Assessment 2 knuth book

Gcd ak 1 ak ak and therefore ak is the desired gcd

Info icon This preview shows pages 2–4. Sign up to view the full content.

View Full Document Right Arrow Icon
gcd (ak-1, ak) = ak, and therefore ak is the desired GCD. This algorithm can be extended [1, p. 302] to yield integers u~ and v~ such that ulal ~- via2 ~- al, i = 1, ... , k. (2) When gcd (al, a~) = 1, it follows that uka~ + vka2 = 1, and therefore uk is an inverse Journal of the Association for Computing Machinery, Vol. 18, No. 4, October 1971
Image of page 2

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
480 w.s. BROWN of al modulo a2, while vk is an inverse of a2 modulo al. If only uk is needed, as in Step (1) of the Chinese remainder algorithm (Section 4.8), then one need not compute v~, • • • , vk ; if al >> a2, the time saved may be substantial. 1.5 THE ALGORITHM FOR POLYNOMIALS. We shall consider two fundamentally different generalizations of Euclid's algorithm (Section 1.4) to domains of poly- nomials. In the classical algorithm (Section 2), we view a multivariate polynomial as a univariate polynomial with polynomial coefficients, and we construct a sequence of polynomials of successively smaller degree. Unfortunately, as the polynomials de- crease in degree, their coefficients (which may themselves be polynomials) tend to grow, so the successive steps tend to become harder as the calculation progresses. If the GCD's of these inflated coefficients are required, the problem is aggravated-- especially in the multivariate case, where the grov¢th may be compounded through several levels of recursion. If the coefficient domain is a field, this same remark ap- plies to any GCD's of numerators and denominators that are required to simplify inflated coefficients. If coefficients in a field are not simplified, then the division steps become harder faster, and the final result, although formally correct, may be prac- tically useless. In the modular algorithm (Section 4) we first project the given polynomials into one or more simpler domains in which images of the GCD can more easily be com- puted. The true GCD is then constructed from these images with the aid of the Chinese remainder algorithm. Since the same method is used for the required GCD computations in the image spaces, it is only necessary to apply Euclid's algorithm to integers and to univariate polynomials with coefficients in a finite field. 1.6 RECENT HISTORY. During the past decade these algorithms have been studied intensively by G. E. Collins, and (mostly in response to Collins' work) by the author. The first major advance was the discovery by Collins [7] of the subresultant PRS algorithm (Section 3.6), which effectively controls coefficient growth without any GCD computations in the coefficient domain or any subdomain thereof. Then, after several years of improvement and consolidation, Collins and the author (working independently but with some communication) discovered the essentials of the modu- lar algorithm (Section 4) which completely eliminates the problem of coefficient growth by using modular arithmetic. With a very few hints from Collins and the author, D. E. Knuth immediately grasped most of the key ideas and published a sketch of a similar algorithm [1, pp. 393-395].
Image of page 3
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern