A divide and conquer method for polynomial zeros

AbstractThe problem of factorising an nth-degree polynomial Pn(x) = xn + a1xn−1 + ⋯ + an−1x + an, where the coefficients ai are real, into two polynomials Q(x) and R(x) of degrees nQ and nR can be posed as a system of n quadratic equations. An efficient implementation of Newton's method for the solution of these equations is achieved by exploiting the block Toeplitz structure of the Jacobian matrix which arises when nQ and nR are restricted to {m − 1, m, m + 1∣m = [12n]}, where [x] is the largest integer ⩽x. This polynomial factorisation permits the development of a divide and conquer method for the simultaneous calculation of all the zeros of a polynomial

BINGO: A code for the efficient computation of the scalar bi-spectrum

We present a new and accurate Fortran code, the BI-spectra and Non-Gaussianity Operator (BINGO), for the efficient numerical computation of the scalar bi-spectrum and the non-Gaussianity parameter f_{NL} in single field inflationary models involving the canonical scalar field. The code can calculate all the different contributions to the bi-spectrum and the parameter f_{NL} for an arbitrary triangular configuration of the wavevectors. Focusing firstly on the equilateral limit, we illustrate the accuracy of BINGO by comparing the results from the code with the spectral dependence of the bi-spectrum expected in power law inflation. Then, considering an arbitrary triangular configuration, we contrast the numerical results with the analytical expression available in the slow roll limit, for, say, the case of the conventional quadratic potential. Considering a non-trivial scenario involving deviations from slow roll, we compare the results from the code with the analytical results that have recently been obtained in the case of the Starobinsky model in the equilateral limit. As an immediate application, we utilize BINGO to examine of the power of the non-Gaussianity parameter f_{NL} to discriminate between various inflationary models that admit departures from slow roll and lead to similar features in the scalar power spectrum. We close with a summary and discussion on the implications of the results we obtain.Comment: v1: 5 pages, 5 figures; v2: 35 pages, 11 figures, title changed, extensively revised; v3: 36 pages, 11 figures, to appear in JCAP. The BINGO code is available online at http://www.physics.iitm.ac.in/~sriram/bingo/bingo.htm

RKSUITE: A Suite of Explicit Runge-Kutta Codes

New software based on explicit Runge-Kutta formulas have been developed to replace well-established, widely-used codes written by the authors (RKF45 and its successors in the SLATEC Library and the NAG Fortran 77 Library Runge-Kutta codes). The new software has greater functionality than its predecessors. Also, it is more efficient, more robust and better documented. 1 Introduction Two of the authors are responsible, in part, for some of the most widely used software based on explicit Runge-Kutta (RK) formulas for solving the initial value problem in ordinary differential equations (ODEs), y 0 = f(x; y); y(a) = ff; a ! x ! b (1) where y; f; ff 2 ! n . Specifically, Shampine is an author of RKF45 [20] (succeeded by DERKF in the SLATEC Library) and Gladwell wrote D02PAF (and its driver routines D02BxF where x has the values A, B, D, G and H) available in the NAG Fortran 77 Library [5]. These very successful codes have been used in a variety of applications, embedded in packages and..

Testing A Fortran 90 Separable Hamiltonian System Solver

We discuss a prototype Fortran 90 separable Hamiltonian system solver and present a template illustrating its use. The solver permits a wide choice of symplectic and nonsymplectic integrators, and fixed and error-controlled integration step sizes. It is designed to permit insertion of new formulas as they become available. We report on numerical experiments which show the clear superiority of the non-symplectic methods for some purposes. 1 Partially supported by NATO Collaborative Research Grant 920037 2 Partially supported by a NASA/Texas Space Grant Fellowship 1995-6 3 Current address: BNR Inc, Richardson, TX, USA 4 Current address: DRA, Malvern, UK 1 Introduction Computing the solution, y ffl ! n , of the initial value problem (IVP) for the special second order ordinary differential equation (ODE) system, y 00 = f(x; y); a x; y(a) = y a ; y 0 (a) = y 0 a ; (1) is an important problem in many applications areas. More or less independently, methods for this problem..

Using shape preserving local interpolation for plotting solutions of ordinary differential equations

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A divide and conquer method of polynomial zeros

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Reliable solution of special event location problems for ODEs

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