@Article{AdelmanO83,
Author = "L. M. Adelman and A. M. Odlyzko",
Title = "Irreducibility Testing and Factorization of Polynomials",
Journal = "Mathematics of Computation",
Volume = 41,
Pages = "699--709",
Year = 1983
}
@Book{Apostol75,
Author = "T. M. Apostol",
Title = "Introduction to Analytic Number Theory",
Publisher = "Springer-Verlag",
Address = "Berlin",
Year = 1975
}
@TechReport{BagnaraPZZ05,
Author = "R. Bagnara and and A. Pescetti and A. Zaccagnini and E. Zaffanella",
Title = "PURRS: Towards Computer Algebra Support for Fully Automatic
Worst-Case Complexity Analysis",
Howpublished = "{\tt arXiv:cs.MS/0512056}
available from \url{http://arxiv.org/}",
Year = 2005,
Abstract = "Fully automatic worst-case complexity analysis has a
number of applications in computer-assisted program
manipulation. A classical and powerful approach to
complexity analysis consists in formally deriving, from
the program syntax, a set of constraints expressing
bounds on the resources required by the program, which
are then solved, possibly applying safe
approximations. In several interesting cases, these
constraints take the form of recurrence relations.
While techniques for solving recurrences are known and
implemented in several computer algebra systems, these
do not completely fulfill the needs of fully automatic
complexity analysis: they only deal with a somewhat
restricted class of recurrence relations, or sometimes
require user intervention, or they are restricted to the
computation of exact solutions that are often so complex
to be unmanageable, and thus useless in practice. In
this paper we briefly describe PURRS, a system and
software library aimed at providing all the computer
algebra services needed by applications performing or
exploiting the results of worst-case complexity
analyses. The capabilities of the system are
illustrated by means of examples derived from the
analysis of programs written in a domain-specific
functional programming language for real-time embedded
systems."
}
@TechReport{BagnaraZZ03TR,
Author = "R. Bagnara and A. Zaccagnini and T. Zolo",
Title = "The Automatic Solution of Recurrence Relations.
{I}. {Linear} Recurrences of Finite Order with Constant
Coefficients",
Number = 334,
Type = "Quaderno",
Institution = "Dipartimento di Matematica, Universit\`a di Parma, Italy",
Year = 2003,
Note = "Available at \url{http://www.cs.unipr.it/Publications/}",
Abstract = "We describe algorithmic techniques for the efficient
solution of a wide class of linear recurrences of finite
order with constant coefficients. We give an outline of
the underlying theory both from an abstract and a more
concrete point of view, in an attempt to convey the
general principles as clearly as possible yet providing
a well marked path for the implementation. In
particular, the presentation is thorough and reasonably
self-contained, covering topics such as the automatic
solution of polynomial equations and efficient, exact
symbolic summation of some special functions."
}
@TechReport{BagnaraZZ07TRa,
Author = "R. Bagnara and A. Zaccagnini and T. Zolo",
Title = "The Automatic Solution of Recurrence Relations.
{II}. ``{Divide} et {Impera}'' Recurrences",
Type = "Quaderno",
Institution = "Dipartimento di Matematica, Universit\`a di Parma, Italy",
Year = 2007,
Note = "To appear."
}
@TechReport{BagnaraZZ07TRb,
Author = "R. Bagnara and A. Zaccagnini and T. Zolo",
Title = "The Automatic Solution of Recurrence Relations.
{III}. {Approximate} Solutions",
Type = "Quaderno",
Institution = "Dipartimento di Matematica, Universit\`a di Parma, Italy",
Year = 2007,
Note = "To appear."
}
@TechReport{BagnaraZZ07TRc,
Author = "R. Bagnara and A. Zaccagnini and T. Zolo",
Title = "The Automatic Solution of Recurrence Relations.
{IV}. {Linear} Recurrences of Finite Order with Variable
Coefficients",
Type = "Quaderno",
Institution = "Dipartimento di Matematica, Universit\`a di Parma, Italy",
Year = 2007,
Note = "In preparation."
