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INTRODUCTION TO COMPUTATIONAL MATHEMATICS

INTRODUCTION TOCOMPUTATIONALMATHEMATICSC ourse Notes forCM 271 / AM 341 / CS 371H. De SterckP. UllrichDepartment of Applied MathematicsUniversity of WaterlooMarch 20th, 2006 These notes have been funded by2 Contents1 Errors and Error Sources of Error .. Floating Point Numbers and Operations .. A Binary Computer .. Standard floating point systems .. Machine Precision .. Floating Point Operations .. Condition of a Mathematical Problem .. Stability of a Numerical Algorithm .. 232 Root INTRODUCTION .. Four Algorithms for Root Finding .. Bisection Method .. Fixed Point Iteration .. Newton s Method .. Secant Method .. Stopping Criteria for Iterative Functions .. Rate of Convergence .. Convergence Theory .. Fixed Point Iteration .. Newton s Method .. Secant Method .. Overview .. 443 Polynomial Interpolation .. The Vandermonde Matrix .. Lagrange Form .. Hermite Interpolation .. Piecewise Polynomial Interpolation.

and the computational or floating-point representation xˆ = fl(x). Since infinite precision cannot be achieved with finite resources, the computational representation is a finite precision approximation of the exact value. Consider, for example, the decimal number x = 0.00012345876543.

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