Numerical analysis studies algorithms that produce approximate values when an exact calculation is unavailable or impractical. Many digits alone are not enough: we need to know how reliable an estimate is, whether rounding errors grow, and how much work the method requires.
Three questions for evaluating a method
- Convergence: do repeated steps approach the desired value, and how quickly?
- Numerical stability: does a small input or rounding error cause only a small change in the answer?
- Computational cost: how many steps and function evaluations are needed for the required accuracy?
A computer always represents a result with finite precision. We therefore compare an estimate with a reference value and observe the absolute error |estimate − reference|.
1. Bisection: halve the interval without losing the root
To approximate √2, find a root of f(x)=x²−2. Because f(1)<0 and f(2)>0, the root lies in [1, 2]. The midpoint 1.5 gives f(1.5)=0.25, so keep [1, 1.5]. The next midpoint 1.25 gives f(1.25)=−0.4375, so keep [1.25, 1.5]. Each step halves the interval while retaining the root, provided the function is continuous and the endpoints have opposite signs.
2. Newton: correct an estimate with a tangent
For √a, the formula becomes xₙ₊₁=(xₙ+a/xₙ)/2. With a=2 and x₀=1.5, the next estimates are about 1.4167 and 1.4142. Near the root, the method can be very fast; an unsuitable starting value or a zero derivative can nevertheless make it unreliable. Speed does not replace checking the conditions.
3. Taylor: replace a function with a sum
The function eˣ can be approximated by 1+x+x²/2!+x³/3!+… . At x=1, the first four terms give 2.6667; e is about 2.7183. Adding terms reduces the error in this case, but it is useful to observe how the sum changes and remember that every computation uses a finite number of digits.
4. Numerical integration
The area under a curve can also be estimated. For example π=∫₀¹4√(1−x²)dx. The midpoint rule uses one height per subinterval; the trapezoidal rule joins consecutive values; Simpson’s rule combines values using parabolas; Monte Carlo uses random samples. Each has different assumptions, costs and error behaviour.
What to remember
Approximation is not guessing. Define a repeatable procedure, check its assumptions and measure the error. The best method depends on the problem and the accuracy required.