Sparse Matrices
***************

The Creation and Manipulation of Sparse Matrices
================================================

The size of mathematical problems that can be treated at any particular time is generally limited by the available computing resources. Both, the speed of the computer and its available memory place limitation on the problem size.

There are many classes mathematical problems which give rise to matrices, where a large number of the elements are zero. In this case it makes sense to have a special matrix type to handle this class of problems where only the non-zero elements of the matrix are stored. Not only done this reduce the amount of memory to store the matrix, but it also means that operations on this type of matrix can take advantage of the a-priori knowledge of the positions of the non-zero elements to accelerate their calculations.

A matrix type that stores only the non-zero elements is generally called sparse. It is the purpose of this document to discuss the basics of the storage and creation of sparse matrices and the fundamental operations on them.

THIS DOCUMENT STILL HAS LARGE BLANKS. PLEASE FILL THEM IN. LOOK FOR THE TAG "WRITE ME"

Storage of Sparse Matrices
--------------------------

It is not strictly speaking necessary for the user to understand how sparse matrices are stored. However, such an understanding will help to get an understanding of the size of sparse matrices. Understanding the storage technique is also necessary for those users wishing to create their own oct-files.

There are many different means of storing sparse matrix data. What all of the methods have in common is that they attempt to reduce the compelxity and storage given a-priori knowledge of the particular class of problems that will be solved. A good summary of the available techniques for storing sparse matrix is given by Saad  (Youcef Saad "SPARSKIT: A basic toolkit for sparse matrix computation", 1994, ftp://ftp.cs.umn.edu/dept/sparse/SPARSKIT2/DOC/paper.ps). With full matrices, knowledge of the point of an element of the matrix within the matrix is implied by its position in the computers memory. However, this is not the case for sparse matrices, and so the positions of the non-zero elements of the matrix must equally be stored.

An obvious way to do this is by storing the elements of the matrix as triplets, with two elements being the of the position in the array (rows and column) and the third being the data itself. This is conceptually easy to grasp, but requires more storage than is strictly needed.

The storage technique used within OCTAVE is compressed column format.  In this format the position of each element in a row and the data are stored as previously. However, if we assume that all elements in the same column are stored adjacent in the computers memory, then we only need to store information on the number of non-zero elements in each column, rather than their positions. Thus assuming that the matrix has more non-zero elements than there are columns in the matrix, we win in terms of the amount of memory used.

In fact, the column index contains one more element than the number of columns, with the first element always being zero. The advantage of this is a simplication in the code, in that their is no special case for the first or last columns. A short example, demonstrating this in C is.

       for (j = 0; j < nc; j++)
         for (i = cidx (j); i < cidx(j+1); i++)
            printf ("non-zero element (%i,%i) is %d\n", ridx(i), j, data(i));

A clear understanding might be had by considering an example of how the above applies to an example matrix. Consider the matrix

         1   2   0  0
         0   0   0  3
         0   0   0  4

The non-zero elements of this matrix are

        (1, 1)  => 1
        (1, 2)  => 2
        (2, 4)  => 3
        (3, 4)  => 4

This will be stored as three vectors CIDX, RIDX and DATA, representing the column indexing, row indexing and data respectively. The contents of these three vectors for the above matrix will be

       CIDX = [0, 2, 2, 4]
       RIDX = [0, 0, 1, 2]
       DATA = [1, 2, 3, 4]

Note that this is the representation of these elements with the first row and column assumed to start as zero. Thus the number of elements in the I-th column is given by CIDX (I + 1) - CIDX (I).

It should be noted that compressed row formats are equally possible. However, in the context of mixed operations between mixed sparse and dense matrices, it makes sense that the elements of the sparse matrices are in the same order as the dense matrices. OCTAVE stores dense matrices in column major ordering, and so sparse matrices are equally stored in this manner.

A further constraint on the sparse matrix storage used by OCTAVE is that all elements in the rows are stored in increasing order of their row index. This makes certain operations, later be faster. However, it imposes the need to sort the elements on the creation of sparse matrices. Having dis-ordered elements is potentially an advantage in that it makes operations such as concatenating two sparse matrices together easier and faster, however it adds complexity and speed problems elsewhere.

