Exact solutions to linear systems of equations using output sensitive lifting

Many methods have been developed to symbolically solve systems of linear equations over the rational numbers. A common approach is to use p-adic lifting or iterative refinement to build a modular or approximate solution, then apply rational number reconstruction. An upper bound can be computed on the number of iterations these algorithms must perform before applying rational reconstruction. In practice such bounds can be conservative. Output sensitive lifting is the technique of performing rational reconstruction at intermediate steps of the algorithm and verifying correctness which allows the possibility of early termination when the solution size is small. In this paper we show how using an appropriate output sensitive lifting strategy can improve several algorithms. We show this procedure to be computationally effective and introduce a variant of the iterative-refinement method that incorporates warm starts into the rational reconstruction procedure.

[1]  Antoine Petitet,et al.  Minimizing development and maintenance costs in supporting persistently optimized BLAS , 2005 .

[2]  Erich Kaltofen,et al.  LINBOX: A GENERIC LIBRARY FOR EXACT LINEAR ALGEBRA , 2002 .

[3]  Carsten Schneider,et al.  A refined difference field theory for symbolic summation , 2008, J. Symb. Comput..

[4]  Victor Y. Pan,et al.  Acceleration of Euclidean Algorithm and Rational Number Reconstruction , 2003, SIAM J. Comput..

[5]  Jack Dongarra,et al.  LAPACK Users' Guide, 3rd ed. , 1999 .

[6]  Carsten Schneider,et al.  Computer proofs of a new family of harmonic number identities , 2003, Adv. Appl. Math..

[7]  Charles L. Lawson,et al.  Basic Linear Algebra Subprograms for Fortran Usage , 1979, TOMS.

[8]  Carsten Schneider,et al.  A Collection of Denominator Bounds To Solve Parameterized Linear Difference Equations in ΠΣ-Fields∗ , 2004 .

[9]  Michael Karr,et al.  Summation in Finite Terms , 1981, JACM.

[10]  John Abbott,et al.  How Tight is Hadamard's Bound? , 2001, Exp. Math..

[11]  Paul S. Wang,et al.  A p-adic algorithm for univariate partial fractions , 1981, SYMSAC '81.

[12]  Carsten Schneider,et al.  Degree Bounds to Find Polynomial Solutions of Parameterized Linear Difference Equations in ΠΣ-Fields , 2005, Applicable Algebra in Engineering, Communication and Computing.

[13]  Manuel Bronstein On Solutions of Linear Ordinary Difference Equations in their Coefficient Field , 2000, J. Symb. Comput..

[14]  Manuel Kauers,et al.  Application of unspecified sequences in symbolic summation , 2006, ISSAC '06.

[15]  Arne Storjohann,et al.  The shifted number system for fast linear algebra on integer matrices , 2005, J. Complex..

[16]  Daniel Lichtblau,et al.  Half-GCD and fast rational recovery , 2005, ISSAC.

[17]  D. H. Lehmer Euclid's Algorithm for Large Numbers , 1938 .

[18]  J. Dixon Exact solution of linear equations usingP-adic expansions , 1982 .

[19]  Erich Kaltofen An output-sensitive variant of the baby steps/giant steps determinant algorithm , 2002, ISSAC '02.

[20]  Joachim von zur Gathen,et al.  Modern Computer Algebra , 1998 .

[21]  Gilles Villard,et al.  Solving sparse rational linear systems , 2006, ISSAC '06.

[22]  Silvio Ursic,et al.  Exact Solution of Systems of Linear Equations with Iterative Methods , 1983 .

[23]  Curtis Bright,et al.  Vector rational number reconstruction , 2011, ISSAC '11.

[24]  Arne Storjohann,et al.  Diophantine linear system solving , 1999, ISSAC '99.

[25]  Arne Storjohann,et al.  A BLAS based C library for exact linear algebra on integer matrices , 2005, ISSAC.

[26]  Arnold Schönhage,et al.  Schnelle Berechnung von Kettenbruchentwicklungen , 1971, Acta Informatica.

[27]  William J. Cook,et al.  Solving Very Sparse Rational Systems of Equations , 2011, TOMS.

[28]  E. V. Krishnamurthy,et al.  p-Adic arithmetic procedures for exact matrix computations , 1975 .

[29]  Liang Chen,et al.  Algorithms for solving linear systems over cyclotomic fields , 2010, J. Symb. Comput..

[30]  Arne Storjohann,et al.  Certified dense linear system solving , 2004, J. Symb. Comput..

[31]  Victor Y. Pan,et al.  On Rational Number Reconstruction and Approximation , 2004, SIAM J. Comput..

[32]  Numerische Mathematik Exact Solution of Linear Equations Using P-Adie Expansions* , 2005 .

[33]  S. Cabay Exact solution of linear equations , 1971, SYMSAC '71.

[34]  Carsten Schneider,et al.  The Summation Package Sigma: Underlying Principles and a Rhombus Tiling Application , 2004, Discret. Math. Theor. Comput. Sci..

[35]  Douglas H. Wiedemann Solving sparse linear equations over finite fields , 1986, IEEE Trans. Inf. Theory.

[36]  Ioannis Z. Emiris,et al.  A Complete Implementation for Computing General Dimensional Convex Hulls , 1998, Int. J. Comput. Geom. Appl..

[37]  Kenneth H. Rosen Elementary Number Theory , 2004 .

[38]  Zhendong Wan,et al.  An algorithm to solve integer linear systems exactly using numerical methods , 2006, J. Symb. Comput..

[39]  Matemática,et al.  Society for Industrial and Applied Mathematics , 2010 .

[40]  Erich Kaltofen,et al.  On Wiedemann's Method of Solving Sparse Linear Systems , 1991, AAECC.

[41]  Carsten Schneider,et al.  Solving parameterized linear difference equations in terms of indefinite nested sums and products , 2005 .

[42]  Jack J. Dongarra,et al.  A set of level 3 basic linear algebra subprograms , 1990, TOMS.

[43]  Jean-Guillaume Dumas,et al.  Dense Linear Algebra over Word-Size Prime Fields: the FFLAS and FFPACK Packages , 2006, TOMS.

[44]  Robert T. Moenck,et al.  Approximate algorithms to derive exact solutions to systems of linear equations , 1979, EUROSAM.

[45]  Manuel Kauers,et al.  Symbolic summation with radical expressions , 2007, ISSAC '07.

[46]  David K. Smith Theory of Linear and Integer Programming , 1987 .

[47]  Michael Karr Theory of Summation in Finite Terms , 1985, J. Symb. Comput..