Algorithms for Computer Algebra is the first comprehensive textbook to be published on the topic of computational symbolic mathematics. The book first develops the foundational material from modern algebra that is required for subsequent topics. It then presents a thorough development of modern computational algorithms for such problems as multivariate polynomial arithmetic and greatest common divisor calculations, factorization of multivariate polynomials, symbolic solution of linear and polynomial systems of equations, and analytic integration of elementary functions.

Links to many different image processing algorithms.

Numerous examples are integrated into the text as an aid to understanding the mathematical development. The algorithms developed for each topic are presented in a Pascal-like computer language. An extensive set of exercises is presented at the end of each chapter. Algorithms for Computer Algebra is suitable for use as a textbook for a course on algebraic algorithms at the third-year, fourth-year, or graduate level. Although the mathematical development uses concepts from modern algebra, the book is self-contained in the sense that a one-term undergraduate course introducing students to rings and fields is the only prerequisite assumed. The book also serves well as a supplementary textbook for a traditional modern algebra course, by presenting concrete applications to motivate the understanding of the theory of rings and fields.

Of an algorithm () for calculating the greatest common divisor (g.c.d.) of two numbers a and b in locations named A and B. The algorithm proceeds by successive subtractions in two loops: IF the test B ≥ A yields 'yes' (or true) (more accurately the number b in location B is greater than or equal to the number a in location A) THEN, the algorithm specifies B ← B − A (meaning the number b − a replaces the old b). Similarly, IF A >B, THEN A ← A − B.

The process terminates when (the contents of) B is 0, yielding the g.c.d. (Algorithm derived from Scott 2009:13; symbols and drawing style from Tausworthe 1977). In and, an algorithm ( ( ) ) is an unambiguous specification of how to solve a class of problems. Algorithms can perform, and tasks. An algorithm is an that can be expressed within a finite amount of space and time and in a well-defined formal language for calculating a.

Starting from an initial state and initial input (perhaps ), the instructions describe a that, when, proceeds through a finite number of well-defined successive states, eventually producing 'output' and terminating at a final ending state. The transition from one state to the next is not necessarily; some algorithms, known as, incorporate random input.

The concept of algorithm has existed for centuries; however, a partial formalization of what would become the modern algorithm began with attempts to solve the (the 'decision problem') posed by in 1928. Subsequent formalizations were framed as attempts to define ' or 'effective method'; those formalizations included the –– of 1930, 1934 and 1935, 's of 1936, 's ' of 1936, and 's of 1936–7 and 1939. Giving a formal definition of algorithms, corresponding to the intuitive notion, remains a challenging problem.

Contents • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Etymology [ ] The word 'algorithm' is a combination of the word algorismus, named after and the word arithmos, i.e. Αριθμός, meaning 'number'.

(: خوارزمی‎, c. 780–850) was a mathematician,,, and scholar in the in, whose name means 'the native of ', a region that was part of and is now in. About 825, he wrote a treatise in the Arabic language, which was translated into in the 12th century under the title Algoritmi de numero Indorum. This title means 'Algoritmi on the numbers of the Indians', where 'Algoritmi' was the translator's Latinization of Al-Khwarizmi's name.

Al-Khwarizmi was the most widely read mathematician in Europe in the late Middle Ages, primarily through his other book, the. In late medieval Latin, algorismus, English ', the corruption of his name, simply meant the 'decimal number system'.

In the 15th century, under the influence of the Greek word ἀριθμός 'number' ( cf. 'arithmetic'), the Latin word was altered to algorithmus, and the corresponding English term 'algorithm' is first attested in the 17th century; the modern sense was introduced in the 19th century.

In English, it was first used in about 1230 and then by in 1391. English adopted the French term, but it wasn't until the late 19th century that 'algorithm' took on the meaning that it has in modern English. Another early use of the word is from 1240, in a manual titled Carmen de Algorismo composed. It begins thus: Haec algorismus ars praesens dicitur, in qua / Talibus Indorum fruimur bis quinque figuris. Which translates as: Algorism is the art by which at present we use those Indian figures, which number two times five. The poem is a few hundred lines long and summarizes the art of calculating with the new style of Indian dice, or Talibus Indorum, or Hindu numerals.

Informal definition [ ]. For a detailed presentation of the various points of view on the definition of 'algorithm', see.