}
@TechReport{BagnaraZ04TR,
Author = "R. Bagnara and A. Zaccagnini",
Title = "Checking and Bounding the Solutions of Some Recurrence Relations",
Type = "Quaderno",
Number = 344,
Institution = "Dipartimento di Matematica, Universit\`a di Parma, Italy",
Year = 2004,
Note = "Available at \url{http://www.cs.unipr.it/Publications/}",
Abstract = "Recurrence relations play an important role in the field
of complexity analysis since complexity measures can
often be elegantly expressed by systems of such
relations. This justifies the interest in automatic,
precise, efficient and correct systems able to solve or
to approximate the solution of systems of recurrence
relations. Assume such a system is built. Since
closed-form solutions for recurrences of even modest
complexity can be so big and complex to be unmanageable,
how can confidence on such a system be gained? How can
we quickly validate or perhaps disprove its results?
And, in those cases where the exact solution is too
complex to be of practical use, how can we trade
precision for efficiency by approximating them from
below and from above? We also concern ourselves with a
problem related to the handling of sets of solutions of
recurrence relations: how can we confine by means of a
lower bound and an upper bound a set of such solutions?
We provide some solutions to these problems where we are
careful to rely, whenever possible, on fast integer
computations and/or conditions that are easy to check in
a completely automatic way. The ongoing experimental
evaluation of these ideas is giving very promising
results, showing order-of-magnitude speedups over the
more traditional methods."
}
@Article{Benzinger01,
Author = "R. Benzinger",
Title = "Automated Complexity Analysis of {Nuprl} Extracted Programs",
Journal = "Journal of Functional Programming",
Volume = 11,
Number = 1,
Year = 2001,
Pages = "3--31",
Abstract = "This paper describes the Automated Complexity Analysis
Prototype (ACAp) system for automated complexity
analysis of functional programs synthesized with the
Nuprl proof development system. We define a simple
abstract cost model for Nuprl's term language based on
the current call-by-name evaluator. The framework uses
abstract functions and abstract lists to facilitate
reasoning about primitive recursive programs with
first-order functions, lazy lists, and a subclass of
higher-order functions. The ACAp system automatically
derives upper bounds on the time complexity of Nuprl
extracts relative to a given profiling
semantics. Analysis proceeds by abstract interpretation
of the extract, where symbolic evaluation rules extend
standard evaluation to terms with free
variables. Symbolic evaluation of recursive programs
generates systems of multi-variable difference
equations, which are solved using the Mathematica
computer algebra system. The use of the system is
exemplified by analyzing a proof extract that computes
the maximum segment sum of a list and a functional
program that determines the minimum of a list via
sorting. For both results, we compare call-by-name to
call-by-value evaluation."
}
@Article{Benzinger04,
Author = "R. Benzinger",
Title = "Automated Higher-Order Complexity Analysis",
Journal = "Theoretical Computer Science",
Volume = 318,
Number = "1--2",
Year = 2004,
Pages = "79--103",
Abstract = "This paper describes the \emph{automated complexity
analysis} (ACA) system for automated higher-order
complexity analysis of functional programs synthesized
with the NImage proof development system. We introduce a
general framework for defining models of computational
complexity for functional programs based on an
annotation of a given operational language
semantics. Within this framework, we use \emph{type
decomposition} and \emph{polynomialization} to express
the complexity of higher-order terms. Symbolic
interpretation of open terms automates complexity
analysis, which involves generating and solving
higher-order recurrence equations. Finally, the use of
the ACA system is demonstrated by analyzing three
different implementations of the pigeonhole principle."
