@c DO NOT EDIT!  Generated automatically by munge-texi.

@c Copyright (C) 2004, 2005 David Bateman
@c This is part of the Octave manual.
@c For copying conditions, see the file gpl.texi.

@node Sparse Matrices 
@chapter Sparse Matrices

@menu
* Basics:: The Creation and Manipulation of Sparse Matrices
* Graph Theory:: Graphs are their use with Sparse Matrices
* Sparse Linear Algebra:: Linear Algebra on Sparse Matrices
* Iterative Techniques:: Iterative Techniques applied to Sparse Matrices
* Oct-Files:: Using Sparse Matrices in Oct-files
* License:: Licensing of Third Party Software
* Function Reference:: Documentation from the Specific Sparse Functions
@end menu

@node Basics, Graph Theory, Sparse Matrices, Sparse Matrices
@section 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"

@menu
* Storage:: Storage of Sparse Matrices
* Creation:: Creating Sparse Matrices
* Operators and Functions:: Basic Operators and Functions on Sparse Matrices
* Information:: Finding out Information about Sparse Matrices
@end menu

@node Storage, Creation, Basics, Basics
@subsection 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 @footnote{Youcef Saad "SPARSKIT: A basic toolkit
for sparse matrix computation", 1994,
@url{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.

@example
  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));
@end example

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

@example
@group
    1   2   0  0
    0   0   0  3
    0   0   0  4
@end group
@end example

The non-zero elements of this matrix are

@example
@group
   (1, 1)  @result{} 1
   (1, 2)  @result{} 2
   (2, 4)  @result{} 3
   (3, 4)  @result{} 4
@end group
@end example

This will be stored as three vectors @var{cidx}, @var{ridx} and @var{data},
representing the column indexing, row indexing and data respectively. The
contents of these three vectors for the above matrix will be

@example
@group
  @var{cidx} = [0, 2, 2, 4]
  @var{ridx} = [0, 0, 1, 2]
  @var{data} = [1, 2, 3, 4]
@end group
@end example

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 
@var{i}-th column is given by @code{@var{cidx} (@var{i} + 1) - 
@var{cidx} (@var{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.

@node Creation, Operators and Functions, Storage, Basics
@subsection Creating Sparse Matrices

There are several means to create sparse matrix.

@table @asis
@item Returned from a function
There are many functions that directly return sparse matrices. These include
@dfn{speye}, @dfn{sprand}, @dfn{spdiag}, etc.
@item Constructed from matrices or vectors
The function @dfn{sparse} allows a sparse matrix to be constructed from 
three vectors representing the row, column and data. ALternatively, the
function @dfn{spconvert} uses a three column matrix format to allow easy
importation of data from elsewhere.
@item Created and then filled
The function @dfn{sparse} or @dfn{spalloc} can be used to create an empty
matrix that is then filled by the user
@item From a user binary program
The user can directly create the sparse matrix within an oct-file.
@end table

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
@code{speye (@var{n})} or @code{speye (@var{r}, @var{c})}, which
creates an @var{n}-by-@var{n} or @var{r}-by-@var{c} sparse identity
matrix.

Another typical sparse matrix that is often needed is a random distribution
of random elements. The functions @dfn{sprand} and @dfn{sprandn} perform
this for uniform and random distributions of elements. They have exactly
the same calling convention as @code{sprand (@var{r}, @var{c}, @var{d})},
which creates an @var{r}-by-@var{c} sparse matrix a density of filled
elements of @var{d}.

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

@c FIXME, when spdiag is overloaded to diag the below won't work. 

@example
s = spdiag (randn(1,n), -1);
@end example

creates a sparse (@var{n}+1)-by-(@var{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

@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);
@end example

creates an @var{r}-by-@var{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 @dfn{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

@example
s = spconvert ([1 2 3 4; 1 3 4 4; 1 2 3 0]')
@result{} Compressed Column Sparse (rows=4, cols=4, nnz=3)
      (1 , 1) -> 1
      (2 , 3) -> 2
      (3 , 4) -> 3
@end example

An example of creating and filling a matrix might be

@example
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
@end example

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
@dfn{spalloc} function ignores the @var{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 @ref{Oct-Files}, to have a full description of the techniques
involved.

@node Operators and Functions, Information, Creation, Basics
@subsection Basic Operators and Functions on Sparse Matrices

@menu
* Functions:: Operators and Functions
* ReturnType:: The Return Types of Operators and Functions
* MathConsiderations:: Mathematical Considerations
@end menu

@node Functions, ReturnType, Operators and Functions, Operators and Functions
@subsubsection Operators and Functions

WRITE ME

@node ReturnType, MathConsiderations, Functions, Operators and Functions
@subsubsection 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

@example
speye(3) + 0
@result{}   1  0  0
  0  1  0
  0  0  1
@end example

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 @code{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 @code{(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
@dfn{diag} function. Whether @dfn{diag} returns a sparse or full matrix
depending on the type of its input arguments. So 

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

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

@node MathConsiderations, , ReturnType, Operators and Functions
@subsubsection 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

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

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

Likewise @code{@var{s} .^ -2} involves terms terms like @code{0 .^ -2} which
is infinity, and so @code{@var{s} .^ -2} is equally a full matrix.

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

@example
 a = 0 ./ [-1, 1; 1, -1];
 b = 1 ./ a
 @result{} -Inf            Inf
     Inf           -Inf
 c = 1 ./ sparse (a)
 @result{}  Inf            Inf
     Inf            Inf
@end example
 
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...

WRITE ME

@node Information, , Operators and Functions, Basics
@subsection Finding out Information about Sparse Matrices

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

WRITE ME

@node Graph Theory, Sparse Linear Algebra, Basics, Sparse Matrices
@section Graphs are their use with Sparse Matrices

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

WRITE ME

@node Sparse Linear Algebra, Iterative Techniques, Graph Theory, Sparse Matrices
@section 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. The cost of determining the matrix type
is small relative to the cost of factorizing the matrix itself, but in any
case the matrix type is cached once it is calculated, so that it is not
re-determined each time it is used in a linear equation.

The selection tree for how the linear equation is solve is

@enumerate 1
@item If the matrix is not square go to 9.

@item If the matrix is diagonal, solve directly and goto 9

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

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

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

@enumerate
@item If the matrix is hermitian, with a positive real diagonal, attempt
      Cholesky factorization using @sc{Lapack} xPTSV.