Creating Sparse Matrices
------------------------

There are several means to create sparse matrix.

Returned from a function
     There are many functions that directly return sparse matrices. These include speye, sprand, spdiag, etc.

Constructed from matrices or vectors
     The function sparse allows a sparse matrix to be constructed from three vectors representing the row, column and data. ALternatively, the function spconvert uses a three column matrix format to allow easy importation of data from elsewhere.

Created and then filled
     The function sparse or spalloc can be used to create an empty matrix that is then filled by the user

From a user binary program
     The user can directly create the sparse matrix within an oct-file.

There are several basic functions to return specific sparse matrices. For example the sparse identity matrix, is a matrix that is often needed. It therefore has its own function to create it as speye (N) or speye (R, C), which creates an N-by-N or R-by-C sparse identity matrix.

Another typical sparse matrix that is often needed is a random distribution of random elements. The functions sprand and sprandn perform this for uniform and random distributions of elements. They have exactly the same calling convention as sprand (R, C, D), which creates an R-by-C sparse matrix a density of filled elements of D.

Other functions of interest that directly creates a sparse matrix, are spdiag or its generalization spdiags, that can take the definition of the diagonals of the matrix and create the sparse matrix that corresponds to this. For example

     s = spdiag (randn(1,n), -1);

creates a sparse (N+1)-by-(N+1) sparse matrix with a single diagonal defined.

The recommended way for the user to create a sparse matrix, is to create two vectors contain the row and column index of the data and a third vector of the same size containing the data to be stored. For example

      function x = foo (r, j)
       idx = randperm (r);
       x = ([zeros(r-2,1); rand(2,1)])(idx);
      endfunction

      ri = [];
      ci = [];
      d = [];

      for j=1:c
         dtmp = foo (r, j);  # Returns vector of length r with (:,j) values
         idx = find (dtmp != 0.);
         ri = [ri; idx];
         ci = [ci; j*ones(length(idx),1)];
         d = [d; dtmp(idx)];
      endfor
      s = sparse (ri, ci, d, r, c);

creates an R-by-C sparse matrix with a random distribution of 2 elements per row. The elements of the vectors do not need to be sorted in any particular order as OCTAVE will store them prior to storing the data. However, per sorting the data will make teh creation of the sparse matrix faster.

The function spconvert takes a three or four column real matrix. The first two columns represent the row and column index respectively and the third and four columns, the real and imaginary parts of the sparse matrix. The matrix can contain zero elements and the elements can be sorted in any order. Adding zero elements is a convenient way to define the size of the sparse matrix. For example

     s = spconvert ([1 2 3 4; 1 3 4 4; 1 2 3 0]')
     => Compressed Column Sparse (rows=4, cols=4, nnz=3)
           (1 , 1) -> 1
           (2 , 3) -> 2
           (3 , 4) -> 3

An example of creating and filling a matrix might be

     k = 5;
     nz = r * k;
     s = spalloc (r, c, nz)
     for j = 1:c
       idx = randperm (r);
       s (:, j) = [zeros(r - k, 1); rand(k, 1)] (idx);
     endfor

It should be noted, that due to the way that the OCTAVE assignment functions are written that the assignment will reallocate the memory used by the sparse matrix at each iteration. Therefore the spalloc function ignores the NZ argument and does not preassign the memory for the matrix. Therefore, it is vitally important that code using to above structure should be as vectorized as possible to minimize the number of assignments and reduce the number of memory allocations.

The above problem can be avoided in oct-files. However, the construction of a sparse matrix from an oct-file is more complex than can be discussed in this brief introduction, and you are referred to section Oct-Files, to have a full description of the techniques involved.

Basic Operators and Functions on Sparse Matrices
------------------------------------------------

Operators and Functions
.......................

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The Return Types of Operators and Functions
...........................................

The two basic reason to use sparse matrices are to reduce the memory usage and to not have to do calculations on zero elements. The two are closely related in that the computation time on a sparse matrix operator or function is roughly linear with the numberof non-zero elements.

