Facultad de Ingenieriacutea UNAM Anaacutelisis Numeacuterico (Semestre 2015-I) Clase 01 (Martes 06 de agosto del 2014)

Paacutegina 1 de 4 Dr Rogelio Hernaacutendez

Introduccioacuten histoacuterica de los meacutetodos numeacutericos

Ejemplo 1 La tabla Babilonia

Numerical analysis is the study of algorithms 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[1] 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] In a square wall section that is two meters by two meters a diagonal beam has to be

meters long[3]

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 equations 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 equations and Markov chains are essential in simulating living cells for medicine and

biologyBefore 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

Facultad de Ingenieriacutea UNAM Anaacutelisis Numeacuterico (Semestre 2015-I) Clase 01 (Martes 06 de agosto del 2014)

Paacutegina 2 de 4 Dr Rogelio Hernaacutendez

Ejemplo 2) El algoritmo numeacuterico para calcular la raiacutez ldquom-eacutesimardquo de un nuacutemero ldquoardquo es

Y para las raiacuteces cuadrada cuacutebica cuarta y quinta correspondientes son

Raiacutez Algoritmo numeacuterico

Cuadrada

Cuacutebica

Cuarta

Quinta

Facultad de Ingenieriacutea UNAM Anaacutelisis Numeacuterico (Semestre 2015-I) Clase 01 (Martes 06 de agosto del 2014)

Paacutegina 3 de 4 Dr Rogelio Hernaacutendez

Aproximacioacuten numeacuterica

Ejemplo) Determine radic120783120785120791120782120787

Presente las aproximaciones numeacutericas en forma tabular y graacutefica

Solucioacuten

El algoritmo numeacuterico correspondiente es

Considerando 1199090 = 15 se obtiene

Comportamiento estable y convergente

Facultad de Ingenieriacutea UNAM Anaacutelisis Numeacuterico (Semestre 2015-I) Clase 01 (Martes 06 de agosto del 2014)

Paacutegina 4 de 4 Dr Rogelio Hernaacutendez

Comportamiento inestable y convergente

Facultad de Ingenieriacutea UNAM Anaacutelisis Numeacuterico (Semestre 2015-I) Clase 01 (Martes 06 de agosto del 2014)

Paacutegina 3 de 4 Dr Rogelio Hernaacutendez

Aproximacioacuten numeacuterica

Ejemplo) Determine radic120783120785120791120782120787

Presente las aproximaciones numeacutericas en forma tabular y graacutefica

Solucioacuten

El algoritmo numeacuterico correspondiente es

Considerando 1199090 = 15 se obtiene

Comportamiento estable y convergente

Facultad de Ingenieriacutea UNAM Anaacutelisis Numeacuterico (Semestre 2015-I) Clase 01 (Martes 06 de agosto del 2014)

Paacutegina 4 de 4 Dr Rogelio Hernaacutendez

Comportamiento inestable y convergente

Facultad de Ingenieriacutea UNAM Anaacutelisis Numeacuterico (Semestre 2015-I) Clase 01 (Martes 06 de agosto del 2014)

Paacutegina 4 de 4 Dr Rogelio Hernaacutendez

Comportamiento inestable y convergente