introduction - Introduction to Numerical Analysis Hector D Ceniceros 1 What is Numerical Analysis This is an introductory course of Numerical Analysis

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Introduction to Numerical Analysis * Hector D. Ceniceros 1 What is Numerical Analysis? This is an introductory course of Numerical Analysis, which comprises the design, analysis, and implementation of constructive methods for the solution of mathematical problems . Numerical Analysis has vast applications both in Mathematics and in modern Science and Technology. In the areas of the Physical and Life Sci- ences, Numerical Analysis plays the role of a virtual laboratory by providing accurate solutions to the mathematical models representing a given physical or biological system in which the system’s parameters can be varied at will, in a controlled way. The applications of Numerical Analysis also extend to more modern areas such as data analysis, web search engines, social networks, and basically anything where computation is involved. 2 An Illustrative Example: Approximating an Integral The 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 are protected by United States Federal Copyright Law, the California Civil Code. The UC Policy 102.23 expressly prohibits students (and all other persons) from recording lectures or discussions and from distributing or selling lectures notes and all other course materials without the prior written permission of the instructor. 1
sider the problem of calculating a definite integral I [ f ] = Z b a f ( x ) dx. (1) In most cases we cannot find an exact value of I [ f ] and very often we only know the integrand f at finite number of points in [ a, b ]. The problem is then to produce an approximation to I [ f ] as accurate as we need and at a reasonable computational cost. 2.1 An Approximation Principle and the Trapezoidal Rule One of the central ideas in Numerical Analysis is to approximate a given function by simpler functions which we can analytically integrate, differenti- ate, etc. For example, we can approximate the integrand f ( x ) in [ a, b ] by the segment of the straight line, a linear polynomial P 1 ( x ), that passes through ( a, f ( a )) and ( b, f ( b )). That is f ( x ) P 1 ( x ) = f ( a ) + f ( b ) - f ( a ) b - a ( x - a ) . (2) and Z b a f ( x ) dx Z b a P 1 ( x ) dx = f ( a )( b - a ) + 1 2 [ f ( b ) - f ( a )]( b - a ) = 1 2 [ f ( a ) + f ( b )]( b - a ) . (3) That is Z b a f ( x ) dx b - a 2 [ f ( a ) + f ( b )] . (4) The right hand side is known as the simple Trapezoidal Rule Quadrature . A quadrature is a method to approximate an integral. How accurate is this approximation? Clearly, if f is a linear polynomial then the Trapezoidal Rule would give us the exact value of the integral, i.e. it would be exact. The underlying question is how well does a linear polynomial P 1 , satisfying P 1 ( a ) = f ( a ) , (5) P 1 ( b ) = f ( b ) , (6) 2
approximates f on the interval [ a, b ]? We can almost guess the answer. The approximation is exact at x = a and x = b because of (5)-(6) and it is exact for all polynomial of degree 1. This suggests that f ( x ) - P 1 ( x ) = Cf 00 ( ξ )( x - a )( x - b ), where C is a constant. But where is f 00 evaluated at?
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