8. Solving equations numerically
Some basic principles
Solving linear equations is generally easy, and quadratic equations are also easy to solve using either the formula, or some factorisation method, however there are many equations for which at A-Level there is no clear method to solving them exactly. Cubic equations that do not factorise can be solved using an equation (like the quadratic equation formula [but more complicated]) however this is inconvenient, and in the case of equations whose solutions are for some practical purpose, there is less need for exact answers (approximations can suffice).
A single value for which an equation is true is called a root of an equation, and the set of all roots of a given equation is the solution.
Many equations are in (or can be rearranged into) the form , and therefore the line for which the intersections with the curve represent the solution of the equation is which is the x-axis. This leads to a useful rule called the sign-change rule.
Consider such that f(a) and f(b) have opposite signs. This implies that there must be at least one intersection with the x-axis between these two values, as the graph (if it is a continuous function) must pass through the axis at least once in order to go from positive y-value to negative y-value, or the other way around.
The sign-change rule is very helpful in finding roots to equations in the sense that it will allow for certain deductions to be made.
Suppose values of "a", and "b" have been found such that:
Now it is possible to refine this value (through a process of trial and error).
It is possible to use a linear interpolation method to make a sensible first estimate when two numbers (integers in many cases) have been found.
Consider a line that joins these two, and where it would cross the axis (approximately).
(Where "y" is the intersection of the line with the axis [not the curve]).
This method can be used to find a good starting point within the found range for the value.
This process can be repeated (with or without the linear interpolation method) to achieve the desired level of accuracy.
Note that if the question calls for 3 decimal places of accuracy, the idea is to find a root of the equation that is 4 decimal places, (the fourth decimal digit being a five) in order to ascertain whether the root is higher (the same as), or lower; and therefore which way to round it.
Finding roots by iteration
Some sequences will diverge, and others will converge to a limit.
If a recursive definition of a sequence can be made such that:
One missing element is the starting number, however, considering a graph of the function can enable a candidate to make an informed decision as to where to start.
Accurate use of a calculator is vital for this method, and generally using any memory function possessed by the calculator is vital for success.
Iterations which go wrong
Some of these rearrangements will no converge to a limit (it should be somewhat obvious that this is happening).
Some recursive definitions involving the exponential function may need to be rearranged in order to make them converge.
The general rule is simply to try rearrangements and see which one will converge.
Choosing convergent iterations
The smaller the modulus of the gradient at this point, the quicker it will converge, and if the modulus of the gradient is above 1 there will be no convergence.
It is often useful to consider the gradient of the curve at the desired root (intersection with the line ) as this enables the candidate to choose whether to use the original expression or the inverse of the expression (before embarking upon the lengthy process of the iterations).