Therefore, there is a certain density of non-zero elements of a matrix where it no longer makes sense to store it as a sparse matrix, but rather as a full matrix. For this reason operators and functions that have a high probability of returning a full matrix will always return one. For example adding a scalar constant to a sparse matrix will almost always make it a full matrix, and so the example

     speye(3) + 0
     =>   1  0  0
       0  1  0
       0  0  1

returns a full matrix as can be seen. Additionally all sparse functions test the amount of memory occupied by the sparse matrix to see if the amount of storage used is larger than the amount used by the full equivalent. Therefore speye (2) * 1 will return a full matrix as the memory used is smaller for the full version than the sparse version.

As all of the mixed operators and functions between full and sparse matrices exist, in general this does not cause any problems. However, one area where it does cause a problem is where a sparse matrix is promoted to a full matrix, where subsequent operations would resparsify the matrix. Such cases as rare, but can be artificially created, for example (fliplr(speye(3)) + speye(3)) - speye(3) gives a full matrix when it should give a sparse one. In general, where such cases occur, they impose only a small memory penalty.

There is however one known case where this behaviour of OCTAVE's sparse matrices will cause a problem. That is in the handling of the diag function. Whether diag returns a sparse or full matrix depending on the type of its input arguments. So

      a = diag (sparse([1,2,3]), -1);

should return a sparse matrix. To ensure this actually happens, the sparse function, and other functions based on it like speye, always returns a sparse matrix, even if the memory used will be larger than its full representation.

Mathematical Considerations
...........................

The attempt has been made to make sparse matrices behave in exactly the same manner as there full counterparts. However, there are certain differences and especially differences with other products sparse implementations.

Firstly, the "./" and ".^" operators must be used with care. Consider what the examples

       s = speye (4);
       a1 = s .^ 2;
       a2 = s .^ s;
       a3 = s .^ -2;
       a4 = s ./ 2;
       a5 = 2 ./ s;
       a6 = s ./ s;

will give. The first example of S raised to the power of 2 causes no problems. However S raised element-wise to itself involves a a large number of terms 0 .^ 0 which is 1. There S .^ S is a full matrix.

Likewise S .^ -2 involves terms terms like 0 .^ -2 which is infinity, and so S .^ -2 is equally a full matrix.

For the "./" operator S ./ 2 has no problems, but 2 ./ S involves a large number of infinity terms as well and is equally a full matrix. The case of S ./ S involves terms like 0 ./ 0 which is a NaN and so this is equally a full matrix with the zero elements of S filled with NaN values.

The above behaviour is consistent with full matrices, but is not consistent with sparse implementations in other products.

A particular problem of sparse matrices comes about due to the fact that as the zeros are not stored, the sign-bit of these zeros is equally not stored... In certain cases the sign-bit of zero is important. For example

      a = 0 ./ [-1, 1; 1, -1];
      b = 1 ./ a
      => -Inf            Inf
          Inf           -Inf
      c = 1 ./ sparse (a)
      =>  Inf            Inf
          Inf            Inf

To correct this behaviour would mean that zero elements with a negative sign-bit would need to be stored in the matrix to ensure that their sign-bit was respected. This is not done at this time, for reasons of efficient, and so the user is warned that calculations where the sign-bit of zero is important must not be done using sparse matrices.

Also discuss issues of fill-in. Discuss symamd etc, and mention randperm that is included  elsewhere in the docs...

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Finding out Information about Sparse Matrices
---------------------------------------------

Talk about the spy, spstats, nnz, spparms, etc function

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Graphs are their use with Sparse Matrices
=========================================

Someone who knows more about this than me should write this...

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Linear Algebra on Sparse Matrices
=================================

OCTAVE includes a poly-morphic solver for sparse matrices, where the exact solver used to factorize the matrix, depends on the properties of the sparse matrix, itself. As the cost of determining the matrix type is small relative to the cost of factorizing the matrix itself, the matrix type is re-determined each time it is used in a linear equation.

The selection tree for how the linear equation is solve is

 1. If the matrix is not square go to 9.

 2. If the matrix is diagonal, solve directly and goto 9

 3. If the matrix is a permuted diagonal, solve directly taking into account the permutations. Goto 9

 4. If the matrix is banded and if the band density is less than that given by spparms ("bandden") continue, else goto 5.

      a. If the matrix is tridiagonal and the right-hand side is not sparse continue, else goto 4b.

           1. If the matrix is hermitian, with a positive real diagonal, attempt Cholesky factorization using LAPACK xPTSV.

           2. If the above failed or the matrix is not hermitian with a positive real diagonal use Gaussian elimination with pivoting using LAPACK xGTSV, and goto 9.

      b. If the matrix is hermitian with a positive real diagonal, attempt Cholesky factorization using LAPACK xPBTRF.

      c. if the above failed or the matrix is not hermitian with a positive real diagonal use Gaussian elimination with pivoting using LAPACK xGBTRF, and goto 9.

 5. If the matrix is upper or lower triangular perform a sparse forward or backward subsitution, and goto 9

 6. If the matrix is a permuted upper or lower triangular matrix, perform a sparse forward or backward subsitution, and goto 9

     FIXME: Detection of permuted triangular matrices not written yet, and so the code itself is not tested either

 7. If the matrix is hermitian with a real positive diagonal, attempt sparse Cholesky factorization.

     FIXME: Detection of positive definite matrices written and tested, but Cholesky factorization isn't yet written

 8. If the sparse Cholesky factorization failed or the matrix is not hermitian with a real positive diagonal, factorize using UMFPACK.

 9. If the matrix is not square, or any of the previous solvers flags a singular or near singular matrix, find a minimum norm solution

     FIXME: QR solvers not yet written.

The band density is defined as the number of non-zero values in the matrix divided by the number of non-zero values in the matrix. The banded matrix solvers can be entirely disabled by using spparms to set bandden to 1 (i.e. spparms ("bandden", 1)).

All of the solvers above, expect the banded solvers, calculate an estimate of the condition number. This can be used to detect numerical stability problems in the solution and force a minimum norm solution to be used. However, for narrow banded matrices, the cost of calculating the condition number is significant, and can in fact exceed the cost of factoring the matrix. Therefore the condition number is not calculated for banded matrices, and therefore unless the factorization is exactly singular, these numerical instabilities won't be detected. In cases where, this might be a problem the user is recommended to disable the banded solvers as above, at a significant cost in terms of speed.

Iterative Techniques applied to sparse matrices
===============================================

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Using Sparse Matrices in Oct-files
==================================

An oct-file is a means of writing an OCTAVE function in a compilable language like C++, rather than as a script file. This results in a significant acceleration in the code.  It is not the purpose of this section to discuss how to write an oct-file, or discuss what they are. There are already two  (Paul Thomas "Dal Segno al Coda - The octave dynamically linked function cookbook", http://perso.wanadoo.fr/prthomas/intro.html, and Cristophe Spiel "Del Coda Al Fine - Pushing Octave's Limits", http://octave.sourceforge.net/coda/coda.pdf) very good references on oct-files themselves. Users who are not familiar with oct-files are urged to read these references to fully understand this chapter. The examples discussed here assume that the oct-file is written entirely in C++.

There are three classes of sparse objects that are of interest to the user.

SparseMatrix
     A double precision sparse matrix class

SparseComplexMatrix
     A Complex sparse matrix class

SparseBoolMatrix
     A boolen sparse matrix class

All of these classes inherit from the Sparse<T> template class, and so all have similar capabilities and usage. The Sparse<T> class was based on OCTAVE Array<T> class, and so users familar with OCTAVE's Array classes will be comfortable with the use of the sparse classes.

The sparse classes will not be entirely described in this section, due to their similar with the existing Array classes. However, there are a few differences due the different nature of sparse objects, and these will be described. Firstly, although it is fundamentally possible to have N-dimensional sparse objects, the OCTAVE sparse classes do not allow them at this time. So all operations of the sparse classes must be 2-dimensional.  This means that in fact SparseMatrix is similar to OCTAVE's Matrix class rather than its NDArray class.

