Numerical Methods

When equations cannot be solved algebraically, numerical methods provide approximate solutions. This lesson covers the change-of-sign method, iteration (fixed-point), Newton-Raphson, and the trapezium rule.

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Locating Roots — Change of Sign

If f(x) is continuous on [a, b] and f(a) and f(b) have opposite signs, then there is at least one root in the interval (a, b).

Worked Example 1

Show that x³ − 3x − 5 = 0 has a root between x = 2 and x = 3.

f(2) = 8 − 6 − 5 = −3 (negative)
f(3) = 27 − 9 − 5 = 13 (positive)

Sign change → root exists in (2, 3). ∎

When It Fails

The method fails if:

• The function is discontinuous in the interval.
• There are multiple roots and the sign changes an even number of times.
• The root is exactly at the boundary.

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Iteration (Fixed-Point Method)

Rearrange f(x) = 0 into the form x = g(x), then use the iteration:

xₙ₊₁ = g(xₙ)

starting from an initial estimate x₀.

Convergence Condition

The iteration converges if |g’(x)| < 1 near the root.

Worked Example 2