Numerical solution of equations

Need to know

What you need to know

• First, write the equation in the form $f(x) = 0$.
• Find two values, $a$ and $b$, such that $f(a)$ and $f(b)$ have opposite signs.
• The function $f(x)$ must be continuous over the interval $[a, b]$.
• Conclude that since there is a change of sign and the function is continuous, at least one root lies between $a$ and $b$.

Explanation

Zeroing In on Solutions

1. First, rearrange your equation from f(x) = 0 into the form x = g(x). This sets up the iterative process.
2. To confirm a root exists, find an interval [a, b] where f(a) and f(b) have opposite signs. This guarantees the curve crosses the x-axis at least once.
3. An iteration can fail if the values move further away from the root. This is called divergence and happens if the rearrangement isn't suitable.
4. To find a root's value, repeat the iteration until consecutive answers are the same to the required number of decimal places.