Pure · Numerical methods
Chapter 1 · 3
The idea
Iterative methods (x = g(x))
Solving f(x) = 0 by rearranging it into x = g(x) and iterating x_(n+1) = g(x_n) from a starting value — why some rearrangements converge and others diverge, and reading staircase and cobweb diagrams.
A full journey — read it, play with it, work it, then earn real exam marks. Everything stays on the timeline below.
In this lesson — start anywhere
Pure · Numerical methods
Iterative methods (x = g(x))
Solving f(x) = 0 by rearranging it into x = g(x) and iterating x_(n+1) = g(x_n) from a starting value — why some rearrangements converge and others diverge, and reading staircase and cobweb diagrams.
Why it works
Fixed points
A change of sign traps a root; an iteration hunts it down to many decimal places. The trick is to rewrite in the form A solution of this is a value that comes out the same as it went in — a fixed point. The idea: guess a starting value near the root, feed it through to get , feed that back to get , and keep going: If this sequence settles down to a limit , then , so is a root of the original equation. You stop when successive values agree to the accuracy you need.Keep reading — free
The rest of the explanation, plus 3 worked examples you step through move by move.
Start freeTakes a minute — no card.