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wiki_3_chunk_131 | American Revolutionary War | In the western theater of the American Revolutionary War, conflicts between settlers and Native Americans led to lingering distrust. In the 1783 Treaty of Paris, Great Britain ceded control of the disputed lands between the Great Lakes and the Ohio River, but the Indian inhabitants were not a part of the peace negotiat... | wikipedia |
wiki_3_chunk_132 | American Revolutionary War | Lord North, Prime Minister since 1770, delegated control of the war in North America to Lord George Germain and the Earl of Sandwich, who was head of the Royal Navy from 1771 to 1782. Defeat at Saratoga in 1777 made it clear the revolt would not be easily suppressed, especially after the Franco-American alliance of Feb... | wikipedia |
wiki_3_chunk_133 | American Revolutionary War | British negotiators now proposed a second peace settlement to Congress. The terms presented by the Carlisle Peace Commission included acceptance of the principle of self-government. Parliament would recognize Congress as the governing body, suspend any objectionable legislation, surrender its right to local colonial ta... | wikipedia |
wiki_3_chunk_134 | American Revolutionary War | When the commissioners returned to London in November 1778, they recommended a change in policy. Sir Henry Clinton, the new British Commander-in-Chief in America, was ordered to stop treating the rebels as enemies, rather than subjects whose loyalty might be regained. Those standing orders would be in effect for three ... | wikipedia |
wiki_3_chunk_135 | American Revolutionary War | North initially backed the Southern strategy attempting to exploit divisions between the mercantile north and slave-owning south, but after the defeat of Yorktown, he was forced to accept the fact that this policy had failed. It was clear the war was lost, although the Royal Navy forced the French to relocate their fle... | wikipedia |
wiki_3_chunk_136 | American Revolutionary War | On February 27, 1782, a Whig motion to end the offensive war in America was carried by 19 votes. North now resigned, obliging the king to invite Lord Rockingham to form a government; a consistent supporter of the Patriot cause, he made a commitment to US independence a condition of doing so. George III reluctantly acce... | wikipedia |
wiki_3_chunk_137 | American Revolutionary War | When Lord Rockingham, the Whig leader and friend of the American cause was elevated to Prime Minister, Congress consolidated its diplomatic consuls in Europe into a peace delegation at Paris. All were experienced in Congressional leadership. The dean of the delegation was Benjamin Franklin of Pennsylvania. He had becom... | wikipedia |
wiki_3_chunk_138 | American Revolutionary War | The Whig negotiators for Lord Rockingham and his successor, Prime Minister Lord Shelburne, included long-time friend of Benjamin Franklin from his time in London, David Hartley and Richard Oswald, who had negotiated Laurens' release from the Tower of London. The Preliminary Peace signed on November 30 met four key Cong... | wikipedia |
wiki_3_chunk_139 | American Revolutionary War | British strategy was to strengthen the US sufficiently to prevent France from regaining a foothold in North America, and they had little interest in these proposals. However, divisions between their opponents allowed them to negotiate separately with each to improve their overall position, starting with the American de... | wikipedia |
wiki_3_chunk_140 | American Revolutionary War | Facing difficulties with Spain over claims involving the Mississippi River, and from France who was still reluctant to agree to American independence until all her demands were met, John Jay promptly told the British that he was willing to negotiate directly with them, cutting off France and Spain, and Prime Minister L... | wikipedia |
wiki_3_chunk_141 | American Revolutionary War | An Anglo-American Preliminary Peace was formally entered into in November 1782, and Congress endorsed the settlement on April 15, 1783. It announced the achievement of peace with independence; the "conclusive" treaty was signed on September 2, 1783, in Paris, effective the next day September 3, when Britain signed its ... | wikipedia |
wiki_3_chunk_142 | American Revolutionary War | Aftermath Washington expressed astonishment that the Americans had won a war against a leading world power, referring to the American victory as "little short of a standing miracle". The conflict between British subjects with the Crown against those with the Congress had lasted over eight years from 1775 to 1783. The l... | wikipedia |
wiki_3_chunk_143 | American Revolutionary War | On April 9, 1783, Washington issued orders that he had long waited to give, that "all acts of hostility" were to cease immediately. That same day, by arrangement with Washington, General Carleton issued a similar order to British troops. British troops, however, were not to evacuate until a prisoner of war exchange occ... | wikipedia |
