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Polynomial Solutions Of P-recursive Equations
In mathematics a P-recursive equation can be solved for polynomial solutions. Sergei A. Abramov in 1989 and Marko Petkovšek in 1992 described an algorithm which finds all polynomial solutions of those recurrence equations with polynomial coefficients. The algorithm computes a ''degree bound'' for the solution in a first step. In a second step an ansatz for a polynomial of this degree is used and the unknown coefficients are computed by a system of linear equations. This article describes this algorithm. In 1995 Abramov, Bronstein and Petkovšek showed that the polynomial case can be solved more efficiently by considering power series solution of the recurrence equation in a specific power basis (i.e. not the ordinary basis (x^n)_). Other algorithms which compute rational or hypergeometric solutions of a linear recurrence equation with polynomial coefficients also use algorithms which compute polynomial solutions. Degree bound Let \mathbb be a field of characteristic zero and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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P-recursive Equation
In mathematics a P-recursive equation is a linear equation of sequences where the coefficient sequences can be represented as polynomials. P-recursive equations are linear recurrence equations (or linear recurrence relations or linear difference equations) with polynomial coefficients. These equations play an important role in different areas of mathematics, specifically in combinatorics. The sequences which are solutions of these equations are called holonomic, P-recursive or D-finite. From the late 1980s on the first algorithms were developed to find solutions for these equations. Sergei A. Abramov, Marko Petkovšek and Mark van Hoeij described algorithms to find polynomial, rational, hypergeometric and d'Alembertian solutions. Definition Let \mathbb be a field of characteristic zero (for example \mathbb = \mathbb), p_k(n) \in \mathbb /math> polynomials for k = 0,\dots,r,f \in \mathbb^ a sequence and y \in \mathbb^ an unknown sequence. The equation\sum_^r p_k(n) \, y (n+k) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Marko Petkovšek
Marko Petkovšek is a Slovenian mathematician, born: 1955, working mainly in symbolic computation. He is a professor of discrete and computational mathematics at the University of Ljubljana. He completed his Ph.D. at Carnegie Mellon University under the supervision of Dana Scott. He is best known for Petkovšek's algorithm. Together with Herbert Wilf and Doron Zeilberger he wrote the book ''A = B A, or a, is the first letter and the first vowel of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''a'' (pronounced ), plural ''aes'' ...''. External links "A = B"* 1955 births Carnegie Mellon University alumni Living people 21st-century Slovenian mathematicians University of Ljubljana faculty 20th-century Slovenian mathematicians {{europe-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can perform automated deductions (referred to as automated reasoning) and use mathematical and logical tests to divert the code execution through various routes (referred to as automated decision-making). Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus". In contrast, a heuristic is an approach to problem solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well-defined correct or optimal result. As an effective method, an algorithm can be expressed within a finite amount of spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Polynomial Ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field. Often, the term "polynomial ring" refers implicitly to the special case of a polynomial ring in one indeterminate over a field. The importance of such polynomial rings relies on the high number of properties that they have in common with the ring of the integers. Polynomial rings occur and are often fundamental in many parts of mathematics such as number theory, commutative algebra, and algebraic geometry. In ring theory, many classes of rings, such as unique factorization domains, regular rings, group rings, rings of formal power series, Ore polynomials, graded rings, have been introduced for generalizing some properties of polynomial rings. A closely related notion is that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Ansatz
In physics and mathematics, an ansatz (; , meaning: "initial placement of a tool at a work piece", plural Ansätze ; ) is an educated guess or an additional assumption made to help solve a problem, and which may later be verified to be part of the solution by its results. Use An ansatz is the establishment of the starting equation(s), the theorem(s), or the value(s) describing a mathematical or physical problem or solution. It typically provides an initial estimate or framework to the solution of a mathematical problem, and can also take into consideration the boundary conditions (in fact, an ansatz is sometimes thought of as a "trial answer" and an important technique in solving differential equations). After an ansatz, which constitutes nothing more than an assumption, has been established, the equations are solved more precisely for the general function of interest, which then constitutes a confirmation of the assumption. In essence, an ansatz makes assumptions about the form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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System Of Linear Equations
In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables. For example, :\begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in the three variables . A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by the ordered triple :(x,y,z)=(1,-2,-2), since it makes all three equations valid. The word "system" indicates that the equations are to be considered collectively, rather than individually. In mathematics, the theory of linear systems is the basis and a fundamental part of linear algebra, a subject which is used in most parts of modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Formal Power Series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.). A formal power series is a special kind of formal series, whose terms are of the form a x^n where x^n is the nth power of a variable x (n is a non-negative integer), and a is called the coefficient. Hence, power series can be viewed as a generalization of polynomials, where the number of terms is allowed to be infinite, with no requirements of convergence. Thus, the series may no longer represent a function of its variable, merely a formal sequence of coefficients, in contrast to a power series, which defines a function by taking numerical values for the variable within a radius of convergence. In a formal power series, the x^n are used only as position-holders for the coefficients, so that the coefficient of x^5 is the fift ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Abramov's Algorithm
In mathematics, particularly in computer algebra, Abramov's algorithm computes all rational solutions of a linear recurrence equation with polynomial coefficients. The algorithm was published by Sergei A. Abramov in 1989. Universal denominator The main concept in Abramov's algorithm is a universal denominator. Let \mathbb be a field of characteristic zero. The ''dispersion'' \operatorname (p,q) of two polynomials p, q \in \mathbb /math> is defined as\operatorname (p,q) =\max \ \cup \,where \N denotes the set of non-negative integers. Therefore the dispersion is the maximum k \in \N such that the polynomial p and the k-times shifted polynomial q have a common factor. It is -1 if such a k does not exist. The dispersion can be computed as the largest non-negative integer root of the resultant \operatorname_n (p(n), q(n+k) ) \in \mathbb /math>. Let \sum_^r p_k(n) \, y (n+k) = f(n) be a recurrence equation of order r with polynomial coefficients p_k \in \mathbb /math>, polynomial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Petkovšek's Algorithm
Petkovšek's algorithm (also Hyper) is a computer algebra algorithm that computes a basis of hypergeometric terms solution of its input linear recurrence equation with polynomial coefficients. Equivalently, it computes a first order right factor of linear difference operators with polynomial coefficients. This algorithm was developed by Marko Petkovšek in his PhD-thesis 1992. The algorithm is implemented in all the major computer algebra systems. Gosper-Petkovšek representation Let \mathbb be a field of characteristic zero. A nonzero sequence y(n) is called hypergeometric if the ratio of two consecutive terms is rational, i.e. y (n+1) /y(n) \in \mathbb(n). The Petkovšek algorithm uses as key concept that this rational function has a specific representation, namely the ''Gosper-Petkovšek normal form''. Let r(n) \in \mathbb /math> be a nonzero rational function. Then there exist monic polynomials a, b, c \in \mathbb /math> and 0 \neq z \in \mathbb such that r(n) = z \frac \f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other result ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Falling And Rising Factorials
In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \end The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, (A reprint of the 1950 edition by Chelsea Publishing Co.) rising sequential product, or upper factorial) is defined as :\begin x^ = x^\overline &= \overbrace^ \\ &= \prod_^n(x+k-1) = \prod_^(x+k) \,. \end The value of each is taken to be 1 (an empty product) when . These symbols are collectively called factorial powers. The Pochhammer symbol, introduced by Leo August Pochhammer, is the notation , where is a non-negative integer. It may represent ''either'' the rising or the falling factorial, with different articles and authors using different conventions. Pochhammer himself actually used with yet another meaning, namely to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |