"6202"^^ . . . . . "Gauss\u2013Legendre algorithm"@en . . "\u30AC\u30A6\u30B9\uFF1D\u30EB\u30B8\u30E3\u30F3\u30C9\u30EB\u306E\u30A2\u30EB\u30B4\u30EA\u30BA\u30E0"@ja . "1115360431"^^ . "Algoritmo di Gauss-Legendre"@it . . . "\u30AC\u30A6\u30B9\uFF1D\u30EB\u30B8\u30E3\u30F3\u30C9\u30EB\u306E\u30A2\u30EB\u30B4\u30EA\u30BA\u30E0\uFF08\u82F1\u8A9E: Gauss\u2013Legendre algorithm\uFF09\u306F\u3001\u5186\u5468\u7387\u3092\u8A08\u7B97\u3059\u308B\u969B\u306B\u7528\u3044\u3089\u308C\u308B\u6570\u5B66\u306E\u53CD\u5FA9\u8A08\u7B97\u30A2\u30EB\u30B4\u30EA\u30BA\u30E0\u3067\u3042\u308B\u3002\u5186\u5468\u7387\u3092\u8A08\u7B97\u3059\u308B\u3082\u306E\u306E\u4E2D\u3067\u306F\u975E\u5E38\u306B\u53CE\u675F\u304C\u901F\u304F\u30012009\u5E74\u306B\u3053\u306E\u5F0F\u3092\u7528\u3044\u30662,576,980,370,000\u6841\uFF08\u7D042\u51466000\u5104\u6841\uFF09\u306E\u8A08\u7B97\u304C\u306A\u3055\u308C\u305F\u3002 \u3053\u306E\u30A2\u30EB\u30B4\u30EA\u30BA\u30E0\u306F\u30AB\u30FC\u30EB\u30FB\u30D5\u30EA\u30FC\u30C9\u30EA\u30D2\u30FB\u30AC\u30A6\u30B9\u3068\u30A2\u30C9\u30EA\u30A2\u30F3\uFF1D\u30DE\u30EA\u30FB\u30EB\u30B8\u30E3\u30F3\u30C9\u30EB\u304C\u305D\u308C\u305E\u308C\u5225\u500B\u306B\u7814\u7A76\u3057\u305F\u3082\u306E\u3067\u3042\u308B\u3002\u3053\u308C\u306F2\u3064\u306E\u6570\u5024\u306E\u7B97\u8853\u5E7E\u4F55\u5E73\u5747\u3092\u6C42\u3081\u308B\u305F\u3081\u306B\u3001\u305D\u308C\u305E\u308C\u306E\u6570\u5024\u3092\u7B97\u8853\u5E73\u5747\uFF08\u76F8\u52A0\u5E73\u5747\uFF09\u3068\u5E7E\u4F55\u5E73\u5747\uFF08\u76F8\u4E57\u5E73\u5747\uFF09\u3067\u7F6E\u304D\u63DB\u3048\u3066\u3044\u304F\u3082\u306E\u3067\u3042\u308B\u3002"@ja . . . . . . "L'algoritmo di Gauss\u2013Legendre \u00E8 un algoritmo per il calcolo di \u03C0. \u00C8 noto per essere rapidamente convergente, 25 iterazioni producono ben 45 milioni di cifre decimali corrette di \u03C0. L'inconveniente \u00E8 un intensivo uso di memoria. Il metodo \u00E8 basato sui lavori di Gauss e Legendre unitamente ai moderni algoritmi per la moltiplicazione e l'estrazione di radice quadrata. Si basa sulla continua sostituzione di due numeri con la loro media aritmetica e geometrica per approssimare la loro media aritmetica-geometrica."@it . . "Het algoritme van Gauss-Legendre is een algoritme om de cijfers van het getal pi te berekenen. Het is genoemd naar de wiskundigen Carl Friedrich Gauss en Adrien-Marie Legendre die het theoretisch uitwerkten, maar toen nog geen computers hadden om het toe te passen. Het is soms ook bekend als algoritme van Brent-Salamin naar en die dit in 1975 toepasten. Van 18 tot 20 september 1999 werden hiermee 206 miljard decimalen van pi berekend en het resultaat werd vergeleken met het algoritme van Borwein. Begin: Herhaal: ,,,. Einde: \u03C0 is dan bij benadering: . De eerste drie benaderingen leveren:"@nl . . . . . . "\u9AD8\u65AF-\u52D2\u8BA9\u5FB7\u7B97\u6CD5\u662F\u4E00\u79CD\u7528\u4E8E\u8BA1\u7B97\u5706\u5468\u7387\uFF08\u03C0\uFF09\u7684\u7B97\u6CD5\u3002\u5B83\u4EE5\u8FC5\u901F\u6536\u655B\u8457\u79F0\uFF0C\u53EA\u970025\u6B21\u8FED\u4EE3\u5373\u53EF\u4EA7\u751F\u03C0\u76844500\u4E07\u4F4D\u6B63\u786E\u6570\u5B57\u3002\u4E0D\u8FC7\uFF0C\u5B83\u7684\u7F3A\u70B9\u662F\u5185\u5B58\u5BC6\u96C6\uFF0C\u56E0\u6B64\u6709\u65F6\u5B83\u4E0D\u5982\u6885\u94A6\u7C7B\u516C\u5F0F\u4F7F\u7528\u5E7F\u6CDB\u3002 \u8BE5\u65B9\u6CD5\u57FA\u4E8E\u5FB7\u570B\u6578\u5B78\u5BB6\u5361\u5C14\u00B7\u5F17\u91CC\u5FB7\u91CC\u5E0C\u00B7\u9AD8\u65AF\uFF08Johann Carl Friedrich Gau\u00DF\uFF0C1777\u20131855\uFF09\u548C\u6CD5\u570B\u6578\u5B78\u5BB6\u963F\u5FB7\u91CC\u5B89-\u9A6C\u91CC\u00B7\u52D2\u8BA9\u5FB7\uFF08Adrien-Marie Legendre\uFF0C1752\u20131833\uFF09\u7684\u4E2A\u4EBA\u6210\u679C\u4E0E\u4E58\u6CD5\u548C\u5E73\u65B9\u6839\u8FD0\u7B97\u4E4B\u73B0\u4EE3\u7B97\u6CD5\u7684\u7ED3\u5408\u3002\u8BE5\u7B97\u6CD5\u53CD\u590D\u66FF\u6362\u4E24\u4E2A\u6570\u503C\u7684\u7B97\u672F\u5E73\u5747\u6570\u548C\u51E0\u4F55\u5E73\u5747\u6570\uFF0C\u4EE5\u63A5\u8FD1\u5B83\u4EEC\u7684\u7B97\u672F-\u51E0\u4F55\u5E73\u5747\u6570\u3002 \u4E0B\u6587\u7684\u7248\u672C\u4E5F\u88AB\u79F0\u4E3A\u9AD8\u65AF-\u6B27\u62C9\uFF0C\u5E03\u4F26\u7279-\u8428\u62C9\u660E\uFF08\u6216\u8428\u62C9\u660E-\u5E03\u4F26\u7279\uFF09\u7B97\u6CD5\uFF1B\u5B83\u4E8E1975\u5E74\u88AB\u548C\u72EC\u7ACB\u53D1\u73B0\u3002\u65E5\u672C\u7B51\u6CE2\u5927\u5B66\u4E8E2009\u5E748\u670817\u65E5\u5BA3\u5E03\u5229\u7528\u6B64\u7B97\u6CD5\u8BA1\u7B97\u51FA\u03C0\u5C0F\u6570\u70B9\u540E2,576,980,370,000\u4F4D\u6570\u5B57\uFF0C\u8BA1\u7B97\u7ED3\u679C\u7528\u68C0\u9A8C\u3002 \u77E5\u540D\u7684\u7535\u8111\u6027\u80FD\u6D4B\u8BD5\u7A0B\u5E8FSuper PI\u4E5F\u4F7F\u7528\u6B64\u7B97\u6CD5\u3002"@zh . . . . "Algoritmo de Gauss-Legendre"@pt . . "\u30AC\u30A6\u30B9\uFF1D\u30EB\u30B8\u30E3\u30F3\u30C9\u30EB\u306E\u30A2\u30EB\u30B4\u30EA\u30BA\u30E0\uFF08\u82F1\u8A9E: Gauss\u2013Legendre algorithm\uFF09\u306F\u3001\u5186\u5468\u7387\u3092\u8A08\u7B97\u3059\u308B\u969B\u306B\u7528\u3044\u3089\u308C\u308B\u6570\u5B66\u306E\u53CD\u5FA9\u8A08\u7B97\u30A2\u30EB\u30B4\u30EA\u30BA\u30E0\u3067\u3042\u308B\u3002\u5186\u5468\u7387\u3092\u8A08\u7B97\u3059\u308B\u3082\u306E\u306E\u4E2D\u3067\u306F\u975E\u5E38\u306B\u53CE\u675F\u304C\u901F\u304F\u30012009\u5E74\u306B\u3053\u306E\u5F0F\u3092\u7528\u3044\u30662,576,980,370,000\u6841\uFF08\u7D042\u51466000\u5104\u6841\uFF09\u306E\u8A08\u7B97\u304C\u306A\u3055\u308C\u305F\u3002 \u3053\u306E\u30A2\u30EB\u30B4\u30EA\u30BA\u30E0\u306F\u30AB\u30FC\u30EB\u30FB\u30D5\u30EA\u30FC\u30C9\u30EA\u30D2\u30FB\u30AC\u30A6\u30B9\u3068\u30A2\u30C9\u30EA\u30A2\u30F3\uFF1D\u30DE\u30EA\u30FB\u30EB\u30B8\u30E3\u30F3\u30C9\u30EB\u304C\u305D\u308C\u305E\u308C\u5225\u500B\u306B\u7814\u7A76\u3057\u305F\u3082\u306E\u3067\u3042\u308B\u3002\u3053\u308C\u306F2\u3064\u306E\u6570\u5024\u306E\u7B97\u8853\u5E7E\u4F55\u5E73\u5747\u3092\u6C42\u3081\u308B\u305F\u3081\u306B\u3001\u305D\u308C\u305E\u308C\u306E\u6570\u5024\u3092\u7B97\u8853\u5E73\u5747\uFF08\u76F8\u52A0\u5E73\u5747\uFF09\u3068\u5E7E\u4F55\u5E73\u5747\uFF08\u76F8\u4E57\u5E73\u5747\uFF09\u3067\u7F6E\u304D\u63DB\u3048\u3066\u3044\u304F\u3082\u306E\u3067\u3042\u308B\u3002"@ja . . . . . "El algoritmo de Gauss-Legendre es un algoritmo para computar los d\u00EDgitos de \u03C0. El m\u00E9todo se basa en los trabajos individuales de Carl Friedrich Gauss (1777-1855) y Adrien-Marie Legendre (1752-1833) combinados con algoritmos modernos para la multiplicaci\u00F3n y la ra\u00EDz cuadrada. Sustituye repetidamente dos n\u00FAmeros por sus medias aritm\u00E9tica y geom\u00E9trica, para obtener una aproximaci\u00F3n a su media aritm\u00E9tico-geom\u00E9trica."@es . . "O algoritmo de Gauss-Legendre \u00E9 um algoritmo para calcular os d\u00EDgitos de \u03C0. \u00C9 not\u00E1vel por ser rapidamente convergente, com 25 itera\u00E7\u00F5es produz 45 milh\u00F5es de d\u00EDgitos corretos do \u03C0. Entretanto, o inconveniente \u00E9 que usa muita mem\u00F3ria e consequentemente n\u00E3o \u00E9 usado em f\u00F3rmulas como a F\u00F3rmula de Machin. O m\u00E9todo \u00E9 baseado no trabalho individual de Carl Friedrich Gauss (1779-1815) e Adrien-Marie Legendre (1799-1855) combinado com os algoritmos modernos para multiplica\u00E7\u00E3o e ra\u00EDzes quadradas. Substitui repetidamente dois n\u00FAmeros pela sua m\u00E9dia aritm\u00E9tica e pela sua m\u00E9dia geom\u00E9trica, a fim de aproximar a sua m\u00E9dia aritm\u00E9tica-geom\u00E9trica."@pt . . . "El algoritmo de Gauss-Legendre es un algoritmo para computar los d\u00EDgitos de \u03C0. El m\u00E9todo se basa en los trabajos individuales de Carl Friedrich Gauss (1777-1855) y Adrien-Marie Legendre (1752-1833) combinados con algoritmos modernos para la multiplicaci\u00F3n y la ra\u00EDz cuadrada. Sustituye repetidamente dos n\u00FAmeros por sus medias aritm\u00E9tica y geom\u00E9trica, para obtener una aproximaci\u00F3n a su media aritm\u00E9tico-geom\u00E9trica. La versi\u00F3n que se presenta aqu\u00ED se conoce tambi\u00E9n como el algoritmo de Brent-Salamin (o Salamin-Brent); que fue descubierto en 1975 y de forma independiente por y . Se us\u00F3 entre el 18 y el 20 de septiembre de 1999 para calcular los primeros 206.158.430.000 d\u00EDgitos decimales de \u03C0, y el resultado se comprob\u00F3 usando el algoritmo de Borwein."@es . . "\u9AD8\u65AF-\u52D2\u8BA9\u5FB7\u7B97\u6CD5"@zh . . "Het algoritme van Gauss-Legendre is een algoritme om de cijfers van het getal pi te berekenen. Het is genoemd naar de wiskundigen Carl Friedrich Gauss en Adrien-Marie Legendre die het theoretisch uitwerkten, maar toen nog geen computers hadden om het toe te passen. Het is soms ook bekend als algoritme van Brent-Salamin naar en die dit in 1975 toepasten. Van 18 tot 20 september 1999 werden hiermee 206 miljard decimalen van pi berekend en het resultaat werd vergeleken met het algoritme van Borwein. Begin: Herhaal: ,,,. Einde: \u03C0 is dan bij benadering: . De eerste drie benaderingen leveren: Het aantal juiste cijfers verdubbelt met elke stap. Het algoritme vraagt wel veel geheugen."