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

1990
*
Journal of symbolic computation
*

We consider

doi:10.1016/s0747-7171(08)80055-7
fatcat:i2phv3jk5raf3f52gmzmxkncny
*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*. ...##
###
Parallel Implementation of Exact Matrix Computation Using Multiple P-adic Arithmetic

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. ...

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

2015
*
International Journal of Networked and Distributed Computing (IJNDC)
*

Based

doi:10.2991/ijndc.2015.3.1.6
fatcat:an27zjvphbca3od53a7yd3gsie
*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. ...##
###
An introduction of Multiple P-adic Data Type and its parallel implementation

2014
*
2014 IEEE/ACIS 13th International Conference on Computer and Information Science (ICIS)
*

Based

doi:10.1109/icis.2014.6912151
dblp:conf/ACISicis/LuL14
fatcat:dlz4iu2tqbbjxh34ov2jhqn6fi
*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. ...##
###
Parallel p-adic method for solving linear systems of equations

1997
*
Parallel Computing
*

We present

doi:10.1016/s0167-8191(97)00062-8
fatcat:xm2f7eaoi5hobp7mgvctkl2shq
*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. ...##
###
Exact reduction of a polynomial matrix to the Smith normal form

1979
*
IEEE Transactions on Automatic Control
*

Since

doi:10.1109/tac.1979.1102104
fatcat:dngu3wn2wbg3rnlvoqbbbtnjda
*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 ...##
###
A dyadic solution of relative pose problems
[article]

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. ...

##
###
Rigid cohomology and p-adic point counting

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. ...

##
###
Total dual dyadicness and dyadic generating sets
[article]

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. ...

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

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). ...

##
###
P-adic numbers and replica symmetry breaking

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. ...

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

2013
*
arXiv
*
pre-print

We present an efficient algorithm for

arXiv:1310.4421v1
fatcat:fni7byydd5dojhvydrxdjtl3ve
*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*.) ...##
###
Efficient Computation of Rankin p-Adic L-Functions
[chapter]

2014
*
Contributions in Mathematical and Computational Sciences
*

We present an efficient algorithm for

doi:10.1007/978-3-319-03847-6_7
fatcat:hsitxztoofcqtky7soude7elwu
*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 . ...##
###
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]

2010
*
arXiv
*
pre-print

Coleman's theory

arXiv:1004.4936v2
fatcat:rz5pym7qlzdg5k6nfyjugnwcny
*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- ...
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