![]() ![]() Pi is computed using the Chudnovsky formula.Binary splitting is used when applicable to subdivide computations of constants.Values of constants below a certain size are cached once computed.Log and inverse trigonometric functions use Taylor series and functional relations.Exponential and trigonometric functions use Taylor series, stable recursion by argument doubling, and functional relations.Bernoulli polynomials are built out of Bernoulli numbers.Bernoulli numbers are computed using recurrence equations or the Fillebrown algorithm depending on the number's index.ClebschGordan and related functions use generalized hypergeometric series.PartitionsP uses Euler's pentagonal formula for small and the non-recursive Hardy –Ramanujan –Rademacher method for larger.Fibonacci uses an iterative method based on the binary digit sequence of.n! uses an algorithm of Sch önhage based on dynamic decomposition to prime powers.Binomial and related functions use a divide-and-conquer algorithm to balance the number of digits in subproducts.Most combinatorial functions use sparse caching and recursion.FromContinuedFraction uses iterated matrix multiplication optimized by a divide-and-conquer method.ContinuedFraction uses recurrence relations to find periodic continued fractions for quadratic irrationals.To find a requested number of terms, ContinuedFraction uses a modification of Lehmer's indirect method, with a self-restarting divide-and-conquer algorithm to reduce the numerical precision required at each step.LatticeReduce uses Storjohann's variant of the Lenstra –Lenstra –Lovasz lattice reduction algorithm.For large, the Lagarias –Miller –Odlyzko algorithm for PrimePi is used, based on asymptotic estimates of the density of primes, and is inverted to give Prime. Prime and PrimePi use sparse caching and sieving.FactorInteger switches between trial division, Pollard, Pollard rho, elliptic curve, and quadratic sieve algorithms.It can return an explicit certificate of primality. ![]() The Primality Proving Package contains a much slower algorithm that has been proved correct for all.There are no known composite numbers that pass this procedure.PrimeQ first tests for divisibility using small primes, then uses the Miller –Rabin strong pseudoprime test base 2 and base 3, and then uses a Lucas test.GCD interleaves the HGCD algorithm, the Jebelean –Sorenson –Weber accelerated GCD algorithm, and a combination of Euclid's algorithm and an algorithm based on iterative removal of powers of 2.The default pseudorandom number generator for functions like RandomReal and RandomInteger uses a cellular automaton-based algorithm.Significance arithmetic is used by default for all arithmetic with approximate numbers not equal to machine precision.Exact roots start from numerical estimates.Reciprocals and rational powers of approximate numbers use Newton's method.Integer powers are found by a left-right binary decomposition algorithm.Machine-code optimization for specific architectures is achieved by using GMP.Multiplication of large integers and high-precision approximate numbers is done using interleaved schoolbook, Karatsuba, three-way Toom –Cook, and number-theoretic transform algorithms.Floor, Ceiling, and related functions use an adaptive procedure similar to N to generate exact results from exact input.N uses an adaptive procedure to increase its internal working precision in order to achieve whatever overall precision is requested.Similar algorithms are used for number input and output. IntegerDigits, RealDigits, and related base conversion functions use recursive divide-and-conquer algorithms.Precision is internally maintained as a floating-point number.Large integers and high-precision approximate numbers are stored as arrays of base or digits, depending on the lengths of machine integers.The series.Numerical and Related Functions Number Representation and Numerical Evaluation The labor involved, but is often helpful in recognizing the general term of In what follows, we will use the summation notation, which not only reduces Return to the main page for the course APMA0340 Return to the main page for the course APMA0330 Return to Mathematica tutorial for the second course APMA0340 Return to computing page for the second course APMA0340 Return to computing page for the first course APMA0330 Laplace transform of discontinuous functions.Series solutions for the second order equations.Part IV: Second and Higher Order Differential Equations.Numerical solution using DSolve and NDSolve.Equations reducible to the separable equations. ![]()
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