Numerical analysis

Numerical analysis designs algorithms that approximate mathematical answers while controlling cost and error. Its central question is how much of a computed answer can be trusted. Finding a zero of a function, evaluating an integral, solving a linear system, and evolving an ordinary differential equation all demand choices about accuracy, work, and the information available. MIT's numerical-methods course explicitly treats performance and accuracy as joint concerns.^[ref]

Three questions organize error analysis. Truncation error measures the approximation introduced by an algorithm even in exact arithmetic; roundoff error comes from finite precision; the condition number measures sensitivity inherent in the problem. Numerical stability asks how the chosen algorithm propagates perturbations. These distinctions suggest different remedies: refine an approximation, change precision or arithmetic, or improve the accuracy of the input.

For example, the square root $s(a)=\sqrt{a}$ has relative condition number $|a{s}^{\prime }(a)/s(a)|=1/2$ for $a>0$: a small relative change in $a$ produces about half that relative change in $s(a)$. In contrast, $g(a)=1/(1-a)$ has relative condition number $|a/(1-a)|$. Changing $a=0.99$ to $0.9901$ changes $g(a)=100$ to about $101.010101$: an input change of about $0.0101\%$ produces an output change of about $1.0101\%$. This sensitivity exists even with exact arithmetic. MIT's conditioning lecture separates the condition number from roundoff error in precisely this way.^[ref]

Numerical analysis is a branch of mathematics that focuses on developing and analyzing numerical methods for solving mathematical problems that cannot be easily solved analytically. This field encompasses various techniques for approximating solutions to problems in areas such as algebra, calculus, differential equations, and optimization. Key aspects of numerical analysis include: 1. **Algorithm Development**: Creating algorithms to obtain numerical solutions to problems. This can involve iterative methods, interpolation, or numerical integration.