@nl . . . "Algoritmo de Gauss-Legendre"@es . . "\u9AD8\u65AF-\u52D2\u8BA9\u5FB7\u7B97\u6CD5\u662F\u4E00\u79CD\u7528\u4E8E\u8BA1\u7B97\u5706\u5468\u7387\uFF08\u03C0\uFF09\u7684\u7B97\u6CD5\u3002\u5B83\u4EE5\u8FC5\u901F\u6536\u655B\u8457\u79F0\uFF0C\u53EA\u970025\u6B21\u8FED\u4EE3\u5373\u53EF\u4EA7\u751F\u03C0\u76844500\u4E07\u4F4D\u6B63\u786E\u6570\u5B57\u3002\u4E0D\u8FC7\uFF0C\u5B83\u7684\u7F3A\u70B9\u662F\u5185\u5B58\u5BC6\u96C6\uFF0C\u56E0\u6B64\u6709\u65F6\u5B83\u4E0D\u5982\u6885\u94A6\u7C7B\u516C\u5F0F\u4F7F\u7528\u5E7F\u6CDB\u3002 \u8BE5\u65B9\u6CD5\u57FA\u4E8E\u5FB7\u570B\u6578\u5B78\u5BB6\u5361\u5C14\u00B7\u5F17\u91CC\u5FB7\u91CC\u5E0C\u00B7\u9AD8\u65AF\uFF08Johann Carl Friedrich Gau\u00DF\uFF0C1777\u20131855\uFF09\u548C\u6CD5\u570B\u6578\u5B78\u5BB6\u963F\u5FB7\u91CC\u5B89-\u9A6C\u91CC\u00B7\u52D2\u8BA9\u5FB7\uFF08Adrien-Marie Legendre\uFF0C1752\u20131833\uFF09\u7684\u4E2A\u4EBA\u6210\u679C\u4E0E\u4E58\u6CD5\u548C\u5E73\u65B9\u6839\u8FD0\u7B97\u4E4B\u73B0\u4EE3\u7B97\u6CD5\u7684\u7ED3\u5408\u3002\u8BE5\u7B97\u6CD5\u53CD\u590D\u66FF\u6362\u4E24\u4E2A\u6570\u503C\u7684\u7B97\u672F\u5E73\u5747\u6570\u548C\u51E0\u4F55\u5E73\u5747\u6570\uFF0C\u4EE5\u63A5\u8FD1\u5B83\u4EEC\u7684\u7B97\u672F-\u51E0\u4F55\u5E73\u5747\u6570\u3002 \u4E0B\u6587\u7684\u7248\u672C\u4E5F\u88AB\u79F0\u4E3A\u9AD8\u65AF-\u6B27\u62C9\uFF0C\u5E03\u4F26\u7279-\u8428\u62C9\u660E\uFF08\u6216\u8428\u62C9\u660E-\u5E03\u4F26\u7279\uFF09\u7B97\u6CD5\uFF1B\u5B83\u4E8E1975\u5E74\u88AB\u548C\u72EC\u7ACB\u53D1\u73B0\u3002\u65E5\u672C\u7B51\u6CE2\u5927\u5B66\u4E8E2009\u5E748\u670817\u65E5\u5BA3\u5E03\u5229\u7528\u6B64\u7B97\u6CD5\u8BA1\u7B97\u51FA\u03C0\u5C0F\u6570\u70B9\u540E2,576,980,370,000\u4F4D\u6570\u5B57\uFF0C\u8BA1\u7B97\u7ED3\u679C\u7528\u68C0\u9A8C\u3002 \u77E5\u540D\u7684\u7535\u8111\u6027\u80FD\u6D4B\u8BD5\u7A0B\u5E8FSuper PI\u4E5F\u4F7F\u7528\u6B64\u7B97\u6CD5\u3002"@zh . . "12916"^^ . "La formule de Brent-Salamin est une formule donnant une bonne approximation de \u03C0. La formule fut trouv\u00E9e ind\u00E9pendamment par Richard P. Brent et (en) en 1976. Elle exploite les liens entre les int\u00E9grales elliptiques et la moyenne arithm\u00E9tico-g\u00E9om\u00E9trique ; sa d\u00E9monstration aurait \u00E9t\u00E9 connue de Gauss, mais la mise en \u0153uvre d'une telle formule est tr\u00E8s difficile sans ordinateur personnel et arithm\u00E9tique multipr\u00E9cision. On l'appelle \u00E9galement la m\u00E9thode de Gauss-Legendre."@fr . . "Algoritme van Gauss-Legendre"@nl . "The Gauss\u2013Legendre algorithm is an algorithm to compute the digits of \u03C0. It is notable for being rapidly convergent, with only 25 iterations producing 45 million correct digits of \u03C0. However, it has some drawbacks (for example, it is computer memory-intensive) and therefore all record-breaking calculations for many years have used other methods, almost always the Chudnovsky algorithm. For details, see Chronology of computation of \u03C0."