Book contents
- Frontmatter
- Contents
- Preface
- 1 Solution of equations by iteration
- 2 Solution of systems of linear equations
- 3 Special matrices
- 4 Simultaneous nonlinear equations
- 5 Eigenvalues and eigenvectors of a symmetric matrix
- 6 Polynomial interpolation
- 7 Numerical integration – I
- 8 Polynomial approximation in the ∞-norm
- 9 Approximation in the 2-norm
- 10 Numerical integration – II
- 11 Piecewise polynomial approximation
- 12 Initial value problems for ODEs
- 13 Boundary value problems for ODEs
- 14 The finite element method
- Appendix A An overview of results from real analysis
- Appendix B WWW-resources
- Bibliography
- Index
11 - Piecewise polynomial approximation
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- 1 Solution of equations by iteration
- 2 Solution of systems of linear equations
- 3 Special matrices
- 4 Simultaneous nonlinear equations
- 5 Eigenvalues and eigenvectors of a symmetric matrix
- 6 Polynomial interpolation
- 7 Numerical integration – I
- 8 Polynomial approximation in the ∞-norm
- 9 Approximation in the 2-norm
- 10 Numerical integration – II
- 11 Piecewise polynomial approximation
- 12 Initial value problems for ODEs
- 13 Boundary value problems for ODEs
- 14 The finite element method
- Appendix A An overview of results from real analysis
- Appendix B WWW-resources
- Bibliography
- Index
Summary
Introduction
Up to now, the focus of our discussion has been the question of approximation of a given function f, defined on an interval [a, b], by a polynomial on that interval either through Lagrange interpolation or Hermite interpolation, or by seeking the polynomial of best approximation (in the ∞-norm or 2-norm). Each of these constructions was global in nature, in the sense that the approximation was defined by the same analytical expression on the whole interval [a, b]. An alternative and more flexible way of approximating a function f is to divide the interval [a, b] into a number of subintervals and to look for a piecewise approximation by polynomials of low degree. Such piecewise-polynomial approximations are called splines, and the endpoints of the subintervals are known as the knots.
More specifically, a spline of degree n, n ≥ 1, is a function which is a polynomial of degree n or less in each subinterval and has a prescribed degree of smoothness. We shall expect the spline to be at least continuous, and usually also to have continuous derivatives of order up to k for some k, 0 ≤ k < n. Clearly, if we require the derivative of order n to be continuous everywhere the spline is just a single polynomial, since if two polynomials have the same value and the same derivatives of every order up to n at a knot, then they must be the same polynomial.
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- Information
- An Introduction to Numerical Analysis , pp. 292 - 309Publisher: Cambridge University PressPrint publication year: 2003