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1.
The solution of Schrödinger's equation leads to a high number N of independent variables. Furthermore, the restriction to (anti)symmetric functions implies some complications. We propose a sparse-grid approximation which leads to a set of non-orthogonal basis. Due to the antisymmetry, scalar products are expressed by sums of N×N-determinants. Because of the sparsity of the sparse-grid approximation, these determinants can be reduced from N×N to a much smaller size K×K. The sums over all permutations reduce to the quantities det K 1,…,α K ):=∑≤i 1,i 2,…,i K Ndet(a ,i β (αβ))α,β=1,…, K to be determined, where a i , j (αβ) are certain one-dimensional scalar products involving (sparse-grid) basis functions ?αβ. We propose a method to evaluate this expression such that the asymptotics of the computational cost with respect to N is O(N 3) for fixed K, while the storage requirements increase only with the factor N 2. Furthermore, we describe a parallel version (N processors) with full speed up.  相似文献   

2.
In the present paper a new method is given for the numerical treatment of the initial problemsy (n)=f(x,y,y′, ...,y (n?1),y (i) (x o )=y o (i) , i=0, 1, ...,n?1. This method is an one-step process of order four. For a class of linear differential equations the exact solution is obtained. Moreover some numerical results are presented.  相似文献   

3.
In many real-life situations, we want to reconstruct the dependencyy=f(x 1,…, xn) from the known experimental resultsx i (k) , y(k). In other words, we want tointerpolate the functionf from its known valuesy (k)=f(x 1 (k) ,…, x n (k) ) in finitely many points $\bar x^{(k)} = (x_1^{(k)} , \ldots ,x_n^{(k)} )$ , 1≤kN There are many functions that go through given points. How to choose one of them? The main goal of findingf is to be able to predicty based onx i. If we getx i from measurements, then usually, we only getintervals that containx i. As a result of applyingf, we get an interval y of possible values ofy. It is reasonable to choosef for which the resulting interval is the narrowest possible. In this paper, we formulate this choice problem in mathematical terms, solve the corresponding problem for several simple cases, and describe the application of these solutions to intelligent control.  相似文献   

4.
In this paper we characterize all algorithms for obtaining the coefficients of (Σn?1i=0xiui)(Σn?1i=0yiui) mod P(u), where P(u) is an irreducible po lynomial of degree n, which use 2n ? 1 multiplications. It is shown that up to equivalence, all such algorithms are obtainable by first obtaining the coefficients of the product of two polynomials, and then reducing modulo the irreducible polynomial.  相似文献   

5.
The problem of locating local maxima and minima of a function from approximate measurement results is vital for many physical applications: inspectral analysis, chemical species are identified by locating local maxima of the spectra; inradioastronomy, sources of celestial radio emission, and their subcomponents, are identified by locating local maxima of the measured brightness of the radio sky;elementary particles are identified by locating local maxima of the experimental curves. Since measurements are never absolutely precise, as a result of the measurements, we have aclass of possible functions. If we measuref(x i ) with interval uncertainty, this class consists of all functionsf for whichf(x i ) ε [y i ??, y i +?], wherey i are the results of measuringf(x i ), andε is the measurement accuracy. For this class, in [2], a linear-time algorithm was described. In real life, a measuring instrument can sometimes malfunction, leading to the so-calledoutliers, i.e., measurementsy i that can be way offf(x i ) (and thus do not restrict the actual valuesf(x i ) at all). In this paper, we describerobust algorithms, i.e., algorithms that find the number of local extrema in the presence of possible outliers. These algorithms solve an important practical problem, but they are not based on any new mathematical results: they simply use algorithms from [2] and [3].  相似文献   

6.
A semi-copula S:[0,1]2→[0,1] is called supermigrative if it is commutative and satisfies S(αx,y)?S(x,αy) for all α∈[0,1] and for all x,y∈[0,1] such that y?x. In this paper, the class of supermigrative semi-copulas is investigated, by focusing, in particular, on the subclass of continuous triangular norms. Some interesting connections with the theory of copulas are also underlined.  相似文献   

