The numerical mathematics , also briefly Numerics called, is engaged in a branch of mathematics in the design and analysis of algorithms for continuous mathematical problems. The main application is the approximate calculation of solutions using approximation algorithms with the help of computers .
Interest in such algorithms usually arises for one of the following reasons:
- There is no explicit solution to the problem (as for example in the Navier-Stokes equations or the three-body problem ) or
- the representation of the solution exists, but is not suitable for calculating the solution quickly, or is in a form in which calculation errors are very noticeable (for example with many power series ).
A distinction is made between two types of procedures: on the one hand direct, which provide the exact solution to a problem after a finite number of exact calculation steps, and on the other hand approximation methods which only provide approximations . A direct method is, for example, the Gaussian elimination method , which provides the solution of a linear system of equations . Approximation methods include quadrature formulas that approximately calculate the value of an integral, or the Newton method , which iteratively provides better approximations to a zero of a function.
Since the solutions are only required for finite accuracy in applications, an iterative method can also be more sensible when a direct method exists if it delivers sufficient accuracy in a shorter time.
Different methods are compared according to runtime , stability and robustness . Occasionally, however, there are also (in contrast to purely numerical procedures) seminumerical procedures that are better suited to solving certain problem classes than unspecialized numerical solutions.
The desire to be able to solve mathematical equations numerically (also approximately) has existed since ancient times . The ancient Greeks already knew problems that they could only solve approximately, such as the calculation of areas ( integral calculus ) or the number of circles . In this sense Archimedes , who provided algorithms for both problems, can be described as the first important numerician.
The names of classic methods clearly show that algorithmic and approximate access to mathematical problems has always been important in order to be able to use purely theoretical statements fruitfully. Concepts such as speed of convergence or stability were also very important when calculating by hand. For example, a high speed of convergence gives hope that the calculation will be done quickly. And even Gauss noticed that his calculation errors in the Gaussian elimination process sometimes had a disastrous effect on the solution and made it completely useless. He therefore preferred the Gauss-Seidel method , in which errors could easily be compensated for by performing a further iteration step .
In order to make the monotonous execution of algorithms easier, mechanical calculating machines were developed in the 19th century , and finally the first computer by Konrad Zuse in the 1930s . The Second World War accelerated the development dramatically and in particular John von Neumann advanced numerics both mathematically and technically as part of the Manhattan Project . The Cold War era was dominated by military applications such as re-entry problems, but the increase in computing power since the 1980s has brought civilian applications to the fore. Furthermore, the need for fast algorithms has increased with the increase in speed. Research has been able to do this for many problems, and so the speed of algorithms has improved by about the same order of magnitude as CPU performance since the mid-1980s . Nowadays numerical methods, for example the finite element method , are present in every technical or scientific field and are everyday tools.
One aspect of the analysis of algorithms in numerics is error analysis . Various types of errors come into play in a numerical calculation : When calculating with floating-point numbers , rounding errors inevitably occur. These errors can be reduced, for example, by increasing the number of digits, but they cannot be completely eliminated, since every computer can in principle only count on a finite number of digits.
How the problem reacts to disturbances in the initial data is measured with the condition . If a problem has a great condition, the solution to the problem depends sensitively on the initial data, which makes a numerical solution difficult, especially since rounding errors can be interpreted as a disruption of the initial data.
The numerical method also replaces the continuous mathematical problem with a discrete, i.e. finite, problem. The so-called discretization error already occurs , which is estimated and evaluated in the context of the consistency analysis. This is necessary because a numerical procedure usually does not provide the exact solution.
The consistency and stability of the algorithm usually lead to convergence (see: Limit value (function) ).
A large number of numerical methods and algorithms exist for many mathematical problems, such as optimization or solving partial differential equations . A commented compilation of selected numerical procedures can be found under List of numerical procedures .
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