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[1] [2] All functions use floating-point numbers in one manner or another. Different C standards provide different, albeit backwards-compatible, sets of functions. Most of these functions are also available in the C++ standard library, though in different headers (the C headers are included as well, but only as a deprecated compatibility feature).
The run-time of this algorithm is at most linear in the number of states. The number of states is at most N times the number of different possible sums. 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 (()).
The following is an incomplete list of some arbitrary-precision arithmetic libraries for C++. GMP [1] [nb 1] MPFR [3] MPIR [4] TTMath [5] Arbitrary Precision Math C++ Package [6] Class Library for Numbers; Number Theory Library; Apfloat [7] C++ Big Integer Library [8] MAPM [9] ARPREC [10] InfInt [11] Universal Numbers [12] mp++ [13] num7 [14]
LIBSVM – C++ support vector machine libraries; mlpack – open-source library for machine learning, exploits C++ language features to provide maximum performance and flexibility while providing a simple and consistent application programming interface (API) Mondrian – data analysis tool using interactive statistical graphics with a link to R
Go: the standard library package math/big implements arbitrary-precision integers (Int type), rational numbers (Rat type), and floating-point numbers (Float type) Guile: the built-in exact numbers are of arbitrary precision. Example: (expt 10 100) produces the expected (large) result. Exact numbers also include rationals, so (/ 3 4) produces 3/4.
If the inputs are all non-negative, then the condition number is 1. Note that the 1 − ε log 2 n {\displaystyle 1-\varepsilon \log _{2}n} denominator is effectively 1 in practice, since ε log 2 n {\displaystyle \varepsilon \log _{2}n} is much smaller than 1 until n becomes of order 2 1/ε , which is roughly 10 10 15 in double precision.
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The Fletcher checksum cannot distinguish between blocks of all 0 bits and blocks of all 1 bits. For example, if a 16-bit block in the data word changes from 0x0000 to 0xFFFF, the Fletcher-32 checksum remains the same. This also means a sequence of all 00 bytes has the same checksum as a sequence (of the same size) of all FF bytes.