Abstract
Since Descartes’s groundbreaking work the exact or numerical solution of polynomial equations has become a frequent task in analytical geometry. We also have to find the roots of orthogonal polynomials for the implementation of the Gauss rules of numerical integration. Based on his differential calculus, Newton invented a powerful method for localizing the roots of twice differentiable functions.
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.
After the removal of its zero elements. For example, the sequence 1, 3, −2, 0, 1, −4 has 3 sign changes.
- 2.
For example, when p is the characteristic polynomial of a symmetric matrix.
- 3.
It can be computed by the Euclidean algorithm.
- 4.
Notice the sign change for the remainders.
- 5.
We consider the usual Euclidean norm on \(\mathbb{R}^{m}\).
- 6.
The matrices H(v) are symmetric, but we do not need this property here.
- 7.
See the remark following the statement of the proposition.
- 8.
The spectrum of a matrix is by definition the set of its eigenvalues.
- 9.
We will find that A has n distinct (real) eigenvalues in this case.
- 10.
See also the second remark following the proof of Proposition 9.4.
- 11.
See Exercise 11.3 for a more general result, p. 281.
- 12.
The more general condition “f′ ≠ 0 and f″ ≠ 0 in [a, b]” may be reduced to this one by replacing f(x) with f(−x), − f(x) or − f(−x).
- 13.
A convergence of this kind is called quadratic.
- 14.
Compare to the example on p. 280.
- 15.
Properties (ii) and (iii) will be improved in Exercise 12.3, p. 295.
- 16.
See Exercise 12.3 for the real case.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer-Verlag London Ltd.
About this chapter
Cite this chapter
Komornik, V. (2017). Finding Roots. In: Topology, Calculus and Approximation. Springer Undergraduate Mathematics Series. Springer, London. https://doi.org/10.1007/978-1-4471-7316-8_11
Download citation
DOI: https://doi.org/10.1007/978-1-4471-7316-8_11
Published:
Publisher Name: Springer, London
Print ISBN: 978-1-4471-7315-1
Online ISBN: 978-1-4471-7316-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)