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Arbitrary-precision arithmetic can also be used to avoid overflow, which is an inherent limitation of fixed-precision arithmetic. Similar to an automobile's odometer display which may change from 99999 to 00000, a fixed-precision integer may exhibit wraparound if numbers grow too large to represent at the fixed level of precision.
Qalculate! supports common mathematical functions and operations, multiple bases, autocompletion, complex numbers, infinite numbers, arrays and matrices, variables, mathematical and physical constants, user-defined functions, symbolic derivation and integration, solving of equations involving unknowns, uncertainty propagation using interval arithmetic, plotting using Gnuplot, unit and currency ...
Hilbert matrix — example of a matrix which is extremely ill-conditioned (and thus difficult to handle) Wilkinson matrix — example of a symmetric tridiagonal matrix with pairs of nearly, but not exactly, equal eigenvalues; Convergent matrix — square matrix whose successive powers approach the zero matrix; Algorithms for matrix multiplication:
Routines for Gauss–Kronrod quadrature are provided by the QUADPACK library, the GNU Scientific Library, the NAG Numerical Libraries, R, [2] the C++ library Boost., [3] as well as the Julia package QuadGK.jl [4] (which can compute Gauss–Kronrod formulas to arbitrary precision).
A variant of the spigot approach uses an algorithm which can be used to compute a single arbitrary digit of the transcendental without computing the preceding digits: an example is the Bailey–Borwein–Plouffe formula, a digit extraction algorithm for π which produces base 16 digits. The inevitable truncation of the underlying infinite ...
PARI/GP performs arbitrary precision calculations (e.g., the significand can be millions of digits long—and billions of digits on 64-bit machines). It can compute factorizations, perform elliptic curve computations and perform algebraic number theory calculations.
Programming languages that support arbitrary precision computations, either built-in, or in the standard library of the language: Ada: the upcoming Ada 202x revision adds the Ada.Numerics.Big_Numbers.Big_Integers and Ada.Numerics.Big_Numbers.Big_Reals packages to the standard library, providing arbitrary precision integers and real numbers.
Another alternative is to use arbitrary-precision arithmetic, which in principle need no rounding at all with a cost of much greater computational effort. A way of performing correctly rounded sums using arbitrary precision is to extend adaptively using multiple floating-point components.