Numerical methods

What numerical methods covers

For equations algebra cannot open, and integrals with no antiderivative. Sign changes trap a root inside an interval, iteration and Newton-Raphson close in on it digit by digit, and the trapezium rule estimates an area from straight-topped strips. Two lessons, Year 13 only, and the marks go as much to naming a failure case as to the arithmetic.

The main ideas

• Trapping a root by a change of sign, with the continuity of the function stated as part of the argument.
• Proving an accuracy claim by choosing the two test values the rounding claim implies, such as 1.25 and 1.35 for one decimal place.
• The two ways the sign change test fails: an even number of roots inside the interval, or a discontinuity that looks like a crossing.
• Fixed-point iteration from a supplied rearrangement, with staircase and cobweb diagrams and the gradient condition behind convergence.
• Newton-Raphson read as a tangent sliding to the axis, and its failure where the tangent is horizontal or nearly so.
• The trapezium rule with a stated number of strips, and deciding from the bend of the curve whether the estimate sits above or below the true value.

The results it turns on

a change of sign in f across [a, b], with f continuous there

the condition that traps a root inside the interval

x_n+1 = g(x_n)

fixed-point iteration, run from the rearrangement the question supplies

x_n+1 = x_n − f(x_n)/f'(x_n)

Newton-Raphson, which fails where f'(x_n) is zero

(h/2)[y₀ + 2(y₁ + … + y_n−1) + y_n], with h = (b − a)/n

the trapezium rule: ends once, middles doubled

Where it usually goes wrong

• A staircase diagram goes with a positive gradient of g at the crossing and a cobweb with a negative one, and convergence from a nearby start needs the size of that gradient below 1.
• Newton-Raphson stops at a horizontal tangent because the tangent then runs parallel to the axis, and saying that is what earns the mark rather than noting the division by zero.
• Miscounting strips against ordinates gives the wrong h, which is the commonest slip in the trapezium work: n strips means n + 1 ordinates.
• Where a question states that an iterative method is required, an exact method does not answer the question that was asked.

Where to start

Locating roots and iteration first, since the sign change argument and the accuracy check are the shortest marks in the unit. Newton-Raphson and the trapezium rule second, and cover differentiation first, because Newton-Raphson calls for f'(x) at every step.