An informal definition could be 'a set of rules that precisely defines a sequence of operations.' Which would include all computer programs, including programs that do not perform numeric calculations. Generally, a program is only an algorithm if it stops eventually. A prototypical example of an algorithm is the to determine the maximum common divisor of two integers; an example (there are others) is described by the above and as an example in a later section. Offer an informal meaning of the word in the following quotation: No human being can write fast enough, or long enough, or small enough† ( †'smaller and smaller without limit.you'd be trying to write on molecules, on atoms, on electrons') to list all members of an enumerably infinite set by writing out their names, one after another, in some notation. But humans can do something equally useful, in the case of certain enumerably infinite sets: They can give explicit instructions for determining the nth member of the set, for arbitrary finite n.

Such instructions are to be given quite explicitly, in a form in which they could be followed by a computing machine, or by a human who is capable of carrying out only very elementary operations on symbols. An 'enumerably infinite set' is one whose elements can be put into one-to-one correspondence with the integers. Thus, Boolos and Jeffrey are saying that an algorithm implies instructions for a process that 'creates' output integers from an arbitrary 'input' integer or integers that, in theory, can be arbitrarily large. Thus an algorithm can be an algebraic equation such as y = m + n – two arbitrary 'input variables' m and n that produce an output y.

But various authors' attempts to define the notion indicate that the word implies much more than this, something on the order of (for the addition example): Precise instructions (in language understood by 'the computer') for a fast, efficient, 'good' process that specifies the 'moves' of 'the computer' (machine or human, equipped with the necessary internally contained information and capabilities) to find, decode, and then process arbitrary input integers/symbols m and n, symbols + and =. And 'effectively' produce, in a 'reasonable' time, output-integer y at a specified place and in a specified format. The concept of algorithm is also used to define the notion of.

That notion is central for explaining how come into being starting from a small set of and rules. In, the time that an algorithm requires to complete cannot be measured, as it is not apparently related with our customary physical dimension. From such uncertainties, that characterize ongoing work, stems the unavailability of a definition of algorithm that suits both concrete (in some sense) and abstract usage of the term. Formalization [ ] Algorithms are essential to the way computers process data. Many computer programs contain algorithms that detail the specific instructions a computer should perform (in a specific order) to carry out a specified task, such as calculating employees' paychecks or printing students' report cards. Thus, an algorithm can be considered to be any sequence of operations that can be simulated by a system.

Authors who assert this thesis include Minsky (1967), Savage (1987) and Gurevich (2000): Minsky: 'But we will also maintain, with Turing... That any procedure which could 'naturally' be called effective, can in fact be realized by a (simple) machine.

Although this may seem extreme, the arguments... In its favor are hard to refute'. Gurevich: '.Turing's informal argument in favor of his thesis justifies a stronger thesis: every algorithm can be simulated by a Turing machine.

According to Savage [1987], an algorithm is a computational process defined by a Turing machine'. Typically, when an algorithm is associated with processing information, data can be read from an input source, written to an output device and stored for further processing. Stored data are regarded as part of the internal state of the entity performing the algorithm. In practice, the state is stored in one or more. For some such computational process, the algorithm must be rigorously defined: specified in the way it applies in all possible circumstances that could arise. That is, any conditional steps must be systematically dealt with, case-by-case; the criteria for each case must be clear (and computable).

Because an algorithm is a precise list of precise steps, the order of computation is always crucial to the functioning of the algorithm. Instructions are usually assumed to be listed explicitly, and are described as starting 'from the top' and going 'down to the bottom', an idea that is described more formally. So far, this discussion of the formalization of an algorithm has assumed the premises of. This is the most common conception, and it attempts to describe a task in discrete, 'mechanical' means.

Unique to this conception of formalized algorithms is the, setting the value of a variable. It derives from the intuition of ' as a scratchpad. There is an example below of such an assignment. For some alternate conceptions of what constitutes an algorithm see and.

Expressing algorithms [ ] Algorithms can be expressed in many kinds of notation, including,,,, or (processed by ). Natural language expressions of algorithms tend to be verbose and ambiguous, and are rarely used for complex or technical algorithms.

Pseudocode, flowcharts, and control tables are structured ways to express algorithms that avoid many of the ambiguities common in natural language statements. Programming languages are primarily intended for expressing algorithms in a form that can be executed by a computer, but are often used as a way to define or document algorithms. There is a wide variety of representations possible and one can express a given program as a sequence of machine tables (see more at, and ), as flowcharts and (see more at ), or as a form of rudimentary or called 'sets of quadruples' (see more at ). Representations of algorithms can be classed into three accepted levels of Turing machine description: 1 High-level description '.prose to describe an algorithm, ignoring the implementation details. At this level we do not need to mention how the machine manages its tape or head.' 2 Implementation description '.prose used to define the way the Turing machine uses its head and the way that it stores data on its tape. At this level we do not give details of states or transition function.'

3 Formal description Most detailed, 'lowest level', gives the Turing machine's 'state table'. For an example of the simple algorithm 'Add m+n' described in all three levels, see. Implementation [ ].

An animation of the sorting an array of randomized values. The red bars mark the pivot element; at the start of the animation, the element farthest to the right hand side is chosen as the pivot. One of the simplest algorithms is to find the largest number in a list of numbers of random order. Finding the solution requires looking at every number in the list. From this follows a simple algorithm, which can be stated in a high-level description English prose, as: High-level description: • If there are no numbers in the set then there is no highest number. • Assume the first number in the set is the largest number in the set.

• For each remaining number in the set: if this number is larger than the current largest number, consider this number to be the largest number in the set. • When there are no numbers left in the set to iterate over, consider the current largest number to be the largest number of the set.

(Quasi-)formal description: Written in prose but much closer to the high-level language of a computer program, the following is the more formal coding of the algorithm in. The example-diagram of Euclid's algorithm from T.L.

Heath (1908), with more detail added. Euclid does not go beyond a third measuring, and gives no numerical examples. Nicomachus gives the example of 49 and 21: 'I subtract the less from the greater; 28 is left; then again I subtract from this the same 21 (for this is possible); 7 is left; I subtract this from 21, 14 is left; from which I again subtract 7 (for this is possible); 7 is left, but 7 cannot be subtracted from 7.' Heath comments that, 'The last phrase is curious, but the meaning of it is obvious enough, as also the meaning of the phrase about ending 'at one and the same number'.'

(Heath 1908:300). 's algorithm to compute the (GCD) to two numbers appears as Proposition II in Book VII ('Elementary Number Theory') of his. Euclid poses the problem thus: 'Given two numbers not prime to one another, to find their greatest common measure'. He defines 'A number [to be] a multitude composed of units': a counting number, a positive integer not including zero. To 'measure' is to place a shorter measuring length s successively ( q times) along longer length l until the remaining portion r is less than the shorter length s. In modern words, remainder r = l − q× s, q being the quotient, or remainder r is the 'modulus', the integer-fractional part left over after the division. For Euclid's method to succeed, the starting lengths must satisfy two requirements: (i) the lengths must not be zero, AND (ii) the subtraction must be “proper”; i.e., a test must guarantee that the smaller of the two numbers is subtracted from the larger (alternately, the two can be equal so their subtraction yields zero).

Euclid's original proof adds a third requirement: the two lengths must not be prime to one another. Euclid stipulated this so that he could construct a proof that the two numbers' common measure is in fact the greatest. While Nicomachus' algorithm is the same as Euclid's, when the numbers are prime to one another, it yields the number '1' for their common measure.

So, to be precise, the following is really Nicomachus' algorithm. 1599 = 650×2 + 299 650 = 299×2 + 52 299 = 52×5 + 39 52 = 39×1 + 13 39 = 13×3 + 0 Computer language for Euclid's algorithm [ ] Only a few instruction types are required to execute Euclid's algorithm—some logical tests (conditional GOTO), unconditional GOTO, assignment (replacement), and subtraction. • A location is symbolized by upper case letter(s), e.g. • The varying quantity (number) in a location is written in lower case letter(s) and (usually) associated with the location's name.

For example, location L at the start might contain the number l = 3009. An inelegant program for Euclid's algorithm [ ]. 'Inelegant' is a translation of Knuth's version of the algorithm with a subtraction-based remainder-loop replacing his use of division (or a 'modulus' instruction). Derived from Knuth 1973:2–4. Depending on the two numbers 'Inelegant' may compute the g.c.d.

In fewer steps than 'Elegant'. The following algorithm is framed as Knuth's four-step version of Euclid's and Nicomachus', but, rather than using division to find the remainder, it uses successive subtractions of the shorter length s from the remaining length r until r is less than s. The high-level description, shown in boldface, is adapted from Knuth 1973:2–4: INPUT: 1 [Into two locations L and S put the numbers l and s that represent the two lengths]: INPUT L, S 2 [Initialize R: make the remaining length r equal to the starting/initial/input length l]: R ← L E0: [Ensure r ≥ s.] 3 [Ensure the smaller of the two numbers is in S and the larger in R]: IF R >S THEN the contents of L is the larger number so skip over the exchange-steps, and: GOTO step ELSE swap the contents of R and S.