@item If the above failed or the matrix is not hermitian with a positive
      real diagonal use Gaussian elimination with pivoting using 
      @sc{Lapack} xGTSV, and goto 9.
@end enumerate

@item If the matrix is hermitian with a positive real diagonal, attempt
      Cholesky factorization using @sc{Lapack} xPBTRF.

@item if the above failed or the matrix is not hermitian with a positive
      real diagonal use Gaussian elimination with pivoting using 
      @sc{Lapack} xGBTRF, and goto 9.
@end enumerate

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

@item If the matrix is a upper triangular matrix with column permutations
or lower triangular matrix with row permutations, perform a sparse forward 
or backward subsitution, and goto 9

@item 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

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

@item 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.

@end enumerate

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 @dfn{spparms} to set @code{bandden}
to 1 (i.e. @code{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.

The user can force the type of the matrix with the @code{matrix_type}
function. This overcomes the cost of discovering the type of the matrix.
However, it should be noted incorrectly identifying the type of the matrix
will lead to unpredictable results, and so @code{matrix_type} should be
use dwith care.

@node Iterative Techniques, Oct-Files, Sparse Linear Algebra, Sparse Matrices
@section Iterative Techniques applied to sparse matrices

WRITE ME

@node Oct-Files, License, Iterative Techniques, Sparse Matrices
@section 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 @footnote{Paul Thomas "Dal Segno al Coda 
- The octave dynamically linked function cookbook", 
@url{http://perso.wanadoo.fr/prthomas/intro.html}, and Cristophe Spiel 
"Del Coda Al Fine - Pushing Octave's Limits", 
@url{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.

@table @asis
@item SparseMatrix
A double precision sparse matrix class
@item SparseComplexMatrix
A Complex sparse matrix class
@item SparseBoolMatrix
A boolen sparse matrix class
@end table

All of these classes inherit from the @code{Sparse<T>} template class,
and so all have similar capabilities and usage. The @code{Sparse<T>}
class was based on Octave @code{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 @code{SparseMatrix} is
similar to Octave's @code{Matrix} class rather than its
@code{NDArray} class.

@menu
* OctDifferences:: The Differences between the Array and Sparse Classes
* OctCreation:: Creating Spare Matrices in Oct-Files
* OctUse:: Using Sparse Matrices in Oct-Files
@end menu

@node OctDifferences, OctCreation, Oct-Files, Oct-Files
@subsection 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

@example
  SparseMatrix sm;
  @dots{}
  int nel = sm.nelem ();
@end example

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 @code{numel} rather than @code{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 @code{numel} can overflow.
An example is @code{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 @code{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

@example
  SparseMatrix sm;
  @dots{}
  for (int j = 0; j < nc; j++)
    for (int i = 0; i < nr; i++)
      std::cerr << " (" << i << "," << j << "): " << sm(i,j) 
                << std::endl;
@end example

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

@example
  SparseMatrix sm;
  @dots{}
  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;
@end example

Finally, as the sparse types aren't just represented as a contiguous
block of memory, the @code{fortran_vec} method of the @code{Array<T>}
is not available. It is however replaced by three seperate methods
@code{ridx}, @code{cidx} and @code{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 @code{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.

@node OctCreation, OctUse, OctDifferences, Oct-Files
@subsection 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

@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);
@end example

which creates the matrix given in section @ref{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 

@example 
  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;
@end example

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

@example 
  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;
@end example

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 @var{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
@dfn{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 @dfn{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 @dfn{maybe_compress} is

@example
  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
@end example

The use of the @dfn{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

@c Note the @verbatim environment is a relatively new addition to texinfo.
@c Therefore use the @example environment and replace @, with @@, 
@c { with @{, etc

@example
  octave_value arg;
  
  @dots{}

  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.
@end example

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 @code{change_capacity} exists to reallocate the sparse memory.
The above example would then be modified as 

@example
  octave_value arg;
  
  @dots{}

  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.
@end example

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.

@node OctUse, , OctCreation, Oct-Files
@subsection 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 @code{octave_value}
and returning them as an @code{octave_value}, can be seen in the
following example

@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 ();

   @dots{}

   retval(2) = sbm;
   retval(1) = scm;
   retval(0) = sm;
@end example

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

@node License, Function Reference, Oct-Files, Sparse Matrices
@section Licensing of Third Party Software

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

@table @asis
@item COLAMD
is used for the @dfn{colamd} and @dfn{symamd} functions.

@table @asis
@item 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.

@item License
Copyright @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. 

@item Availability
@url{http://www.cise.ufl.edu/research/sparse/colamd/}
@end table

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

@table @asis
@item License
UMFPACK Version 4.4 (Jan. 28, 2005), Copyright @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.

@item Availability
@url{http://www.cise.ufl.edu/research/sparse/umfpack/}
@end table
@end table

