∂g

∂x

, x

)ε

∂g

∂x

, x

)ε

. (C.2)

The expansions (C.1) and (C.2) can be put toge ther in matrix-vector form as

g(x + ε) ≈ g(x) +

∂g

∂x

(x) ε, (C.3)

where ∂g/∂x denotes the Jacobian of g:

∂g

∂x







∂g

∂x

∂g

∂x

∂g

∂x

∂g

∂x







. (C.4)

In Chapter 15 we also looked at the derivatives of an ODE solution with

respect to the initial conditions. Here, we brieﬂy explain how the relevant varia-

tional equations can be derived. We will restrict ourselves to the case of two au-

tonomous ODEs, x

(t) = f (x(t)), for x ∈ R

, with initial conditions x

(0) = a

and x

(0) = b. We write this more explicitly as







, x

)

, x

)



, x

(0) = a, x

(0) = b. (C.5)

Our aim is to derive an ODE for the collection of partial derivatives ∂x

/∂a,

∂x

/∂b, ∂x

/∂a and ∂x

/∂b. Diﬀerentiating ∂x

/∂a with respect to time, in-

terchanging the order of the derivatives, and using the chain rule, we ﬁnd that



∂x

∂a



∂

∂a





∂

∂a

, x

)) =

∂f

∂x

∂a

∂f

∂x

∂a

Analogously, we have



∂x

∂b



∂f

∂x

∂b

∂f

∂x

∂b



∂x

∂a



∂f

∂x

∂a

∂f

∂x

∂a



∂x

∂b



∂f

∂x

∂b

∂f

∂x

∂b

Putting the four partial derivatives into a matrix, these relations may be written







∂x

∂a

∂x

∂b

∂x

∂a

∂x

∂b













∂f

∂x

∂f

∂x

∂f

∂x

∂f

∂x













∂x

∂a

∂x

∂b

∂x

∂a

∂x

∂b







. (C.6)

C. Jacobians and Variational Equations 253

Here, the matrix







∂f

∂x

∂f

∂x

∂f

∂x

∂f

∂x







, (C.7)

which is known as the Jacobian of the ODE system, is to be evaluated along

the solution x

(t), x

(t) of (C.5).

Constant-Coeﬃcient Diﬀerence Equations

We will describe here means of solving simple linear constant-coeﬃcient diﬀer-

ence equations (4Es for short). For a more in-depth introduction we recom-

mend the books by Dahlquist and Bj¨ork [17, Section 3.3.5] or Elaydi [18].

A kth order diﬀerence is a relationship between k + 1 consecutive terms of a

sequence x

, x

, . . . , x

, . . . . In a kth-order linear constant-coeﬃcient 4E this

relationship is of the form

n+k

+ a

k−1

n+k−1

+ ··· + a

= f

(D.1)

in which the coeﬃcients are a

, a

k−1

, . . . , a

, and f

is a given sequence of

numbers. We shall assume throughout that n runs consecutively through the

non-negative integers: n = 0, 1, . . . . We shall also assume that neither a

nor

is zero, for otherwise this could be written as a 4E of order lower than k.

Our objective is: given k > 0, a set of coeﬃcients and the sequence f

obtain a formula for the nth term of the s equence satisfying (D.1). We shall

focus mainly on ﬁrst (k = 1) and second-order (k = 2) 4Es and also consider

only the cases when either f

≡ 0 for all n (called the homogeneous case) or

when the forcing term f

has a particularly simple form.

As for linear c onstant-coeﬃ cient ODEs, the general solution of 4Es of the

form (D.1) may be composed of the sum of a complementary function (CF—the

general solution of the homogeneous 4E) and a particular solution (PS—any

solution of the given 4E). The arbitrary constants in a general solution may be

ﬁxed by specifying the appropriate number (k) of starting conditions: x

= η

for j = 0 : k − 1.