Mathematical Methods for Physics and Engineering : A Comprehensive Guide

NUMERICAL METHODS

27.8 A possible rule for obtaining an approximation to an integral is themid-point
rule,givenby
∫x 0 +∆x

x 0

f(x)dx=∆xf(x 0 +^12 ∆x)+O(∆x^3 ).

Writinghfor ∆x, and evaluating all derivates at the mid-point of the interval
(x, x+∆x), use a Taylor series expansion to find, up to O(h^5 ), the coefficients of
the higher-order errors in both the trapezium and mid-point rules. Hence find a
linear combination of these two rules that gives O(h^5 ) accuracy for each step ∆x.
27.9 Although it can easily be shown, by direct calculation, that
∫∞

0

e−xcos(kx)dx=

1

1+k^2

,

the form of the integrand is appropriate for Gauss–Laguerre numerical integra-
tion. Using a 5-point formula, investigate the range of values ofkfor which the
formula gives accurate results. At about what value ofkdo the results become
inaccurate at the 1% level?
27.10 Using the points and weights given in table 27.9, answer the following questions.

(a) A table of unnormalised Hermite polynomialsHn(x) has been spattered with

ink blots and givesH 5 (x)as32x^5 −?x^3 + 120xandH 4 (x)as?x^4 −?x^2 + 12,

where the coefficients marked? cannot be read. What should they read?

(b) What is the value of the integral

I=

∫∞

−∞

e−^2 x

2

4 x^2 +3x+1

dx,

as given by a 7-point integration routine?

27.11 Consider the integralsIpdefined by

Ip=

∫ 1

− 1

x^2 p

√

1 −x^2

dx.

(a) By settingx=sinθand using the results given in exercise 2.42, show thatIp

has the value

Ip=2

2 p− 1

2 p

2 p− 3

2 p− 2

···

1

2

π

2

.

(b) EvaluateIpforp=1, 2 ,... ,6 using 5- and 6-point Gauss–Chebyshev inte-

gration(convenientlyrunonaspreadsheetsuchasExcel) and compare the

results with those in (a). In particular, show that, as expected, the 5-point

scheme first fails to be accurate when the order of the polynomial numerator

(2p) exceeds (2×5)−1 = 9. Likewise, verify that the 6-point scheme evaluates

I 5 accurately but is in error forI 6.

27.12 In normal use only a single application ofn-point Gaussian quadrature is made,
using a value ofnthat is estimated from experience to be ‘safe’. However, it is
instructive to examine what happens whennis changed in a controlled way.
(a) Evaluate the integral

In=

∫ 5

2

√

7 x−x^2 − 10 dx

usingn-point Gauss–Legendre formulae forn=2, 3 ,... ,6. Estimate (to 4

s.f.) the valueI∞you would obtain for very largenand compare it with the

resultIobtained by exact integration. Explain why the variation ofInwith

nis monotonically decreasing.