Numerical solution of equations
Locating a root
- Many equations can't be solved exactly. A root is a solution.
- Sign change: if $f(a)$ and $f(b)$ have opposite signs (and no break between them), a root lies between $a$ and $b$.
Practice
If f(a) and f(b) have opposite signs (and f is continuous between them), then between a and b there is:
A sign change of a continuous function guarantees a root between a and b.
Iteration
- Rearrange the equation into the form $x = F(x)$.
- Use the iterative formula $x_{n+1} = F(x_n)$ from a first guess $x_0$.
- If the values settle, they converge to a root. Keep going until steady to the asked accuracy.
Practice
An iterative formula has the form:
Iteration repeatedly applies x_{n+1} = F(x_n) from a first guess.
Practice
If the iteration values settle down to a steady number, they have converged to a root.
Convergence means the sequence approaches a fixed value — a root of the equation.
You've got it
Key idea
- a sign change of $f$ between $a$ and $b$ traps a root in between
- iteration: $x_{n+1} = F(x_n)$ from a starting guess
- if the values converge (settle), they approach a root