@node Function Reference, , License, Sparse Matrices
@section Function Reference

@iftex
@subsection Functions by Category
@subsubsection Generate sparse matrix
@table @asis
@item spdiags
A generalization of the function `spdiag'.
@item speye
Returns a sparse identity matrix.
@item sprand
Generate a random sparse matrix.
@item sprandn
Generate a random sparse matrix.
@item sprandsym
@emph{Not implemented}
@end table
@subsubsection Sparse matrix conversion
@table @asis
@item full
returns a full storage matrix from a sparse one See also: sparse
@item sparse
SPARSE: create a sparse matrix
@item spconvert
This function converts for a simple sparse matrix format easily produced by other programs into Octave's internal sparse format.
@item spfind
SPFIND: a sparse version of the find operator 1.
@end table
@subsubsection Manipulate sparse matrices
@table @asis
@item issparse
Return 1 if the value of the expression EXPR is a sparse matrix.
@item nnz
returns number of non zero elements in SM See also: sparse
@item nonzeros
Returns a vector of the non-zero values of the sparse matrix S
@item nzmax
Returns the amount of storage allocated to the sparse matrix SM.
@item spalloc
Returns an empty sparse matrix of size R-by-C.
@item spfun
Compute `f(X)' for the non-zero values of X This results in a sparse matrix with the same structure as X.
@item spones
Replace the non-zero entries of X with ones.
@item spy
Plot the sparsity pattern of the sparse matrix X
@end table
@subsubsection Graph Theory
@table @asis
@item etree
Returns the elimination tree for the matrix S.
@item etreeplot
@emph{Not implemented}
@item gplot
@emph{Not implemented}
@item treelayout
@emph{Not implemented}
@item treeplot
@emph{Not implemented}
@end table
@subsubsection Sparse matrix reordering
@table @asis
@item colamd
Column approximate minimum degree permutation.
@item colperm
Returns the column permutations such that the columns of `S (:, P)' are ordered in terms of increase number of non-zero elements.
@item dmperm
Perform a Deulmage-Mendelsohn permutation on the sparse matrix S.
@item symamd
For a symmetric positive definite matrix S, returns the permutation vector p such that `S (P, P)' tends to have a sparser Cholesky factor than S.
@item symrcm
Returns the Reverse Cuthill McKee reordering of the sparse matrix S.
@end table
@subsubsection Linear algebra
@table @asis
@item cholinc
@emph{Not implemented}
@item condest
@emph{Not implemented}
@item eigs
@emph{Not implemented}
@item matrix_type
Identify the matrix type or mark a matrix as a particular type.
@item normest
@emph{Not implemented}
@item spdet
Compute the determinant of sparse matrix A using UMFPACK.
@item spinv
Compute the inverse of the square matrix A.
@item spkron
Form the kronecker product of two sparse matrices.
@item splu
Compute the LU decomposition of the sparse matrix A, using subroutines from UMFPACK.
@item sprank
@emph{Not implemented}
@item svds
@emph{Not implemented}
@end table
@subsubsection Iterative techniques
@table @asis
@item bicg
@emph{Not implemented}
@item bicgstab
@emph{Not implemented}
@item cgs
@emph{Not implemented}
@item gmres
@emph{Not implemented}
@item luinc
Produce the incomplete LU factorization of the sparse matrix A.
@item lsqr
@emph{Not implemented}
@item minres
@emph{Not implemented}
@item pcg
@emph{Not implemented}
@item pcr
@emph{Not implemented}
@item qmr
@emph{Not implemented}
@item symmlq
@emph{Not implemented}
@end table
@subsubsection Miscellaneous
@table @asis
@item spaugment
@emph{Not implemented}
@item spparms
Sets or displays the parameters used by the sparse solvers and factorization functions.
@item symbfact
Performs a symbolic factorization analysis on the sparse matrix S.
@item spstats
Return the stats for the non-zero elements of the sparse matrix S COUNT is the number of non-zeros in each column, MEAN is the mean of the non-zeros in each column, and VAR is the variance of the non-zeros in each column
@item spprod
Product of elements along dimension DIM.
@item spcumprod
Cumulative product of elements along dimension DIM.
@item spcumsum
Cumulative sum of elements along dimension DIM.
@item spsum
Sum of elements along dimension DIM.
@item spsumsq
Sum of squares of elements along dimension DIM.
@item spmin
For a vector argument, return the minimum value.
@item spmax
For a vector argument, return the maximum value.
@item spatan2
Compute atan (Y / X) for corresponding sparse matrix elements of Y and X.
@item spdiag
Return a diagonal matrix with the sparse vector V on diagonal K.
@item spreshape
Return a sparse matrix with M rows and N columns whose elements are taken from the sparse matrix A.
@end table

@subsection Functions Alphabetically
@end iftex

@menu
* colamd::	Column approximate minimum degree permutation.
* colperm::	Returns the column permutations such that the columns of `S
		(:, P)' are ordered in terms of increase number of non-zero
		elements.
* dmperm::	Perfrom a Deulmage-Mendelsohn permutation on the sparse
		matrix S.
* etree::	Returns the elimination tree for the matrix S.
* full::	returns a full storage matrix from a sparse one See also:
		sparse
* issparse::	Return 1 if the value of the expression EXPR is a sparse
		matrix.
* luinc::	Produce the incomplete LU factorization of the sparse 
		A.
* matrix_type:: Identify the matrix type or mark a matrix as a particular
		type.
* nnz:: 	returns number of non zero elements in SM See also: sparse
* nonzeros::	Returns a vector of the non-zero values of the sparse
		matrix S
* nzmax::	Returns the amount of storage allocated to the sparse
		matrix SM.
* spalloc::	Returns an empty sparse matrix of size R-by-C.
* sparse::	SPARSE: create a sparse matrix
* spatan2::	Compute atan (Y / X) for corresponding sparse matrix
		elements of Y and X.
* spconvert::	This function converts for a simple sparse matrix format
		easily produced by other programs into Octave's internal
		sparse format.
* spcumprod::	Cumulative product of elements along dimension DIM.
* spcumsum::	Cumulative sum of elements along dimension DIM.
* spdet::	Compute the determinant of sparse matrix A using UMFPACK.
* spdiag::	Return a diagonal matrix with the sparse vector V on
		diagonal K.
* spdiags::	A generalization of the function `spdiag'.
* speye::	Returns a sparse identity matrix.
* spfind::	SPFIND: a sparse version of the find operator 1.
* spfun::	Compute `f(X)' for the non-zero values of X This results in
		a sparse matrix with the same structure as X.
* spinv::	Compute the inverse of the square matrix A.
* spkron::	Form the kronecker product of two sparse matrices.
* splu::	Compute the LU decomposition of the sparse matrix A, using
		subroutines from UMFPACK.
* spmax::	For a vector argument, return the maximum value.
* spmin::	For a vector argument, return the minimum value.
* spones::	Replace the non-zero entries of X with ones.
* spparms::	Sets or displays the parameters used by the sparse solvers
		and factorization functions.
* spprod::	Product of elements along dimension DIM.
* sprand::	Generate a random sparse matrix.
* sprandn::	Generate a random sparse matrix.
* spreshape::	Return a sparse matrix with M rows and N columns whose
		elements are taken from the sparse matrix A.
* spstats::	Return the stats for the non-zero elements of the sparse
		matrix S COUNT is the number of non-zeros in each column,
		MEAN is the mean of the non-zeros in each column, and VAR
		is the variance of the non-zeros in each column
* spsum::	Sum of elements along dimension DIM.
* spsumsq::	Sum of squares of elements along dimension DIM.
* spy:: 	Plot the sparsity pattern of the sparse matrix X
* symamd::	For a symmetric positive definite matrix S, returns the
		permutation vector p such that `S (P, P)' tends to have a
		sparser Cholesky factor than S.
* symbfact::	Performs a symbolic factorization analysis on the sparse
		matrix S.
* symrcm::	Returns the Reverse Cuthill McKee reordering of the sparse
		matrix S.
@end menu

@node colamd, colperm, , Function Reference
@subsubsection colamd

@anchor{doc-colamd}
@deftypefn {Loadable Function} {@var{p} =} colamd (@var{s})
@deftypefnx {Loadable Function} {@var{p} =} colamd (@var{s}, @var{knobs})
@deftypefnx {Loadable Function} {[@var{p}, @var{stats}] =} colamd (@var{s})
@deftypefnx {Loadable Function} {[@var{p}, @var{stats}] =} colamd (@var{s}, @var{knobs})

Column approximate minimum degree permutation. @code{@var{p} = colamd
(@var{s})} returns the column approximate minimum degree permutation
vector for the sparse matrix @var{s}. For a non-symmetric matrix @var{s},
@code{@var{s} (:,@var{p})} tends to have sparser LU factors than @var{s}.
The Cholesky factorization of @code{@var{s} (:,@var{p})' * @var{s}
(:,@var{p})} also tends to be sparser than that of @code{@var{s}' *
@var{s}}.