7.
A pipelined computer architecture for rapid consecutive evaluation of several elementary functions (x/y, √x, sin x, cos, x, ex, ln x, …) using basic CORDIC algorithms is proposed. Continued products iterations of the form (1 + σim 2?k) allow linking n-identical ALU structures to permit n different function evaluations. New algorithms for sin?1, cos?1, cot?1, sinh?1, cosh?1 and xv are developed. Lastly, a new functional efficiency is defined for pipeline architectures which compares favorably to iterative arrays.Index terms—Digital Arithmetic, Pipeline, Unified Elementary Functions, Iterative Algorithms, CORDIC  相似文献   

8.
In this paper, we introduce “approximate solutions" to solve the following problem: given a polynomial F(x, y) over Q, where x represents an n -tuple of variables, can we find all the polynomials G(x) such that F(x, G(x)) is identically equal to a constant c in Q ? We have the following: let F(x, y) be a polynomial over Q and the degree of y in F(x, y) be n. Either there is a unique polynomial g(x)   Q [ x ], with its constant term equal to 0, such that F(x, y)  = j = 0ncj(y  g(x))jfor some rational numbers cj, hence, F(x, g(x)  + a)   Q for all a  Q, or there are at most t distinct polynomials g1(x),⋯ , gt(x), t  n, such that F(x, gi(x))   Q for 1   i  t. Suppose that F(x, y) is a polynomial of two variables. The polynomial g(x) for the first case, or g1(x),⋯ , gt(x) for the second case, are approximate solutions of F(x, y), respectively. There is also a polynomial time algorithm to find all of these approximate solutions. We then use Kronecker’s substitution to solve the case of F(x, y).  相似文献   

9.
Let f(x) be a member of a set of functions over a probability space. Samples of f(x) are 2-tuples (xi,f(xi) where xi is a sample of the random variable X and f(xi) is a sample of f(x) at x = xi. Some procedures and analysis are presented for the approximation of such functions by systems of orthonormal functions. The approximations are based on the data samples. The analysis includes the case of error in the measurement of f(xi). The properties of the expected square error in the approximation are examined for a number of different estimators for the coefficients in the expansion and these well-behaved and easily analyzed estimators are compared to those obtained using the method of least squares. The effectiveness of different sets of basis functions, those involved in the Karhunen-Loeve expansion and others, can be compared and an approach is suggested to adaptive basis selection in order to select that basis which is most efficient in approximating the particular function under examination. The connection between results and applications are discussed in the introduction and conclusion.  相似文献   

10.
We apply results on extracting randomness from independent sources to “extract” Kolmogorov complexity. For any α,?>0, given a string x with K(x)>α|x|, we show how to use a constant number of advice bits to efficiently compute another string y, |y|=Ω(|x|), with K(y)>(1-?)|y|. This result holds for both unbounded and space-bounded Kolmogorov complexity.We use the extraction procedure for space-bounded complexity to establish zero-one laws for the strong dimensions of complexity classes within ESPACE. The unbounded extraction procedure yields a zero-one law for the constructive strong dimensions of Turing degrees.  相似文献   

11.
Approximation algorithms for covering/packing integer programs   总被引:1,自引:0,他引:1  
Given matrices A and B and vectors a, b, c and d, all with non-negative entries, we consider the problem of computing . We give a bicriteria-approximation algorithm that, given ε∈(0,1], finds a solution of cost O(ln(m)/ε2) times optimal, meeting the covering constraints (Ax?a) and multiplicity constraints (x?d), and satisfying Bx?(1+ε)b+β, where β is the vector of row sums βi=∑jBij. Here m denotes the number of rows of A.This gives an O(lnm)-approximation algorithm for CIP—minimum-cost covering integer programs with multiplicity constraints, i.e., the special case when there are no packing constraints Bx?b. The previous best approximation ratio has been O(ln(maxjiAij)) since 1982. CIP contains the set cover problem as a special case, so O(lnm)-approximation is the best possible unless P=NP.  相似文献   

12.
In this paper, we propose a simple general form of high-order approximation of O(c2+ch2+h4) to solve the two-dimensional parabolic equation αuxx+βuyy=F(x,y,t,u,ux,uy,ut), where α and β are positive constants. We apply the compact form for solving diffusion-convection equation. The results of numerical experiments are presented and compared with analytical solutions to confirm the higher accuracy of the presented scheme.  相似文献   

