91 lines
3.8 KiB
Plaintext
91 lines
3.8 KiB
Plaintext
<HTML><HEAD>
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<TITLE> Sparse LU Decomposition Methods </TITLE>
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</HEAD><BODY>
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<H1> Sparse LU Decomposition Methods </H1>
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<P>The sparse modulo-2 matrix LU decomposition routine <A
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HREF="mod2sparse.html#decomp"><TT>mod2sparse_decomp</TT></A> (which
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is used by the <A HREF="encoding.html#make-gen"><TT>make-gen</TT></A>
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program when it is asked to create a sparse generator matrix) tries to
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find an sub-matrix of a matrix (for <TT>make-gen</TT>, the parity
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check matrix), and an ordering of rows and columns for this
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sub-matrix, that leads to the lower-triangular matrix <B>L</B> and the
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upper-triangular matrix <B>U</B> making up the LU decomposition being
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as sparse as possible. Finding an optimal solution is too difficult,
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so instead a heuristic strategy is used.
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<P>The overall algorithm finds <B>L</B> and <B>U</B> a column at a
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time, from left to right (as reordered, in the case of <B>U</B>). As
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this is done, a copy, <B>B</B>, of the original matrix is modified.
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To create column <I>i</I> of <B>L</B> and <B>U</B>, some element with
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value 1 in <B>B</B> whose row and column indexes, after reordering,
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are both greater than <I>i</I> is found. The row and column of this
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element are considered to come next in the reordering, and the
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contents of the column containing this element is copied to <B>L</B>
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and <B>U</B> (upper elements going to <B>U</B>, lower to <B>L</B>).
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The row containing this element is then added to some later rows so as
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to clear the lower part of this column to zeros.
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<P>At the first step of this process - selecting an element with value
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1 from the later rows and columns - there will often be several
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possibilities. Different choices can lead to the final result being
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more or less sparse. The possible strategies for picking an element
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are identified by the constants <TT>Mod2sparse_first</TT>,
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<TT>Mod2sparse_mincol</TT>, and <TT>Mod2sparse_minprod</TT>. These
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strategies operate as follows:
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<P><TT>Mod2sparse_first</TT>
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<BLOCKQUOTE>
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Select the first element with value 1 that is encountered in a top
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to bottom, left to right search.
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</BLOCKQUOTE>
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<P><TT>Mod2sparse_mincol</TT>
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<BLOCKQUOTE>
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Select the first element with value 1 that is contained in a column
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of <B>B</B> that has the smallest number of 1s of any column.
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</BLOCKQUOTE>
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<P><TT>Mod2sparse_minprod</TT>
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<BLOCKQUOTE>
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Select an element with value 1 for which the product of the number of
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1s in that row of <B>B</B> minus one times the number of 1s in that
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column of <B>B</B> minus one is as small as possible.
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</BLOCKQUOTE>
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<P>The <B>abandon_number</B> and <B>abandon_when</B> parameters can
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modify the basic strategy. If <B>abandon_number</B> is greater than
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zero, then after <B>abandon_when</B> columns have been selected,
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<B>abandon_number</B> of the remaining columns are abandoned as
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candidates for possible future selection, the abandoned columns being
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those with the greatest number of entries. Abandoning such columns
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saves space and time, but may make the final result less sparse than
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it would otherwise be, and can possibly result in the matrix appearing
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to have lower rank than it actually has.
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<P>The methods described here are fairly straightforward adaptations
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of standard methods for sparse square matrices of reals, as described, for
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example, in
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<BLOCKQUOTE>
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I. S. Duff, A. M. Erisman, J. K. Reid (1986) <I>Direct Methods for
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Sparse Matrices</I>, Oxford: Clarendon Press.
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</BLOCKQUOTE>
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In the coding context, however, we are interested in matrices of
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modulo-2 elements, and it is enough to find a sparse LU decomposition
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of any square sub-matrix that can be obtained by selecting columns of
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the rectangular parity check matrix. I talked about the application
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of sparse matrix methods to encoding of LDPC codes at the 1999 IMA
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workshop on Codes, Systems and Graphical Models (see the <A
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HREF="refs.html">references</A>).
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