@var{knobs} is an optional two-element input vector. If @var{s} is
m-by-n, then rows with more than @code{(@var{knobs} (1)) * @var{n}}
entries are ignored.  Columns with more than @code{(@var{knobs} (2)) *
@var{m}} entries are removed prior to ordering, and ordered last in the
output permutation @var{p}. If the knobs parameter is not present, then
0.5 is used instead, for both @code{@var{knobs} (1)} and
@code{@var{knobs} (2)}. @code{@var{knobs} (3)} controls the printing of
statistics and error messages.

@var{stats} is an optional 20-element output vector that provides data
about the ordering and the validity of the input matrix @var{s}. Ordering
statistics are in @code{@var{stats} (1:3)}. @code{@var{stats} (1)} and
@code{@var{stats} (2)} are the number of dense or empty rows and columns
ignored by COLAMD and @code{@var{stats} (3)} is the number of garbage
collections performed on the internal data structure used by COLAMD
(roughly of size @code{2.2 * nnz(@var{s}) + 4 * @var{m} + 7 * @var{n}}
integers).

Octave built-in functions are intended to generate valid sparse matrices,
with no duplicate entries, with ascending row indices of the nonzeros
in each column, with a non-negative number of entries in each column (!)
and so on.  If a matrix is invalid, then COLAMD may or may not be able
to continue.  If there are duplicate entries (a row index appears two or
more times in the same column) or if the row indices in a column are out
of order, then COLAMD can correct these errors by ignoring the duplicate
entries and sorting each column of its internal copy of the matrix
@var{s} (the input matrix @var{s} is not repaired, however). If a matrix
is invalid in other ways then COLAMD cannot continue, an error message is
printed, and no output arguments (@var{p} or @var{stats}) are returned.
COLAMD is thus a simple way to check a sparse matrix to see if it's
valid.

@code{@var{stats} (4:7)} provide information if COLAMD was able to
continue. The matrix is OK if @code{@var{stats} (4)} is zero, or 1 if
invalid. @code{@var{stats} (5)} is the rightmost column index that is
unsorted or contains duplicate entries, or zero if no such column exists.
@code{@var{stats} (6)} is the last seen duplicate or out-of-order row
index in the column index given by @code{@var{stats} (5)}, or zero if no
such row index exists. @code{@var{stats} (7)} is the number of duplicate
or out-of-order row indices. @code{@var{stats} (8:20)} is always zero in
the current version of COLAMD (reserved for future use).

The ordering is followed by a column elimination tree post-ordering.

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. (see
@url{http://www.cise.ufl.edu/research/sparse/colamd})
@end deftypefn
@seealso{colperm, symamd}


@node colperm, dmperm, colamd, Function Reference
@subsubsection colperm

@anchor{doc-colperm}
@deftypefn {Function File} {@var{p} =} colperm (@var{s})
Returns the column permutations such that the columns of
@code{@var{s} (:, @var{p})} are ordered in terms of increase number
of non-zero elements. If @var{s} is symmetric, then @var{p} is chosen
such that @code{@var{s} (@var{p}, @var{p})} orders the rows and
columns with increasing number of non zeros elements.
@end deftypefn


@node dmperm, etree, colperm, Function Reference
@subsubsection dmperm



@node etree, full, dmperm, Function Reference
@subsubsection etree

@anchor{doc-etree}
@deftypefn {Loadable Function} {@var{p} =} etree (@var{s})
@deftypefnx {Loadable Function} {@var{p} =} etree (@var{s}, @var{typ})
@deftypefnx {Loadable Function} {[@var{p}, @var{q}] =} etree (@var{s}, @var{typ})

Returns the elimination tree for the matrix @var{s}. By default @var{s}
is assumed to be symmetric and the symmetric elimination tree is
returned. The argument @var{typ} controls whether a symmetric or
column elimination tree is returned. Valid values of @var{typ} are
'sym' or 'col', for symmetric or column elimination tree respectively

Called with a second argument, @dfn{etree} also returns the postorder
permutations on the tree.
@end deftypefn


@node full, issparse, etree, Function Reference
@subsubsection full

@anchor{doc-full}
@deftypefn {Loadable Function} {@var{FM} =} full (@var{SM})
 returns a full storage matrix from a sparse one
@seealso{sparse}
@end deftypefn


@node issparse, luinc, full, Function Reference
@subsubsection issparse

@anchor{doc-issparse}
@deftypefn {Loadable Function} {} issparse (@var{expr})
Return 1 if the value of the expression @var{expr} is a sparse matrix.
@end deftypefn


@node luinc, matrix_type, issparse, Function Reference
@subsubsection luinc

@anchor{doc-luinc}
@deftypefn {Loadable Function} {[@var{l}, @var{u}, @var{p}, @var{q}] =} luinc (@var{a}, '0')
@deftypefnx {Loadable Function} {[@var{l}, @var{u}, @var{p}, @var{q}] =} luinc (@var{a}, @var{droptol})
@deftypefnx {Loadable Function} {[@var{l}, @var{u}, @var{p}, @var{q}] =} luinc (@var{a}, @var{opts})
@cindex LU decomposition
Produce the incomplete LU factorization of the sparse matrix @var{a}.
Two types of incomplete factorization are possible, and the type
is determined by the second argument to @dfn{luinc}.

Called with a second argument of '0', the zero-level incomplete
LU factorization is produced. This creates a factorization of @var{a}
where the position of the non-zero arguments correspond to the same
positions as in the matrix @var{a}.

Alternatively, the fill-in of the incomplete LU factorization can
be controlled through the variable @var{droptol} or the structure
@var{opts}. The UMFPACK multifrontal factorization code by Tim A.
Davis is used for the incomplete LU factorication, (availability
@url{http://www.cise.ufl.edu/research/sparse/umfpack/})

@var{droptol} determines the values below which the values in the LU
factorization are dropped and replaced by zero. It must be a positive
scalar, and any values in the factorization whose absolute value are
less than this value are dropped, expect if leaving them increase the
sparsity of the matrix. Setting @var{droptol} to zero results in a
complete LU factorization which is the default.

@var{opts} is a structure containing one or more of the fields

@table @code
@item droptol
The drop tolerance as above. If @var{opts} only contains @code{droptol}
then this is equivalent to using the variable @var{droptol}.

@item milu
A logical variable flagging whether to use the modified incomplete LU
factorization. In the case that @code{milu} is true, the dropped values
are subtract from the diagonal of the matrix U of the factorization.
The default is @code{false}.

@item udiag
A logical variable that flags whether zero elements on the diagonal of U
should be replaced with @var{droptol} to attempt to avoid singular
factors. The default is @code{false}.

@item thresh
Defines the pivot threshold in the interval [0,1]. Values outside that
range are ignored.
@end table

All other fields in @var{opts} are ignored. The outputs from @dfn{luinc}
are the same as for @dfn{lu}.
@end deftypefn
@seealso{sparse, lu, cholinc}


@node matrix_type, nnz, luinc, Function Reference
@subsubsection matrix_type

@anchor{doc-matrix_type}
@deftypefn {Loadable Function} {@var{type} =} matrix_type (@var{a})
@deftypefnx {Loadable Function} {@var{a} =} matrix_type (@var{a}, @var{type})
@deftypefnx {Loadable Function} {@var{a} =} matrix_type (@var{a}, 'upper', @var{perm})
@deftypefnx {Loadable Function} {@var{a} =} matrix_type (@var{a}, 'lower', @var{perm})
@deftypefnx {Loadable Function} {@var{a} =} matrix_type (@var{a}, 'banded', @var{nl}, @var{nu})
Identify the matrix type or mark a matrix as a particular type. This allows rapid
for solutions of linear equations involving @var{a} to be performed. Called with a
single argument, @code{matrix_type} returns the type of the matrix and caches it for
future use. Called with more than one argument, @code{matrix_type} allows the type
of the matrix to be defined.

The possible matrix types depend on whether the matrix is full or sparse, and can be
one of the following

@table @asis
@item 'unknown'
Remove any previously cached matrix type, and mark type as unknown

@item 'full'
Mark the matrix as full.

@item 'positive definite'
Full positive definite matrix.

@item 'diagonal'
Diagonal Matrix. (Sparse matrices only)

@item 'permuted diagonal'
Permuted Diagonal matrix. The permutation does not need to be specifically
indicated, as the structure of the matrix explicitly gives this. (Sparse matrices
only)

@item 'upper'
Upper triangular. If the optional third argument @var{perm} is given, the matrix is
assumed to be a permuted upper triangular with the permutations defined by the
vector @var{perm}.

@item 'lower'
Lower triangular. If the optional third argument @var{perm} is given, the matrix is
assumed to be a permuted lower triangular with the permutations defined by the
vector @var{perm}.

@item 'banded'
@itemx 'banded positive definite'
Banded matrix with the band size of @var{nl} below the diagonal and @var{nu} above
it. If @var{nl} and @var{nu} are 1, then the matrix is tridiagonal and treated
with specialized code. In addition the matrix can be marked as positive definite
(Sparse matrices only)

@item 'singular'
The matrix is assumed to be singular and will be treated with a minimum norm solution

@end table

Note that the matrix type will be discovered automatically on the first attempt to
solve a linear equation involving @var{a}. Therefore @code{matrix_type} is only
useful to give Octave hints of the matrix type. Incorrectly defining the
matrix type will result in incorrect results from solutions of linear equations,
and so it is entirely the responsibility of the user to correctly indentify the
matrix type.
@end deftypefn


@node nnz, nonzeros, matrix_type, Function Reference
@subsubsection nnz

@anchor{doc-nnz}
@deftypefn {Loadable Function} {@var{scalar} =} nnz (@var{SM})
returns number of non zero elements in SM
@seealso{sparse}
@end deftypefn


@node nonzeros, nzmax, nnz, Function Reference
@subsubsection nonzeros

@anchor{doc-nonzeros}
@deftypefn {Function File} {} nonzeros (@var{s})
Returns a vector of the non-zero values of the sparse matrix @var{s}.
@end deftypefn


@node nzmax, spalloc, nonzeros, Function Reference
@subsubsection nzmax

@anchor{doc-nzmax}
@deftypefn {Loadable Function} {@var{scalar} =} nzmax (@var{SM})
Return the amount of storage allocated to the sparse matrix @var{SM}.
Note that Octave tends to crop unused memory at the first oppurtunity
for sparse objects. There are some cases of user created sparse objects
where the value returned by @dfn{nzmaz} will not be the same as @dfn{nnz},
but in general they will give the same result.
@seealso{sparse, spalloc}
@end deftypefn


@node spalloc, sparse, nzmax, Function Reference
@subsubsection spalloc

@anchor{doc-spalloc}
@deftypefn {Function File} {@var{s} =} spalloc (@var{r}, @var{c}, @var{nz})
Returns an empty sparse matrix of size @var{r}-by-@var{c}. As Octave
resizes sparse matrices at the first opportunity, so that no additional 
space is needed, the argument @var{nz} is ignored. This function is 
provided only for compatiability reasons.

It should be noted that this means that code like

@example
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
@end example

will reallocate memory at each step. It is therefore vitally important
that code like this is vectorized as much as possible.
@end deftypefn
@seealso{sparse, nzmax}


@node sparse, spatan2, spalloc, Function Reference
@subsubsection sparse

@anchor{doc-sparse}
@deftypefn {Loadable Function} {@var{sparse_val} =} sparse (...)
SPARSE: create a sparse matrix

sparse can be called in the following ways:

@enumerate
@item @var{S} = sparse(@var{A})  where @var{A} is a full matrix

@item @var{S} = sparse(@var{A},1)  where @var{A} is a full matrix, result
is forced back to a full matrix is resulting matrix is sparse

@item @var{S} = sparse(@var{i},@var{j},@var{s},@var{m},@var{n},@var{nzmax})  where
   @itemize @w 
@var{i},@var{j}   are integer index vectors (1 x nnz) @* 
@var{s}     is the vector of real or complex entries (1 x nnz) @* 
@var{m},@var{n}   are the scalar dimentions of S @* 
@var{nzmax} is ignored (here for compatability with Matlab) @* 

        if multiple values are specified with the same @var{i},@var{j}
        position, the corresponding values in @var{s} will be added
   @end itemize

