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The ''chakravala'' method ((サンスクリット:चक्रवाल विधि)) is a cyclic algorithm to solve indeterminate quadratic equations, including Pell's equation. It is commonly attributed to Bhāskara II, (c. 1114 – 1185 CE)〔 Hoiberg & Ramchandani – Students' Britannica India: Bhaskaracharya II, page 200〕〔Kumar, page 23〕 although some attribute it to Jayadeva (c. 950 ~ 1000 CE).〔Plofker, page 474〕 Jayadeva pointed out that Brahmagupta's approach to solving equations of this type could be generalized, and he then described this general method, which was later refined by Bhāskara II in his ''Bijaganita'' treatise. He called it the Chakravala method: ''chakra'' meaning "wheel" in Sanskrit, a reference to the cyclic nature of the algorithm.〔Goonatilake, page 127 – 128〕 E. O. Selenius held that no European performances at the time of Bhāskara, nor much later, exceeded its marvellous height of mathematical complexity.〔〔 This method is also known as the cyclic method and contains traces of mathematical induction.〔Cajori (1918), p. 197 "The process of reasoning called "Mathematical Induction" has had several independent origins. It has been traced back to the Swiss Jakob (James) Bernoulli, the Frenchman B. Pascal and P. Fermat, and the Italian F. Maurolycus. () By reading a little between the lines one can find traces of mathematical induction still earlier, in the writings of the Hindus and the Greeks, as, for instance, in the "cyclic method" of Bhaskara, and in Euclid's proof that the number of primes is infinite."〕 == History == ''Chakra'' in Sanskrit means cycle. As per popular legend, Chakravala indicates a mythical range of mountains which orbits around the earth like a wall and not limited by light and darkness. Brahmagupta in 628 CE studied indeterminate quadratic equations, including Pell's equation : for minimum integers ''x'' and ''y''. Brahmagupta could solve it for several ''N'', but not all. Jayadeva (9th century) and Bhaskara (12th century) offered the first complete solution to the equation, using the ''chakravala'' method to find for the solution : This case was notorious for its difficulty, and was first solved in Europe by Brouncker in 1657–58 in response to a challenge by Fermat, using continued fractions. A method for the general problem was first completely described rigorously by Lagrange in 1766. Lagrange's method, however, requires the calculation of 21 successive convergents of the continued fraction for the square root of 61, while the ''chakravala'' method is much simpler. Selenius, in his assessment of the ''chakravala'' method, states :"The method represents a best approximation algorithm of minimal length that, owing to several minimization properties, with minimal effort and avoiding large numbers automatically produces the best solutions to the equation. The ''chakravala'' method anticipated the European methods by more than a thousand years. But no European performances in the whole field of algebra at a time much later than Bhaskara's, nay nearly equal up to our times, equalled the marvellous complexity and ingenuity of ''chakravala''."〔〔 Hermann Hankel calls the ''chakravala'' method :"the finest thing achieved in the theory of numbers before Lagrange."〔Kaye (1919), p. 337.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Chakravala method」の詳細全文を読む スポンサード リンク
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