@en . . . . . . "L'algoritmo di Gauss\u2013Legendre \u00E8 un algoritmo per il calcolo di \u03C0. \u00C8 noto per essere rapidamente convergente, 25 iterazioni producono ben 45 milioni di cifre decimali corrette di \u03C0. L'inconveniente \u00E8 un intensivo uso di memoria. Il metodo \u00E8 basato sui lavori di Gauss e Legendre unitamente ai moderni algoritmi per la moltiplicazione e l'estrazione di radice quadrata. Si basa sulla continua sostituzione di due numeri con la loro media aritmetica e geometrica per approssimare la loro media aritmetica-geometrica."@it . . . . . . . . "O algoritmo de Gauss-Legendre \u00E9 um algoritmo para calcular os d\u00EDgitos de \u03C0. \u00C9 not\u00E1vel por ser rapidamente convergente, com 25 itera\u00E7\u00F5es produz 45 milh\u00F5es de d\u00EDgitos corretos do \u03C0. Entretanto, o inconveniente \u00E9 que usa muita mem\u00F3ria e consequentemente n\u00E3o \u00E9 usado em f\u00F3rmulas como a F\u00F3rmula de Machin."@pt . . . . "La formule de Brent-Salamin est une formule donnant une bonne approximation de \u03C0. La formule fut trouv\u00E9e ind\u00E9pendamment par Richard P. Brent et (en) en 1976. Elle exploite les liens entre les int\u00E9grales elliptiques et la moyenne arithm\u00E9tico-g\u00E9om\u00E9trique ; sa d\u00E9monstration aurait \u00E9t\u00E9 connue de Gauss, mais la mise en \u0153uvre d'une telle formule est tr\u00E8s difficile sans ordinateur personnel et arithm\u00E9tique multipr\u00E9cision. On l'appelle \u00E9galement la m\u00E9thode de Gauss-Legendre."@fr . . . "The Gauss\u2013Legendre algorithm is an algorithm to compute the digits of \u03C0. It is notable for being rapidly convergent, with only 25 iterations producing 45 million correct digits of \u03C0. However, it has some drawbacks (for example, it is computer memory-intensive) and therefore all record-breaking calculations for many years have used other methods, almost always the Chudnovsky algorithm. For details, see Chronology of computation of \u03C0. The method is based on the individual work of Carl Friedrich Gauss (1777\u20131855) and Adrien-Marie Legendre (1752\u20131833) combined with modern algorithms for multiplication and square roots. It repeatedly replaces two numbers by their arithmetic and geometric mean, in order to approximate their arithmetic-geometric mean. The version presented below is also known as the Gauss\u2013Euler, Brent\u2013Salamin (or Salamin\u2013Brent) algorithm; it was independently discovered in 1975 by Richard Brent and Eugene Salamin. It was used to compute the first 206,158,430,000 decimal digits of \u03C0 on September 18 to 20, 1999, and the results were checked with Borwein's algorithm."@en . . . . "Formule de Brent-Salamin"@fr .