Numerical Methods

Numerical methods are used to obtain approximate solutions when equations or integrals cannot be solved conveniently by exact algebraic methods.

A root of an equation $f(x)=0$ may be located by finding an interval in which $f(x)$ changes sign, provided the function is sufficiently well-behaved on that interval.

If $f(a)$ and $f(b)$ have opposite signs, a root may lie between $a$ and $b$, and repeatedly narrowing the interval can improve the approximation.

A change-of-sign method can fail when a root touches the x-axis without crossing it or when assumptions about the behaviour of the function are not satisfied.

Iteration maths uses a recurrence relation of the form $x_{n+1}=g(x_n)$ to generate increasingly accurate approximations to a solution.

For a simple iterative method, a starting value $x_0$ is substituted into $g(x)$ repeatedly to obtain $x_1,x_2,x_3,\ldots$ until the values converge.

Cobweb and staircase diagrams provide graphical ways of showing how an iterative sequence approaches, moves away from or cycles around a fixed point.

An iterative method may fail if the sequence diverges, oscillates or approaches a value that is not the required solution.

The Newton Raphson method is an iterative root-finding technique that uses the tangent to a curve to generate successive approximations to a root.

The Newton Raphson formula is $x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$, where $x_n$ is the current approximation and $x_{n+1}$ is the next approximation.

A Newton Raphson equation is applied repeatedly from a suitable starting value until successive approximations agree to the required degree of accuracy.

The Newton Raphson technique can fail when the starting value is unsuitable, when $f'(x)$ is zero or very small, or when the iteration does not converge to the required root.

Numerical integration estimates the value of a definite integral and therefore gives an approximate area under a curve.

The trapezium rule approximates the area under a curve by dividing the interval into trapezia and adding their areas; the trapezium rule formula is $\frac{h}{2}[y_0+y_n+2(y_1+\cdots+y_{n-1})]$ for equal-width intervals.

Numerical methods, including root finding, iteration, the Newton Raphson method and the trapezium rule, can be applied to contextual problems, and the accuracy and limitations of each approximation should be considered.