The Differences between the Array and Sparse Classes
----------------------------------------------------

The number of elements in a sparse matrix is considered to be the number of non-zero elements rather than the product of the dimensions. Therefore

       SparseMatrix sm;
       ...
       int nel = sm.nelem ();

returns the number of non-zero elements. If the user really requires the number of elements in the matrix, including the non-zero elements, they should use numel rather than nelem. Note that for very large matrices, where the product of the two dimensions is large that the representation of the an unsigned int, then numel can overflow. An example is speye(1e6) which will create a matrix with a million rows and columns, but only a million non-zero elements. Therefore the number of rows by the number of columns in this case is more than two hundred times the maximum value that can be represented by an unsigned int. The use of numel should therefore be avoided useless it is known it won't overflow.

Extreme care must be take with the elem method and the "()" operator, which perform basically the same function. The reason is that if a sparse object is non-const, then OCTAVE will assume that a request for a zero element in a sparse matrix is in fact a request to create this element so it can be filled. Therefore a piece of code like

       SparseMatrix sm;
       ...
       for (int j = 0; j < nc; j++)
         for (int i = 0; i < nr; i++)
           std::cerr << " (" << i << "," << j << "): " << sm(i,j)
                     << std::endl;

is a great way of turning the sparse matrix into a dense one, and a very slow way at that since it reallocates the sparse object at each zero element in the matrix.

An easy way of preventing the above from hapening is to create a temporary constant version of the sparse matrix. Note that only the container for the sparse matrix will be copied, while the actual representation of the data will be shared between the two versions of the sparse matrix. So this is not a costly operation. For example, the above would become

       SparseMatrix sm;
       ...
       const SparseMatrix tmp (sm);
       for (int j = 0; j < nc; j++)
         for (int i = 0; i < nr; i++)
           std::cerr << " (" << i << "," << j << "): " << tmp(i,j)
                     << std::endl;

Finally, as the sparse types aren't just represented as a contiguous block of memory, the fortran_vec method of the Array<T> is not available. It is however replaced by three seperate methods ridx, cidx and data, that access the raw compressed column format that the OCTAVE sparse matrices are stored in. Additionally, these methods can be used in a manner similar to elem, to allow the matrix to be accessed or filled. However, in that case it is up to the user to repect the sparse matrix compressed column format discussed previous.

Creating Spare Matrices in Oct-Files
------------------------------------

The user has several alternatives in how to create a sparse matrix. They can first create the data as three vectors representing the row and column indexes and the data, and from those create the matrix. Or alternatively, they can create a sparse matrix with the appropriate amount of space and then fill in the values. Both techniques have their advantages and disadvantages.

An example of how to create a small sparse matrix with the first technique might be seen the example

       int nz = 4, nr = 3, nc = 4;
       ColumnVector ridx (nz);
       ColumnVector cidx (nz);
       ColumnVector data (nz);

       ridx(0) = 0; ridx(1) = 0; ridx(2) = 1; ridx(3) = 2;
       cidx(0) = 0; cidx(1) = 1; cidx(2) = 3; cidx(3) = 3;
       data(0) = 1; data(1) = 2; data(2) = 3; data(3) = 4;

       SparseMatrix sm (data, ridx, cidx, nr, nc);

which creates the matrix given in section Storage. Note that the compressed matrix format is not used at the time of the creation of the matrix itself, however it is used internally.

As previously mentioned, the values of the sparse matrix are stored in increasing column-major ordering. Although the data passed by the user does not need to respect this requirement, the pre-sorting the data significantly speeds up the creation of the sparse matrix.

The disadvantage of this technique of creating a sparse matrix is that there is a brief time where two copies of the data exists. Therefore for extremely memory constrained problems this might not be the right technique to create the sparse matrix.

The alternative is to first create the sparse matrix with the desired number of non-zero elements and then later fill those elements in. The easiest way to do this is

       int nz = 4, nr = 3, nc = 4;
       SparseMatrix sm (nr, nc, nz);
       sm(0,0) = 1; sm(0,1) = 2; sm(1,3) = 3; sm(2,3) = 4;

That creates the same matrix as previously. Again, although it is not strictly necessary, it is significantly faster if the sparse matrix is created in this manner that the elements are added in column-major ordering. The reason for this is that if the elements are inserted at the end of the current list of known elements then no element in the matrix needs to be moved to allow the new element to be inserted. Only the column indexes need to be updated.

