First Hurwitz triplet
An integer relation between a set of real numbers x1, x2, ..., xn is a set of integers a1, a2, ..., an, not all 0, such that
An integer relation algorithm is an algorithm for finding integer relations. Specifically, given a set of real numbers known to a given precision, an integer relation algorithm will either find an integer relation between them, or will determine that no integer relation exists with coefficients whose magnitudes are less than a certain upper bound.[1]
History
For the case n = 2, an extension of the Euclidean algorithm can determine whether there is an integer relation between any two real numbers x1 and x2. The algorithm generates successive terms of the continued fraction expansion of x1/x2; if there is an integer relation between the numbers then their ratio is rational and the algorithm eventually terminates.
- The Ferguson–Forcade algorithm was published in 1979 by Helaman Ferguson and R.W. Forcade.[2] Although the paper treats general n, it is not clear if the paper fully solves the problem because it lacks detailed steps, proofs, and a precision bound; crucial for a reliable implementation.
- The first algorithm with complete proofs was the LLL algorithm, developed by Arjen Lenstra, Hendrik Lenstra and László Lovász in 1982.[3]
- The HJLS algorithm, developed by Johan Håstad, Bettina Just, Jeffrey Lagarias, and Claus-Peter Schnorr in 1986.[4][5]
- The PSOS algorithm, developed by Ferguson in 1988.[6]
- The PSLQ algorithm, developed by Ferguson and Bailey in 1992 and substantially simplified by Ferguson, Bailey, and Arno in 1999.[7][8] In 2000 the PSLQ algorithm was selected as one of the "Top Ten Algorithms of the Century" by Jack Dongarra and Francis Sullivan[9] even though it is essentially equivalent to HJLS.[10]
- The LLL algorithm has been improved by numerous authors. Modern implementations can solve integer relation problems with n well above 500.
Applications
Integer relation algorithms have numerous applications. The first application is to determine whether a given real number x is likely to be algebraic, by searching for an integer relation between a set of powers of x {1, x, x2, ..., xn}. The second application is to search for an integer relation between a real number x and a set of mathematical constants such as e, π and ln(2), which will lead to an expression for x as a linear combination of these constants.
A typical approach in experimental mathematics is to use numerical methods and arbitrary precision arithmetic to find an approximate value for an infinite series, infinite product or an integral to a high degree of precision (usually at least 100 significant figures), and then use an integer relation algorithm to search for an integer relation between this value and a set of mathematical constants. If an integer relation is found, this suggests a possible closed-form expression for the original series, product or integral. This conjecture can then be validated by formal algebraic methods. The higher the precision to which the inputs to the algorithm are known, the greater the level of confidence that any integer relation that is found is not just a numerical artifact.
A notable success of this approach was the use of the PSLQ algorithm to find the integer relation that led to the Bailey-Borwein-Plouffe formula for the value of π. PSLQ has also helped find new identities involving multiple zeta functions and their appearance in quantum field theory; and in identifying bifurcation points of the logistic map. For example, where B4 is the logistic map's fourth bifurcation point, the constant α=-B4(B4-2) appears to be a root of a 120th-degree polynomial whose largest coefficient is 25730 or about 2·1072.[11] Integer relation algorithms are combined with tables of high precision mathematical constants and heuristic search methods in applications such as the Inverse Symbolic Calculator or Plouffe's Inverter.
Integer relation finding can be used to factor polynomials over the rationals of high degree.[12]
References
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External links
- Recognizing Numerical Constants by David H. Bailey and Simon Plouffe
- Ten Problems in Experimental Mathematics by David H. Bailey, Jonathan M. Borwein, Vishaal Kapoor, and Eric W. Weisstein
Template:Number theoretic algorithms
- ↑ Since the set of real numbers can only be specified up to a finite precision, an algorithm that did not place limits on the size of its coefficients would always find an integer relation for sufficiently large coefficients. Results of interest occur when the size of the coefficients in an integer relation is small compared to the precision with which the real numbers are specified.
- ↑
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Hostgator tutorials on how to install Wordpress need not be complicated, especially when you will be dealing with a web hosting service that is friendly for novice webmasters and a blogging platform that is as intuitive as riding a bike. After that you can get Hostgator to host your domain and use the wordpress to do the blogging. Once you start site flipping, trust me you will not be able to stop. I cut my webmaster teeth on Control Panel many years ago, but since had left for other hosting companies with more commercial (cough, cough) interfaces. If you don't like it, you can chalk it up to experience and go on. First, find a good starter template design. When I signed up, I did a search for current "HostGator codes" on the web, which enabled me to receive a one-word entry for a discount. Your posts, comments, and pictures will all be imported into your new WordPress blog. - ↑
I had like 17 domains hosted on single account, and never had any special troubles. If you are not happy with the service you will get your money back with in 45 days, that's guaranteed. But the Search Engine utility inside the Hostgator account furnished an instant score for my launched website. Fantastico is unable to install WordPress in a directory which already have any file i.e to install WordPress using Fantastico the destination directory must be empty and it should not have any previous installation files. When you share great information, others will take note. Once your hosting is purchased, you will need to setup your domain name to point to your hosting. Money Back: All accounts of Hostgator come with a 45 day money back guarantee. If you have any queries relating to where by and how to use Hostgator Discount Coupon, you can make contact with us at our site. If you are starting up a website or don't have too much website traffic coming your way, a shared plan is more than enough. Condition you want to take advantage of the worldwide web you prerequisite a HostGator web page, -1 of the most trusted and unfailing web suppliers on the world wide web today. Since, single server is shared by 700 to 800 websites, you cannot expect much speed.