wiki_3_chunk_144 | American Revolutionary War | As directed by a Congressional resolution of May 26, 1783, all non-commissioned officers and enlisted were furloughed "to their homes" until the "definitive treaty of peace", when they would be automatically discharged. The US armies were directly disbanded in the field as of Washington's General Orders on Monday, June... | wikipedia |
wiki_3_chunk_145 | American Revolutionary War | The expanse of territory that was now the United States was ceded from its colonial Mother country alone. It included millions of sparsely settled acres south of the Great Lakes Line between the Appalachian Mountains and the Mississippi River. The tentative colonial migration west became a flood during the years of the... | wikipedia |
wiki_3_chunk_146 | American Revolutionary War | Britain's extended post-war policy for the US continued to try to establish an Indian buffer state below the Great Lakes as late as 1814 during the War of 1812. The formally acquired western American lands continued to be populated by a dozen or so American Indian tribes that had been British allies for the most part. ... | wikipedia |
wiki_3_chunk_147 | American Revolutionary War | While tribes were not consulted by the British for the treaty, in practice the British refused to abandon the forts on territory they formally transferred. Instead, they provisioned military allies for continuing frontier raids and sponsored the Northwest Indian War (1785–1795), including erecting an additional British... | wikipedia |
wiki_3_chunk_148 | American Revolutionary War | Of the European powers with American colonies adjacent to the newly created United States, Spain was most threatened by American independence, and it was correspondingly the most hostile to it. Its territory adjacent to the US was relatively undefended, so Spanish policy developed a combination of initiatives. Spanish ... | wikipedia |
wiki_3_chunk_149 | American Revolutionary War | Casualties and losses The total loss of life throughout the conflict is largely unknown. As was typical in wars of the era, diseases such as smallpox claimed more lives than battle. Between 1775 and 1782, a smallpox epidemic broke out throughout North America, killing an estimated 130,000 among all its populations duri... | wikipedia |
wiki_3_chunk_150 | American Revolutionary War | Up to 70,000 American Patriots died during active military service. Of these, approximately 6,800 were killed in battle, while at least 17,000 died from disease. The majority of the latter died while prisoners of war of the British, mostly in the prison ships in New York Harbor. The number of Patriots seriously wounded... | wikipedia |
wiki_3_chunk_151 | American Revolutionary War | A British report in 1781 puts their total Army deaths at 6,046 in North America (1775–1779). Approximately 7,774 Germans died in British service in addition to 4,888 deserters; of the former, it is estimated 1,800 were killed in combat. Legacy | wikipedia |
wiki_3_chunk_152 | American Revolutionary War | The American Revolution established the United States with its numerous civil liberties and set an example to overthrow both monarchy and colonial governments. The United States has the world's oldest written constitution, and the constitutions of other free countries often bear a striking resemblance to the US Constit... | wikipedia |
wiki_3_chunk_153 | American Revolutionary War | Although the Revolution eliminated many forms of inequality, it did little to change the status of women, despite the role they played in winning independence. Most significantly, it failed to end slavery which continued to be a serious social and political issue and caused divisions that would ultimately end in civil ... | wikipedia |
wiki_3_chunk_154 | American Revolutionary War | In 1782, Virginia passed a law permitting manumission and over the next eight years more than 10,000 slaves were given their freedom. With support from Benjamin Franklin, in 1790 the Quakers petitioned Congress to abolish slavery; the number of abolitionist movements greatly increased, and by 1804 all the northern stat... | wikipedia |
wiki_3_chunk_155 | American Revolutionary War | Historiography
A large historiography concerns the reasons the Americans revolted and successfully broke away. The "Patriots", an insulting term used by the British that was proudly adopted by the Americans, stressed the constitutional rights of Englishmen, especially "No taxation without representation." Contemporari... | wikipedia |
wiki_3_chunk_156 | American Revolutionary War | The Atlantic history view places the American story in a broader context, including revolutions in France and Haiti. It tends to reintegrate the historiographies of the American Revolution and the British Empire. The "new social history" approach looks at community social structure to find cleavages that were magnified... | wikipedia |
