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    Padé Approximant

    Continued fraction definition

    Given a function f with associated Taylor series A(x) = sum_(j = 0)^∞ a_j x^j, the Padé approximants to f are a collection of rational approximations devised to provide accurate estimations of f by way of matching A as long as is mathematically feasible and deviating onward in order to avoid perpetuation of error. In particular, the [L, M] Padé approximant to f is defined to be the rational function P_L(x)/Q_M(x), where P_L(x) = p_0 + p_1 x + ... + p_L x^L and Q_M(x) = q_0 + q_1 x + ... + q_M x^M are polynomials of degree at most L and M, respectively, which satisfies the asymptotic relation A(x) - P_L(x)/Q_M(x) = O(x^(L + M + 1)). This asymptotic relation uniquely determines the coefficients p_i and q_j, i = 0, 1, ..., L, j = 0, 1, ..., M, the association of which can be written out algorithmically as follows: Define a_n congruent 0 if n<0, q_j congruent 0 if j>M, and a_0 | = | p_0 a_1 + a_0 q_1 | = | p_1 a_2 + a_1 q_1 + a_0 q_2 | = | p_2 ⋮ | | ⋮ a_L + a_(L - 1) q_1 + ... + a_0 q_L | = | p_L a_(L + 1) + a_L q_1 + ... + a_(L - M + 1) q_M | = | 0 ⋮ | | ⋮ a_(L + M) + a_(L + M - 1) q_1 + ... + a_L q_M | = | 0. Note that the above procedure is what remains when the normalization assumption Q_M(0) = 1 is made; this is assumed in several modern contexts though is often omitted in classical literature on the subject.

    Details

    Padé approximant column | Padé approximant denominator | Padé approximant diagonal | Padé approximant numerator | Padé approximant row | complex rational function | complex number | complex poles

    Padé table | sequence | continued fraction | generalized continued fraction | polynomial | polynomial degree | partial denominator | partial numerator | meromorphic continued fraction | Padé approximant denominator | Padé conjecture

    Timeline

    Timeline

    References

    Oskar Perron. Die Lehre von den Kettenbrüchen. pp. 419, 420, 421, 422, and 423, 1913. (in German) Annie A.M. Cuyt, Vigdis Petersen, Brigitte Verdonk, Haakon Waadeland, and William B. Jones. Handbook of Continued Fractions for Special Functions. pp. 59, 60, 61, and 62, 2008. George Allen Baker, Jr., John L. Gammel, and John G. Willis.

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