100%(1)1 out of 1 people found this document helpful
This preview shows page 1 - 4 out of 13 pages.
Introduction to Numerical Analysis*Hector D. Ceniceros1What is Numerical Analysis?This is an introductory course of Numerical Analysis, whichcomprises thedesign, analysis, and implementation of constructive methods for the solutionof mathematical problems.Numerical Analysis has vast applications both in Mathematics and inmodern Science and Technology. In the areas of the Physical and Life Sci-ences, Numerical Analysis plays the role of a virtual laboratory by providingaccurate solutions to the mathematical models representing a given physicalor biological system in which the system’s parameters can be varied at will, ina controlled way. The applications of Numerical Analysis also extend to moremodern areas such as data analysis, web search engines, social networks, andbasically anything where computation is involved.2An Illustrative Example:Approximatingan IntegralThe main principles and objectives of Numerical Analysis are better illus-trated with concrete examples and this is the purpose of this chapter. Con-*These are lecture notes for Math 104 A. These notes and all course materials areprotected by United States Federal Copyright Law, the California Civil Code. The UCPolicy 102.23 expressly prohibits students (and all other persons) from recording lecturesor discussions and from distributing or selling lectures notes and all other course materialswithout the prior written permission of the instructor.1
sider the problem of calculating a definite integralI[f] =Zbaf(x)dx.(1)In most cases we cannot find an exact value ofI[f] and very often we onlyknow the integrandfat finite number of points in [a, b].The problem isthen to produce an approximation toI[f] as accurate as we need and at areasonable computational cost.2.1An Approximation Principle and the TrapezoidalRuleOne of the central ideas in Numerical Analysis is to approximate a givenfunction by simpler functions which we can analytically integrate, differenti-ate, etc. For example, we can approximate the integrandf(x) in [a, b] by thesegment of the straight line, a linear polynomialP1(x), that passes through(a, f(a)) and (b, f(b)). That isf(x)≈P1(x) =f(a) +f(b)-f(a)b-a(x-a).(2)andZbaf(x)dx≈ZbaP1(x)dx=f(a)(b-a) +12[f(b)-f(a)](b-a)=12[f(a) +f(b)](b-a).(3)That isZbaf(x)dx≈b-a2[f(a) +f(b)].(4)The right hand side is known as thesimple Trapezoidal Rule Quadrature. Aquadrature is a method to approximate an integral.How accurate is thisapproximation?Clearly, iffis a linear polynomial then the TrapezoidalRule would give us the exact value of the integral, i.e.it would be exact.The underlying question is how well does a linear polynomialP1, satisfyingP1(a) =f(a),(5)P1(b) =f(b),(6)2
approximatesfon the interval [a, b]? We can almost guess the answer. Theapproximation is exact atx=aandx=bbecause of (5)-(6) and it isexact for all polynomial of degree≤1. This suggests thatf(x)-P1(x) =Cf00(ξ)(x-a)(x-b), whereCis a constant. But where isf00evaluated at?