wiki_3_chunk_157 | American Revolutionary War | The ideological approach that centers on republicanism in the United States. Republicanism dictated there would be no royalty, aristocracy or national church but allowed for continuation of the British common law, which American lawyers and jurists understood and approved and used in their everyday practice. Historians... | wikipedia |
wiki_3_chunk_158 | American Revolutionary War | Commemorations of the Revolutionary War
After the first U.S. postage stamp was issued in 1849, the U.S. Post Office frequently issued commemorative stamps celebrating the various people and events of the Revolutionary War. However, it would be more than 140 years after the Revolution before any stamp commemorating that... | wikipedia |
wiki_3_chunk_159 | American Revolutionary War | Topics of the Revolution
Committee of safety (American Revolution)
Financial costs of the American Revolutionary War
Flags of the American Revolution
Naval operations in the American Revolutionary War Social history of the Revolution
Black Patriot
Christianity in the United States#American Revolution
The Colored... | wikipedia |
wiki_3_chunk_160 | American Revolutionary War | Lists of Revolutionary military
List of American Revolutionary War battles
List of British Forces in the American Revolutionary War
List of Continental Forces in the American Revolutionary War
List of infantry weapons in the American Revolution
List of United States militia units in the American Revolutionary War ... | wikipedia |
wiki_3_chunk_161 | American Revolutionary War | Legacy and related
American Revolution Statuary
Commemoration of the American Revolution
Founders Online
Independence Day (United States)
The Last Men of the Revolution
List of plays and films about the American Revolution
Museum of the American Revolution
Tomb of the Unknown Soldier of the American Revolution
... | wikipedia |
wiki_3_chunk_162 | American Revolutionary War | – Highly regarded examination of British strategy and leadership. An introduction by John W. Shy with his biographical sketch of Mackesy. Robinson Library (See also:British Warships in the Age of Sail) Websites without authors
Canada's Digital Collections Program
History.org
Maryland State House
The History Pl... | wikipedia |
wiki_3_chunk_163 | American Revolutionary War | Allison, David, and Larrie D. Ferreiro, eds. The American Revolution: A World War (Smithsonian, 2018) excerpt
Volumes committed to the American Revolution: Vol. 7; Vol. 8; Vol. 9; Vol. 10
Bobrick, Benson. Angel in the Whirlwind: The Triumph of the American Revolution. Penguin, 1998 (paperback reprint)
Char... | wikipedia |
wiki_3_chunk_164 | American Revolutionary War | Primary sources
In addition to this selection, many primary sources are available at the Princeton University Law School Avalon Project and at the Library of Congress Digital Collections (previously LOC webpage, American Memory). Original editions for titles related to the American Revolutionary War can be found open-s... | wikipedia |
wiki_3_chunk_165 | American Revolutionary War | Bibliographies online
Library of Congress Guide to the American Revolution
Bibliographies of the War of American Independence compiled by the United States Army Center of Military History
Political bibliography from Omohundro Institute of Early American History and Culture Conflicts in 1775
Conflicts in 1776
Conflic... | wikipedia |
wiki_4_chunk_0 | Algorithm | In mathematics and computer science, an algorithm () is a finite sequence of well-defined 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. By making use of artificial intelligence, algor... | wikipedia |
wiki_4_chunk_1 | Algorithm | 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. | wikipedia |
wiki_4_chunk_2 | Algorithm | As an effective method, an algorithm can be expressed within a finite amount of space and time, and in a well-defined formal language for calculating a function. Starting from an initial state and initial input (perhaps empty), the instructions describe a computation that, when executed, proceeds through a finite numbe... | wikipedia |
wiki_4_chunk_3 | Algorithm | History
The concept of algorithm has existed since antiquity. Arithmetic algorithms, such as a division algorithm, were used by ancient Babylonian mathematicians c. 2500 BC and Egyptian mathematicians c. 1550 BC. Greek mathematicians later used algorithms in 240 BC in the sieve of Eratosthenes for finding prime number... | wikipedia |
wiki_4_chunk_4 | Algorithm | The word algorithm is derived from the name of the 9th-century Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī, whose nisba (identifying him as from Khwarazm) was Latinized as Algoritmi (Arabized Persian الخوارزمی c. 780–850).
Muḥammad ibn Mūsā al-Khwārizmī was a mathematician, astronomer, geographer, and scholar ... | wikipedia |
wiki_4_chunk_5 | Algorithm | Indian mathematics was predominantly algorithmic.