13.
In the present paper a new exponentially fitted one-step method is given for the numerical treatment of the initial value problemy (n)=f(x, y, y′, ..., y (n?1)),y (j) (x 0)=y 0 (j) j=0, 1, ...n?1. The method is given by a local linearisation off(x, y, y′, ..., y (n?1)). Using new functions the solution of a special linear differential equation of then-th order with constant coefficients is transformed in such a way so that it no longer contains numerical singularities. The efficiency of the method is demonstrated by several numerical stiff-examples.  相似文献   

14.
A double fixed-point theorem is applied to obtain the existence of at least two positive solutions for the boundary value problem, (−1)my(2m)(t) = f(y(t)), t ϵ [0, 1], y(2i)(0) = y(2i+1)(1) = 0, 0 ≤ im−1. It is later applied to obtain the existence of at least two positive solutions for the analogous discrete boundary value problem, (−1)mΔ2mu(k) = g(u(k)), k ϵ {0, …, N}, Δ2iu(0) = Δ2i+1u(N + 1) = 0, 0 ⩽ m − 1.  相似文献   

15.
We study the problem of minimizing the number of late jobs on a single machine where job processing times are known precisely and due dates are uncertain. The uncertainty is captured through a set of scenarios. In this environment, an appropriate criterion to select a schedule is to find one with the best worst-case performance, which minimizes the maximum number of late jobs over all scenarios. For a variable number of scenarios and two distinct due dates over all scenarios, the problem is proved NP-hard in the strong sense and non-approximable in pseudo-polynomial time with approximation ratio less than 2. It is polynomially solvable if the number s of scenarios and the number v of distinct due dates over all scenarios are given constants. An O(nlog?n) time s-approximation algorithm is suggested for the general case, where n is the number of jobs, and a polynomial 3-approximation algorithm is suggested for the case of unit-time jobs and a constant number of scenarios. Furthermore, an O(n s+v?2/(v?1) v?2) time dynamic programming algorithm is presented for the case of unit-time jobs. The problem with unit-time jobs and the number of late jobs not exceeding a given constant value is solvable in polynomial time by an enumeration algorithm. The obtained results are related to a min-max assignment problem, an exact assignment problem and a multi-agent scheduling problem.  相似文献   

16.
A. Bachem  B. Korte 《Computing》1979,23(2):189-198
Given a nonnegative real (m, n) matrixA and positive vectorsu, v, then the biproportional constrained matrix problem is to find a nonnegative (m, n) matrixB such thatB=diag (x) A diag (y) holds for some vectorsx ∈ ? m andy ∈ ? n and the row (column) sums ofB equalu i (v j )i=1,...,m(j=1,..., n). A solution procedure (called the RAS-method) was proposed by Bacharach [1] to solve this problem. The main disadvantage of this algorithm is, that round-off errors slow down the convergence. Here we present a modified RAS-method which together with several other improvements overcomes this disadvantage.  相似文献   

17.
LetL p be the plane with the distanced p (A 1 ,A 2 ) = (¦x 1 ?x 2¦ p + ¦y1 ?y 2¦p)/1p wherex i andy i are the cartesian coordinates of the pointA i . LetP be a finite set of points inL p . We consider Steiner minimal trees onP. It is proved that, for 1 <p < ∞, each Steiner point is of degree exactly three. Define the Steiner ratio ? p to be inf{L s (P)/L m (P)¦P?L p } whereL s (P) andL m (P) are lengths of the Steiner minimal tree and the minimal spanning tree onP, respectively. Hwang showed ?1 = 2/3. Chung and Graham proved ?2 > 0.842. We prove in this paper that ?{∞} = 2/3 and √(√2/2)?1?2 ≤ ?p ≤ √3/2 for anyp.  相似文献   

18.
We prove that there is a polynomial time substitution (y1,…,yn):=g(x1,…,xk) with k?n such that whenever the substitution instance A(g(x1,…,xk)) of a 3DNF formula A(y1,…,yn) has a short resolution proof it follows that A(y1,…,yn) is a tautology. The qualification “short” depends on the parameters k and n.  相似文献   

19.
F. Dubeau 《Computing》1996,57(4):365-369
Fromf(x)=x n ?r and a polynomialQ p (y)=∑ i=0 p a i y i , we consider Newton's method to solveF p (x)=Q p (f(x))=0. We obtain convergent iterative methods of orderp+1 to findr 1/n for arbitraryp.  相似文献   

20.
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