There are a few further points to note about this technique of creating a sparse matrix. Firstly, it is not illegal to create a sparse matrix with fewer elements than are actually inserted in the matrix. Therefore

       int nz = 4, nr = 3, nc = 4;
       SparseMatrix sm (nr, nc, 0);
       sm(0,0) = 1; sm(0,1) = 2; sm(1,3) = 3; sm(2,3) = 4;

is perfectly legal. However it is a very bad idea. The reason is that as each new element is added to the sparse matrix the space allocated to it is increased by reallocating the memory. This is an expensive operation, that will significantly slow this means of creating a sparse matrix. Furthermore, it is not illegal to create a sparse matrix with too much storage, so having NZ above equaling 6 is also legal. The disadvantage is that the matrix occupies more memory than strictly needed.

It is not always easy to known the number of non-zero elements prior to filling a matrix. For this reason the additional storage for the sparse matrix can be removed after its creation with the maybe_compress function. Furthermore, the maybe_compress can deallocate the unused storage, but it can equally remove zero elements from the matrix.  The removal of zero elements from the matrix is controlled by setting the argument of the maybe_compress function to be 'true'. However, the cost of removing the zeros is high because it implies resorting the elements. Therefore, if possible it is better is the user doesn't add the zeros in the first place. An example of the use of maybe_compress is

       int nz = 6, nr = 3, nc = 4;
       SparseMatrix sm1 (nr, nc, nz);
       sm1(0,0) = 1; sm1(0,1) = 2; sm1(1,3) = 3; sm1(2,3) = 4;
       sm1.maybe_compress ();  // No zero elements were added

       SparseMatrix sm2 (nr, nc, nz);
       sm2(0,0) = 1; sm2(0,1) = 2; sm(0,2) = 0; sm(1,2) = 0;
       sm1(1,3) = 3; sm1(2,3) = 4;
       sm2.maybe_compress (true);  // Zero elements were added

The use of the maybe_compress function should be avoided if possible, as it will slow the creation of the matrices.

A third means of creating a sparse matrix is to work directly with the data in compressed row format. An example of this technique might be

       octave_value arg;

       ...

       int nz = 6, nr = 3, nc = 4;   // Assume we know the max no nz
       SparseMatrix sm (nr, nc, nz);
       Matrix m = arg.matrix_value ();

       int ii = 0;
       sm.cidx (0) = 0;
       for (int j = 1; j < nc; j++)
         {
           for (int i = 0; i < nr; i++)
             {
               double tmp = foo (m(i,j));
               if (tmp != 0.)
                 {
                   sm.data(ii) = tmp;
                   sm.ridx(ii) = i;
                   ii++;
                 }
             }
           sm.cidx(j+1) = ii;
        }
       sm.maybe_mutate ();  // If don't know a-priori the final no of nz.

which is probably the most efficient means of creating the sparse matrix.

Finally, it might sometimes arise that the amount of storage initially created is insufficient to completely store the sparse matrix. Therefore, the method change_capacity exists to reallocate the sparse memory. The above example would then be modified as

       octave_value arg;

       ...

       int nz = 6, nr = 3, nc = 4;   // Assume we know the max no nz
       SparseMatrix sm (nr, nc, nz);
       Matrix m = arg.matrix_value ();

       int ii = 0;
       sm.cidx (0) = 0;
       for (int j = 1; j < nc; j++)
         {
           for (int i = 0; i < nr; i++)
             {
               double tmp = foo (m(i,j));
               if (tmp != 0.)
                 {
                   if (ii == nz)
                     {
                       nz += 2;   // Add 2 more elements
                       sm.change_capacity (nz);
                     }
                   sm.data(ii) = tmp;
                   sm.ridx(ii) = i;
                   ii++;
                 }
             }
           sm.cidx(j+1) = ii;
        }
       sm.maybe_mutate ();  // If don't know a-priori the final no of nz.

Note that both increasing and decreasing the number of non-zero elements in a sparse matrix is expensive, as it involves memory reallocation. Also as parts of the matrix, though not its entirety, exist as the old and new copy at the same time, additional memory is needed. Therefore if possible this should be avoided.