Algorithms that are representative of the Indian mathematical tradition range from the ancient Śulbasūtrās to the medieval texts of the Kerala School. In English, the word algorithm was first used in about 1230 and then by Chaucer in 1391. English adopted the French ter... | wikipedia |
wiki_4_chunk_6 | Algorithm | The poem is a few hundred lines long and summarizes the art of calculating with the new styled Indian dice (Tali Indorum), or Hindu numerals. | wikipedia |
wiki_4_chunk_7 | Algorithm | A partial formalization of the modern concept of algorithm began with attempts to solve the Entscheidungsproblem (decision problem) posed by David Hilbert in 1928. Later formalizations were framed as attempts to define "effective calculability" or "effective method". Those formalizations included the Gödel–Herbrand–Kl... | wikipedia |
wiki_4_chunk_8 | Algorithm | 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), and (for example) any prescribed bureaucratic procedure
or cook-book recipe. In general, a program is only an algorit... | wikipedia |
wiki_4_chunk_9 | Algorithm | offer an informal meaning of the word "algorithm" in the following quotation: | wikipedia |
wiki_4_chunk_10 | Algorithm | 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 ... | wikipedia |
wiki_4_chunk_11 | Algorithm | 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. ... | wikipedia |
wiki_4_chunk_12 | Algorithm | The concept of algorithm is also used to define the notion of decidability—a notion that is central for explaining how formal systems come into being starting from a small set of axioms and rules. In logic, the time that an algorithm requires to complete cannot be measured, as it is not apparently related to the custom... | wikipedia |
wiki_4_chunk_13 | Algorithm | Most algorithms are intended to be implemented as computer programs. However, algorithms are also implemented by other means, such as in a biological neural network (for example, the human brain implementing arithmetic or an insect looking for food), in an electrical circuit, or in a mechanical device. Formalization | wikipedia |
wiki_4_chunk_14 | Algorithm | 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 ... | wikipedia |
wiki_4_chunk_15 | Algorithm | 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".Turing machines can define computational processes that do not terminate. Th... | wikipedia |
wiki_4_chunk_16 | Algorithm | 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 dat... | wikipedia |
wiki_4_chunk_17 | Algorithm | 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 by flow o... | wikipedia |
wiki_4_chunk_18 | Algorithm | So far, the discussion on the formalization of an algorithm has assumed the premises of imperative programming. This is the most common conception—one which attempts to describe a task in discrete, "mechanical" means. Unique to this conception of formalized algorithms is the assignment operation, which sets the value o... | wikipedia |
wiki_4_chunk_19 | Algorithm | Expressing algorithms
Algorithms can be expressed in many kinds of notation, including natural languages, pseudocode, flowcharts, drakon-charts, programming languages or control tables (processed by interpreters). Natural language expressions of algorithms tend to be verbose and ambiguous, and are rarely used for compl... | wikipedia |
wiki_4_chunk_20 | Algorithm | There is a wide variety of representations possible and one can express a given Turing machine program as a sequence of machine tables (see finite-state machine, state transition table and control table for more), as flowcharts and drakon-charts (see state diagram for more), or as a form of rudimentary machine code or ... | wikipedia |
wiki_4_chunk_21 | Algorithm | Representations of algorithms can be classed into three accepted levels of Turing machine description, as follows:
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 des... | wikipedia |
wiki_4_chunk_22 | Algorithm | For an example of the simple algorithm "Add m+n" described in all three levels, see Examples. Design Algorithm design refers to a method or a mathematical process for problem-solving and engineering algorithms. The design of algorithms is part of many solution theories of operation research, such as dynamic programming... | wikipedia |
wiki_4_chunk_23 | Algorithm | One of the most important aspects of algorithm design is resource (run-time, memory usage) efficiency; the big O notation is used to describe e.g. an algorithm's run-time growth as the size of its input increases. Typical steps in the development of algorithms:
Problem definition
Development of a model
Specification... | wikipedia |
wiki_4_chunk_24 | Algorithm | "Elegant" (compact) programs, "good" (fast) programs : The notion of "simplicity and elegance" appears informally in Knuth and precisely in Chaitin:
Knuth: " ... we want good algorithms in some loosely defined aesthetic sense. One criterion ... is the length of time taken to perform the algorithm .... Other criteria ar... | wikipedia |
wiki_4_chunk_25 | Algorithm | Chaitin prefaces his definition with: "I'll show you can't prove that a program is 'elegant—such a proof would solve the Halting problem (ibid). Algorithm versus function computable by an algorithm: For a given function multiple algorithms may exist. This is true, even without expanding the available instruction set av... | wikipedia |
