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Pairwise summation is the default summation algorithm in NumPy [9] and the Julia technical-computing language, [10] where in both cases it was found to have comparable speed to naive summation (thanks to the use of a large base case).
Unless Newton's method is initialized at the exact root 0, it is seen that the sequence of iterates will fail to converge. For example, even if initialized at the reasonably accurate guess of 0.001, the first several iterates are −0.002, 0.004, −0.008, 0.016, reaching 1048.58, −2097.15, ... by the 20th iterate.
Indeterminate form is a mathematical expression that can obtain any value depending on circumstances. In calculus, it is usually possible to compute the limit of the sum, difference, product, quotient or power of two functions by taking the corresponding combination of the separate limits of each respective function.
A running total or rolling total is the summation of a sequence of numbers which is updated each time a new number is added to the sequence, by adding the value of the new number to the previous running total. Another term for it is partial sum. The purposes of a running total are twofold.
Condition numbers can also be defined for nonlinear functions, and can be computed using calculus.The condition number varies with the point; in some cases one can use the maximum (or supremum) condition number over the domain of the function or domain of the question as an overall condition number, while in other cases the condition number at a particular point is of more interest.
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 ...
Analogously to the case of the sum of a series, if α = 0, the result is convergence of the improper integral. In the case α = 1 , (C, 1) convergence is equivalent to the existence of the limit lim λ → ∞ 1 λ ∫ 0 λ ∫ 0 x f ( y ) d y d x {\displaystyle \lim _{\lambda \to \infty }{\frac {1}{\lambda }}\int _{0}^{\lambda }\int _{0}^{x}f ...
Express each term of the final sequence y 0, y 1, y 2, ... as the sum of up to two terms of these intermediate sequences: y 0 = x 0, y 1 = z 0, y 2 = z 0 + x 2, y 3 = w 1, etc. After the first value, each successive number y i is either copied from a position half as far through the w sequence, or is the previous value added to one value in the ...