chapter 1-1 - Notes for Numerical Analysis Math 5466 by S...

This preview shows page 1 - 7 out of 61 pages.

Notes for Numerical Analysis Math 5466 by S. Adjerid Virginia Polytechnic Institute and State University (A Rough Draft) 1
2
Contents 1 Polynomial Interpolation 5 1.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.1 Basic results from calculus . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.2 Pointwise versus uniform convergence . . . . . . . . . . . . . . . . . . 6 1.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Taylor interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Lagrange Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4.1 Standard form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4.2 Lagrange form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4.3 Newton form and divided differences . . . . . . . . . . . . . . . . . . 10 1.5 Interpolation error and convergence . . . . . . . . . . . . . . . . . . . . . . . 14 1.5.1 Interpolation error . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.5.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.6 Interpolation at Chebyshev points . . . . . . . . . . . . . . . . . . . . . . . . 20 1.7 Hermite interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.7.1 Lagrange form of Hermite interpolation polynomials . . . . . . . . . . 26 1.7.2 Newton form of Hermite interpolation polynomial . . . . . . . . . . . 28 1.7.3 Hermite interpolation error . . . . . . . . . . . . . . . . . . . . . . . . 30 1.8 Spline Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.8.1 Piecewise Lagrange interpolation . . . . . . . . . . . . . . . . . . . . 32 1.8.2 Cubic spline interpolation . . . . . . . . . . . . . . . . . . . . . . . . 33 1.8.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.8.4 Convergence of cubic splines . . . . . . . . . . . . . . . . . . . . . . . 43 1.8.5 B-splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 1.8.6 Interpolation for parametric curves . . . . . . . . . . . . . . . . . . . 54 1.9 Interpolation in multiple dimensions . . . . . . . . . . . . . . . . . . . . . . . 54 1.10 Least-squares Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . 55 1.10.1 Discrete least-squares . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 1.10.2 Continuous least-squares . . . . . . . . . . . . . . . . . . . . . . . . . 57 3
4
Chapter 1 Polynomial Interpolation 1.1 Review We start this chapter with a brief review of calculus including the main theorems that will need in our analysis and convergence of sequences of functions. 1.1.1 Basic results from calculus Theorem 1.1.1. Mean-value theorem: Let f C [ a, b ] and differentiable on ( a, b ) then there exists c ( a, b ) such that f 0 ( c ) = f ( b ) - f ( a ) b - a . (1.1) Proof. Consult [4] Theorem 1.1.2. Weighted Mean-value theorem: If f C [ a, b ] and g ( x ) > 0 on [ a, b ] , then there exists c ( a, b ) such that Z b a f ( x ) g ( x ) dx = f ( c ) Z b a g ( x ) dx. (1.2) Proof. Consult [4] Theorem 1.1.3. Rolle’s theorem: If f C [ a, b ] , differentiable on ( a, b ) and f ( a ) = f ( b ) , then there exists c ( a, b ) such that f 0 ( c ) = 0 . Proof. Consult [4] Theorem 1.1.4. Generalized Rolle’s theorem: If f C n +1 ( a, b ) , i.e., f is n + 1 times differentiable on ( a, b ) , and admits n + 2 zeros in [ a, b ] , then there exists c ( a, b ) such that f ( n +1) ( c ) = 0 . Proof. Consult [4] 5
Theorem 1.1.5. Intermediate value theorem: If f C [ a, b ] such that f ( a ) 6 = f ( b ) , then for each y between f ( a ) and f ( b ) there exists c ( a, b ) such that f ( c ) = y . Proof. Consult [4] 1.1.2 Pointwise versus uniform convergence Here we consider a sequence of functions f n ( x ) , n = 0 , 1 , · · · for a x b and define pointwise and uniform convergence for sequences of functions. Definition 1. A sequence f n ( x ) , n = 0 , 1 , · · · converges pointwise to a function f on [ a, b ] if and only if for each x [ a, b ] , lim n →∞ f n ( x ) = f ( x ) . (1.3) Definition 2. The sequence f n ( x ) converges uniformly to a function f on an interval [ a, b ] if and only if lim n →∞ max x [ a,b ] | f n ( x ) - f ( x ) | = 0 . (1.4) Example 1: Let us consider the sequence of functions f n ( x ) = sin ( x/n ) , n = 0 , 1 , · · · , on [0 , ). One can check that f n converges pointwise to f = 0 on [0 , ).

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture