Metodi Multistep con accelerazione per la risoluzione numerica dei sistemi di equazioni differenziali ordinarie

In this paper a procedure for the acceleration of the convergence is given. It allows the doubling of the order of the multistep methods for the numerical solution of the systems of ordinary differential equations: $$Y' = F(x,Y); Y_0 = Y(x_0 ) \begin{array}{*{20}c} x \\ {x_0 } \\ \end{array} \in [a,b]$$ whereY andF(x,Y) aret-vectors.     

Sulle formule di quadratura di Tschebyscheff

For the Tschebyscheff quadrature formula: $$\int\limits_{ - 1}^1 {\left( {1 - x^2 } \right)^{\lambda - 1/2} f(x) dx} = K_n \sum\limits_{k = 1}^n {f(x_{n,k} )} + R_n (f), \lambda > 0$$ it is shown that the degre,N, of exactness is bounded by: $$N \leqslant C(\lambda )n^{1/(2\lambda + 1)} $$ whereC(λ) is a convenient function of λ. For λ=1 the complete solution of Tschebyscheff's problem is given.     

Metodi pseudo Runge-Kutta ottimali

For the numerical integration of the problem with initial value $$y' = f(x,y),y(x_0 ) = y_0 ,\begin{array}{*{20}c} {\begin{array}{*{20}c} x \\ {x_0 } \\ \end{array} \in [a,b],} \\ \end{array} $$ the pseudo R. K. methods of second kind are taken again and approximations are drawn, that in particular casef(x, y)≡f(x) are reduced to quadrature formulae of Radau and Lobatto. The limits of the trancation's error and the stability's intervals of the pseudo R. K. methods of the first and second species with the approximations of the same order of R. K. are determined and compared. At the end of that, a numerical example is taken.     

Convergence,stability and truncation error estimation of a method for the numerical integration of the initial value problemY″=F(X,Y)

In this paper a detailed study of the convergence and stability of a numerical method for the differential problem $$\left\{ \begin{gathered} y'' = f(x,y) \hfill \\ y(x_0 ) = y_0 \hfill \\ y'(x_0 ) = y_0 ^\prime \hfill \\ \end{gathered} \right.$$ has carried out and its truncation error estimated. Some numerical experiments are described.     

Alcuni problemi relativi alle formule di quadratura del tipo di tchebycheff

In this paper we study quadrature formulas of the form $$\int\limits_{ - 1}^1 {(1 - x)^a (1 + x)^\beta f(x)dx = \sum\limits_{i = 0}^{r - 1} {[A_i f^{(i)} ( - 1) + B_i f^{(i)} (1)] + K_n (\alpha ,\beta ;r)\sum\limits_{i = 1}^n {f(x_{n,i} ),} } } $$ (α>?1, β>?1), with realA _i ,B _i ,K _n and real nodesx _n,i in (?1,1), valid for prolynomials of degree ≤2n+2r?1. In the first part we prove that there is validity for polynomials exactly of degree2n+2r?1 if and only if α=β=?1/2 andr=0 orr=1. In the second part we consider the problem of the existence of the formula $$\int\limits_{ - 1}^1 {(1 - x^2 )^{\lambda - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} f(x)dx = A_n f( - 1) + B_n f(1) + C\sum\limits_{i = 1}^n {f(x_{n,i} )} }$$ for polynomials of degree ≤n+2. Some numerical results are given when λ=1/2.     

Formule di quadratura «chiuse» di tipo gaussiano: Tabulzione dei nodi e dei coefficienti

The coefficientsA _hi and the nodesx _mi for «closed” Gaussian-type quadrature formulae $$\int\limits_{ - 1}^1 {f(x)dx = \sum\limits_{h = 0}^{2_8 } {\sum\limits_{i = 0}^{m + 1} {A_{hi} f^{(h)} (x_{mi} ) + R\left[ {f(x)} \right]} } } $$ withx _m0 =?1,x _{m, m+1} =1 andR[f(x)]=0 iff(x) is a polinomial of degree at most2m(s+1)+2(2s+1)?1, have been tabulated for the cases: $$\left\{ \begin{gathered} s = 1,2 \hfill \\ m = 2,3,4,5 \hfill \\ \end{gathered} \right.$$ .     

Sul grado di precisione di formule di quadratura del tipo di tchebycheff

In this paper we study quadrature formulas of the types (1) $$\int\limits_{ - 1}^1 {(1 - x^2 )^{\lambda - 1/2} f(x)dx = C_n^{ (\lambda )} \sum\limits_{i = 1}^n f (x_{n,i} ) + R_n \left[ f \right]} ,$$ (2) $$\int\limits_{ - 1}^1 {(1 - x^2 )^{\lambda - 1/2} f(x)dx = A_n^{ (\lambda )} \left[ {f\left( { - 1} \right) + f\left( 1 \right)} \right] + K_n^{ (\lambda )} \sum\limits_{i = 1}^n f (\bar x_{n,i} ) + \bar R_n \left[ f \right]} ,$$ with 0<λ<1, and we obtain inequalities for the degreeN of their polynomial exactness. By using such inequalities, the non-existence of (1), with λ=1/2,N=n+1 ifn is even andN=n ifn is odd, is directly proved forn=8 andn≥10. For the same value λ=1/2 andN=n+3 ifn is evenN=n+2 ifn is odd, the formula (2) does not exist forn≥12. Some intermediary results regarding the first zero and the corresponding Christoffel number of ultraspherical polynomialP _n ^(λ) (x) are also obtained.     

Problemi al contorno per equazioni del tipoy″=f(x,y) trattati come equazioni integrali con metodi alle differenze finite di elevata accuratezza

This work faces the problem of the numerical treatment on digital computers of boundary value problems of the type: $$y'' = f(x,y); y(a) = A,y(b) = B$$ through the reduction into the equivalent integral equation: $$y(x) = \mathop \smallint \limits_a^b g_K (x,\xi )[K^2 y(\xi ) - f(\xi ,y(\xi ))]d\xi $$ whereg _K (x,ξ) is the Green function associated to the differential operator \(\frac{{d^2 }}{{dx^2 }} - K^2 \) . I have extended to this problem a discrete analogue of higher accuracy introduced in [1] with reference to a boundary value problem analised under a differential point of view: this extention costitutes the original part of the work. The above problem is analysed with reference to the discretization error and the convergence of the discrete analogue solution algorithm; the actnal numerical treatment of a few systems follows.     

Quadrature rules for Prandtl's integral equation

In this paper we construct an interpolatory quadrature formula of the type $$\mathop {\rlap{--} \smallint }\limits_{ - 1}^1 \frac{{f'(x)}}{{y - x}}dx \approx \sum\limits_{i = 1}^n {w_{ni} (y)f(x_{ni} )} ,$$ wheref(x)=(1?x)^α(1+x)^β f _o(x), α, β>0, and {x _ni} are then zeros of then-th degree Chebyshev polynomial of the first kind,T _n (x). We also give a convergence result and examine the behavior of the quantity \( \sum\limits_{i = 1}^n {|w_{ni} (y)|} \) asn→∞.     

Un metodo del terzo ordine per l'integrazione numerica dell'equazione differenziale ordinaria

For the numerical integration of the ordinary differential equation $$\frac{{dy}}{{dx}} = F(x,y) y(x_0 ) = y_0 \begin{array}{*{20}c} x \\ {x_0 } \\ \end{array} \varepsilon [a,b]$$ a third method utilizing only two points for every step, is determined different from the analogous Runge-Kutta method employing three points; it is useless take the first step as the «pseudo Runge-Kutta method». The truncation error is given, the convergence is proved and finally a numerical exercise is given.     

Generalizzazione di un metodo alle differenze finite per una classe di problemi al contorno di tipo ellittico

In a previous paper the numerical solution of a particular boundary-value problem for the «weakly linear» equation $$\Delta \left[ {u(P)} \right] = f(P,u)$$ was obtained and the convergence of a suitable finite-difference scheme was proved. This paper is concerned with the more general equation $$L\left[ {u(P)} \right] = f(P,u)$$ where $$L \equiv - \left[ {a\frac{\partial }{{\partial x^2 }} + c\frac{\partial }{{\partial y^2 }} + d\frac{\partial }{{\partial x}} + e\frac{\partial }{{\partial y}}} \right]$$ ; the solution is obtained using the same finite-difference scheme as in the previous paper, and sufficient condition for its convergence are given for this new case     

Interval methods that are guaranteed to underestimate (and the resulting new justification of Kaucher arithmetic)

One of the main objectives of interval computations is, given the functionf(x ₁, ...,x _n ), andn intervals $\bar x_1 ,...,\bar x_n$ , to compute the range $\bar y = f(\bar x_1 ,...,\bar x_n )$ . Traditional methods of interval arithmetic compute anenclosure $Y \supseteq \bar y$ for the desired interval $\bar y$ , an enclosure that is often an overestimation. It is desirable to know how close this enclosure is to the desired range interval. For that purpose, we develop a new interval formalism that produces not only the enclosure, but also theinner estimate for the desired range $\bar y$ , i.e., an interval y such that $y \subseteq \bar y$ . The formulas for this new method turn out to be similar to the formulas of Kaucher arithmetic. Thus, we get a new justification for Kaucher arithmetic.     

Deterministic function computation with chemical reaction networks

Chemical reaction networks (CRNs) formally model chemistry in a well-mixed solution. CRNs are widely used to describe information processing occurring in natural cellular regulatory networks, and with upcoming advances in synthetic biology, CRNs are a promising language for the design of artificial molecular control circuitry. Nonetheless, despite the widespread use of CRNs in the natural sciences, the range of computational behaviors exhibited by CRNs is not well understood. CRNs have been shown to be efficiently Turing-universal (i.e., able to simulate arbitrary algorithms) when allowing for a small probability of error. CRNs that are guaranteed to converge on a correct answer, on the other hand, have been shown to decide only the semilinear predicates (a multi-dimensional generalization of “eventually periodic” sets). We introduce the notion of function, rather than predicate, computation by representing the output of a function \({f:{\mathbb{N}}^k\to{\mathbb{N}}^l}\) by a count of some molecular species, i.e., if the CRN starts with \(x_1,\ldots,x_k\) molecules of some “input” species \(X_1,\ldots,X_k, \) the CRN is guaranteed to converge to having \(f(x_1,\ldots,x_k)\) molecules of the “output” species \(Y_1,\ldots,Y_l\) . We show that a function \({f:{\mathbb{N}}^k \to {\mathbb{N}}^l}\) is deterministically computed by a CRN if and only if its graph \({\{({\bf x, y}) \in {\mathbb{N}}^k \times {\mathbb{N}}^l | f({\bf x}) = {\bf y}\}}\) is a semilinear set. Finally, we show that each semilinear function f (a function whose graph is a semilinear set) can be computed by a CRN on input x in expected time \(O(\hbox{polylog} \|{\bf x}\|_1)\) .     

Converging multistep methods for initial value problems involving multivalued maps

Every consistent and strongly stable multistep method of stepnumberk yields a solution, of the setvalued initial value problem \(\dot y \in F(t,y),y(t_0 ) = y_0 \) . The setF(t, z) is assumed to be nonvoid, convex and closed. Upper semicontinuity of F with respect to both variables is not required everywhere. If the initial value problem is uniquely solvable, the solutions of the multistep method will converge to the solution of the continuous problem. These results carry over to functional differential equations \(\dot y \in F(t,M_t y)\) of Volterra type and to discontinuous problems \(\dot y(t) = f(t,M_t y)\) in the sense of A.F. Filippov. A difference method is applied to the discontinuous delay equation \(\ddot x(t) + 2D\dot x(t) + \omega ^2 x(t) = = - \operatorname{sgn} (x(t - \tau ) + \dot x(t - \tau ))\) . In the limit τ→0 we obtain results for the problem \(\ddot x + 2D\dot x + \omega ^2 x = = - \operatorname{sgn} (x + \dot x)\) which cannot be solved classically everywhere.     

Metodi ad un passo fortemente stabili per equazioni integrali di Volterra di seconda specie di tipo stiff

In this paper we propose some implicit methods for stiff Volterra integral equations of second kind. The methods are constructed on the integro-differential equation obtained by differentiation of the Volterra equation. The numerical schemes are derived using a class of A-stable and L-stable methods for ordinary differential equations (proposed by Liniger and Willoughby in [3]) associated with the Gregory quadrature formula. Related to the test equation: $$y(t) = 1 + \int_0^t {[\lambda + \mu (t - s)] y(s)ds \lambda , \mu< 0} $$ we give the definition ofA-stability and ?-stability for the proposed numerical methods how natural extension of the A-stability and L-stability for the schemes for solving ordinary differential equations. We show how we have to choose the parameters of the methods in order to obtainA-stability and ?-stability schemes. Because of these properties the proposed schemes are particularly advantageous in the case of stiff Volterra integral equations.     

Un metodo per l’integrazione numerica della equazione differenziale ordinariaY″=F(X,Y) con condizioni iniziali

Numerical integration of problem: $$y'' = f(x,y),y(a) = y_0 ,y'(a) = y'_0 $$ is purposed. A method withouty′(x) calculation is attained.     

Valutazione dell'errore per una formula di quadratura alla Tchebycheff di tipo chiuso

In this paper we study the remainder term of a quadrature formula of the form $$\int\limits_{ - 1}^1 {f(x)dx = A_n \left[ {f( - 1) + f(1)} \right] + C_n \sum\limits_{i = 1}^n {f(x_{n,i} ) + R_n \left[ f \right],} } $$ , withx _x,i ∈^-1,1, andR _n [f]=0 whenf(x) is a polynomial of degree ≤n+3 ifn is even, or ≤n+2 ifn is odd. Such a formula exists only forn=1(1)11. It is shown that, iff(x)∈ ^C(h+1) [-1,1], (h=n+3 orn+2), thenR _n [f]=f ^h+1 (τ)·± _n . The values α _n are given.     

Compact Numerical Quadrature Formulas for Hypersingular Integrals and Integral Equations

In the first part of this work, we derive compact numerical quadrature formulas for finite-range integrals $I[f]=\int^{b}_{a}f(x)\,dx$ , where f(x)=g(x)|x?t| ^?? , ?? being real. Depending on the value of ??, these integrals are defined either in the regular sense or in the sense of Hadamard finite part. Assuming that g??C ^??[a,b], or g??C ^??(a,b) but can have arbitrary algebraic singularities at x=a and/or x=b, and letting h=(b?a)/n, n an integer, we derive asymptotic expansions for ${T}^{*}_{n}[f]=h\sum_{1\leq j\leq n-1,\ x_{j}\neq t}f(x_{j})$ , where x _j =a+jh and t??{x ₁,??,x _n?1}. These asymptotic expansions are based on some recent generalizations of the Euler?CMaclaurin expansion due to the author (A.?Sidi, Euler?CMaclaurin expansions for integrals with arbitrary algebraic endpoint singularities, in Math. Comput., 2012), and are used to construct our quadrature formulas, whose accuracies are then increased at will by applying to them the Richardson extrapolation process. We pay particular attention to the case in which ??=?2 and f(x) is T-periodic with T=b?a and $f\in C^{\infty}(-\infty,\infty)\setminus\{t+kT\}^{\infty}_{k=-\infty}$ , which arises in the context of periodic hypersingular integral equations. For this case, we propose the remarkably simple and compact quadrature formula $\widehat{Q}_{n}[f]=h\sum^{n}_{j=1}f(t+jh-h/2)-\pi^{2} g(t)h^{-1}$ , and show that $\widehat{Q}_{n}[f]-I[f]=O(h^{\mu})$ as h??0 ???>0, and that it is exact for a class of singular integrals involving trigonometric polynomials of degree at most n. We show how $\widehat{Q}_{n}[f]$ can be used for solving hypersingular integral equations in an efficient manner. In the second part of this work, we derive the Euler?CMaclaurin expansion for integrals $I[f]=\int^{b}_{a} f(x)dx$ , where f(x)=g(x)(x?t) ^?? , with g(x) as before and ??=?1,?3,?5,??, from which suitable quadrature formulas can be obtained. We revisit the case of ??=?1, for which the known quadrature formula $\widetilde{Q}_{n}[f]=h\sum^{n}_{j=1}f(t+jh-h/2)$ satisfies $\widetilde{Q}_{n}[f]-I[f]=O(h^{\mu})$ as h??0 ???>0, when f(x) is T-periodic with T=b?a and $f\in C^{\infty}(-\infty,\infty)\setminus\{t+kT\}^{\infty}_{k=-\infty}$ . We show that this formula too is exact for a class of singular integrals involving trigonometric polynomials of degree at most n?1. We provide numerical examples involving periodic integrands that confirm the theoretical results.     

A note on tensor chain approximation

This paper deals with the approximation of \(d\) -dimensional tensors, as discrete representations of arbitrary functions \(f(x_1,\ldots ,x_d)\) on \([0,1]^d\) , in the so-called tensor chain format. The main goal of this paper is to show that the construction of a tensor chain approximation is possible using skeleton/cross approximation type methods. The complete algorithm is described, computational issues are discussed in detail and the complexity of the algorithm is shown to be linear in \(d\) . Some numerical examples are given to validate the theoretical results.     

On the evaluation of one-dimensional Cauchy principal value integrals by rules based on cubic spline interpolation

In this note we consider the numerical evaluation of one dimensional Cauchy principal value integrals of the form $$\rlap{--} \smallint _a^b \frac{{k(x)f(x)}}{{x - \lambda }}dx, a< \lambda< b,$$ by rules obtained by “subtracting out” the singularity and then applying product quadratures based on cubic spline interpolation at equally spaced nodes. Convergence results are established for Hölder continuous functions of order, μ, 0<μ≤1, and asymptotic rates are obtained for functionsf≠C ^k [a, b],k=1, 2, 3 or 4. Some comparisons with other methods and numerical examples are also given.