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Topic: Recursion

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Recursion definition

Ackermann function

Computable function

Tail recursion

Factorial function

Lambda calculus

Recursive function

Factorial

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Dynamic programming

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	Recursion - Wikipedia, the free encyclopedia
		So a recursive meal might be potato skins, baby greens salad, chicken parmesan, and for dessert, a four course meal, consisting of crab cakes, Caesar salad, for an entrée, a four course meal, and chocolate cake for dessert, so on until each of the meals within the meals is completed.
		This one is also the second-shortest possible example of an erroneous recursive definition of an object, the error being the absence of the termination condition (or lack of the initial state, if to look at it from an opposite point of view).
		The function calls itself recursively on a smaller version of the input (n - 1) and multiplies the result of the recursive call by n, until reaching the base case, analogously to the mathematical definition of factorial.
	en.wikipedia.org /wiki/Recursion (1812 words)

	Recursion
		Recursion can be direct when an entity refers to itself directly or indirect when it refers to other entities which refer to it.
		Recursive routines are often simple and elegant and their correctness easily seen.
		The term ancestor is recursive: A parent is an ancestor and an ancestor of a parent is an ancestor.
	www.csse.monash.edu.au /~lloyd/tildeAlgDS/Recn (969 words)

	Cprogramming.com Tutorial: Recursion
		Recursion is a programming technique that allows the programmer to express operations in terms of themselves.
		A useful way to think of recursive functions is to imagine them as a process being performed where one of the instructions is to "repeat the process".
		On the other hand, recursion makes it easier to express ideas in which the result of the recursive call is necessary to complete the task.
	www.cprogramming.com /tutorial/lesson16.html (742 words)

	What's Recursion? (Site not responding. Last check: 2007-10-22)
		That is, recursion is a sequential process, and much of its power lies in the precise nature of that sequentiality.
		Recursion is the same thing, but it breaks the problem up into an "outside-to-inside" sequence rather than a "beginning-to-end" sequence.
		The way to understand recursion is to think of it as a way of solving a problem by breaking it down into smaller parts, but they're organized "outside-to-inside" rather than "beginning-to-end".
	www.erasmatazz.com /library/Lilan/recursion.html (546 words)

	Recursion
		Recursion is a technique that allows us to break down a problem into one or more subproblems that are similar in form to the original problem.
		Tail recursive functions are often said to "return the value of the last recursive call as the value of the function." Tail recursion is very desirable because the amount of information which must be stored during the computation is independent of the number of recursive calls.
		A recursive function is said to be tree recursive (or non-linearly recursive) when the pending operation does involve another recursive call to the function.
	triton.towson.edu /~akayabas/COSC455_Spring2000/Recursion_Iteration.htm (894 words)

	recursion
		Recursion is a fundamental concept in mathemetics and computer science and many practical co mputations can be fitted to a recursive framework.
		In recursive algorithms, many levels of function calls can be initiated resulting in many copies of th e function being currently active and copies of the different functions information residing in the memory spaces.
		Two keys to each recursive function are: a test of a "termination condition" which controls if the function will call itself or stop the recursion; and, each call to itself (the recursive function) must modify the parameters to pass to the next iteration of the function.
	et.nmsu.edu /~etti/winter98/computers/jenkins/jenkins.html (1214 words)

	Recursion
		Recursion is usually less computationally efficient, compiler generated procedure calls introduce overhead when storing all variables before a procedure call and loading them back upon entering the procedure.
		The characteristic property of recursion is that different invocation of the same procedure are active at the same time, and the variables associated with each invocation must be kept separately, so they may hold different values.
		Remember that the recursive algorithm may be equivalent to a simple iterative algorithm, and imitating a compiler may hide that simple algorithm behind redundant complexity.
	www.ibiblio.org /pub/languages/fortran/ch1-12.html (1641 words)

	Recursion
		We use a recursive solution which is based on the following idea: To move N disks from pole A to pole C, we first move the top N-1 disks from pole A to pole B, then move disk N to pole C, then move the N-1 disks from B to C (onto disk N).
		We bottom out of the recursion when s ≤ 1/2 by drawing a spot (x, y) in the color that is average of the four endpoints.
		Recursive functions are especially useful when either the underlying data structure (binary trees, Unix directories) or the underlying algorithmic paradigm (divide-and-conquer) are naturally self-referential.
	www.cs.princeton.edu /introcs/27recursion (6224 words)

	Insights Into Algebra 1 . Workshop 5
		Recursion is the process of describing the next term in a sequence in relation to preceding terms.
		Recursive formulas can model population growth patterns, the distance traveled on a trip, the interest earned on a bank account, and other real-life events.
		A recursion (or a recursive function) is an expression, such as a polynomial, each term of which is determined by application of a formula to preceding terms.
	www.learner.org /channel/workshops/algebra/workshop5/index2.html (1203 words)

	Zend Technologies - Articles - Recursion In PHP: Tapping Unharnessed Power
		Because this particular type of recursion is performing the recursive function call as the second part of the return line, it is called tail recursion.
		As we have seen, implementing a factorial function recursively is slightly more elegant in terms of the number of variables involved and the way you get to think about the problem.
		Approaching the problem recursively also means thinking about the factorial problem in much the same way that a mathematician might define a factorial.
	www.zend.com /zend/art/recursion.php (2281 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		Seymour Papert, who is responsible for the teaching of the recursive programming language Logo to grade school children, has reported that children readily learn the concept.
		The additional power of recursion arises from your ability to start a clone in the middle of a task and then complete the task when the clone is finished.
		Before writing the recursion initializing procedure, it is necessary to notice that both negative integers and 0 have yet to be handled correctly and that neither case is difficult with the tools now in hand.
	www.mapfree.com /sbf/tips/recurs.html (1659 words)

	Java Programming: Section 11.1
		And when recursion is applied during the solution of a problem, it must be applied to a problem that is in some sense smaller -- that is, closer to the base cases -- than the original problem.
		A common error in writing recursive subroutines is to violate one of the two rules: There must be one or more base cases, and when the subroutine is applied recursively, it must be applied to a problem that is smaller than the original problem.
		In the recursive subroutine that solves the problem, the stacks that serve as the source and destination of the disks have to be specified.
	math.hws.edu /javanotes/c11/s1.html (4511 words)

	recursion
		Although recursion may not be the best way to write some of these functions, it is good practice.
		See dynamic algorithms for an example of one trade-off between speed and clarity for a recursive vs. an iterative implementation.
		Paul E. Black and Patrick Rodgers, "recursion", from Dictionary of Algorithms and Data Structures, Paul E. Black, ed., NIST.
	www.nist.gov /dads/HTML/recursion.html (388 words)

	Use recursion effectively in XSL
		Through the use of recursive functions, declarative languages are able to provide the same functionality as their imperative counterparts.
		Instead of stacking numbers to add at the various recursive steps, the addition is done at each stage and the result passed along as a parameter to the next recursion step.
		Recursion can be a difficult concept to understand at first, but its usefulness and elegance becomes clearer with use.
	www.ibm.com /developerworks/linux/library/x-xslrecur (3653 words)

	Recursion
		A recursive algorithm for solving a problem is an algorithm that works by dividing the problem into several problems of a smaller size and by applying the same algorithm to at least one of the smaller problems.
		All recursive algorithms are based on the idea that if we keep dividing a problem into problems of a smaller size, then eventually we'll get to a problem (or problems) which is so small that the solution for it is trivial.
		A recursive function must have a base case (or base cases), otherwise it will be looping forever.
	cs-people.bu.edu /dbuzan/cs112/lab5/lab5.html (985 words)

	Mastering recursive programming
		Recursion is a tool not often used by imperative language developers, because it is thought to be slow and to waste space, but as the author demonstrates, there are several techniques that can be used to minimize or eliminate these problems.
		With recursive programs that recurse instead of change state, the number of occasions of state change is small and the program variables maintain self-consistency by setting all of the recursion variables at once.
		Recursion is a great art, enabling programs for which it is easy to verify correctness without sacrificing performance, but it requires the programmer to look at programming in a new light.
	www-128.ibm.com /developerworks/linux/library/l-recurs.html?ca=dgr-lnxw07Recursion (5941 words)

	fUSION Anomaly. Recursion
		Recursion enables certain algorithms to be implemented with small, simple routines, but it does not guarantee speed or efficiency.
		Erroneous use of recursion can cause a program to run out of stack space during execution, causing the program, and sometimes the entire system, to crash.
		oscillators which indicates the onset of this self-aware condition is clearly coherence optimized by recursion.
	fusionanomaly.net /recursion.html (408 words)

