x = g(x) Iteration (AQA A-Level Mathematics): Flashcards

Simultaneous equations: $y = x$ and $y = g(x)$

Choose initial $x_0$, find $y$ on $y = g(x)$ curve

Move to $y = x$ line, use this as new $x$-value

When suitably close to root (if converges)

Root exists in the interval

One side of equation = 0 and function continuous

$x_{n+1} = g(x_n)$

$x = \frac{3 - x^3}{4}$

$x = \frac{\ln(x + 3)}{\ln 3}$

How iteration converges visually