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Numerical analysis is the study of Algorithm s for the problems of ''continuous mathematics'' (as distinguished from Discrete Mathematics ). One of the earliest mathematical writing is the Babylonian tablet YBC 7289, which gives a Sexagesimal numerical approximation of , the length of the diagonal in a unit square.The approximation of the Square Root Of 2 is four Sexagesimal figures, which is about six Decimal figures. 1 + 24/60 + 51/602 + 10/603 = 1.41421296... Photograph, illustration, and description of the ''root(2)'' tablet from the Yale Babylonian Collection 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 constructionThe New Zealand Qualification authority specifically mentions this skill in document 13004 version 2, dated 17 October 2003 titled CARPENTRY THEORY: Demonstrate knowledge of setting out a building . In a square wall section that is two meters by two meters, a diagonal beam has to be meters long.By the Pythagorean Theorem , a square whose side is 2 meters has a diagonal of length meters. Numerical analysis continues this long tradition of practical mathematical calculations. Much like the Babylonian approximation to , modern numerical analysis does not seek exact answers, because exact answers are impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors. 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 Equation s appear in the movement of heavenly bodies (planets, stars and galaxies); Optimization occurs in portfolio management; Numerical Linear Algebra is essential to quantitative psychology; Stochastic Differential Equation s and Markov Chain s 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. Nowadays (after mid 20th Century ) these tables have fallen into disuse, because computers can calculate the required functions. The Interpolation Algorithms nevertheless may be used as part of the software for solving Differential Equations and the like. GENERAL INTRODUCTION We will now outline several important themes of numerical analysis. The overall goal is the design and analysis of techniques to give approximate solutions to hard problems. To fix ideas, the reader might consider the following problems and methods:
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 , Lagrange Interpolation 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 estimates of some functions. The canonical work in the field is the NIST publication edited by Abramowitz And Stegun , a 1000-plus page book of a very large number of commonly used formulas and functions and their values at many points. The function values are no longer very useful when a computer is available, but the large listing of formulas can still be very handy. The Mechanical Calculator was also developed as a tool for hand computation. These calculators evolved into electronic computers in the 1940s , and it was then found that these computers were also useful for administrative purposes. But the invention of the computer also influenced the field of numerical analysis, since now longer and more complicated calculations could be done. Direct and iterative methods For the iterative method, apply the Bisection Method to ''f''(''x'') = 3''x''3 + 4. The initial values are ''a'' = 0, ''b'' = 3, ''f''(''a'') = 4, ''f''(''b'') = 85. We conclude from this table that the solution is between 1.875 and 2.0625. The algorithm might return any number in that range with an error less than 0.2. Discretization and numerical integration In a two hour race, we have measured the speed of the car at three instants and recorded them in the following table. Time 0:20 1:00 1:40 km/h 140 150 180 A discretization would be to say that the speed of the car was constant from 0:00 to 0:40, then from 0:40 to 1:20 and finally from 1:20 to 2:00. For instance, the total distance traveled in the first 40 minutes is approximately (2/3h x 140Km/h)=93.3Km. This would allow us to estimate the total distance traveled as 93.3Km + 100Km + 120Km = 313.3Km, which is an example of '''numerical integration''' (see below) using a Riemann Sum , because displacement is the Integral of velocity. Ill posed problem: Take the function ''f''(''x'') = 1/(''x'' − 1). Note that ''f''(1.1) = 10 and ''f''(1.001) = 1000: a change in ''x'' of less than 0.1 turns into a change in ''f''(''x'') of nearly 1000. Evaluating ''f''(''x'') near ''x'' = 1 is an ill-conditioned problem. Well-posed problem: By contrast, the function is continuous and so evaluating it is well-posed.
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