(Each of the four coloured regions shows you the initial estimates from which Newton-Raphson converges to one of the four 4th-roots of 1 among complex numbers)
LESSON 12: NUMERICAL APPROXIMATIONS: INTERVAL BISECTION
Calculate √2 on your calculator. Write down the answer on a whiteboard (1.414213562). Clear the calculator and key the answer back in. Square the number you’ve just keyed in.
Why is the answer different from 2?
Why do you get exactly 2 if you calculate √2 on your calculator and then square the result immediately, rather than writing it down and keying it back in again? (Hint: this only works on modern scientific calculators, not on the cheapest pocket calculators).
We know that √2 is more than 1.414213562 and less than 1.414213563 because the function f(x)=x2 changes sign between those two values, and so the graph of the function goes from below the x-axis to above the x-axis. It’s not the sort of graph which makes sudden jumps, so it must cross the x-axis, in other words there must be a x where x2=0, somewhere between the two values. It’s the same sort of argument as shows that if you travel (within London) from south London to north London, then you must cross the river at some point on your journey.
You’re trying to get a better approximation to √2 than “between 1 and 2”. What is your best strategy? Work in groups, and write your strategy on a whiteboard.
This is also called “bracketing” or “splitting the difference”.
In the exam you’re often asked to explain why a function f(x) has a root in an interval. You should work out the values of f(x) at the two ends of the interval, show that at one f(x) is positive and at the other it is negative, and write “Change of sign, ∴ root in the interval”.
Use two rounds of interval bisection to work out the following square roots to within ¼
√5, starting off with 2 too small, 3 too big
√17, starting off with 4 too small, 5 too big
√222, starting off with 14 too small, 15 too big
√380, starting off with 19 too small, 20 too big
Write down your working neatly and comprehensibly, and check your answers with your calculator.
Do not do exercises from chapter 2 of the book on interval bisection. They ask for insane numbers of rounds of interval-halving, and in the exam never asks for more than two rounds. We will do past exam questions instead. Do not use the way of writing down your working shown in the book, with a table. It’s far too complicated.
LESSON 13: NUMERICAL APPROXIMATIONS: LINEAR INTERPOLATION
Starter activity: √17 is between 4 and 5. But 42 is only one less than 17, and 52 is 8 more than 17.
Write on a whiteboard:
- Which do you think √17 is closer to, 4 or 5?
- Your best guess of how much closer it is to 4 than to 5.
We will work through the pdf on linear interpolation
Practise linear interpolation.
We will work through the pdf on Newton-Raphson
Do this question sheet on interval bisection, linear interpolation, and Newton-Raphson. A second sheet has answers to check your work.
Why three different methods for the same problem? Interval bisection always works, and predictably: you can expect the error to be halved at each bisection. Newton-Raphson sometimes doesn’t work, but when it does work, it converges to the answer much faster. If you have a guess accurate to n decimal places, then another round of Newton-Raphson will give you an answer accurate to 2n decimal places. (Linear interpolation is somewhere in the middle).