Using Sparse Matrices in Oct-Files
----------------------------------

Most of the same operators and functions on sparse matrices that are available from the OCTAVE are equally available with oct-files. The basic means of extracting a sparse matrix from an octave_value and returning them as an octave_value, can be seen in the following example

        octave_value_list retval;

        SparseMatrix sm = args(0).sparse_matrix_value ();
        SparseComplexMatrix scm = args(1).sparse_complex_matrix_value ();
        SparseBoolMatrix sbm = args(2).sparse_bool_matrix_value ();

        ...

        retval(2) = sbm;
        retval(1) = scm;
        retval(0) = sm;

The conversion to an octave-value is handled by the sparse octave_value constructors, and so no special care is needed.

Licensing of Third Party Software
=================================

There are several third party software packages used by the OCTAVE sparse matrix.

COLAMD
     is used for the colamd and symamd functions.

     Authors
          The authors of the code itself are Stefan I. Larimore and Timothy A. Davis (davis@cise.ufl.edu), University of Florida.  The algorithm was developed in collaboration with John Gilbert, Xerox PARC, and Esmond Ng, Oak Ridge National Laboratory.

     License
          Copyright © 1998-2003 by the University of Florida. All Rights Reserved.

          THIS MATERIAL IS PROVIDED AS IS, WITH ABSOLUTELY NO WARRANTY EXPRESSED OR IMPLIED.  ANY USE IS AT YOUR OWN RISK.

          Permission is hereby granted to use, copy, modify, and/or distribute this program, provided that the Copyright, this License, and the Availability of the original version is retained on all copies and made accessible to the end-user of any code or package that includes COLAMD or any modified version of COLAMD.

     Availability
          http://www.cise.ufl.edu/research/sparse/colamd/

UMFPACK
     is used in various places with the sparse types, including the LU decomposition and solvers.

     License
          UMFPACK Version 4.4 (Jan. 28, 2005), Copyright © 2005 by Timothy A. Davis.  All Rights Reserved.

          Your use or distribution of UMFPACK or any modified version of UMFPACK implies that you agree to this License.

          THIS MATERIAL IS PROVIDED AS IS, WITH ABSOLUTELY NO WARRANTY EXPRESSED OR IMPLIED.  ANY USE IS AT YOUR OWN RISK.

          Permission is hereby granted to use or copy this program, provided that the Copyright, this License, and the Availability of the original version is retained on all copies.  User documentation of any code that uses UMFPACK or any modified version of UMFPACK code must cite the Copyright, this License, the Availability note, and 'Used by permission.' Permission to modify the code and to distribute modified code is granted, provided the Copyright, this License, and the Availability note are retained, and a notice that the code was modified is included.  This software was developed with support from the National Science Foundation, and is provided to you free of charge.

     Availability
          http://www.cise.ufl.edu/research/sparse/umfpack/

Function Reference
==================

colamd
......

(colamd)

colperm
.......

(colperm)

dmperm
......

(dmperm)

etree
.....

(etree)

full
....

(full)

issparse
........

(issparse)

luinc
.....

(luinc)

nnz
...

(nnz)

nonzeros
........

(nonzeros)

nzmax
.....

(nzmax)

spalloc
.......

(spalloc)

sparse
......

(sparse)

spatan2
.......

(spatan2)

spconvert
.........

(spconvert)

spcumprod
.........

(spcumprod)

spcumsum
........

(spcumsum)

spdet
.....

(spdet)

spdiag
......

(spdiag)

spdiags
.......

(spdiags)

speye
.....

(speye)

spfind
......

(spfind)

spfun
.....

(spfun)

spinv
.....

(spinv)

splu
....

(splu)

spmax
.....

(spmax)

spmin
.....

(spmin)

spones
......

(spones)

spparms
.......

(spparms)

spprod
......

(spprod)

sprand
......

(sprand)

sprandn
.......

(sprandn)

spreshape
.........

(spreshape)

spstats
.......

(spstats)

spsum
.....

(spsum)

spsumsq
.......

(spsumsq)

spy
...

(spy)

symamd
......

(symamd)

symbfact
........

(symbfact)

symrcm
......

(symrcm)