wiki_4_chunk_26 | Algorithm | Unfortunately, there may be a tradeoff between goodness (speed) and elegance (compactness)—an elegant program may take more steps to complete a computation than one less elegant. An example that uses Euclid's algorithm appears below. | wikipedia |
wiki_4_chunk_27 | Algorithm | Computers (and computors), models of computation: A computer (or human "computor") is a restricted type of machine, a "discrete deterministic mechanical device" that blindly follows its instructions. Melzak's and Lambek's primitive models reduced this notion to four elements: (i) discrete, distinguishable locations, (i... | wikipedia |
wiki_4_chunk_28 | Algorithm | Minsky describes a more congenial variation of Lambek's "abacus" model in his "Very Simple Bases for Computability". Minsky's machine proceeds sequentially through its five (or six, depending on how one counts) instructions unless either a conditional IF-THEN GOTO or an unconditional GOTO changes program flow out of se... | wikipedia |
wiki_4_chunk_29 | Algorithm | Simulation of an algorithm: computer (computor) language: Knuth advises the reader that "the best way to learn an algorithm is to try it . . . immediately take pen and paper and work through an example". But what about a simulation or execution of the real thing? The programmer must translate the algorithm into a langu... | wikipedia |
wiki_4_chunk_30 | Algorithm | This means that the programmer must know a "language" that is effective relative to the target computing agent (computer/computor). | wikipedia |
wiki_4_chunk_31 | Algorithm | But what model should be used for the simulation? Van Emde Boas observes "even if we base complexity theory on abstract instead of concrete machines, arbitrariness of the choice of a model remains. It is at this point that the notion of simulation enters". When speed is being measured, the instruction set matters. For ... | wikipedia |
wiki_4_chunk_32 | Algorithm | Structured programming, canonical structures: Per the Church–Turing thesis, any algorithm can be computed by a model known to be Turing complete, and per Minsky's demonstrations, Turing completeness requires only four instruction types—conditional GOTO, unconditional GOTO, assignment, HALT. Kemeny and Kurtz observe tha... | wikipedia |
wiki_4_chunk_33 | Algorithm | Canonical flowchart symbols: The graphical aide called a flowchart, offers a way to describe and document an algorithm (and a computer program of one). Like the program flow of a Minsky machine, a flowchart always starts at the top of a page and proceeds down. Its primary symbols are only four: the directed arrow showi... | wikipedia |
wiki_4_chunk_34 | Algorithm | Examples Algorithm example
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 in English prose, as: | wikipedia |
wiki_4_chunk_35 | Algorithm | 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.
W... | wikipedia |
wiki_4_chunk_36 | Algorithm | (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 pseudocode or pidgin code: Input: A list of numbers L.
Output: The largest number in the list L. if L.size = 0 return null
largest ← L[0]
fo... | wikipedia |
wiki_4_chunk_37 | Algorithm | Euclid's algorithm
In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers (numbers), the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first describe... | wikipedia |
wiki_4_chunk_38 | Algorithm | 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 lo... | wikipedia |
wiki_4_chunk_39 | Algorithm | 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 (or the two can be equal so their subtraction yields zero). | wikipedia |
wiki_4_chunk_40 | Algorithm | 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 reductio ad absurdum 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... | wikipedia |
wiki_4_chunk_41 | Algorithm | 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. S, A, etc.
The varying quantity (number) in a lo... | wikipedia |
wiki_4_chunk_42 | Algorithm | An inelegant program for Euclid's algorithm 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 descri... | wikipedia |
wiki_4_chunk_43 | Algorithm | INPUT:
[Into two locations L and S put the numbers l and s that represent the two lengths]:
INPUT L, S
[Initialize R: make the remaining length r equal to the starting/initial/input length l]:
R ← L | wikipedia |
wiki_4_chunk_44 | Algorithm | E0: [Ensure r ≥ s.]
[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 4, 5 and 6:
GOTO step 7
ELSE
swap the contents of R and S.
L ← R (this first step is redundant, but is useful for later... | wikipedia |
wiki_4_chunk_45 | Algorithm | 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.
IF S > R THEN
done measuring so
GOTO 10
ELSE
measure again,
R ← R − S
[Remainder-loop]:
GOTO 7. | wikipedia |
wiki_4_chunk_46 | Algorithm | 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.
IF R = 0 THEN
done so
GOTO step 15
ELSE
CONTINUE TO step 11, | wikipedia |
wiki_4_chunk_47 | Algorithm | 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.
L ← R
R ← S
S ← L
[Repeat the measuring process]:
GOTO 7 OUTPUT: [Done. S contains the greatest common divisor]:
PRINT S DONE:
HALT, END, ST... | wikipedia |
wiki_4_chunk_48 | Algorithm | An elegant program for Euclid's algorithm
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 ←.
