The Chemistry Maths Book, Second Edition

562 Chapter 20Numerical methods

20.3 Solution of ordinary equations

An ordinary equation (an equation not involving derivatives or integrals) in one

variable can be written in the form

f(x) 1 = 10 (20.4)

wheref(x)is a function of x. The solutions of the equation are those values of xfor

which (20.4) is true; they are the zerosof the functionf(x). For example, a quadratic

equation is a special case of the algebraic (polynomial) equation

f(x) 1 = 1 a

0

1 + 1 a

1

x 1 + 1 a

2

x

2

1 +1-1+ 1 a

n

x

n

1 = 10 (20.5)

and the solutions are the roots of the polynomial. The general formula for the roots of

the quadratic function was discussed in Chapter 2 and Example 20.3 but, although

exact formulas exist for the roots of cubics and quartics, the general algebraic

equation must be solved numerically. Similarly, the solutions of a transcendental

equation (one that involves transcendental functions) such as

e

−x

1 = 1 tan 1 x or f(x) 1 = 1 e

−x

1 − 1 tan 1 x 1 = 10 (20.6)

cannot normally be written in terms of a finite number of known functions, and the

equation must be solved numerically.

All numerical methods of solving equations proceed by iteration whereby, given

an initial approximate solution,x

0

say, an algorithm exists that usesx

0

to give a new,

hopefully more accurate solutionx

1

. The same algorithm is then applied tox

1

to give

x

2

, and so on:

(20.7)

The iterative process is terminated when a given accuracy has been achieved; for

example, when the change|x

n+ 1

1 − 1 x

n

|is less than a predetermined value. The success

of a numerical procedure often depends on a good choice of initial approximation,

and this can usually be obtained from a graph of the function or from a tabulation of

values of the function.

We consider here two simple methods that are widely used, and that also form

the basis for more sophisticated methods. We note that methods for equations in

one variable can be generalized for several variables and, sometimes, for systems of

simultaneous equations.

The bisection method

The starting point for this ancient method are two values of xfor which the function

is known to lie on opposite sides of zero. As in Figure 20.1, letf(x

1

) 1 < 10 andf(x

2

) 1 > 10 ,

and let the function be continuous in the intervalx

1

tox

2

. We evaluate the function

at the midpoint of the interval,

(20.8)

xxx

312

1

2

=+()

xxn

nn

numerical

algorithm

→

+

,=,,,

1

0123 ...