Introduction to Numerical Analysis

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1、IntroductiontoNumericalAnalysisDoronLevyDepartmentofMathematicsandCenterforScientificComputationandMathematicalModeling(CSCAMM)UniversityofMarylandSeptember21,2010D.LevyPrefaceiD.LevyCONTENTSContentsPrefacei1Introduction12MethodsforSolvingNonlinearProblems22.1PreliminaryDiscussion..................

2、..........22.1.1Arethereanyrootsanywhere?....................32.1.2Examplesofroot-findingmethods..................52.2IterativeMethods...............................62.3TheBisectionMethod............................82.4Newton’sMethod...............................112.5TheSecantMethod...............

3、...............153Interpolation193.1WhatisInterpolation?............................193.2TheInterpolationProblem..........................203.3Newton’sFormoftheInterpolationPolynomial..............223.4TheInterpolationProblemandtheVandermondeDeterminant......233.5TheLagrangeFormoftheInterpolationP

4、olynomial............253.6DividedDifferences..............................283.7TheErrorinPolynomialInterpolation...................313.8InterpolationattheChebyshevPoints...................333.9HermiteInterpolation.............................403.9.1Divideddifferenceswithrepetitions.................4

5、23.9.2TheLagrangeformoftheHermiteinterpolant...........443.10SplineInterpolation..............................473.10.1Cubicsplines.............................493.10.2Whatisnaturalaboutthenaturalspline?.............534Approximations564.1Background..................................564.2TheMinimaxAp

6、proximationProblem...................614.2.1Existenceoftheminimaxpolynomial................624.2.2Boundsontheminimaxerror....................644.2.3Characterizationoftheminimaxpolynomial............654.2.4Uniquenessoftheminimaxpolynomial...............654.2.5Thenear-minimaxpolynomial..............

7、......664.2.6Constructionoftheminimaxpolynomial..............674.3Least-squaresApproximations........................694.3.1Theleast-squaresapproximationproblem..............694.3.2Solvingtheleast-squaresproblem: