## The Yacas Book of Algorithms

### BibTeX

@MISC{Team_theyacas,

author = {The Yacas Team and Yacas Version and Symbolic Algebra Algorithms},

title = {The Yacas Book of Algorithms},

year = {}

}

### OpenURL

### Abstract

September 27, 2007 This book is a detailed description of the algorithms used in the Yacas system for exact symbolic and arbitrary-precision numerical computations. Very few of these algorithms are new, and most are well-known. The goal of this book is to become a compendium of all relevant issues of design and implementation of these algorithms.

### Citations

2269 |
The Art of Computer Programming
- Knuth
- 1981
(Show Context)
Citation Context ...ger powers Integer powers x n with integer n are computed by a fast algorithm of “repeated squaring”. This algorithm is well known (see, for example, the famous book, The art of computer programming [=-=Knuth 1973-=-]). The algorithm is based on the following trick: if n is even, say n = 2k, then x n = ( x k) 2 ; and if n is odd, n = 2k + 1, then x n = x ( x k) 2 n . Thus we can reduce the calculation of x to the... |

465 |
Introduction to Asymptotics and Special Functions
- Olver
- 1974
(Show Context)
Citation Context ...m of G (s) to obtain an asymptotic estimate of Qn at large k. The asymptotic growth of the sequence Qn can be estimated by the method of steepest descent, also known as Laplace’s method. (See, e.g., [=-=Olver 1974-=-], ch. 3, sec. 7.5.) This method is somewhat complicated but quite powerful. The method requires that we find an integral representation for Qn (usually a contour integral in the complex plane). Then ... |

238 |
Monte Carlo methods for index computation (mod p
- Pollard
- 1978
(Show Context)
Citation Context ...ower if and only if p is itself a prime power. If we find no integer roots of orders s ≤ s0, then n is not a prime power. If the number n is not a prime power, the Pollard “rho” algorithm is applied [=-=Pollard 1978-=-]. The Pollard “rho” algorithm 11takes an irreducible polynomial, e.g. p (x) = x 2 + 1 and builds a sequence of integers xk+1 ≡ p (xk) mod n, starting from x0 = 2. For each k, the value x2k − xk is a... |

195 |
Probabilistic algorithm for testing primality
- Rabin
- 1980
(Show Context)
Citation Context ...er words, the Miller-Rabin test could sometimes flag a large number n as prime when in fact n is composite; but the probability for this to happen can be made extremely small. The basic reference is [=-=Rabin 1980-=-]. We also implemented some of the improvements suggested in [Davenport 1992]. The idea of the Miller-Rabin algorithm is to improve the Fermat primality test. If n is prime, then for any x we have Gcd... |

117 |
Mathematical Functions and Their Approximations
- Luke
- 1975
(Show Context)
Citation Context ...y a few short multiplications and divisions. Therefore this computation costs O ( n 2) short operations. and finally + f (x) = X0q0 + X1 (q1 − A1q1) N∑ Xn (qn − Anqn−1 − Bnqn−2) = X0q0. n=2 The book [=-=Luke 1975-=-] warns that the recurrence relation for Xn is not always numerically stable. Note that in the book there seems to be some confusion as to how the coefficient A1 is defined. (It is not defined explici... |

60 | Multiple-precision zero-finding methods and the complexity of elementary function evaluation, in Analytic Computational Complexity (edited by
- Brent
- 1975
(Show Context)
Citation Context ...total cost is 2M (4P ) + M (3P ) + M (2P ) + M (P ). The asymptotic cost of finding the root x of the equation f (x) = 0 with P digits of precision is usually the same as the cost of computing f (x) [=-=Brent 1975-=-]. The main argument can be summarized by the following simple example. To get the result to P digits, we need O (ln P ) Newton’s iterations. At each iteration we shall have to compute the function f ... |

58 | A Fortran Multiple-Precision Arithmetic Package
- Brent
- 1978
(Show Context)
Citation Context ... x can be obtained through MathExpThreshold(), and set through SetMathExpThreshold(threshold) in stdfuncs. A modification of the squaring reduction allows to significantly reduce the round-off error [=-=Brent 1978-=-]. Instead of exp (x) = ( exp ( )) x 2, we use the identity 2 exp (x) − 1 = ( exp ( ) ) ( x − 1 exp 2 and reduce exp (x)−1 directly to exp ( x 2 then exp (x) − 1 = 2y + y 2 . Method 3: inverse logarit... |

48 | Computational strategies for the Riemann zeta function - Borwein, Bradley, et al. - 2000 |

