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If the inputs are all non-negative, then the condition number is 1. Note that the denominator is effectively 1 in practice, since is much smaller than 1 until n becomes of order 2 1/ε, which is roughly 10 10 15 in double precision.
2.1 Low-order polylogarithms. ... 7.2 Sum of reciprocal of factorials. 7.3 Trigonometry and ... The first few values are:
It is always true that the left-hand side is at most the right-hand side (max–min inequality) but equality only holds under certain conditions identified by minimax theorems. The first theorem in this sense is von Neumann's minimax theorem about two-player zero-sum games published in 1928, [2] which is considered the starting point of game ...
Sum of sets The Minkowski sum of two sets A {\displaystyle A} and B {\displaystyle B} of real numbers is the set A + B := { a + b : a ∈ A , b ∈ B } {\displaystyle A+B~:=~\{a+b:a\in A,b\in B\}} consisting of all possible arithmetic sums of pairs of numbers, one from each set.
exists and is finite (Titchmarsh 1948, §1.15). The value of this limit, should it exist, is the (C, α) sum of the integral. 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
Because the sum in the second line has only eleven 1's after the decimal, the difference when 1 is subtracted from this displayed value is three 0's followed by a string of eleven 1's. However, the difference reported by Excel in the third line is three 0's followed by a string of thirteen 1's and two extra erroneous digits.
Let A be the sum of the negative values and B the sum of the positive values; the number of different possible sums is at most B-A, so the total runtime is in (()). For example, if all input values are positive and bounded by some constant C , then B is at most N C , so the time required is O ( N 2 C ) {\displaystyle O(N^{2}C)} .
For example, for the array of values [−2, 1, −3, 4, −1, 2, 1, −5, 4], the contiguous subarray with the largest sum is [4, −1, 2, 1], with sum 6. Some properties of this problem are: If the array contains all non-negative numbers, then the problem is trivial; a maximum subarray is the entire array.