}
@Article{BurrowsT84,
Author = "B. L. Burrows and R. F. Talbot",
Title = "Sums of Powers of Integers",
Journal = "American Mathematical Monthly",
Volume = 91,
Number = 7,
Pages = "394--403",
Year = 1984
}
@Book{Cassels57,
Author = "J. W. S. Cassels",
Title = "An Introduction to Diophantine Approximation",
Publisher = "Cambridge University Press",
Address = "Cambridge, UK",
Year = 1957
}
@Article{Celis84,
Author = "P. Celis",
Title = "Remark: Corrections and Errors in {John Ivie}'s Some
{MACSYMA} Programs for Solving Recurrence Relations",
Journal = "ACM Transactions on Mathematical Software",
Volume = 10,
Number = 4,
Pages = "477--478",
Year = 1984,
ISSN = "0098-3500",
Note = "See \cite{Ivie78}"
}
@Book{Chandrasekharan68,
Author = "K. Chandrasekharan",
Title = "Introduction to Analytic Number Theory",
Publisher = "Springer-Verlag",
Address = "Berlin",
Year = 1968
}
@Book{Childs79,
Author = "L. Childs",
Title = "A Concrete Introduction to Higher Algebra",
Publisher = "Springer-Verlag",
Address = "Berlin",
Year = 1979
}
@Article{CohenK77,
Author = "J. Cohen and J. Katcoff",
Title = "Symbolic Solution of Finite-Difference Equations",
Journal = "ACM Transactions on Mathematical Software",
Volume = 3,
Number = 3,
Pages = "261--271",
Year = 1977,
ISSN = "0098-3500",
Abstract = "An interactive computer program which has some capability
for solving systems of finite difference equations
is described. Although this capability is limited to
linear systems, a knowledgeable user can, with the help
of the program, solve a wider class of equations.
Over 100 examples, covering a variety of cases,
have been solved by using the program.
Some representative examples are presented.
Additional features which would improve the versatility
of the program are also discussed."
}
@Article{Edwards86,
Author = "A. W. F. Edwards",
Title = "A Quick Route to Sums of Powers",
Journal = "American Mathematical Monthly",
Volume = 93,
Number = 6,
Pages = "451--455",
Year = 1986
}
@Book{GrahamKP89,
Author = "R. L. Graham and D. E. Knuth and O. Patashnik",
Title = "Concrete Mathematics. {A} Foundation for Computer Science",
Publisher = "Addison-Wesley",
Address = "Reading, MA",
Year = 1989,
}
@Book{HardyW79,
Author = "G. H. Hardy and E. M. Wright",
Title = "An Introduction to the Theory of Numbers",
Publisher = "Oxford University Press",
Address = "London, UK",
Year = 1979,
Note = "Fifth Edition"
}
@Book{Immink84,
Author = "G. K. Immink",
Title = "Asymptotics of Analytic Difference Equations",
Publisher = "Springer-Verlag",
Address = "Berlin",
Series = "Lecture Notes in Computer Mathematics",
Volume = 1085,
Pages = "134",
Year = 1984,
ISBN = "3-540-13867-6",
}
@Article{Ivie78,
Author = "J. Ivie",
Title = "Some {MACSYMA} Programs for Solving Recurrence Relations",
Journal = "ACM Transactions on Mathematical Software",
Volume = 4,
Number = 1,
Pages = "24--33",
Year = 1978,
ISSN = "0098-3500",
Abstract = "A set of programs to find closed-form solutions to
recurrence relations is described. These programs
are written in the MACSYMA programming language,
a LISP-based system implemented at the Massachusetts
Institute of Technology on the PDP-10 mc computer.
Single variable recurrence relations with constant
coefficients are solved as well as some variable
coefficient recurrences. The use of these programs
is illustrated with several examples taken from textbooks.",
Note = "See also \cite{Celis84}"
}
@Book{KelleyP91,
Author = "W. G. Kelley and A. C. Peterson",
Title = "Difference Equations. {An} Introduction with Applications",
Publisher = "Academic Press",
Address = "New York",
Year = 1991
}
@Article{LenstraLL82,
Author = "A. Lenstra and H. Lenstra Jr. and L. Lovasz",
Title = "Factoring polynomials with rational coefficients",
Journal = "Mathematische Annalen",
Volume = 261,
Number = 4,
Pages = "515--534",
Year = 1982
}
@Article{Levy70,
Author = "L. S. Levy",
Title = "Summation of the Series $1^n+2^n+\cdots+x^n$
Using Elementary Calculus",
Journal = "American Mathematical Monthly",
Volume = 77,
Year = 1970,
Pages = "840--847",
Note = "See also \cite{Levy71}"
}
@Article{Levy71,
Author = "L. S. Levy",
Title = "Corrigendum",
Journal = "American Mathematical Monthly",
Volume = 78,
Year = 1971,
Pages = "987",
Note = "See also \cite{Levy70}"
}
@Article{Lueker80,
Author = "G. S. Lueker",
Title = "Some Techniques for Solving Recurrences",
Journal = "ACM Computing Surveys",
Volume = 12,
Number = 4,
Pages = "419--436",
Year = 1980,
Abstract = "Some techniques for solving recurrences are presented,
along with examples of how these recurrences arise
in the analysis of algorithms. In addition, probability
generating functions are discussed and used to analyze
problems in computer science."