13 | An efficient algorithm for the riemann zeta function
- Borwein
- 1995
(Show Context)
Citation Context ...any results concerning the properties of ζ (s). For the numerical evaluation of Riemann’s Zeta function with arbitrary precision to become feasible, one needs special algorithms. Recently P. Borwein [=-=Borwein 1995-=-] gave a simple and quick approximation algorithm for Re (s) > 0. See also [Borwein et al. 1999] for a review of methods. It is the “third” algorithm (the simplest one) from P. Borwein’s paper which i... |

11 |
E–cient Multiple-Precision Evaluation of Elementary Functions
- Smith
- 1989
(Show Context)
Citation Context ... 2) . Method 3: “rectangular” or “baby step/giant step” We can organize the calculation much more efficiently if we are able to estimate the necessary number of terms and to afford some storage (see [=-=Smith 1989-=-]). The “rectangular” algorithm uses 2 √ N long multiplications (assuming that the coefficients of the series are short rational numbers) and √ N units of storage. For high-precision floating-point x,... |

10 |
The complexity of multiple-precision arithmetic, in The Complexity of Computational Problem Solving (edited by
- Brent
- 1976
(Show Context)
Citation Context ...ethod of summation, the total cost is O (√ P M (P ) ) . Method 7: binary reduction This method is based on the binary splitting technique and is described in [Haible et al. 1998] with a reference to [=-=Brent 1976-=-]. The method shall compute ln (1 + x) for real x such that |x| < 1 2 . For other x, some sort of argument reduction needs to be applied. (So this method is a replacement for the Taylor series that is... |

5 | Primality testing revisited
- Davenport
- 1992
(Show Context)
Citation Context ... prime when in fact n is composite; but the probability for this to happen can be made extremely small. The basic reference is [Rabin 1980]. We also implemented some of the improvements suggested in [=-=Davenport 1992-=-]. The idea of the Miller-Rabin algorithm is to improve the Fermat primality test. If n is prime, then for any x we have Gcd (n, x) = 1. Then by Fermat’s “little theorem”, x n−1 ≡ 1 mod n. (This is re... |

2 |
Computation of Catalan's constant using Ramanujan's formula
- Fee
- 1990
(Show Context)
Citation Context ...eries. The running time of this formula seems to be a few times slower than the previous method. This method combined with Brent’s summation trick (see the section on the Euler constant) was used in [=-=Fee 1990-=-]. Brent’s trick allows to avoid a separate computation of the harmonic sum and all long multiplications. Catalan’s constant is obtained as a limit of Gk where G0 = B0 = 1 2 and Bk = k 2k + 1 Bk−1, Gk... |

2 | Algorithm 814: Fortran 90 software for floating-point multiple precision arithmetic, gamma and related functions - Smith |

2 |
Handbook of special functions and definite integrals: algorithms and programs for calculators, Radio and communications (publisher
- Tsimring
- 1988
(Show Context)
Citation Context ...m-up with estimated reMethod 3: mainders There is an improvement over the bottom-up method that can sometimes increase the achieved precision without computing more terms. This trick is suggested in [=-=Tsimring 1988-=-], sec. 2.4, where it is also claimed that the remainder estimates improve convergence. The idea is that the starting value of the backward recurrence should be chosen not as an but as another number ... |

1 | The Rapid Computation of Various Polylogarithmic - Plouffe - 1997 |

1 | Average Case Error Estimates for the Strong Probable - Pomerance - 1993 |

1 | Algebra, systems and algorithms for algebraic computation - Tournier - 1989 |

1 | 2001) (unpublished text): http://winnie.fit.edu/~gabdo/gamma.txt - Godfrey |

1 |
Algorithm 650: Efficient square root implementation on the 68000
- Johnson
- 1987
(Show Context)
Citation Context ...ethod 1: bisection The square root can be computed by using the bisection method, which works well for integers (if only the integer part of the square root is needed). The algorithm is described in [=-=Johnson 1987-=-]. The general approach is to scan each bit of the input number and to see if a certain bit should be set in the resulting integer. The time is linear in the number of decimals, or logarithmic in the ... |

1 |
Algorithm 180, Error function for large real
- Thacher
- 1963
(Show Context)
Citation Context ...l be slower because it requires at least as many or more long multiplications per term. Also, in most cases the Taylor series can be computed much more efficiently using the 5 This method is used by [=-=Thacher 1963-=-], who refers to a suggestion by Hans Maehly. 28rectangular scheme. (See, e.g., the section on arctan x for a more detailed consideration.) However, there are some functions for which a Taylor series... |

1 | von zur Gathen et al. 1999] J. von zur Gathen and - Gerhard - 1999 |