Secular function
In a numerical analysis, a secular function has zeros where the eigenvalues of the matrix lie like the characteristic polynomial. So, finding the roots of the characteristic equation is a popular way to obtain the eigenvalues that can describe the property of the matrix well.
The characteristic polynomial is defined by the determinant of the matrix with a shift. It is usually has only zeros without any root. 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, we can iterate the eigenvalues in a recursive way. According to the methods of partitioning, the variant forms of the secular function can be built up. However, they are all of the form of the series of the simple ration finctions, which have poles at the eigenvalues of the partitioned matrices.
Recently, the secular function has been utilized in the signal processing. The estimatation problem with uncertainty involves a sort of an 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 the 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 can not gurrantee the stability. The fine search algorithm such bisectoin steps is still required for accuracy.