Misplaced Pages

#860139

51-584: Numerical Recipes is the generic title of a series of books on algorithms and numerical analysis by William H. Press , Saul A. Teukolsky , William T. Vetterling and Brian P. Flannery . In various editions, the books have been in print since 1986. The most recent edition was published in 2007. The Numerical Recipes books cover a range of topics that include both classical numerical analysis ( interpolation , integration , linear algebra , differential equations , and so on), signal processing ( Fourier methods , filtering ), statistical treatment of data, and

102-595: A binary search algorithm (with cost ⁠ O ( log ⁡ n ) {\displaystyle O(\log n)} ⁠ ) outperforms a sequential search (cost ⁠ O ( n ) {\displaystyle O(n)} ⁠ ) when used for table lookups on sorted lists or arrays. The analysis, and study of algorithms is a discipline of computer science . Algorithms are often studied abstractly, without referencing any specific programming language or implementation. Algorithm analysis resembles other mathematical disciplines as it focuses on

153-468: A flowchart offers a way to describe and document an algorithm (and a computer program corresponding to it). It has four primary symbols: arrows showing program flow, rectangles (SEQUENCE, GOTO), diamonds (IF-THEN-ELSE), and dots (OR-tie). Sub-structures can "nest" in rectangles, but only if a single exit occurs from the superstructure. It is often important to know how much time, storage, or other cost an algorithm may require. Methods have been developed for

204-535: A programming language , with the code printed in the book. Each variant of the book is keyed to a specific language. According to the publisher, Cambridge University Press , the Numerical Recipes books are historically the all-time best-selling books on scientific programming methods. In recent years, Numerical Recipes books have been cited in the scientific literature more than 3000 times per year according to ISI Web of Knowledge (e.g., 3962 times in

255-625: A class of specific problems or to perform a computation . Algorithms are used as specifications for performing calculations and data processing . More advanced algorithms can use conditionals to divert the code execution through various routes (referred to as automated decision-making ) and deduce valid inferences (referred to as automated reasoning ). In contrast, a heuristic is an approach to solving problems that do not have well-defined correct or optimal results. For example, although social media recommender systems are commonly called "algorithms", they actually rely on heuristics as there

306-680: A computer-executable form, but are also used to define or document algorithms. There are many possible representations and Turing machine programs can be expressed 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), as a form of rudimentary machine code or assembly code called "sets of quadruples", and more. Algorithm representations can also be classified into three accepted levels of Turing machine description: high-level description, implementation description, and formal description. A high-level description describes qualities of

357-719: A computing machine or a human who could only carry out specific elementary operations on symbols . 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 performing arithmetic or an insect looking for food), in an electrical circuit , or a mechanical device. Step-by-step procedures for solving mathematical problems have been recorded since antiquity. This includes in Babylonian mathematics (around 2500 BC), Egyptian mathematics (around 1550 BC), Indian mathematics (around 800 BC and later),

408-527: A few topics in machine learning ( hidden Markov model , support vector machines ). The writing style is accessible and has an informal tone. The emphasis is on understanding the underlying basics of techniques, not on the refinements that may, in practice, be needed to achieve optimal performance and reliability. Few results are proved with any degree of rigor, although the ideas behind proofs are often sketched, and references are given. Importantly, virtually all methods that are discussed are also implemented in

459-479: A final ending state. The transition from one state to the next is not necessarily deterministic ; some algorithms, known as randomized algorithms , incorporate random input. Around 825 AD, Persian scientist and polymath Muḥammad ibn Mūsā al-Khwārizmī wrote kitāb al-ḥisāb al-hindī ("Book of Indian computation") and kitab al-jam' wa'l-tafriq al-ḥisāb al-hindī ("Addition and subtraction in Indian arithmetic"). In

510-525: A programmer can write structured programs using only these instructions; on the other hand "it is also possible, and not too hard, to write badly structured programs in a structured language". Tausworthe augments the three Böhm-Jacopini canonical structures : SEQUENCE, IF-THEN-ELSE, and WHILE-DO, with two more: DO-WHILE and CASE. An additional benefit of a structured program is that it lends itself to proofs of correctness using mathematical induction . By themselves, algorithms are not usually patentable. In

561-477: A sequence of operations", which would include all computer programs (including programs that do not perform numeric calculations), and any prescribed bureaucratic procedure or cook-book recipe . In general, a program is an algorithm only if it stops eventually —even though infinite loops may sometimes prove desirable. Boolos, Jeffrey & 1974, 1999 define an algorithm to be an explicit set of instructions for determining an output, that can be followed by

