Numerical analysis

Definition

Numerical analysis is the branch of mathematics that studies algorithms for computing approximate values of quantities defined by problems of continuous mathematics. Its objects are the following.

A problem: a map $f$ from data $x$ to an exact answer $y = f(x)$ , such as a root or an integral.
A numerical method: a finite procedure, or a sequence of them, that computes an approximation $\tilde y$ of $y$ from $x$ by arithmetic operations.
An error analysis: a bound on the error $\tilde y - y$ , or a proof that the approximations have limit $y$ , together with the cost of the method.

Remarks

Example: the problem is to compute $\sqrt{2}$ , and the method is the iteration $x_{k+1} = (x_k + 2/x_k)/2$ from $x_0 = 1$ . The error analysis shows that $x_k \to \sqrt{2}$ and that the error is roughly squared at each step. Non-example: the symbolic simplification $\int_0^1 2x \, dx = 1$ by an antiderivative is exact computation, not an approximation, so it is not a numerical method in this sense. The error of a method has two parts: the error of the method in exact arithmetic, and the error caused by rounding in a computer.

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Limit of a sequence concept
Sequence concept
Numerical analysis definition

Tag	`J0NR3YRW07`
URI	`m2://J0NR3YRW07`
Kind	definition
Writing	W0 stub
MSC	65-01, 65G50
Defines	Numerical analysis
Uses	Sequence, Limit of a sequence

Numerical analysis

Definition

Remarks

Reading path

Further reading

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