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Padé approximation is a useful tool for extracting singularity information from a power series. A linear Padé approximant is a rational function and can provide estimates of pole and zero locations in the complex plane. A quadratic Padé approximant has square root singularities and can, therefore, provide additional information such as estimates of branch point locations. In this paper, we discuss numerical aspects of computing quadratic Padé approximants as well as some applications. Two algorithms for computing the coefficients in the approximant are discussed: a direct method involving the solution of a linear system (well-known in the mathematics community) and a recursive method (well-known in the physics community). We compare the accuracy of these two methods when implemented in floating-point arithmetic and discuss their pros and cons. In addition, we extend Luke’s perturbation analysis of linear Padé approximation to the quadratic case and identify the problem of spurious branch points in the quadratic approximant, which can cause a significant loss of accuracy. A possible remedy for this problem is suggested by noting that these troublesome points can be identified by the recursive method mentioned above. Another complication with the quadratic approximant arises in choosing the appropriate branch. One possibility, which is to base this choice on the linear approximant, is discussed in connection with an example due to Stahl. It is also known that the quadratic method is capable of providing reasonable approximations on secondary sheets of the Riemann surface, a fact we illustrate here by means of an example. Two concluding applications show the superiority of the quadratic approximant over its linear counterpart: one involving a special function (the Lambert $W$-function) and the other a nonlinear PDE (the continuation of a solution of the inviscid Burgers equation into the complex plane).

Padé approximation is a useful tool for extracting singularity information from a power series. A linear Padé approximant is a rational function and can provide estimates of pole and zero locations in the complex plane. A quadratic Padé approximant has square root singularities and can, therefore, provide additional information such as estimates of branch point locations. In this paper, we discuss numerical aspects of computing quadratic Padé approximants as well as some applications. Two algorithms for computing the coefficients in the approximant are discussed: a direct method involving the solution of a linear system (well-known in the mathematics community) and a recursive method (well-known in the physics community). We compare the accuracy of these two methods when implemented in floating-point arithmetic and discuss their pros and cons. In addition, we extend Luke’s perturbation analysis of linear Padé approximation to the quadratic case and identify the problem of spurious branch points in the quadratic approximant, which can cause a significant loss of accuracy. A possible remedy for this problem is suggested by noting that these troublesome points can be identified by the recursive method mentioned above. Another complication with the quadratic approximant arises in choosing the appropriate branch. One possibility, which is to base this choice on the linear approximant, is discussed in connection with an example due to Stahl. It is also known that the quadratic method is capable of providing reasonable approximations on secondary sheets of the Riemann surface, a fact we illustrate here by means of an example. Two concluding applications show the superiority of the quadratic approximant over its linear counterpart: one involving a special function (the Lambert $W$-function) and the other a nonlinear PDE (the continuation of a solution of the inviscid Burgers equation into the complex plane).

В рамках актуальной проблемы кометно-астероидной опасности численно исследуются физические процессы, вызывающие разрушение и фрагментацию метеорных тел в атмосфере Земли. На основе разработанной физико-математической модели, определяющей движение космических объектов естественного происхождения в атмосфере и их взаимодействия с ней, рассмотрено падение трех одних из самых крупных и по некоторым показателям необычных болидов в истории метеоритики: Тунгусского, Витимского и Челябинского. Их необычность заключается в отсутствии каких-либо материальных метеоритных останков и кратеров в районе предполагаемого места падения для двух первых тел и необнаружении, как предполагается, основного материнского тела для третьего тела (из-за слишком малого количества массы выпавших осколков по сравнению с оценочной массой). Изучено воздействие аэродинамических нагрузок и тепловых потоков на эти тела, приводящее к интенсивному поверхностному уносу массы и возможной фрагментации. Скорости изучаемых небесных тел, изменение их масс определяются из модернизированной системы уравнений теории метеорной физики. Важный фактор, который здесь учитывается, — это переменность параметра уноса массы метеорита под действием тепловых потоков (радиационных и конвективных) вдоль траектории полета. Процесс фрагментации болидов в настоящей работе рассматривается в рамках модели прогрессивного дробления на основе статистической теории прочности с учетом влияния масштабного фактора на предел прочности объектов. Выявлены явления и эффекты, возникающие при различных кинематических и физических параметрах каждого из этих тел. В частности, изменение баллистики их полета в более плотных слоях атмосферы, заключающееся в переходе от режима падения к режиму подъема. При этом возможна реализация следующих сценариев события: первый— возврат тела обратно в космическое пространство при его остаточной скорости, большей второй космической; второй — переход тела на орбиту спутника Земли при остаточной скорости, большей первой космической; третий — при меньших значениях остаточной скорости тела возвращение его через некоторое время к режиму падения и выпадение на значительном расстоянии от предполагаемого места падения. Именно реализация одного из этих трех сценариев события объясняет, например, отсутствие материальных следов, в том числе и кратеров в случае Тунгусского болида в окрестности вывала леса. Предположения о возможности таких сценариев события высказывались и ранее другими авторами, а в настоящей работе их реализация подтверждена результатами численных расчетов.

Within the framework of the actual problem of comet-asteroid danger, the physical processes causing the destruction and fragmentation of meteor bodies in the Earth’s atmosphere are numerically investigated. Based on the developed physicalmathematical models that determines the movements of space objects of natural origin in the atmosphere and their interaction with it, the fall of three, one of the largest and by some parameters unusual bolides in the history of meteoritics, are considered: Tunguska, Vitim and Chelyabinsk. Their singularity lies in the absence of any material meteorite remains and craters in the area of the alleged crash site for the first two bodies and the non-detection, as it is assumed, of the main mother body for the third body (due to the too small amount of mass of the fallen fragments compared to the estimated mass). The effect of aerodynamic loads and heat flows on these bodies are studied, which leads to intensive surface mass loss and possible mechanical destruction. The velocities of the studied celestial bodies and the change in their masses are determined from the modernized system of equations of the theory of meteoric physics. An important factor that is taken into account here is the variability of the meteorite mass entrainment parameter under the action of heat fluxes (radiation and convective) along the flight path. The process of fragmentation of meteoroids in this paper is considered within the framework of a progressive crushing model based on the statistical theory of strength, taking into account the influence of the scale factor on the ultimate strength of objects. The phenomena and effects arising at various kinematic and physical parameters of each of these bodies are revealed. In particular, the change in the ballistics of their flight in the denser layers of the atmosphere, consisting in the transition from the fall mode to the ascent mode. At the same time, the following scenarios of the event can be realized: 1) the return of the body back to outer space at its residual velocity greater than the second cosmic one; 2) the transition of the body to the orbit of the Earth satellite at a residual velocity greater than the first cosmic one; 3) at lower values of the residual velocity of the body, its return after some time to the fall mode and falling out at a considerable distance from the intended crash site. It is the implementation of one of these three scenarios of the event that explains, for example, the absence of material traces, including craters, in the case of the Tunguska bolide in the vicinity of the forest collapse. Assumptions about the possibility of such scenarios have been made earlier by other authors, and in this paper their implementation is confirmed by the results of numerical calculations.