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The direct sum of matrices is a special type of block matrix. In particular, the direct sum of square matrices is a block diagonal matrix. The adjacency matrix of the union of disjoint graphs (or multigraphs) is the direct sum of their adjacency matrices. Any element in the direct sum of two vector spaces of matrices can be represented as a ...
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).
Graphs of functions commonly used in the analysis of algorithms, showing the number of operations versus input size for each function. The following tables list the computational complexity of various algorithms for common mathematical operations.
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 ...
Conversely, in an eager language the above definition for ifThenElse a b c would evaluate (a), (b), and (c) regardless of the value of (a). This is not the desired behavior, as (b) or (c) may have side effects, take a long time to compute, or throw errors. It is usually possible to introduce user-defined lazy control structures in eager ...
The computer may also offer facilities for splitting a product into a digit and carry without requiring the two operations of mod and div as in the example, and nearly all arithmetic units provide a carry flag which can be exploited in multiple-precision addition and subtraction. This sort of detail is the grist of machine-code programmers, and ...
In mathematical analysis, Cesàro summation (also known as the Cesàro mean [1] [2] or Cesàro limit [3]) assigns values to some infinite sums that are not necessarily convergent in the usual sense. The Cesàro sum is defined as the limit, as n tends to infinity, of the sequence of arithmetic means of the first n partial sums of the series.
The following is an incomplete list of some arbitrary-precision arithmetic libraries for C++ ... MPIR [4] TTMath [5] Arbitrary Precision Math C++ Package [6] Class ...