Abstract
In this chapter we consider three problems originating from:
-
cubic spline interpolation,
-
a two point boundary value problem,
-
an eigenvalue problem for a two point boundary value problem.
Each of these problems leads to a linear algebra problem with a matrix which is diagonally dominant and tridiagonal. Taking advantage of structure we can show existence, uniqueness and characterization of a solution, and derive efficient and stable algorithms based on LU factorization to compute a numerical solution.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
This is due to the fact that the sites are uniformly spaced. High degree interpolation converges uniformly to the function being interpolated when a sequence consisting of the extrema of the Chebyshev polynomial of increasing degree is used as sites. This is not true for any continuous function (the Faber theorem), but holds if the function is Lipschitz continuous.
- 2.
We show in Sect. 3.3.2 that Gaussian elimination on a full n × n system is an O(n 3) process.
- 3.
Hint: consider an arbitrary polynomial of degree n and expanded it in Taylor series around x i.
- 4.
Hint, use Theorem 2.2.
- 5.
The name spline is inherited from a“physical analogue”, an elastic ruler that is used to draw smooth curves. Heavy weights, called ducks, are used to force the ruler to pass through, or near given locations. The ruler will take a shape that minimizes its potential energy. Since the potential energy is proportional to the integral of the square of the curvature, and the curvature can be approximated by the second derivative it follows from Theorem 2.5 that the mathematical N-spline approximately models the physical spline.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Lyche, T. (2020). Diagonally Dominant Tridiagonal Matrices; Three Examples. In: Numerical Linear Algebra and Matrix Factorizations. Texts in Computational Science and Engineering, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-030-36468-7_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-36468-7_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-36467-0
Online ISBN: 978-3-030-36468-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)