4 L ← R (this first step is redundant, but is useful for later discussion). 5 R ← S 6 S ← L E1: [Find remainder]: Until the remaining length r in R is less than the shorter length s in S, repeatedly subtract the measuring number s in S from the remaining length r in R. 7 IF S >R THEN done measuring so GOTO ELSE measure again, 8 R ← R − S 9 [Remainder-loop]: GOTO.

E2: [Is the remainder zero?]: EITHER (i) the last measure was exact, the remainder in R is zero, and the program can halt, OR (ii) the algorithm must continue: the last measure left a remainder in R less than measuring number in S. 10 IF R = 0 THEN done so GOTO ELSE CONTINUE TO, E3: [Interchange s and r]: The nut of Euclid's algorithm. Use remainder r to measure what was previously smaller number s; L serves as a temporary location. 11 L ← R 12 R ← S 13 S ← L 14 [Repeat the measuring process]: GOTO OUTPUT: 15 [Done. S contains the ]: PRINT S DONE: 16 HALT, END, STOP. An elegant program for Euclid's algorithm [ ] The following version of Euclid's algorithm requires only six core instructions to do what thirteen are required to do by 'Inelegant'; worse, 'Inelegant' requires more types of instructions.

The flowchart of 'Elegant' can be found at the top of this article. In the (unstructured) Basic language, the steps are numbered, and the instruction LET [] = [] is the assignment instruction symbolized by ←. Main article: It is frequently important to know how much of a particular resource (such as time or storage) is theoretically required for a given algorithm. Methods have been developed for the to obtain such quantitative answers (estimates); for example, the sorting algorithm above has a time requirement of O( n), using the with n as the length of the list. At all times the algorithm only needs to remember two values: the largest number found so far, and its current position in the input list. Therefore, it is said to have a space requirement of O(1), if the space required to store the input numbers is not counted, or O( n) if it is counted. Different algorithms may complete the same task with a different set of instructions in less or more time, space, or ' than others.

For example, a algorithm (with cost O(log n) ) outperforms a sequential search (cost O(n) ) when used for on sorted lists or arrays. Formal versus empirical [ ]. Main articles:,, and The is a discipline of, and is often practiced abstractly without the use of a specific or implementation. In this sense, algorithm analysis resembles other mathematical disciplines in that it focuses on the underlying properties of the algorithm and not on the specifics of any particular implementation. Usually is used for analysis as it is the simplest and most general representation. However, ultimately, most algorithms are usually implemented on particular hardware / software platforms and their is eventually put to the test using real code. For the solution of a 'one off' problem, the efficiency of a particular algorithm may not have significant consequences (unless n is extremely large) but for algorithms designed for fast interactive, commercial or long life scientific usage it may be critical.

Scaling from small n to large n frequently exposes inefficient algorithms that are otherwise benign. Empirical testing is useful because it may uncover unexpected interactions that affect performance.

May be used to compare before/after potential improvements to an algorithm after program optimization. Execution efficiency [ ]. Main article: To illustrate the potential improvements possible even in well established algorithms, a recent significant innovation, relating to algorithms (used heavily in the field of image processing), can decrease processing time up to 1,000 times for applications like medical imaging. In general, speed improvements depend on special properties of the problem, which are very common in practical applications. Speedups of this magnitude enable computing devices that make extensive use of image processing (like digital cameras and medical equipment) to consume less power. Classification [ ] There are various ways to classify algorithms, each with its own merits.

By implementation [ ] One way to classify algorithms is by implementation means. Recursion A is one that invokes (makes reference to) itself repeatedly until a certain condition (also known as termination condition) matches, which is a method common to. Algorithms use repetitive constructs like and sometimes additional data structures like to solve the given problems. Some problems are naturally suited for one implementation or the other.

For example, is well understood using recursive implementation. Every recursive version has an equivalent (but possibly more or less complex) iterative version, and vice versa. Logical An algorithm may be viewed as controlled.

This notion may be expressed as: Algorithm = logic + control. The logic component expresses the axioms that may be used in the computation and the control component determines the way in which deduction is applied to the axioms. This is the basis for the paradigm.

In pure logic programming languages the control component is fixed and algorithms are specified by supplying only the logic component. The appeal of this approach is the elegant: a change in the axioms has a well-defined change in the algorithm. Serial, parallel or distributed Algorithms are usually discussed with the assumption that computers execute one instruction of an algorithm at a time. Those computers are sometimes called serial computers. An for such an environment is called a serial algorithm, as opposed to. Parallel algorithms take advantage of computer architectures where several processors can work on a problem at the same time, whereas distributed algorithms utilize multiple machines connected with a.

Parallel or distributed algorithms divide the problem into more symmetrical or asymmetrical subproblems and collect the results back together. The resource consumption in such algorithms is not only processor cycles on each processor but also the communication overhead between the processors. Some sorting algorithms can be parallelized efficiently, but their communication overhead is expensive. Iterative algorithms are generally parallelizable. Some problems have no parallel algorithms, and are called inherently serial problems. Deterministic or non-deterministic solve the problem with exact decision at every step of the algorithm whereas solve problems via guessing although typical guesses are made more accurate through the use of.

Exact or approximate While many algorithms reach an exact solution, seek an approximation that is closer to the true solution. Approximation can be reached by either using a deterministic or a random strategy. Such algorithms have practical value for many hard problems. One of the examples of an approximate algorithm is the Knapsack problem. The Knapsack problem is a problem where there is a set of given items. The goal of the problem is to pack the knapsack to get the maximum total value. Each item has some weight and some value.

Total weight that we can carry is no more than some fixed number X. So, we must consider weights of items as well as their value. They run on a realistic model of. The term is usually used for those algorithms which seem inherently quantum, or use some essential feature of quantum computation such as. By design paradigm [ ] Another way of classifying algorithms is by their design methodology or paradigm. There is a certain number of paradigms, each different from the other.

Furthermore, each of these categories include many different types of algorithms. Some common paradigms are: or exhaustive search This is the naive method of trying every possible solution to see which is best. Divide and conquer A repeatedly reduces an instance of a problem to one or more smaller instances of the same problem (usually ) until the instances are small enough to solve easily. One such example of divide and conquer is. Sorting can be done on each segment of data after dividing data into segments and sorting of entire data can be obtained in the conquer phase by merging the segments.

A simpler variant of divide and conquer is called a decrease and conquer algorithm, that solves an identical subproblem and uses the solution of this subproblem to solve the bigger problem. Divide and conquer divides the problem into multiple subproblems and so the conquer stage is more complex than decrease and conquer algorithms. An example of decrease and conquer algorithm is the. Search and enumeration Many problems (such as playing ) can be modeled as problems on. A specifies rules for moving around a graph and is useful for such problems. This category also includes, enumeration and. Such algorithms make some choices randomly (or pseudo-randomly).

They can be very useful in finding approximate solutions for problems where finding exact solutions can be impractical (see heuristic method below). For some of these problems, it is known that the fastest approximations must involve some.

Whether randomized algorithms with can be the fastest algorithms for some problems is an open question known as the. There are two large classes of such algorithms: • return a correct answer with high-probability. Is the subclass of these that run in. • always return the correct answer, but their running time is only probabilistically bound, e.g..

This technique involves solving a difficult problem by transforming it into a better known problem for which we have (hopefully) algorithms. The goal is to find a reducing algorithm whose is not dominated by the resulting reduced algorithm's. For example, one for finding the median in an unsorted list involves first sorting the list (the expensive portion) and then pulling out the middle element in the sorted list (the cheap portion).

This technique is also known as. Optimization problems [ ] For there is a more specific classification of algorithms; an algorithm for such problems may fall into one or more of the general categories described above as well as into one of the following: When searching for optimal solutions to a linear function bound to linear equality and inequality constraints, the constraints of the problem can be used directly in producing the optimal solutions. There are algorithms that can solve any problem in this category, such as the popular. Problems that can be solved with linear programming include the for directed graphs. If a problem additionally requires that one or more of the unknowns must be an then it is classified in. A linear programming algorithm can solve such a problem if it can be proved that all restrictions for integer values are superficial, i.e., the solutions satisfy these restrictions anyway.

In the general case, a specialized algorithm or an algorithm that finds approximate solutions is used, depending on the difficulty of the problem. When a problem shows – meaning the optimal solution to a problem can be constructed from optimal solutions to subproblems – and, meaning the same subproblems are used to solve many different problem instances, a quicker approach called dynamic programming avoids recomputing solutions that have already been computed. For example,, the shortest path to a goal from a vertex in a weighted can be found by using the shortest path to the goal from all adjacent vertices. Dynamic programming and go together. The main difference between dynamic programming and divide and conquer is that subproblems are more or less independent in divide and conquer, whereas subproblems overlap in dynamic programming. The difference between dynamic programming and straightforward recursion is in caching or memoization of recursive calls. When subproblems are independent and there is no repetition, memoization does not help; hence dynamic programming is not a solution for all complex problems.

