Fast algorithms

Fast algorithms is the field of computational mathematics that studies algorithms of evaluation of a given function with a given accuracy, using as few bit operations as possible.

Bit operation

We assume that numbers are written in the binary form, the signs of which 0 and 1 are called bits.

Def.1. Writing down of one of the symbols $0,1,+,-,(,)$ , putting together, subtraction and multiplication of two bits is called an elementary operation or a bit operation.

Complexity of computation (bit)

To estimate the quality of a fast method or algorithm is used the function of complexity of computation (bit) which is denoted by $s_{f}(n)=s_{f,x_{0}}(n).$

The function of complexity of multiplication has the special notation $M(n)$ .

The best known (at present) upper bound for $M(n),$ is

M(n)=O(n\log n\log \log n).\,

Fast algorithm of computation of a function

An algorithm of computation of a function $f=f(x)$ is said to be fast if, assuming the best bound for $M(n)$ , for this algorithm

s_{f}(n)=O\left(M(n)\log ^{c}n\right)\ ,

where $c$ is a constant.

History of the problem

The field fast algorithms was born in 1960 ^[1], when the first fast method --- the Karatsuba multiplication --- was found. Later the Karatsuba method was called “Divide and Conquer” (sometimes any method of computation with subdivisions is called with the same name), other names which people use for the method invented by Karatsuba are “Binary Splitting”,"Dichotomy Principle" etc. After the Karatsuba method, many other fast methods were constructed^[2], including the Strassen fast matrix multiplication method^[3] (the generalization of the Karatsuba idea for matrices), the Schönhage-Strassen multiplication method^[4]^[5], the FEE method ^[6]^[7] for evaluation elementary and higher transcendental functions etc . Some old methods become fast computational methods with use of one of the fast multiplication algorithms, such as the Newton method for calculation of elementary algebraic functions and the AGM method of Gauss for evaluation of elementary transcendental functions.

References

^ A.A. Karatsuba, The Complexity of Computations. Proceedings of the Steklov Institute of Mathematics, Vol.211 (1995)
^ D.E. Knuth, The art of computer programming. Vol.2 Addison-Wesley Publ.Co., Reading (1969).
^ V. Strassen, Gaussian elimination is not optimal. J. Numer. Math., N 13 (1969)
^ A. Schönhage und V. Strassen, Schnelle Multiplikation grosser Zahlen. Computing, Vol.7 (1971)
^ A. Schönhage, A.F.W. Grotefeld and E. Vetter, Fast Algorithms. BI-Wiss.-Verl., Zürich (1994).
^ E.A. Karatsuba,Fast evaluations of transcendental functions. Probl. Peredachi Informat., Vol. 27, N 4 (1991).
^ D.W. Lozier and F.W.J. Olver, Numerical Evaluation of Special Functions. Mathematics of Computation 1943-1993: A Half -Century of Computational Mathematics, W.Gautschi,eds., Proc. Sympos. Applied Mathematics, AMS, Vol.48 (1994).

External links

[1] A.A. Karatsuba, The Complexity of Computations. Proceedings of the Steklov Institute of Mathematics, Vol.211 (1995)

[2] D.E. Knuth, The art of computer programming. Vol.2 Addison-Wesley Publ.Co., Reading (1969).

[3] V. Strassen, Gaussian elimination is not optimal. J. Numer. Math., N 13 (1969)

[4] A. Schönhage und V. Strassen, Schnelle Multiplikation grosser Zahlen. Computing, Vol.7 (1971)

[5] A. Schönhage, A.F.W. Grotefeld and E. Vetter, Fast Algorithms. BI-Wiss.-Verl., Zürich (1994).

[6] E.A. Karatsuba,Fast evaluations of transcendental functions. Probl. Peredachi Informat., Vol. 27, N 4 (1991).

[7] D.W. Lozier and F.W.J. Olver, Numerical Evaluation of Special Functions. Mathematics of Computation 1943-1993: A Half -Century of Computational Mathematics, W.Gautschi,eds., Proc. Sympos. Applied Mathematics, AMS, Vol.48 (1994).

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