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On p-adic computation of the rational form of a matrix

Marie-Hélène Mathieu, David Ford
1990 Journal of symbolic computation  
We consider the problem of bringing a given matrix into "cyclic form," from which the rational form can be computed easily.  ...  Matrices are taken to have p-adic integer entries, and computations are done with rational integer approximations to p-adic integers.  ...  Introduction In his recent book Liineberg (1987) remarks on the absence of an effective modular algorithm for computing the rational normal form of a matrix.  ... 
doi:10.1016/s0747-7171(08)80055-7 fatcat:i2phv3jk5raf3f52gmzmxkncny

Parallel Implementation of Exact Matrix Computation Using Multiple P-adic Arithmetic

Xinkai Li, Chao Lu
2013 International Journal of Networked and Distributed Computing (IJNDC)  
A P-adic Exact Scientific Computational Library (ESCL) for rational matrix operations has been developed over the past few years.  ...  In this paper, we report our progress on parallel implementation of P-adic arithmetic by means of a multiple modulus rational system related to the Chinese remainder theorem.  ...  Acknowledgements This research project is supported by the Air Force Office of Scientific Research, FA9550-11-1-0315.  ... 
doi:10.2991/ijndc.2013.1.3.1 fatcat:nszwcnnrozh7zozh55kagjcz4m

An Introduction of Multiple P-adic Data Type and Its Parallel Implementation

Chao Lu, Xinkai Li
2015 International Journal of Networked and Distributed Computing (IJNDC)  
Based on the Chinese Remainder theorem and the Hensel code a new data type, called Multiple P-adic Data Type, has been established to realize rational calculation.  ...  Our research group has been working on the P-adic theory and its implementation.  ...  The Main Properties of Multiple P-adic Data Type Error-free Computing in Rational Number Field Each rational number is represented by a finite sequence of integers.  ... 
doi:10.2991/ijndc.2015.3.1.6 fatcat:an27zjvphbca3od53a7yd3gsie

An introduction of Multiple P-adic Data Type and its parallel implementation

Chao Lu, Xinkai Li
2014 2014 IEEE/ACIS 13th International Conference on Computer and Information Science (ICIS)  
Based on the Chinese Remainder theorem and the Hensel code a new data type, called Multiple P-adic Data Type, has been established to realize rational calculation.  ...  Our research group has been working on the P-adic theory and its implementation.  ...  The Main Properties of Multiple P-adic Data Type Error-free Computing in Rational Number Field Each rational number is represented by a finite sequence of integers.  ... 
doi:10.1109/icis.2014.6912151 dblp:conf/ACISicis/LuL14 fatcat:dlz4iu2tqbbjxh34ov2jhqn6fi

Parallel p-adic method for solving linear systems of equations

Ç.K. Koç
1997 Parallel Computing  
We present a parallel algorithm for exact solution of an integer linear system of equations using the single modulus p-adic expansion technique.  ...  The parallel algorithm presented here can be used together with the multiple moduli algorithms and parallel Chinese remainder algorithms for fast computation of the exact solution of a system of linear  ...  n verting to the rational form from p-adic approximation, which depends on the magnitude of the numerator in the exact solution.  ... 
doi:10.1016/s0167-8191(97)00062-8 fatcat:xm2f7eaoi5hobp7mgvctkl2shq

Exact reduction of a polynomial matrix to the Smith normal form

V. Ramachandran
1979 IEEE Transactions on Automatic Control  
Since the Smith normal form of a matrix depends on the roots of its polynomial elements, it need not be a continuous function of its elements.  ...  Any rational number a / b (where a and b are integers) is written in the form Arithmetic operations within the p-adic system are similar to those within a residue system and are explained in detail in  ... 
doi:10.1109/tac.1979.1102104 fatcat:dngu3wn2wbg3rnlvoqbbbtnjda

A dyadic solution of relative pose problems [article]

Patrick Erik Bradley
2009 arXiv   pre-print
A hierarchical interval subdivision is shown to lead to a p-adic encoding of image data.  ...  This allows in the case of the relative pose problem in computer vision and photogrammetry to derive equations having 2-adic numbers as coefficients, and to use Hensel's lifting method to their solution  ...  Acknowledgements Sven Wursthorn is thanked for the introduction into this fascinating topic, and Boris Jutzi for multiple fruitful discussions.  ... 
arXiv:0908.1919v3 fatcat:q4vpokgd5jegfg4wwxhhahyolm

Rigid cohomology and p-adic point counting

Alan G.B. Lauder
2005 Journal de Théorie des Nombres de Bordeaux  
Forms in A † are p-adic limits of rational forms, and since derivation is continuous we can reduce elements in A † to the limits of reduced rational forms.  ...  However, one can calculate bounds on the domain of holomorphy of the entries in a matrix for Frob p (Γ), as p-adic holomorphic functions in the sense of Krasner.  ...  In [10] the local expansion actually converged on the closed p-adic unit disk, which was a helpful simplification.  ... 
doi:10.5802/jtnb.484 fatcat:5p4hzlvfdrcs7ceghgk57sxwv4

Total dual dyadicness and dyadic generating sets [article]

Ahmad Abdi and Gérard Cornuéjols and Bertrand Guenin and Levent Tunçel
2022 arXiv   pre-print
A vector is dyadic if each of its entries is a dyadic rational number, i.e. of the form a/2^k for some integers a,k with k≥ 0.  ...  Along the way, we see some surprising turn of events when compared to total dual integrality, primarily led by the density of the dyadic rationals.  ...  Acknowledgement We would like to thank the referees whose comments on an earlier draft improved the final presentation. Bertrand Guenin was supported by NSERC grant 238811.  ... 
arXiv:2111.05749v2 fatcat:wps3fchsuzaqzgq6pkd3x7vvey

Exact solution of linear systems over rational numbers by parallel p-adic arithmetic [chapter]

Carla Limongelli, Roberto Pirastu
1994 Lecture Notes in Computer Science  
The rationals are represented by truncated p-adic expansion.  ...  We describe a parallel implementation of an algorithm for solving systems of linear equations over the eld of rational numbers based on Gaussian elimination.  ...  Each of them computes the image of the problem w.r.t. one prime in p-adic representation of the rational entries, i.e., H pi;r (A) and H pi;r (b).  ... 
doi:10.1007/3-540-58430-7_28 fatcat:xlxyr3wo3fgptil7ibdk6vlbty

P-adic numbers and replica symmetry breaking

G. Parisi, N. Sourlas
2000 European Physical Journal B : Condensed Matter Physics  
The p-adic formulation of replica symmetry breaking is presented. In this approach ultrametricity is a natural consequence of the basic properties of the p-adic numbers.  ...  Many properties can be simply derived in this approach and p-adic Fourier transform seems to be an promising tool.  ...  Closing the rational field with respect to the previously defined norm one obtains the p-adic field. Continuity of a p-adic function can be defined as usual.  ... 
doi:10.1007/s100510051063 fatcat:euxpjzlva5aqtcnzmjenz56xmm

Efficient computation of Rankin p-adic L-functions [article]

Alan G.B. Lauder
2013 arXiv   pre-print
We present an efficient algorithm for computing certain special values of Rankin triple product p-adic L-functions and give an application of this to the explicit construction of rational points on elliptic  ...  One can now compute a matrix A ord over Z/(p m ) for the U p operator on this basis by explicitly computing with q-expansions.  ...  of the matrix A.)  ... 
arXiv:1310.4421v1 fatcat:fni7byydd5dojhvydrxdjtl3ve

Efficient Computation of Rankin p-Adic L-Functions [chapter]

Alan G. B. Lauder
2014 Contributions in Mathematical and Computational Sciences  
We present an efficient algorithm for computing certain special values of Rankin triple product p-adic L-functions and give an application of this to the explicit construction of rational points on elliptic  ...  One can now compute a matrix A ord over Z/(p m ) for the U p operator on this basis by explicitly computing with q-expansions.  ...  Hence we can iterate U p on H by iterating the finite matrix A on a finite vector of length .  ... 
doi:10.1007/978-3-319-03847-6_7 fatcat:hsitxztoofcqtky7soude7elwu

Page 4351 of Mathematical Reviews Vol. , Issue 94h [page]

1994 Mathematical Reviews  
On the basis of the concept presented a statistical interpretation of p-adic field theory is given; here t is the p-adic topology on Q. Evgenii I.  ...  The theoretical justification for our algorithm is based on a study of the differential equation y?~-") + y? = 0 of order p — 1 in the rational function field F,(x).  ... 

Explicit Coleman integration for hyperelliptic curves [article]

Jennifer S. Balakrishnan, Robert W. Bradshaw, Kiran S. Kedlaya
2010 arXiv   pre-print
Coleman's theory of p-adic integration figures prominently in several number-theoretic applications, such as finding torsion and rational points on curves, and computing p-adic regulators in K-theory (  ...  including p-adic heights on elliptic curves).  ...  The authors thank William Stein for access to his com-  ... 
arXiv:1004.4936v2 fatcat:rz5pym7qlzdg5k6nfyjugnwcny
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