By using memoization or maintaining a of subproblems already solved, dynamic programming reduces the exponential nature of many problems to polynomial complexity. The greedy method A is similar to a dynamic programming algorithm in that it works by examining substructures, in this case not of the problem but of a given solution. Such algorithms start with some solution, which may be given or have been constructed in some way, and improve it by making small modifications.

For some problems they can find the optimal solution while for others they stop at, that is, at solutions that cannot be improved by the algorithm but are not optimum. The most popular use of greedy algorithms is for finding the minimal spanning tree where finding the optimal solution is possible with this method.,,, are greedy algorithms that can solve this optimization problem.

The heuristic method In, can be used to find a solution close to the optimal solution in cases where finding the optimal solution is impractical. These algorithms work by getting closer and closer to the optimal solution as they progress. In principle, if run for an infinite amount of time, they will find the optimal solution. Their merit is that they can find a solution very close to the optimal solution in a relatively short time. Such algorithms include,,, and. Some of them, like simulated annealing, are non-deterministic algorithms while others, like tabu search, are deterministic. When a bound on the error of the non-optimal solution is known, the algorithm is further categorized as an.

By field of study [ ]. See also: Every field of science has its own problems and needs efficient algorithms. Related problems in one field are often studied together. Some example classes are,,,,,,,,,,, algorithms and. Fields tend to overlap with each other, and algorithm advances in one field may improve those of other, sometimes completely unrelated, fields. For example, dynamic programming was invented for optimization of resource consumption in industry, but is now used in solving a broad range of problems in many fields.

By complexity [ ]. See also: and Algorithms can be classified by the amount of time they need to complete compared to their input size: • Constant time: if the time needed by the algorithm is the same, regardless of the input size. An access to an element. • Linear time: if the time is proportional to the input size.

The traverse of a list. • Logarithmic time: if the time is a logarithmic function of the input size. • Polynomial time: if the time is a power of the input size. The algorithm has quadratic time complexity.

• Exponential time: if the time is an exponential function of the input size. Some problems may have multiple algorithms of differing complexity, while other problems might have no algorithms or no known efficient algorithms.

There are also mappings from some problems to other problems. Owing to this, it was found to be more suitable to classify the problems themselves instead of the algorithms into equivalence classes based on the complexity of the best possible algorithms for them. Continuous algorithms [ ] The adjective 'continuous' when applied to the word 'algorithm' can mean: • An algorithm operating on data that represents continuous quantities, even though this data is represented by discrete approximations—such algorithms are studied in; or • An algorithm in the form of a that operates continuously on the data, running on an. Legal issues [ ]. • 'Any classical mathematical algorithm, for example, can be described in a finite number of English words' (Rogers 1987:2). • Well defined with respect to the agent that executes the algorithm: 'There is a computing agent, usually human, which can react to the instructions and carry out the computations' (Rogers 1987:2). • 'an algorithm is a procedure for computing a function (with respect to some chosen notation for integers).

This limitation (to numerical functions) results in no loss of generality', (Rogers 1987:1). • 'An algorithm has or more inputs, i.e., which are given to it initially before the algorithm begins' (Knuth 1973:5).

• 'A procedure which has all the characteristics of an algorithm except that it possibly lacks finiteness may be called a 'computational method' (Knuth 1973:5). • 'An algorithm has one or more outputs, i.e.

Quantities which have a specified relation to the inputs' (Knuth 1973:5). • Whether or not a process with random interior processes (not including the input) is an algorithm is debatable. Rogers opines that: 'a computation is carried out in a discrete stepwise fashion, without use of continuous methods or analogue devices... Carried forward deterministically, without resort to random methods or devices, e.g. Black Magic Rs422 Serial Port. , dice' Rogers 1987:2. • Kleene 1943 in Davis 1965:274 • Rosser 1939 in Davis 1965:225 • Moschovakis, Yiannis N.

'What is an algorithm?' In Engquist, B.; Schmid, W.. Pp. 919–936 (Part II).. Chambers Dictionary. Retrieved December 13, 2016.

• Hogendijk, Jan P. Archived from on April 12, 2009. • Oaks, Jeffrey A... Retrieved May 30, 2008. • Brezina, Corona (2006)..

The Rosen Publishing Group.. •, according to. •, Third Edition, 2012 • Stone 1973:4 • Stone simply requires that 'it must terminate in a finite number of steps' (Stone 1973:7–8). • Boolos and Jeffrey 1974,1999:19 • cf Stone 1972:5 • Knuth 1973:7 states: 'In practice we not only want algorithms, we want good algorithms. One criterion of goodness is the length of time taken to perform the algorithm. Other criteria are the adaptability of the algorithm to computers, its simplicity and elegance, etc.' • cf Stone 1973:6 • Stone 1973:7–8 states that there must be, '.a procedure that a robot [i.e., computer] can follow in order to determine precisely how to obey the instruction.'