}
@Book{MignotteS99,
Author = "M. Mignotte and D. {\c S}tef{\u a}nescu",
Title = "Polynomials. {An} Algorithmic Approach",
Publisher = "Springer-Verlag",
Address = "Berlin",
Year = 1999
}
@MastersThesis{Pescetti04,
Author = "A. Pescetti",
Title = "L'algoritmo di {Zeilberger} per la risoluzione automatica
di ricorrenze",
Type = "{M.Sc.} dissertation",
School = "University of Parma",
Month = oct,
Year = 2004,
Note = "In Italian",
}
@Book{PetkovsekWZ96,
Author = "M. Petkov{\v s}ek and H. S. Wilf and D. Zeilberger",
Title = "$A=B$",
Publisher = "A. K. Peters",
Address = "Natick, MA",
Year = 1996,
}
@Article{Roura01,
Author = "S. Roura",
Title = "Improved Master Theorems for Divide-and-Conquer Recurrences",
Journal = "Journal of the ACM",
Volume = 48,
Number = 2,
Year = 2001,
Pages = "170--205",
Abstract = "This paper presents new theorems to analyze
divide-and-conquer recurrences, which improve other
similar ones in several aspects. In particular, these
theorems provide more information, free us almost
completely from technicalities like floors and ceilings,
and cover a wider set of toll functions and weight
distributions, stochastic recurrences included."
}
@Book{SloaneP95,
Author = "N. J. A. Sloane and S. Plouffe",
Title = "The Encyclopedia of Integer Sequences",
Publisher = "Academic Press",
Address = "New York",
Year = 1995,
}
@Misc{OEIS10,
Title = "The On-Line Encyclopedia of Integer Sequences",
Year = 2010,
Howpublished = "Published electronically at \url{http://oeis.org}"
}
@Book{Titchmarsh88,
Author = "E. C. Titchmarsh",
Title = "The Theory of Functions",
Publisher = "Oxford University Press",
Address = "London, UK",
Year = 1988
}
@Incollection{VasconcelosH04,
Author = "P. B. Vasconcelos and K. Hammond",
Title = "Inferring Cost Equations for Recursive, Polymorphic
and Higher-Order Functional Programs",
Editor = "P. Trinder and G. Michaelson and R. {Pe\~na}",
Booktitle = "Implementation of Functional Languages:
15th International Workshop, IFL 2003, Edinburgh, UK,
September 8--11, 2003. Revised Papers",
Publisher = "Springer-Verlag, Berlin",
Series = "Lecture Notes in Computer Science",
Volume = 3145,
Pages = "86--101",
Year = 2004,
ISBN = "3-540-23727-5",
Abstract = "This paper presents a type-based analysis for inferring
size- and cost-equations for recursive, higher-order and
polymorphic functional programs without requiring user
annotations or unusual syntax. Our type reconstruction
algorithm is capable of inferring first-order cost
equations for a non-trivial subset of higher-order,
recursive and polymorphic functions. We illustrate the
approach with reference to some standard examples of
recursive programs."
}
@Misc{Weisstein,
Author = "E. W. Weisstein",
Title = "The World of Mathematics",
Howpublished = "\url{http://mathworld.wolfram.com/}"
}
@Book{Wilf94,
Author = "H. S. Wilf",
Title = "Generatingfunctionology",
Publisher = "Academic Press",
Address = "New York",
Year = 1994,
}