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In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. It is also called Abel's lemma or Abel transformation , named after Niels Henrik Abel who introduced it in 1826.
For example, summation of [1, 2, 4, 2] is denoted 1 + 2 + 4 + 2, and results in 9, that is, 1 + 2 + 4 + 2 = 9. Because addition is associative and commutative, there is no need for parentheses, and the result is the same irrespective of the order of the summands. Summation of a sequence of only one summand results in the summand itself.
If a and b are integers in the range [0, N − 1], then their sum is in the range [0, 2N − 2] and their difference is in the range [−N + 1, N − 1], so determining the representative in [0, N − 1] requires at most one subtraction or addition (respectively) of N. However, the product ab is in the range [0, N 2 − 2N + 1].
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The algorithm performs summation with two accumulators: sum holds the sum, and c accumulates the parts not assimilated into sum, to nudge the low-order part of sum the next time around. Thus the summation proceeds with "guard digits" in c , which is better than not having any, but is not as good as performing the calculations with double the ...
Abel's summation formula – Integration by parts version of Abel's method for summation by parts; Nachbin resummation – Theorem bounding the growth rate of analytic functions; Summation by parts – Theorem to simplify sums of products of sequences
Abel's summation formula can be generalized to the case where is only assumed to be continuous if the integral is interpreted as a Riemann–Stieltjes integral: ∑ x < n ≤ y a n ϕ ( n ) = A ( y ) ϕ ( y ) − A ( x ) ϕ ( x ) − ∫ x y A ( u ) d ϕ ( u ) . {\displaystyle \sum _{x<n\leq y}a_{n}\phi (n)=A(y)\phi (y)-A(x)\phi (x)-\int _{x ...
The immediate future is up in the air for Jacksonville Jaguars quarterback Trevor Lawrence. According to NFL Network's Ian Rapoport, Lawrence is unlikely to play Sunday against the Minnesota ...