Stone adds finiteness of the process, and definiteness (having no ambiguity in the instructions) to this definition. • Knuth, loc. Cit •, p. 105 • Gurevich 2000:1, 3 • Sipser 2006:157 • Knuth 1973:7 • Chaitin 2005:32 • Rogers 1987:1–2 • In his essay 'Calculations by Man and Machine: Conceptual Analysis' Seig 2002:390 credits this distinction to Robin Gandy, cf Wilfred Seig, et al., 2002 Reflections on the foundations of mathematics: Essays in honor of Solomon Feferman, Association for Symbolic Logic, A. K Peters Ltd, Natick, MA. • cf Gandy 1980:126, Robin Gandy Church's Thesis and Principles for Mechanisms appearing on pp. 123–148 in et al. 1980 The Kleene Symposium, North-Holland Publishing Company.

• A 'robot': 'A computer is a robot that performs any task that can be described as a sequence of instructions.' Cf Stone 1972:3 • Lambek's 'abacus' is a 'countably infinite number of locations (holes, wires etc.) together with an unlimited supply of counters (pebbles, beads, etc). The locations are distinguishable, the counters are not'. The holes have unlimited capacity, and standing by is an agent who understands and is able to carry out the list of instructions' (Lambek 1961:295). Lambek references Melzak who defines his Q-machine as 'an indefinitely large number of locations...

An indefinitely large supply of counters distributed among these locations, a program, and an operator whose sole purpose is to carry out the program' (Melzak 1961:283). Cit.) add the stipulation that the holes are 'capable of holding any number of stones' (p. Both Melzak and Lambek appear in The Canadian Mathematical Bulletin, vol. 3, September 1961.

• If no confusion results, the word 'counters' can be dropped, and a location can be said to contain a single 'number'. • 'We say that an instruction is effective if there is a procedure that the robot can follow in order to determine precisely how to obey the instruction.' (Stone 1972:6) • cf Minsky 1967: Chapter 11 'Computer models' and Chapter 14 'Very Simple Bases for Computability' pp. 255–281 in particular • cf Knuth 1973:3. • But always preceded by IF–THEN to avoid improper subtraction. • However, a few different assignment instructions (e.g.

DECREMENT, INCREMENT and ZERO/CLEAR/EMPTY for a Minsky machine) are also required for Turing-completeness; their exact specification is somewhat up to the designer. The unconditional GOTO is a convenience; it can be constructed by initializing a dedicated location to zero e.g. The instruction ' Z ← 0 '; thereafter the instruction IF Z=0 THEN GOTO xxx is unconditional.

• Knuth 1973:4 • Stone 1972:5. Methods for extracting roots are not trivial: see. • Leeuwen, Jan (1990).. • and 1985 Back to Basic: The History, Corruption, and Future of the Language, Addison-Wesley Publishing Company, Inc. Reading, MA,. • Tausworthe 1977:101 • Tausworthe 1977:142 • Knuth 1973 section 1.2.1, expanded by Tausworthe 1977 at pages 100ff and Chapter 9.1 • cf Tausworthe 1977 • Heath 1908:300; Hawking's Dover 2005 edition derives from Heath.

• ' 'Let CD, measuring BF, leave FA less than itself.' This is a neat abbreviation for saying, measure along BA successive lengths equal to CD until a point F is reached such that the length FA remaining is less than CD; in other words, let BF be the largest exact multiple of CD contained in BA' (Heath 1908:297) • For modern treatments using division in the algorithm, see Hardy and Wright 1979:180, Knuth 1973:2 (Volume 1), plus more discussion of Euclid's algorithm in Knuth 1969:293–297 (Volume 2). • Euclid covers this question in his Proposition 1. Retrieved May 20, 2012.

• Knuth 1973:13–18. He credits 'the formulation of algorithm-proving in terms of assertions and induction' to R. Floyd, Peter Naur, C.

Goldstine and J. Tausworth 1977 borrows Knuth's Euclid example and extends Knuth's method in section 9.1 Formal Proofs (pages 288–298). • Tausworthe 1997:294 • cf Knuth 1973:7 (Vol. I), and his more-detailed analyses on pp. 1969:294–313 (Vol II).

• Breakdown occurs when an algorithm tries to compact itself. Success would solve the. • Gillian Conahan (January 2013).. • Haitham Hassanieh,, Dina Katabi, and Eric Price, ' July 4, 2013, at the., Kyoto, January 2012.

See also the. • Kowalski 1979 •. • Carroll, Sue; Daughtrey, Taz (July 4, 2007).. American Society for Quality. Pp. 282 et seq.. • For instance, the of a (described using a membership oracle) can be approximated to high accuracy by a randomized polynomial time algorithm, but not by a deterministic one: see Dyer, Martin; Frieze, Alan; Kannan, Ravi (January 1991), 'A Random Polynomial-time Algorithm for Approximating the Volume of Convex Bodies', J. ACM, New York, NY, USA: ACM, 38 (1): 1–17,,:.

• and Mukund N. Linear Programming 2: Theory and Extensions. • Tsypkin (1971).. Academic Press. • Tutt, Andrew (March 15, 2016).

'An FDA for Algorithms'. Administrative Law Review.

• Algorithmic accountability. Applying the concept to different country contexts.: The Web Foundation.

• ^ Cooke, Roger L. The History of Mathematics: A Brief Course.

John Wiley & Sons.. • (2001), Episodes from the Early History of Astronomy, New York: Springer, pp. 40–62, • Davis 2000:18 • Bolter 1984:24 • Bolter 1984:26 • Bolter 1984:33–34, 204–206. • All quotes from W. Stanley Jevons 1880 Elementary Lessons in Logic: Deductive and Inductive, Macmillan and Co., London and New York. Republished as a googlebook; cf Jevons 1880:199–201. Louis Couturat 1914 the Algebra of Logic, The Open Court Publishing Company, Chicago and London. Republished as a googlebook; cf Couturat 1914:75–76 gives a few more details; interestingly he compares this to a typewriter as well as a piano.

Jevons states that the account is to be found at Jan. 20, 1870 The Proceedings of the Royal Society. Halstead Wickes Combi 102 Manual Dexterity on this page. • Jevons 1880:199–200 • All quotes from John Venn 1881 Symbolic Logic, Macmillan and Co., London. Republished as a googlebook. Cf Venn 1881:120–125.

The interested reader can find a deeper explanation in those pages. • Bell and Newell diagram 1971:39, cf. Davis 2000 • * Melina Hill, Valley News Correspondent, A Tinkerer Gets a Place in History, Valley News West Lebanon NH, Thursday March 31, 1983, page 13. • Davis 2000:14 • van Heijenoort 1967:81ff • van Heijenoort's commentary on Frege's Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought in van Heijenoort 1967:1 • Dixon 1906, cf. Kleene 1952:36–40 • cf.

Footnote in Alonzo Church 1936a in Davis 1965:90 and 1936b in Davis 1965:110 • Kleene 1935–6 in Davis 1965:237ff, Kleene 1943 in Davis 1965:255ff • Church 1936 in Davis 1965:88ff • cf. 'Formulation I', Post 1936 in Davis 1965:289–290 • Turing 1936–7 in Davis 1965:116ff • Rosser 1939 in Davis 1965:226 • Kleene 1943 in Davis 1965:273–274 • Kleene 1952:300, 317 • Kleene 1952:376 • Turing 1936–7 in Davis 1965:289–290 • Turing 1936 in Davis 1965, Turing 1939 in Davis 1965:160 • Hodges, p. 96 • Turing 1936–7:116 • ^ Turing 1936–7 in Davis 1965:136 • Turing 1939 in Davis 1965:160 References [ ]. • Jean Luc Chabert (1999). A History of Algorithms: From the Pebble to the Microchip.

Springer Verlag.. • Algorithmics.: The Spirit of Computing. Stanford, California: Center for the Study of Language and Information. • Knuth, Donald E.

Stanford, California: Center for the Study of Language and Information. • Berlinski, David (2001).

The Advent of the Algorithm: The 300-Year Journey from an Idea to the Computer. Harvest Books.. Cormen; Charles E.

Leiserson; Ronald L. Rivest; Clifford Stein (2009).

Introduction To Algorithms, Third Edition. External links [ ] Look up in Wiktionary, the free dictionary. Wikibooks has a book on the topic of: At, you can learn more and teach others about Algorithm at the •, ed. (2001) [1994],,, Springer Science+Business Media B.V. / Kluwer Academic Publishers, • at Curlie (based on ) •..

• — • Algorithm repositories • —OpenGenus Foundation • — • — and • — • — • — and • —previously from Lecture notes •. Jeff Erickson..