SECTION 10

#1732797310861

612-550: Is "full of bugs" . They attributed this to people using outdated versions of the code, bugs in other parts of the code and misuse of routines which require some understanding to use correctly. The rebuttal does not, however, cover criticisms regarding lack of mentions to code limitations, boundary conditions, and more modern algorithms, another theme in Snyder's comment compilation. A precision issue in Bessel functions has persisted to

663-416: Is a method or mathematical process for problem-solving and engineering algorithms. The design of algorithms is part of many solution theories, such as divide-and-conquer or dynamic programming within operation research . Techniques for designing and implementing algorithm designs are also called algorithm design patterns, with examples including the template method pattern and the decorator pattern. One of

714-581: 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: One of the simplest algorithms finds 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 described in plain English as: High-level description: (Quasi-)formal description: Written in prose but much closer to

765-441: Is a single volume that covers a very broad range of algorithms. Unfortunately that format skewed the choice of algorithms towards simpler and shorter early algorithms which were not as accurate, efficient or stable as later more complex algorithms. The first edition had also some minor bugs, which were fixed in later editions; however according to the authors for years they were encountering on the internet rumors that Numerical Recipes

816-460: Is no truly "correct" recommendation. 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 number of well-defined successive states, eventually producing "output" and terminating at

867-453: Is useful for uncovering 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 non-trivial to perform fairly. To illustrate the potential improvements possible even in well-established algorithms, a recent significant innovation, relating to FFT algorithms (used heavily in

918-1107: The Entscheidungsproblem (decision problem) posed by David Hilbert . Later formalizations were framed as attempts to define " effective calculability " or "effective method". Those formalizations included the Gödel – Herbrand – Kleene recursive functions of 1930, 1934 and 1935, Alonzo Church 's lambda calculus of 1936, Emil Post 's Formulation 1 of 1936, and Alan Turing 's Turing machines of 1936–37 and 1939. 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 complex or technical algorithms. Pseudocode, flowcharts, drakon-charts, and control tables are structured expressions of algorithms that avoid common ambiguities of natural language. Programming languages are primarily for expressing algorithms in

969-629: The Jacquard loom , a precursor to Hollerith cards (punch cards), and "telephone switching technologies" led to the development of the first computers. By the mid-19th century, the telegraph , the precursor of the telephone, was in use throughout the world. By the late 19th century, the ticker tape ( c.  1870s ) was in use, as were Hollerith cards (c. 1890). Then came the teleprinter ( c.  1910 ) with its punched-paper use of Baudot code on tape. Telephone-switching networks of electromechanical relays were invented in 1835. These led to

1020-467: The 1980s were fertile years for the "black box" side, yielding important libraries such as BLAS and LAPACK , and integrated environments like MATLAB and Mathematica . By the early 1990s, when Second Edition versions of Numerical Recipes (with code in C, Fortran-77, and Fortran-90) were published, it was clear that the constituency for Numerical Recipes was by no means the majority of scientists doing computation, but only that slice that lived between

1071-792: The Ifa Oracle (around 500 BC), Greek mathematics (around 240 BC), and Arabic mathematics (around 800 AD). The earliest evidence of algorithms is found in ancient Mesopotamian mathematics. A Sumerian clay tablet found in Shuruppak near Baghdad and dated to c.  2500 BC describes the earliest division algorithm . During the Hammurabi dynasty c.  1800  – c.  1600 BC , Babylonian clay tablets described algorithms for computing formulas. Algorithms were also used in Babylonian astronomy . Babylonian clay tablets describe and employ algorithmic procedures to compute

SECTION 20

#1732797310861

1122-536: The NR routines run out of steam. Problems will occur because [...] The code listings are copyrighted and commercially licensed by the Numerical Recipes authors. A license to use the code is given with the purchase of a book, but the terms of use are highly restrictive. For example, programmers need to make sure NR code cannot be extracted from their finished programs and used – a difficult requirement with dubious enforceability. However, Numerical Recipes does include

1173-596: The United States, a claim consisting solely of simple manipulations of abstract concepts, numbers, or signals does not constitute "processes" (USPTO 2006), so algorithms are not patentable (as in Gottschalk v. Benson ). However practical applications of algorithms are sometimes patentable. For example, in Diamond v. Diehr , the application of a simple feedback algorithm to aid in the curing of synthetic rubber

1224-454: The algorithm itself, ignoring how it is implemented on the Turing machine. An implementation description describes the general manner in which the machine moves its head and stores data in order to carry out the algorithm, but does not give exact states. In the most detail, a formal description gives the exact state table and list of transitions of the Turing machine. The graphical aid called

1275-588: The algorithm's properties, not implementation. Pseudocode is typical for analysis as it is a simple and general representation. Most algorithms are implemented on particular hardware/software platforms and their algorithmic efficiency is tested using real code. The efficiency of a particular algorithm may be insignificant for many "one-off" problems but it may be critical for algorithms designed for fast interactive, commercial or long life scientific usage. Scaling from small n to large n frequently exposes inefficient algorithms that are otherwise benign. Empirical testing

