Secular function
 | It has been suggested that this article be merged into characteristic polynomial. (Discuss) Proposed since February 2008. |
The term secular function has been used for what mathematicians now call a characteristic function of a linear operator (in some literature the term secular function is still used). The term comes from the fact that these functions were used to calculate secular perturbations (on a time scale of a century, i.e slow compared to annual motion) of planetary orbits, according to Lagrange's theory of oscillations.
In linear algebra, zeros of a secular function are the eigenvalues of a matrix. Characteristic polynomials also have eigenvalues as roots.
The characteristic polynomial is defined by the determinant of the matrix with a shift. It has zeros only, without any pole. Commonly, the secular function implies the characteristic polynomial. But, in the strict sense, the secular function has poles as well. Interestingly, the poles are located in the eigenvalues of its sub-matrices. Thus, if the information of the sub-matrices is available, the eigenvalues of the matrix can be described using that kind of information. Furthermore, by partitioning the matrix like matrix tearing or gruing, we can iterate the eigenvalues in a recursive way. According to the methods of partitioning, the variant forms of the secular functions can be built up. However, they are all of the form of a series of the simple rational functions, which have poles at the eigenvalues of the partitioned matrices. For example, we can find a form of secular function in the divide-and-conquer eigenvalue algorithm.
Recently, the secular function has been utilized in signal processing. The estimation problem with uncertainty involves a sort of eigenvalue problem, such as a bounded data uncertainty, total least squares, data least squares, partial least squares, errors-in-variables model, etc. Many cases have been solved using their own secular equations. Some are still trying to find the unique secular equation that can resolve a given uncertainty estimation problem.
As for a numerical aspect, it is known that Newton's method is delicate when finding the roots of the secular equation. The higher-order interpolations are recommended. Among them, a simple rational approximation is a good choice considering the balance between the stability and the computational complexity. It is because the secular equation itself consists of a series of simple rational functions. However, using only interpolation cannot guarantee the stability. Thus fine search algorithms such as bisection steps are still required for accuracy.
See also
- secular equation
- characteristic polynomial
- characteristic equation
- Divide-and-conquer eigenvalue algorithm