Analisis Numerico

Numerical analysis

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Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). One of the earliest mathematical writings is the Babylonian tablet BC 7289, which gives a sexagesimal numerical approximation of , thelength of the diagonal in a unit square. Being able to compute the sides of a triangle (and hence, being able to compute square roots) is extremely important, for instance, in carpentry and construction.[2] Numerical analysis continues this long tradition of practical mathematical calculations. Much like the Babylonian approximation of , modern numerical

analysis does not seek exact answers,because exact answers are often impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors.

Babylonian clay tablet BC 7289 (c. 1800–1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 + [1] 51/602 +10/603 = 1.41421296...

Numerical analysis naturally finds applications in all fields of engineering and the physical sciences, but in the 21st century, the life sciences and even the arts have adopted elements of scientific computations. Ordinary differential equations appear in the movement of heavenly bodies (planets, stars and galaxies); optimization occurs in portfolio management; numericallinear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology. Before the advent of modern computers numerical methods often depended on hand interpolation in large printed tables. Since the mid 20th century, computers calculate the required functions instead. The interpolation algorithmsnevertheless may be used as part of the software for solving differential equations.

General introduction
The overall goal of the field of numerical analysis is the design and analysis of techniques to give approximate but accurate solutions to hard problems, the variety of which is suggested by the following. • Advanced numerical methods are essential in making numerical weather prediction feasible. •Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of ordinary differential equations. • Car companies can improve the crash safety of their vehicles by using computer simulations of car crashes. Such simulations essentially consist of solving partial differential equations numerically. • Hedge funds (private investment funds) use tools from all fields ofnumerical analysis to calculate the value of stocks and derivatives more precisely than other market participants. • Airlines use sophisticated optimization algorithms to decide ticket prices, airplane and crew assignments and fuel needs. This field is also called operations research. • Insurance companies use numerical programs for actuarial analysis. The rest of this section outlines severalimportant themes of numerical analysis.

Numerical analysis

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History
The field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was already in use more than 2000 years ago. Many great mathematicians of the past were preoccupied by numerical analysis, as is obvious from the names of important algorithms like Newton's method, Lagrangeinterpolation polynomial, Gaussian elimination, or Euler's method. To facilitate computations by hand, large books were produced with formulas and tables of data such as interpolation points and function coefficients. Using these tables, often calculated out to 16 decimal places or more for some functions, one could look up values to plug into the formulas given and achieve very good numerical...