Search results for "Recurrence relation"
showing 10 items of 17 documents
Exact treatment of linear difference equations with noncommutative coefficients
2007
The exact solution of a Cauchy problem related to a linear second-order difference equation with constant noncommutative coefficients is reported.
Evaluation of legendre functions of argument greater than one
1997
Abstract In this paper we present an algorithm to evaluate Legendre functions of the first and second kinds ( P v , Q v ) for integraland half-integral order and argument greater than one. The code is based on the calculation of the continued fraction for the Q 's. the Wronskian relating P 's and Q 's and the application of forward recurrence relations for the P 's and backward recurrence for the Q 's. We also show an application of these algorithms to the evaluation of the electrostatic field due to a charged toroidal conductor at potential V .
A code to evaluate prolate and oblate spheroidal harmonics
1998
Abstract We present a code to evaluate prolate ( P n m ( x ), Q n m ( x ); n ≥ m , x > 1) and oblate ( P n m ( ix ), Q n m ( ix ); n ≥ m , x > 0) spheroidal harmonics, that is, spherical harmonics ( n and m integers) for real arguments larger than one and for purely imaginary arguments. We start from the known values (in closed form) of P m m and P m +1 m and we apply the forward recurrence relation over n up to a given degree n = N Max . The Wronskian relating P 's and Q 's, together with the evaluation of the continued fraction for Q m+N staggeredMax m / Q m+N staggeredMax -1 m , allows the calculation of Q m+N staggeredMax m and Q m+N staggeredMax -1 m . Backward recurrence is then appli…
A new formulation of the loop-tree duality at higher loops
2019
We present a new formulation of the loop-tree duality theorem for higher loop diagrams valid both for massless and massive cases. $l$-loop integrals are expressed as weighted sum of trees obtained from cutting $l$ internal propagators of the loop graph. In addition, the uncut propagators gain a modified $i \delta$-prescription, named dual-propagators. In this new framework one can go beyond graphs and calculate the integrand of loop amplitudes as a weighted sum of tree graphs, which form a tree-like object. These objects can be computed efficiently via recurrence relations.
Enumeration of L-convex polyominoes by rows and columns
2005
In this paper, we consider the class of L-convex polyominoes, i.e. the convex polyominoes in which any two cells can be connected by a path of cells in the polyomino that switches direction between the vertical and the horizontal at most once.Using the ECO method, we prove that the number fn of L-convex polyominoes with perimeter 2(n + 2) satisfies the rational recurrence relation fn = 4fn-1 - 2fn-2, with f0 = 1, f1 = 2, f2 = 7. Moreover, we give a combinatorial interpretation of this statement. In the last section, we present some open problems.
A comparison of efficient methods for the computation of Born gluon amplitudes
2006
We compare four different methods for the numerical computation of the pure gluonic amplitudes in the Born approximation. We are in particular interested in the efficiency of the various methods as the number n of the external particles increases. In addition we investigate the numerical accuracy in critical phase space regions. The methods considered are based on (i) Berends-Giele recurrence relations, (ii) scalar diagrams, (iii) MHV vertices and (iv) BCF recursion relations.
Integrands of loop amplitudes within loop-tree duality
2020
Using loop-tree duality, we relate a renormalised $n$-point $l$-loop amplitude in a quantum field theory to a phase-space integral of a regularised $l$-fold forward limit of a UV-subtracted $(n+2l)$-point tree-amplitude-like object. We show that up to three loops the latter object is easily computable from recurrence relations. This defines an integrand of the loop amplitude with a global definition of the loop momenta. Field and mass renormalisation are performed in the on-shell scheme.
Recurrence relations for rational cubic methods I: The Halley method
1990
In this paper we present a system of a priori error bounds for the Halley method in Banach spaces. Our theorem supplies sufficient conditions on the initial point to ensure the convergence of Halley iterates, by means of a system of “recurrence relations”, analogous to those given for the Newton method by Kantorovich, improving previous results by Doring [4]. The error bounds presented are optimal for second degree polynomials. Other rational cubic methods, as the Chebyshev method, will be treated in a subsequent paper.
Recurrence and genericity
2003
We prove a C^1-connecting lemma for pseudo-orbits of diffeomorphisms on compact manifolds. We explore some consequences for C^1-generic diffeomorphisms. For instance, C^1-generic conservative diffeomorphisms are transitive. Nous montrons un lemme de connexion C^1 pour les pseudo-orbites des diffeomorphismes des varietes compactes. Nous explorons alors les consequences pour les diffeomorphismes C^1-generiques. Par exemple, les diffeomorphismes conservatifs C^1-generiques sont transitifs.
New special function recurrences giving new indefinite integrals
2018
ABSTRACTSequences of new recurrence relations are presented for Bessel functions, parabolic cylinder functions and associated Legendre functions. The sequences correspond to values of an integer variable r and are generalizations of each conventional recurrence relation, which correspond to r=1. The sequences can be extended indefinitely, though the relations become progressively more intricate as r increases. These relations all have the form of a first-order linear inhomogeneous differential equation, which can be solved by an integrating factor. This gives a very general indefinite integral for each recurrence. The method can be applied to other special functions which have conventional …