Hostgator tutorials on how to install Wordpress need not be complicated, especially when you will be dealing with a web hosting service that is friendly for novice webmasters and a blogging platform that is as intuitive as riding a bike. After that you can get Hostgator to host your domain and use the wordpress to do the blogging. Once you start site flipping, trust me you will not be able to stop. I cut my webmaster teeth on Control Panel many years ago, but since had left for other hosting companies with more commercial (cough, cough) interfaces. If you don't like it, you can chalk it up to experience and go on. First, find a good starter template design. When I signed up, I did a search for current "HostGator codes" on the web, which enabled me to receive a one-word entry for a discount. Your posts, comments, and pictures will all be imported into your new WordPress blog. - ↑ Johan Håstad, Bettina Just, Jeffrey Lagarias, Claus-Peter Schnorr: Polynomial time algorithms for finding integer relations among real numbers. Preliminary version: STACS 1986 (Symposium Theoret. Aspects Computer Science) Lecture Notes Computer Science 210 (1986), p. 105–118. SIAM J. Computing, Vol. 18 (1989), p. 859–881
- ↑
I had like 17 domains hosted on single account, and never had any special troubles. If you are not happy with the service you will get your money back with in 45 days, that's guaranteed. But the Search Engine utility inside the Hostgator account furnished an instant score for my launched website. Fantastico is unable to install WordPress in a directory which already have any file i.e to install WordPress using Fantastico the destination directory must be empty and it should not have any previous installation files. When you share great information, others will take note. Once your hosting is purchased, you will need to setup your domain name to point to your hosting. Money Back: All accounts of Hostgator come with a 45 day money back guarantee. If you have any queries relating to where by and how to use Hostgator Discount Coupon, you can make contact with us at our site. If you are starting up a website or don't have too much website traffic coming your way, a shared plan is more than enough. Condition you want to take advantage of the worldwide web you prerequisite a HostGator web page, -1 of the most trusted and unfailing web suppliers on the world wide web today. Since, single server is shared by 700 to 800 websites, you cannot expect much speed.
Hostgator tutorials on how to install Wordpress need not be complicated, especially when you will be dealing with a web hosting service that is friendly for novice webmasters and a blogging platform that is as intuitive as riding a bike. After that you can get Hostgator to host your domain and use the wordpress to do the blogging. Once you start site flipping, trust me you will not be able to stop. I cut my webmaster teeth on Control Panel many years ago, but since had left for other hosting companies with more commercial (cough, cough) interfaces. If you don't like it, you can chalk it up to experience and go on. First, find a good starter template design. When I signed up, I did a search for current "HostGator codes" on the web, which enabled me to receive a one-word entry for a discount. Your posts, comments, and pictures will all be imported into your new WordPress blog. - ↑
I had like 17 domains hosted on single account, and never had any special troubles. If you are not happy with the service you will get your money back with in 45 days, that's guaranteed. But the Search Engine utility inside the Hostgator account furnished an instant score for my launched website. Fantastico is unable to install WordPress in a directory which already have any file i.e to install WordPress using Fantastico the destination directory must be empty and it should not have any previous installation files. When you share great information, others will take note. Once your hosting is purchased, you will need to setup your domain name to point to your hosting. Money Back: All accounts of Hostgator come with a 45 day money back guarantee. If you have any queries relating to where by and how to use Hostgator Discount Coupon, you can make contact with us at our site. If you are starting up a website or don't have too much website traffic coming your way, a shared plan is more than enough. Condition you want to take advantage of the worldwide web you prerequisite a HostGator web page, -1 of the most trusted and unfailing web suppliers on the world wide web today. Since, single server is shared by 700 to 800 websites, you cannot expect much speed.
Hostgator tutorials on how to install Wordpress need not be complicated, especially when you will be dealing with a web hosting service that is friendly for novice webmasters and a blogging platform that is as intuitive as riding a bike. After that you can get Hostgator to host your domain and use the wordpress to do the blogging. Once you start site flipping, trust me you will not be able to stop. I cut my webmaster teeth on Control Panel many years ago, but since had left for other hosting companies with more commercial (cough, cough) interfaces. If you don't like it, you can chalk it up to experience and go on. First, find a good starter template design. When I signed up, I did a search for current "HostGator codes" on the web, which enabled me to receive a one-word entry for a discount. Your posts, comments, and pictures will all be imported into your new WordPress blog. - ↑ A Polynomial Time, Numerically Stable Integer Relation Algorithm by Helaman R. P. Ferguson and David H. Bailey; RNR Technical Report RNR-91-032; July 14, 1992
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- ↑ David H. Bailey and David J. Broadhurst, "Parallel Integer Relation Detection: Techniques and Applications," Mathematics of Computation, vol. 70, no. 236 (Oct 2000), pg. 1719-1736; LBNL-44481.
- ↑ M. van Hoeij: Factoring polynomials and the knapsack problem. J. of Number Theory, 95, 167-189, (2002).