27.9 EXERCISES
27.9 Exercises27.1 Use an iteration procedure to find the root of the equation 40x=expxto four
significant figures.
27.2 Using the Newton–Raphson procedure find, correct to three decimal places, the
root nearest to 7 of the equation 4x^3 +2x^2 − 200 x−50 = 0.
27.3 Show the following results about rearrangement schemes for polynomial equa-
tions.
(a) That if a polynomial equationg(x)≡xm−f(x)=0,wheref(x)isa
polynomial of degree less thanmand for whichf(0)= 0, is solved using
a rearrangement iteration schemexn+1=[f(xn)]^1 /m, then, in general, the
scheme will have only first-order convergence.
(b) By considering the cubic equation
x^3 −ax^2 +2abx−(b^3 +ab^2 )=0
for arbitrary non-zero values ofaandb, demonstrate that, in special cases, the
same rearrangement scheme can give second- (or higher-) order convergence.
27.4 The square root of a numberNis to be determined by means of the iteration
scheme
xn+1=xn
[
1 −
(
N−x^2 n)
f(N)]
.
Determine how to choosef(N) so that the process has second-order convergence.
Given that√
7 ≈ 2 .65, calculate√
7 as accurately as a single application of the
formula will allow.
27.5 Solve the following set of simultaneous equations using Gaussian elimination
(including interchange where it is formally desirable):
x 1 +3x 2 +4x 3 +2x 4 =0,
2 x 1 +10x 2 − 5 x 3 +x 4 =6,
4 x 2 +3x 3 +3x 4 =20,
− 3 x 1 +6x 2 +12x 3 − 4 x 4 =16.
27.6 The following table of values of a polynomialp(x) of low degree contains an
error. Identify and correct the erroneous value and extend the table up tox=1.2.
xp(x) xp(x)
0.0 0.000 0.5 0.165
0.1 0.011 0.6 0.216
0.2 0.040 0.7 0.245
0.3 0.081 0.8 0.256
0.4 0.128 0.9 0.24327.7 Simultaneous linear equations that result in tridiagonal matrices can sometimes
be treated as three-term recurrence relations, and their solution may be found
in a similar manner to that described in chapter 15. Consider the tridiagonal
simultaneous equations
xi− 1 +4xi+xi+1=3(δi+1, 0 −δi− 1 , 0 ),i=0,± 1 ,± 2 ,....
Prove that, fori>0, the equations have a general solution of the formxi=
αpi+βqi,wherepandqare the roots of a certain quadratic equation. Show that
a similar result holds fori<0. In each case expressx 0 in terms of the arbitrary
constantsα, β,.. ..
Now impose the condition thatxiis bounded asi→±∞and obtain a unique
solution.