@item The following usages are equivalent to (2) above:
   @itemize @w 
@var{S} = sparse(@var{i},@var{j},@var{s},@var{m},@var{n})@*
@var{S} = sparse(@var{i},@var{j},@var{s},@var{m},@var{n},'summation')@*
@var{S} = sparse(@var{i},@var{j},@var{s},@var{m},@var{n},'sum')@*
   @end itemize

@item @var{S} = sparse(@var{i},@var{j},@var{s},@var{m},@var{n},'unique')@*

   @itemize @w 
same as (2) above, except that rather than adding,
if more than two values are specified for the same @var{i},@var{j}
position, then the last specified value will be kept
   @end itemize

@item @var{S}=  sparse(@var{i},@var{j},@var{sv})          uses @var{m}=max(@var{i}), @var{n}=max(@var{j})

@item @var{S}=  sparse(@var{m},@var{n})            does sparse([],[],[],@var{m},@var{n},0)

@var{sv}, and @var{i} or @var{j} may be scalars, in
which case they are expanded to all have the same length
@end enumerate
@seealso{full}
@end deftypefn


@node spatan2, spconvert, sparse, Function Reference
@subsubsection spatan2

@anchor{doc-spatan2}
@deftypefn {Loadable Function} {} spatan2 (@var{y}, @var{x})
Compute atan (Y / X) for corresponding sparse matrix elements of Y and X.
The result is in range -pi to pi.
@end deftypefn



@node spconvert, spcumprod, spatan2, Function Reference
@subsubsection spconvert

@anchor{doc-spconvert}
@deftypefn {Function File} {@var{x} =} spconvert (@var{m})
This function converts for a simple sparse matrix format easily
produced by other programs into Octave's internal sparse format. The
input @var{x} is either a 3 or 4 column real matrix, containing
the row, column, real and imaginary parts of the elements of the
sparse matrix. An element with a zero real and imaginay part can
be used to force a particular matrix size.
@end deftypefn


@node spcumprod, spcumsum, spconvert, Function Reference
@subsubsection spcumprod

@anchor{doc-spcumprod}
@deftypefn {Loadable Function} {@var{y} =} spcumprod (@var{x},@var{dim})
Cumulative product of elements along dimension @var{dim}.  If @var{dim}
is omitted, it defaults to 1 (column-wise cumulative products).
@end deftypefn
@seealso{spcumsum}


@node spcumsum, spdet, spcumprod, Function Reference
@subsubsection spcumsum

@anchor{doc-spcumsum}
@deftypefn {Loadable Function} {@var{y} =} spcumsum (@var{x},@var{dim})
Cumulative sum of elements along dimension @var{dim}.  If @var{dim}
is omitted, it defaults to 1 (column-wise cumulative sums).
@end deftypefn
@seealso{spcumprod}


@node spdet, spdiag, spcumsum, Function Reference
@subsubsection spdet

@anchor{doc-spdet}
@deftypefn {Loadable Function} {[@var{d}, @var{rcond}] = } spdet (@var{a})
Compute the determinant of sparse matrix @var{a} using UMFPACK.  Return
an estimate of the reciprocal condition number if requested.
@end deftypefn


@node spdiag, spdiags, spdet, Function Reference
@subsubsection spdiag

@anchor{doc-spdiag}
@deftypefn {Loadable Function} {} spdiag (@var{v}, @var{k})
Return a diagonal matrix with the sparse vector @var{v} on diagonal
@var{k}. The second argument is optional. If it is positive, the vector is
placed on the @var{k}-th super-diagonal. If it is negative, it is placed
on the @var{-k}-th sub-diagonal.  The default value of @var{k} is 0, and
the vector is placed on the main diagonal.  For example,

@example
spdiag ([1, 2, 3], 1)
ans =

Compressed Column Sparse (rows=4, cols=4, nnz=3)
  (1 , 2) -> 1
  (2 , 3) -> 2
  (3 , 4) -> 3
@end example

@end deftypefn
@seealso{diag}


@node spdiags, speye, spdiag, Function Reference
@subsubsection spdiags

@anchor{doc-spdiags}
@deftypefn {function File} {[@var{b}, @var{c}]} = spdiags (@var{a})
@deftypefnx {function File} {@var{b}} = spdiags (@var{a}, @var{c})
@deftypefnx {function File} {@var{b}} = spdiags (@var{v}, @var{c}, @var{a})
@deftypefnx {function File} {@var{b}} = spdiags (@var{v}, @var{c}, @var{m}, @var{n})
A generalization of the function @code{spdiag}. Called with a single
input argument, the non-zero diagonals @var{c} of @var{A} are extracted.
With two arguments the diagonals to extract are given by the vector 
@var{c}.

The other two forms of @code{spdiags} modify the input matrix by
replacing the diagonals. They use the columns of @var{v} to replace
the columns represented by the vector @var{c}. If the sparse matrix
@var{a} is defined then the diagonals of this matrix are replaced.
Otherwise a matrix of @var{m} by @var{n} is created with the
diagonals given by @var{v}.

Negative values of @var{c} representive diagonals below the main
diagonal, and positive values of @var{c} diagonals above the main
diagonal.

For example

@example
@group
spdiags (reshape (1:12, 4, 3), [-1 0 1], 5, 4)
@result{}    5 10  0  0
      1  6 11  0
      0  2  7 12
      0  0  3  8
      0  0  0  4
@end group
@end example

@end deftypefn


@node speye, spfind, spdiags, Function Reference
@subsubsection speye

@anchor{doc-speye}
@deftypefn {Function File} {@var{y} =} speye (@var{m})
@deftypefnx {Function File} {@var{y} =} speye (@var{m}, @var{n})
@deftypefnx {Function File} {@var{y} =} speye (@var{sz})
Returns a sparse identity matrix. This is significantly more
efficient than @code{sparse (eye (@var{m}))} as the full matrix
is not constructed.

Called with a single argument a square matrix of size @var{m} by
@var{m} is created. Otherwise a matrix of @var{m} by @var{n} is
created. If called with a single vector argument, this argument 
is taken to be the size of the matrix to create.
@end deftypefn


@node spfind, spfun, speye, Function Reference
@subsubsection spfind

@anchor{doc-spfind}
@deftypefn {Loadable Function} {[...] =} spfind (...)
SPFIND: a sparse version of the find operator
@enumerate
    @item
@var{x }= spfind( @var{a })
    @itemize @w
is analagous to @var{x}= find(@var{A}(:))@*
where @var{A}= full(@var{a})
    @end itemize
    @item
[@var{i},@var{j},@var{v},@var{nr},@var{nc}] = spfind( @var{a} )
    @itemize @w
returns column vectors @var{i},@var{j},@var{v} such that@*
@var{a}= sparse(@var{i},@var{j},@var{v},@var{nr},@var{nc})
    @end itemize
@end enumerate
@seealso{sparse}
@end deftypefn


@node spfun, spinv, spfind, Function Reference
@subsubsection spfun

@anchor{doc-spfun}
@deftypefn {Function File} {@var{y} =} spfun (@var{f},@var{x})
Compute @code{f(@var{x})} for the non-zero values of @var{x}.
This results in a sparse matrix with the same structure as 
@var{x}. The function @var{f} can be passed as a string, a
function handle or an inline function.
@end deftypefn


@node spinv, spkron, spfun, Function Reference
@subsubsection spinv

@anchor{doc-spinv}
@deftypefn {Loadable Function} {[@var{x}, @var{rcond}] = } spinv (@var{a}, @var{Q})
@deftypefnx {Loadable Function} {[@var{x}, @var{rcond}, @var{Q}] = } spinv (@var{a}, @var{Q})
Compute the inverse of the square matrix @var{a}.  Return an estimate
of the reciprocal condition number if requested, otherwise warn of an
ill-conditioned matrix if the reciprocal condition number is small.

An optional second input argument @var{Q} is the optional pre-ordering of
the matrix, such that @code{@var{x} = inv (@var{a} (:, @var{Q}))}. @var{Q}
can equally be a matrix, in which case @code{@var{x} = inv (@var{a} *
@var{Q}))}.

If a third output argument is given then the permuations to achieve a sparse
inverse are returned. It is not required that the return column permutations
@var{Q} and the same as the user supplied permutations
@end deftypefn


@node spkron, splu, spinv, Function Reference
@subsubsection spkron

@anchor{doc-spkron}
@deftypefn {Function File} {} spkron (@var{a}, @var{b})
Form the kronecker product of two sparse matrices. This is defined
block by block as

@example
x = [a(i, j) b]
@end example

For example,

@example
@group
kron (1:4, ones (3, 1))
      @result{}  1  2  3  4
          1  2  3  4
          1  2  3  4
@end group
@end example
@end deftypefn


@node splu, spmax, spkron, Function Reference
@subsubsection splu

@anchor{doc-splu}
@deftypefn {Loadable Function} {[@var{l}, @var{u}] =} splu (@var{a})
@deftypefnx {Loadable Function} {[@var{l}, @var{u}, @var{P}] =} splu (@var{a})
@deftypefnx {Loadable Function} {[@var{l}, @var{u}, @var{P}, @var{Q}] =} splu (@var{a})
@deftypefnx {Loadable Function} {[@var{l}, @var{u}, @var{P}, @var{Q}] =} splu (@dots{}, @var{thres})
@deftypefnx {Loadable Function} {[@var{l}, @var{u}, @var{P}] =} splu (@dots{}, @var{Q})
@cindex LU decomposition
Compute the LU decomposition of the sparse matrix @var{a}, using
subroutines from UMFPACK.  The result is returned in a permuted
form, according to the optional return values @var{P} and @var{Q}.

Called with two or three output arguments and a single input argument,
@dfn{splu} is a replacement for @dfn{lu}, and therefore the sparsity
preserving column permutations @var{Q} are not performed. Called with
a fourth output argument, the sparsity preserving column transformation
@var{Q} is returned, such that @code{@var{P} * @var{a} * @var{Q} =
@var{l} * @var{u}}.

An additional input argument @var{thres}, that defines the pivoting
threshold can be given. Alternatively, the desired sparsity preserving
column permutations @var{Q} can be passed. Note that @var{Q} is assumed
to be fixed if three are fewer than four output arguments. Otherwise,
the updated column permutations are returned as the fourth argument.

With two output arguments, returns the permuted forms of the upper and
lower triangular matrices, such that @code{@var{a} = @var{l} * @var{u}}.
With two or three output arguments, if a user-defined @var{Q} is given,
then @code{@var{u} * @var{Q}'} is returned. The matrix is not required to
be square.
@end deftypefn
@seealso{sparse, spinv, colamd, symamd}


@node spmax, spmin, splu, Function Reference
@subsubsection spmax

@anchor{doc-spmax}
@deftypefn {Mapping Function} {} spmax (@var{x}, @var{y}, @var{dim})
@deftypefnx {Mapping Function} {[@var{w}, @var{iw}] =} spmax (@var{x})
@cindex Utility Functions
For a vector argument, return the maximum value.  For a matrix
argument, return the maximum value from each column, as a row
vector, or over the dimension @var{dim} if defined. For two matrices
(or a matrix and scalar), return the pair-wise maximum.
Thus,

@example
max (max (@var{x}))
@end example

@noindent
returns the largest element of @var{x}, and

@example
@group
max (2:5, pi)
    @result{}  3.1416  3.1416  4.0000  5.0000
@end group
@end example
@noindent
compares each element of the range @code{2:5} with @code{pi}, and
returns a row vector of the maximum values.

For complex arguments, the magnitude of the elements are used for
comparison.

If called with one input and two output arguments,
@code{max} also returns the first index of the
maximum value(s). Thus,

@example
@group
[x, ix] = max ([1, 3, 5, 2, 5])
    @result{}  x = 5
        ix = 3
@end group
@end example
@end deftypefn


@node spmin, spones, spmax, Function Reference
@subsubsection spmin

@anchor{doc-spmin}
@deftypefn {Mapping Function} {} spmin (@var{x}, @var{y}, @var{dim})
@deftypefnx {Mapping Function} {[@var{w}, @var{iw}] =} spmin (@var{x})
@cindex Utility Functions
For a vector argument, return the minimum value.  For a matrix
argument, return the minimum value from each column, as a row
vector, or over the dimension @var{dim} if defined. For two matrices
(or a matrix and scalar), return the pair-wise minimum.
Thus,

@example
min (min (@var{x}))
@end example

@noindent
returns the smallest element of @var{x}, and

@example
@group
min (2:5, pi)
    @result{}  2.0000  3.0000  3.1416  3.1416
@end group
@end example
@noindent
compares each element of the range @code{2:5} with @code{pi}, and
returns a row vector of the minimum values.

For complex arguments, the magnitude of the elements are used for
comparison.

