Numeric-symbolic exact rational linear system solver

An iterative refinement approach is taken to rational linear system solving. Such methods produce, for each entry of the solution vector, a rational approximation with denominator a power of 2. From this the correct rational entry can be reconstructed. Our iteration is a numeric-symbolic hybrid in that it uses an approximate numeric solver at each step together with a symbolic (exact arithmetic) residual computation and symbolic rational reconstruction. The rational solution may be checked symbolically (exactly). However, there is some possibility of failure of convergence, usually due to numeric ill-conditioning. Alternatively, the algorithm may be used to obtain an extended precision floating point approximation of any specified precision. In this case we cannot guarantee the result by rational reconstruction and an exact solution check, but the approach gives evidence (not proof) that the probability of error is extremely small. The chief contributions of the method and implementation are (1) confirmed continuation, (2) improved rational reconstruction, and (3) faster and more robust performance.

[1]  Patrizia Gianni.  Symbolic and Algebraic Computation, International Symposium ISSAC'88, Rome, Italy, July 4-8, 1988, Proceedings , 1989, ISSAC.

[2]  Michael B. Monagan,et al.  Maximal quotient rational reconstruction: an almost optimal algorithm for rational reconstruction , 2004, ISSAC '04.

[3]  James Hardy Wilkinson,et al.  Rounding errors in algebraic processes , 1964, IFIP Congress.

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

[5]  Basic Numerical Mathematics , 1977 .

[6]  Daniel E. Steffy.  Exact solutions to linear systems of equations using output sensitive lifting , 2011, ACCA.

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

[8]  S. Barnett,et al.  Brownian matrices: properties and extensions , 1986 .

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

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

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

[13]  Alicja Smoktunowicz,et al.  Binary cascades iterative refinement in doubled-mantissa arithmetics , 1984 .

[14]  Daniel E. Stey.  Exact Solutions to Linear Systems of Equations using Output Sensitive Lifting , 2010 .

[15]  Keith O. Geddes,et al.  Exploiting fast hardware floating point in high precision computation , 2003, ISSAC '03.

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

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

[18]  Mark van Hoeij,et al.  A modular GCD algorithm over number fields presented with multiple extensions , 2002, ISSAC '02.

[19]  Bernhard Beckermann,et al.  The condition number of real Vandermonde, Krylov and positive definite Hankel matrices , 2000, Numerische Mathematik.

[20]  Andrzej Kiełbasiński,et al.  Iterative refinement for linear systems in variable-precision arithmetic , 1981 .

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

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

[23]  Basic Numerical Mathematics: Numerical Algebra , 1980 .

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

[25]  Victor Y. Pan,et al.  Acceleration of Euclidean algorithm and extensions , 2002, ISSAC '02.

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

[27]  B. David Saunders,et al.  Computing the smith forms of integer matrices and solving related problems , 2005 .

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