5 REM Euclid's algorithm for greatest common divisor
6 PRINT "Ty... | wikipedia |
wiki_4_chunk_49 | Algorithm | The following version can be used with programming languages from the C-family:
// Euclid's algorithm for greatest common divisor
int euclidAlgorithm (int A, int B){
A=abs(A);
B=abs(B);
while (B!=0){
while (A>B) A=A-B;
B=B-A;
}
return A;
} | wikipedia |
wiki_4_chunk_50 | Algorithm | Testing the Euclid algorithms
Does an algorithm do what its author wants it to do? A few test cases usually give some confidence in the core functionality. But tests are not enough. For test cases, one source uses 3009 and 884. Knuth suggested 40902, 24140. Another interesting case is the two relatively prime numbers 1... | wikipedia |
wiki_4_chunk_51 | Algorithm | But "exceptional cases" must be identified and tested. Will "Inelegant" perform properly when R > S, S > R, R = S? Ditto for "Elegant": B > A, A > B, A = B? (Yes to all). What happens when one number is zero, both numbers are zero? ("Inelegant" computes forever in all cases; "Elegant" computes forever when A = 0.) What... | wikipedia |
wiki_4_chunk_52 | Algorithm | Proof of program correctness by use of mathematical induction: Knuth demonstrates the application of mathematical induction to an "extended" version of Euclid's algorithm, and he proposes "a general method applicable to proving the validity of any algorithm". Tausworthe proposes that a measure of the complexity of a pr... | wikipedia |
wiki_4_chunk_53 | Algorithm | Measuring and improving the Euclid algorithms
Elegance (compactness) versus goodness (speed): With only six core instructions, "Elegant" is the clear winner, compared to "Inelegant" at thirteen instructions. However, "Inelegant" is faster (it arrives at HALT in fewer steps). Algorithm analysis indicates why this is the... | wikipedia |
wiki_4_chunk_54 | Algorithm | Can the algorithms be improved?: Once the programmer judges a program "fit" and "effective"—that is, it computes the function intended by its author—then the question becomes, can it be improved? | wikipedia |
wiki_4_chunk_55 | Algorithm | The compactness of "Inelegant" can be improved by the elimination of five steps. But Chaitin proved that compacting an algorithm cannot be automated by a generalized algorithm; rather, it can only be done heuristically; i.e., by exhaustive search (examples to be found at Busy beaver), trial and error, cleverness, insig... | wikipedia |
wiki_4_chunk_56 | Algorithm | The speed of "Elegant" can be improved by moving the "B=0?" test outside of the two subtraction loops. This change calls for the addition of three instructions (B = 0?, A = 0?, GOTO). Now "Elegant" computes the example-numbers faster; whether this is always the case for any given A, B, and R, S would require a detailed... | wikipedia |
wiki_4_chunk_57 | Algorithm | 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 analysis of algorithms to obtain such quantitative answers (estimates); for example, an algorithm which adds up the elements of a list of n ... | wikipedia |
wiki_4_chunk_58 | Algorithm | Different algorithms may complete the same task with a different set of instructions in less or more time, space, or 'effort' than others. For example, a binary search algorithm (with cost O(log n)) outperforms a sequential search (cost O(n) ) when used for table lookups on sorted lists or arrays. Formal versus empiric... | wikipedia |
wiki_4_chunk_59 | Algorithm | The analysis, and study of algorithms is a discipline of computer science, and is often practiced abstractly without the use of a specific programming language or implementation. In this sense, algorithm analysis resembles other mathematical disciplines in that it focuses on the underlying properties of the algorithm a... | wikipedia |
wiki_4_chunk_60 | Algorithm | Empirical testing is useful because it may uncover unexpected interactions that affect performance. Benchmarks may be used to compare before/after potential improvements to an algorithm after program optimization.
Empirical tests cannot replace formal analysis, though, and are not trivial to perform in a fair manner. E... | wikipedia |
wiki_4_chunk_61 | Algorithm | To illustrate the potential improvements possible even in well-established algorithms, a recent significant innovation, relating to FFT 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 ... | wikipedia |
wiki_4_chunk_62 | Algorithm | Recursion
A recursive algorithm 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 functional programming. Iterative algorithms use repetitive constructs like loops and sometimes additional data structures like... | wikipedia |
wiki_4_chunk_63 | Algorithm | 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 includes many different types of algorithms. Some common paradigms are: | wikipedia |
wiki_4_chunk_64 | Algorithm | Brute-force or exhaustive search
This is the naive method of trying every possible solution to see which is best.
Divide and conquer
A divide and conquer algorithm repeatedly reduces an instance of a problem to one or more smaller instances of the same problem (usually recursively) until the instances are small enou... | wikipedia |
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