Numerical solution of equations

Need to know

What you need to know

• To show a root of $f(x)=0$ exists in the interval $[a, b]$:
• 1. Ensure $f(x)$ is a continuous function in that interval (polynomials, exponentials, and trig functions usually are).
• 2. Calculate $f(a)$ and $f(b)$.
• 3. State that there is a change of sign (one is positive, one is negative).
• 4. Conclude that a root must lie between $a$ and $b$.

Explanation

Zeroing In on Solutions

1. Change of sign indicates root in interval [a, b].
2. Iteration x_{n+1} = g(x_n) — converges if |g′(x)| < 1 near root.
3. Newton–Raphson: x_{n+1} = x_n − f(x_n)/f′(x_n).
4. Show convergence failure when gradient small or wrong start.