1326-403: The analysis of algorithms to obtain such quantitative answers (estimates); for example, an algorithm that adds up the elements of a list of n numbers would have a time requirement of ⁠ O ( n ) {\displaystyle O(n)} ⁠ , using big O notation . The algorithm only needs to remember two values: the sum of all the elements so far, and its current position in

1377-640: The authors significantly expanded the scope of the book, and significantly rewrote a large part of the text. They continued to include code, still printed in the book, now in C++, for every method discussed. The Third Edition was also released as an electronic book, eventually made available on the Web for free (with nags) or by paid or institutional subscription (with faster, full access and no nags). In 2015 Numerical Recipes sold its historic two-letter domain name nr.com and became numerical.recipes instead. Numerical Recipes

1428-521: The earliest codebreaking algorithm. Bolter credits the invention of the weight-driven clock as "the key invention [of Europe in the Middle Ages ]," specifically the verge escapement mechanism producing the tick and tock of a mechanical clock. "The accurate automatic machine" led immediately to "mechanical automata " in the 13th century and "computational machines"—the difference and analytical engines of Charles Babbage and Ada Lovelace in

1479-523: The early 12th century, Latin translations of said al-Khwarizmi texts involving the Hindu–Arabic numeral system and arithmetic appeared, for example Liber Alghoarismi de practica arismetrice , attributed to John of Seville , and Liber Algorismi de numero Indorum , attributed to Adelard of Bath . Hereby, alghoarismi or algorismi is the Latinization of Al-Khwarizmi's name; the text starts with

1530-427: 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. Algorithm design

1581-411: The following statement regarding copyrights on computer programs: Copyright does not protect ideas, but only the expression of those ideas in a particular form. In the case of a computer program, the ideas consist of the program's methodology and algorithm, including the necessary sequence of steps adopted by the programmer. The expression of those ideas is the program source code   ... If you analyze

Numerical Recipes - Misplaced Pages Continue

1632-456: The format of the book because of space limitations and for readability. The books differ by edition (1st, 2nd, and 3rd) and by the computer language in which the code is given. The books are published by Cambridge University Press . Algorithm In mathematics and computer science , an algorithm ( / ˈ æ l ɡ ə r ɪ ð əm / ) is a finite sequence of mathematically rigorous instructions, typically used to solve

1683-429: The high-level language of a computer program, the following is the more formal coding of the algorithm in pseudocode or pidgin code : GNU Scientific Library The GSL project was initiated in 1996 by physicists Mark Galassi and James Theiler of Los Alamos National Laboratory . They aimed at writing a modern replacement for widely used but somewhat outdated Fortran libraries such as Netlib . They carried out

1734-617: The ideas contained in a program, and then express those ideas in your own completely different implementation, then that new program implementation belongs to you. One early motivation for the GNU Scientific Library was that a free library was needed as a substitute for Numerical Recipes . Another line of criticism centers on the coding style of the books, which strike some modern readers as "Fortran-ish", though written in contemporary, object-oriented C++. The authors have defended their very terse coding style as necessary to

1785-450: The input list. If the space required to store the input numbers is not counted, it has a space requirement of ⁠ O ( 1 ) {\displaystyle O(1)} ⁠ , otherwise ⁠ O ( n ) {\displaystyle O(n)} ⁠ is required. 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,

1836-490: The invention of the digital adding device by George Stibitz in 1937. While working in Bell Laboratories, he observed the "burdensome" use of mechanical calculators with gears. "He went home one evening in 1937 intending to test his idea... When the tinkering was over, Stibitz had constructed a binary adding device". In 1928, a partial formalization of the modern concept of algorithms began with attempts to solve

1887-463: The library expanded only slowly; as the documentation stated, the maintainers were more interested in stability than in additional functionality. Major version 1 ended with release 1.16 of July 2013; this was the only public activity in the three years 2012–2014. Vigorous development resumed with publication of version 2.0 in October 2015, which included user contributed patches. The latest version 2.8

1938-429: The mid-19th century. Lovelace designed the first algorithm intended for processing on a computer, Babbage's analytical engine, which is the first device considered a real Turing-complete computer instead of just a calculator . Although a full implementation of Babbage's second device was not realized for decades after her lifetime, Lovelace has been called "history's first programmer". Bell and Newell (1971) write that

1989-426: The more mathematical numerical analysts and the larger community using integrated environments. The Second Edition versions occupied a stable role in this niche environment. By the mid-2000s, the practice of scientific computing had been radically altered by the mature Internet and Web. Recognizing that their Numerical Recipes books were increasingly valued more for their explanatory text than for their code examples,

2040-627: 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. Per the Church–Turing thesis , any algorithm can be computed by any Turing complete model. Turing completeness only requires four instruction types—conditional GOTO, unconditional GOTO, assignment, HALT. However, Kemeny and Kurtz observe that, while "undisciplined" use of unconditional GOTOs and conditional IF-THEN GOTOs can result in " spaghetti code ",

2091-430: The motivations and impatience of the book's intended audience. The declared premise of the NR authors is that you will come to grief one way or the other if you use numerical routines you do not understand. They attempt to give you enough mathematical detail that you understand the routines they present, in enough depth that you can diagnose problems when they occur, and make more sophisticated choices about replacements when

Numerical Recipes - Misplaced Pages Continue

2142-678: The official note in that book says 1986.) Supplemental editions followed with code in Pascal, BASIC, and C. Numerical Recipes took, from the start, an opinionated editorial position at odds with the conventional wisdom of the numerical analysis community: If there is a single dominant theme in this book, it is that practical methods of numerical computation can be simultaneously efficient, clever, and — important — clear. The alternative viewpoint, that efficient computational methods must necessarily be so arcane and complex as to be useful only in "black box" form, we firmly reject. However, as it turned out,

2193-405: The overall design and wrote early modules; with that ready they recruited other scientists to contribute. The "overall development of the library and the design and implementation of the major modules" was carried out by Brian Gough and Gerard Jungman. Other major contributors were Jim Davies , Reid Priedhorsky, M. Booth, and F. Rossi. Version 1.0 was released in 2001. In the following years,

2244-564: The phrase Dixit Algorismi , or "Thus spoke Al-Khwarizmi". Around 1230, the English word algorism is attested and then by Chaucer in 1391, English adopted the French term. In the 15th century, under the influence of the Greek word ἀριθμός ( arithmos , "number"; cf. "arithmetic"), the Latin word was altered to algorithmus . One informal definition is "a set of rules that precisely defines

2295-411: The third edition according to Pavel Holoborodko. Despite criticism by numerical analysts, engineers and scientists generally find the book conveniently broad in scope. Norman Gray concurs in the following quote: Numerical Recipes [nr] does not claim to be a numerical analysis textbook, and it makes a point of noting that its authors are (astro-)physicists and engineers rather than analysts, and so share

2346-675: The time and place of significant astronomical events. Algorithms for arithmetic are also found in ancient Egyptian mathematics , dating back to the Rhind Mathematical Papyrus c.  1550 BC . Algorithms were later used in ancient Hellenistic mathematics . Two examples are the Sieve of Eratosthenes , which was described in the Introduction to Arithmetic by Nicomachus , and the Euclidean algorithm , which

2397-502: The type of pointer to member function is different from pointer to function . Instead, pointers to static functions have to be used. Another common workaround is using a functor . C++ wrappers for GSL are available. Not all of these are regularly maintained. They do offer access to matrix and vector classes without having to use GSL's interface to malloc and free functions. Some also offer support for also creating workspaces that behave like Smart pointer classes. Finally, there

2448-429: The year 2008). And as of the end of 2017, the book had over 44000 citations on Google Scholar . The first publication was in 1986 with the title,”Numerical Recipes, The Art of Scientific Computing”, containing code in both Fortran and Pascal; an accompanying book, “Numerical Recipes Example Book (Pascal)” was first published in 1985. (A preface note in “Examples" mentions that the main book was also published in 1985, but

2499-449: Was deemed patentable. The patenting of software is controversial, and there are criticized patents involving algorithms, especially data compression algorithms, such as Unisys 's LZW patent . Additionally, some cryptographic algorithms have export restrictions (see export of cryptography ). Another way of classifying algorithms is by their design methodology or paradigm . Some common paradigms are: For optimization problems there

2550-692: Was first described in Euclid's Elements ( c.  300 BC ). Examples of ancient Indian mathematics included the Shulba Sutras , the Kerala School , and the Brāhmasphuṭasiddhānta . The first cryptographic algorithm for deciphering encrypted code was developed by Al-Kindi , a 9th-century Arab mathematician, in A Manuscript On Deciphering Cryptographic Messages . He gave the first description of cryptanalysis by frequency analysis ,

2601-780: Was released in May 2024. The following example program calculates the value of the Bessel function of the first kind and order zero for 5: The example program has to be linked to the GSL library upon compilation: The output is shown below and should be correct to double-precision accuracy: The software library provides facilities for: Since the GSL is written in C, it is straightforward to provide wrappers for other programming languages. Such wrappers currently exist for The GSL can be used in C++ classes, but not using pointers to member functions, because

SECTION 50

#1732797310861
#860139