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MathGroup Archive 1992

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Re: LatticeReduce

  • To: mathgroup at yoda.physics.unc.edu
  • Subject: Re: LatticeReduce
  • From: Brian Evans <evans at gauss.eedsp.gatech.edu>
  • Date: Wed, 11 Mar 1992 16:53:30 GMT

	In response to Shiv Gupta's question about reducing rectangular
	  integer matrices (that represent lattices), Mr. Grayson of Wolfram
	  Research Inc. correctly pointed out that he can reduce the integer
	  matrix to column Hermite form and discard the zero rows.

        This kind of operation is very common in linear systems and signal
          processing.  In linear systems, coupled systems represented
          as matrices have Laplace transforms that are matrices of
          polynomials.  These are often reduced using gcd and modulo
          operations with polynomials into a column Hermite form.
          In signal processing, lattice theory forms the basis for
          many multidimensional operations on non-rectangular sampling
          grids (FFT, etc.).  Lattice theory also forms the basis
          for multidimensional multirate signal processing (filter banks,
          wavelets, etc.).

        Since lattice operations are commonly performed in signal
          processing, I encoded a comprehensive set of lattice operations
          for the signal processing packages, including row and column
          Hermite forms, left and right least common multiples, left
          and right greatest common divisors, and various Smith form
          decompositions.  The routines can show intermediate calculations.
	  They are defined by the file "SignalProcessing/Support/Multirate.m".
	  These have only been a part of the signal processing packages for
	  the last two months or so.

        The signal processing packages are available via anonymous ftp
          to gauss.eedsp.gatech.edu (IP #130.207.226.24) in the compressed
          tar file "Mathematica/SigProc2.0.tar.Z".

	I hope that his helps.

Brian Evans
evans at eedsp.gatech.edu

P.S.  Although I am not a fan of Maple, Maple does provide many
      lattice operations in its standard library.





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