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WOLFRAMALPHA POWER SERIES SERIES
You can compute power series at a symbolic variable point to find a formula for the derivative, so there is theoretically no reason why Wolfram should be so much slower at finding n-th derivatives.Wolfram Data Framework Semantic framework for real-world data. Note that computing a power series at a point z=z0 is the same as finding all derivatives at z=z0, so the two problems are equivalent. This gives you a O(nk) time algorithm to find the first k terms in the power series of 1/p(z). From a state you can compute the next term in the power series and the next state in time O(n). You can store the coefficients of those terms in an array and use that as your state. When you do this for a general polynomial of degree n you always have a numerator with less than n terms. Going on like that you find 1/(1 - z^2) = 1 + z^2 + z^4 + z^6 and so on. The algorithm is basically the same as long division for polynomials. You can extend this to division by any polynomial. So if f_n is the series for f, then the series for g(z) = f(z)/(1 - z) is given by g_n = f_n + g_(n-1). If f(z) has a power series, then coefficients of the power series of f(z)/(1 - z) are simply the partial sums of the power series of f(z).
In special cases there are more efficient methods. Using these methods you can compute the power series of practically any function in polynomial time. Given power series for f,g you can find a recurrence relation for power series of f(z)+g(z), f(z)g(z), f(z)/g(z), f(g(z)), and even f^-1(z). You can do each operation on formal power series. In particular, a dynamic programming approach would be able to calculate coefficients in linear time and space. This is a restricted version of the partition problem, but the same ideas would still hold. In the end, the coefficient of x N is the number of ways that N can be written as a sum of 5s, 7s, and 11s.
For example, Wolfram shows that the coefficient of x 40 is 4, which is from (x 5♱)(x 7♵)(x 0♱1)+(x 5♰)(x 7♱)(x 11♳)+(x 5♳)(x 7♲)(x 11♱)+(x 5♸)(x 7♰)(x 11♰).īut if we only need to add the exponents, then we don’t need to care about the coefficients or the variable x. But all of the coefficients are 1, so we only need to look at the exponents, which add together. But we can replace x with x n, and the relation will still hold. InverseSeries can be applied to any SeriesData object with the appropriate structure, whether or not it has been generated by Series. Given a series, InverseSeries s, x gives a series for such that. I’m not sure how Wolfram does it, but for this function, it is possible to more efficiently compute the coefficients (using techniques you would see at the end of your first year in calculus). InverseSeries performs 'reversion' of series. Tl dr: The coefficient of x N is the number of ways that N can be partitioned using only 5, 7, and 11.
WOLFRAMALPHA POWER SERIES CODE
Suppose my code is inefficient, but still it does not seem like Wolfram is using the same technique they show you if the first year calculus class.įor comparison Wolfram takes quite a bit more time to just compute tenth derivative of the same function than it takes to get the first 1000 terms of polynomial, which, if done naively, would require differentiating the function 1000 times. This program crashes with exception after computing first seven, or so, coefficients, because numerator and denominator of the fraction become enormously long polynomials. Print("Next coefficient is: " + function(0)/factorial) In pseudocode it would be something like: BigInt next=1 Out of curiosity I wrote a very naive java program to do the same using BigInteger for polynomial coefficients. WolframAlpha computes first 1000 elements of the MacLaurin series expansion in a few seconds:
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