Numerical Methods

Topic overview

Numerical methods must not only give an approximate value but also justify the location of the root and whether the iteration is reasonable, and record the process to the precision specified in the question.

Key concepts

If a continuous function satisfies $f(a)f(b)<0$ , then there is at least one root in $(a,b)$ . Uniqueness is usually proved by showing that $f'$ has the same sign throughout the interval. Write fixed-point iteration as $x_{n+1}=g(x_n)$ , and retain enough digits until the end.

Key points

◆First state the continuity
◆Uniqueness and existence are two different things
◆Display the value of each iteration in a list

Worked examples

Try each example yourself before checking the explanation.

1Prove that there is a unique root

Show that $x+\ln x=3$ has a unique root in $(2,3)$ .

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