	Section 3 Chapter 3 Recursion (Site not responding. Last check: 2007-10-22)
		This situation is known as infinite recursion and is similar to the fault in indefinite loops where a faulty loop control condition causes the loop to iterate infinitely.
		To avoid infinite recursion, the recursive subprogram has to be carefully constructed to ensure that at some stage the subprogram terminates without calling itself.
		Where the recursive component is simpler there comes a stage in the recursion where the description is trivial and the possibility of an infinitely recursive description can be avoided.
	www.scism.sbu.ac.uk /law/Section3/chapter3/s3c3int.html (208 words)

	Recursion (Site not responding. Last check: 2007-10-22)
		Recursion is a simple but very powerful concept.
		Recursion means an instance (or instances) is derived from an exactly identical instance.
		Recursion is used extensively to solve computer and math problems.
	www.cs.ucla.edu /~klinger/dorene/math4.htm (158 words)

	Getting Started
		Recursion is a simple concept: the application of a procedure from within that procedure.
		Recursion is more general and eliminates the need for the variable assignments required by many other languages' iteration constructs, resulting in code that is more reliable and easier to follow.
		Some recursion is essentially iteration and executes as such; Section 3.2 has more to say about this.
	www.scheme.com /tspl2d/start.html (7692 words)

	Solving Problems with Recursion
		Recursion is a way to solve a problem by...reducing it to the same problem.
		Recursion is quite counter-intuitive, and you will certainly find it weird at first, but once grasped, it becomes a seriously powerful weapon in your programming arsenal.
		The recursive way of solving a problem is to reduce the problem into another problem that is exactly of the same type, and solving the new one instead.
	www.devshed.com /c/a/Practices/Solving-Problems-with-Recursion (665 words)

	Recursion at opensource encyclopedia (Site not responding. Last check: 2007-10-22)
		Recursion is a way of specifying a process by means of itself.
		More precisely (and to dispel the appearance of circularity in the definition), "complicated" instances of the process are defined in terms of "simpler" instances, and the "simplest" instances are given explicitly.
		Examples of mathematical objects often defined recursively are functions and sets.
	www.wiki.tatet.com /Recursion.html (979 words)

	Recursion Software - Evaluation (Site not responding. Last check: 2007-10-22)
		Recursion has granted you ("Licensee") a license to this software upon the terms and conditions set forth in this Software Evaluation License Agreement (this "Agreement") and by installing the software you agree to all of the terms and conditions set forth in this Agreement.
		Recursion reserves the right to assign or transfer this Agreement or any of its rights, duties and obligations to any direct or indirect subsidiary or affiliate of Recursion.
		Furthermore, it is the intention of the parties that in lieu of such illegal, invalid, or unenforceable provision, there automatically be added as a part of this Agreement a provision as similar in terms to such illegal, invalid, or unenforceable provision as may be possible and be legal, valid, and enforceable.
	www.recursionsw.com /evaluation.htm (1672 words)

	Recursion
		When a recursive call is made, the marker in the old copy of the code is just after the call; the marker in the "cloned" copy is at the beginning of the method.
		Here are a few examples where recursion makes things a little bit clearer, though in the second case, the efficiency problem makes it a bad choice.
		Recursion is often simple and elegant, can be efficient, and tends to be underutilized.
	www.cs.wisc.edu /~vernon/cs367/notes/6.RECURSION.html (1769 words)

	Recursion
		When referring to computer programming a recursion refers to a function or subroutine that calls itself purposely or by mistake.
		Recursion is commonly performed to solve problems capable of being resolved by deduction, many numeric problems can often be resolved by recursion.
		Microsoft's DNS server recursion can be disabled by setting the below registry key to a value of 1.
	www.computerhope.com /jargon/r/recursio.htm (101 words)

	Roundabout, A Pattern Language for Recursive Programming
		As a general rule, we can follow the advice of Wirth (1976) and rule out recursion as potential solution only in cases where there is an obvious iterative solution.
		Perhaps you are using an Interface Procedure (2), Mutual Recursion (3), an Accumulator Variable (5), or a Syntax Procedure (5).
		Woolf (1998) describes recursion in object-oriented programming, wherein an object computes an answer to a message by delegating the same message to its instance variables and then combining their answers.
	www.cs.uni.edu /~wallingf/patterns/recursion.html (4225 words)

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