| dc.description.abstract | A rational function is defined as the ratio of two polynomials. The Pade approximant (PM,N) is a special kind of rational function. Basically, the Pade approximant of the power series expansion of a function f(z), (z ∈ φ) see pdf for handwritten formula is formed by approximating the power series via the ratio of two polynomials (degree M in numerator, degree N in the denominator). The coefficients of the two polynomials which forms the ratio are called Pade coefficients.
When working with approximations, it is necessary to determine how accurate the approximations are. This is expecially true when working with Pade approximants since the denominator of the approximant has N zeros (N is the degree of the polynomial in the denominator). There is not a formula which bounds the error like the remainder term of the Taylor approximation. Once the poles of the Pade approximant are found, it is important to know how they migrate in the complex plane as the degrees of the numerator and denominator increase. If, as the degrees increase, the poles do not move away from the origin rapidly, then the Pade approximant of the given function is of little use because, PM,N becomes unbounded for small value of z near the poles.
It is helpful to find the disk centered at the origin, which contains no zeros of the denominator polynomial. This is accomplished by using a minimum modulus theorem. Inside the disk of radius, ρ, (i.e., |z| < ρ) the theorem guarantees no zeros of the denominator. This assures that f(z) remains bounded inside the disk of the radius, ρ.
A Fortran program was written to accomplish this task. Using Gaussian elimination, the Pade coefficients are computed. Knowing the Pade coefficients of the denominator, the subroutine MULLER (1) is used to find the roots of the polynomial (MULLER finds complex roots if they exist). The values of the zeros are evaluated in f(z) to determine the accuracy of MULLER. The modulus of the zero is computed to determine its distance from the origin of the complex plane. The minimum modulus theorem is incorporated into another subroutine. The output is printed in tabular form.
As is common in research, the mistakes and unexpected results are as important, if sometimes not more important, than the expected results. This research project was no exception. The many mathematical and computer-related mistakes are fully described in the second part of this report. | en |