If called with one input and two output arguments,
@code{min} also returns the first index of the
minimum value(s). Thus,

@example
@group
[x, ix] = min ([1, 3, 0, 2, 5])
    @result{}  x = 0
        ix = 3
@end group
@end example
@end deftypefn


@node spones, spparms, spmin, Function Reference
@subsubsection spones

@anchor{doc-spones}
@deftypefn {Function File} {@var{y} =} spones (@var{x})
Replace the non-zero entries of @var{x} with ones. This creates a
sparse matrix with the same structure as @var{x}.
@end deftypefn


@node spparms, spprod, spones, Function Reference
@subsubsection spparms

@anchor{doc-spparms}
@deftypefn {Loadable Function} { } spparms ()
@deftypefnx {Loadable Function} {@var{vals} =} spparms ()
@deftypefnx {Loadable Function} {[@var{keys}, @var{vals}] =} spparms ()
@deftypefnx {Loadable Function} {@var{val} =} spparms (@var{key})
@deftypefnx {Loadable Function} { } spparms (@var{vals})
@deftypefnx {Loadable Function} { } spparms ('defaults')
@deftypefnx {Loadable Function} { } spparms ('tight')
@deftypefnx {Loadable Function} { } spparms (@var{key}, @var{val})
Sets or displays the parameters used by the sparse solvers and factorization
functions. The first four calls above get information about the current
settings, while the others change the current settings. The parameters are
stored as pairs of keys and values, where the values are all floats and the
keys are one of the strings

@itemize
@item spumoni
Printing level of debugging information of the solvers (default 0)
@item ths_rel
?? (default 1)
@item ths_abs
?? (default 1)
@item exact_d
?? (default 0)
@item supernd
?? (default 3)
@item rreduce
?? (default 3)
@item wh_frac
?? (default 0.5)
@item autommd
Flag whether the LU/QR and the '\' and '/' operators will automatically
use the sparsity preserving mmd functions (default 1)
@item autoamd
Flag whether the LU and the '\' and '/' operators will automatically
use the sparsity preserving amd functions (default 1)
@item piv_tol
The pivot tolerance of the UMFPACK solvers (default 0.1)
@item bandden
?? (default 0.5)
@item umfpack
Flag whether the UMFPACK or mmd solvers are used for the LU, '\' and
'/' operations (default 1)
@end itemize

The value of individual keys can be set with @code{spparms (@var{key},
@var{val})}. The default values can be restored with the special keyword
'defaults'. The special keyword 'tight' can be used to set the mmd solvers
to attempt for a sparser solution at the potetial cost of longer running
time.
@end deftypefn


@node spprod, sprand, spparms, Function Reference
@subsubsection spprod

@anchor{doc-spprod}
@deftypefn {Loadable Function} {@var{y} =} spprod (@var{x},@var{dim})
Product of elements along dimension @var{dim}.  If @var{dim} is omitted,
it defaults to 1 (column-wise products).
@end deftypefn
@seealso{spsum, spsumsq}


@node sprand, sprandn, spprod, Function Reference
@subsubsection sprand

@anchor{doc-sprand}
@deftypefn {Function File} {} sprand (@var{m}, @var{n}, @var{d})
@deftypefnx {Function File} {} sprand (@var{s})
Generate a random sparse matrix. The size of the matrix will be
@var{m} by @var{n}, with a density of values given by @var{d}.
@var{d} should be between 0 and 1. Values will be normally
distributed with mean of zero and variance 1.

Note: sometimes the actual density  may be a bit smaller than @var{d}. 
This is unlikely to happen for large really sparse matrices.

If called with a single matrix argument, a random sparse matrix is
generated wherever the matrix @var{S} is non-zero.
@end deftypefn
@seealso{sprandn}


@node sprandn, spreshape, sprand, Function Reference
@subsubsection sprandn

@anchor{doc-sprandn}
@deftypefn {Function File} {} sprand (@var{m}, @var{n}, @var{d})
@deftypefnx {Function File} {} sprand (@var{s})
Generate a random sparse matrix. The size of the matrix will be
@var{m} by @var{n}, with a density of values given by @var{d}.
@var{d} should be between 0 and 1. Values will be normally
distributed with mean of zero and variance 1.

Note: sometimes the actual density  may be a bit smaller than @var{d}. 
This is unlikely to happen for large really sparse matrices.

If called with a single matrix argument, a random sparse matrix is
generated wherever the matrix @var{S} is non-zero.
@end deftypefn
@seealso{sprandn}


@node spreshape, spstats, sprandn, Function Reference
@subsubsection spreshape



@node spstats, spsum, spreshape, Function Reference
@subsubsection spstats

@anchor{doc-spstats}
@deftypefn {Function File} {[@var{count}, @var{mean}, @var{var}]} = spstats (@var{s})
@deftypefnx {Function File} {[@var{count}, @var{mean}, @var{var}]} = spstats (@var{s}, @var{j})
Return the stats for the non-zero elements of the sparse matrix @var{s}.
@var{count} is the number of non-zeros in each column, @var{mean}
is the mean of the non-zeros in each column, and @var{var} is the  
variance of the non-zeros in each column.

Called with two output arguments, if @var{s} is the data and @var{j}
is the bin number for the data, compute the stats for each bin.  In 
this case, bins can contain data values of zero, whereas with 
@code{spstats (@var{s})} the zeros may disappear.
@end deftypefn


@node spsum, spsumsq, spstats, Function Reference
@subsubsection spsum

@anchor{doc-spsum}
@deftypefn {Loadable Function} {@var{y} =} spsum (@var{x},@var{dim})
Sum of elements along dimension @var{dim}.  If @var{dim} is omitted, it
defaults to 1 (column-wise sum).
@end deftypefn
@seealso{spprod, spsumsq}


@node spsumsq, spy, spsum, Function Reference
@subsubsection spsumsq

@anchor{doc-spsumsq}
@deftypefn {Loadable Function} {@var{y} =} spsumsq (@var{x},@var{dim})
Sum of squares of elements along dimension @var{dim}.  If @var{dim}
is omitted, it defaults to 1 (column-wise sum of squares).
This function is equivalent to computing
@example
spsum (x .* spconj (x), dim)
@end example
but it uses less memory and avoids calling @code{spconj} if @var{x} is
real.
@end deftypefn
@seealso{spprod, spsum}


@node spy, symamd, spsumsq, Function Reference
@subsubsection spy

@anchor{doc-spy}
@deftypefn {Function File} {} spy (@var{x})
Plot the sparsity pattern of the sparse matrix @var{x}.
@end deftypefn


@node symamd, symbfact, spy, Function Reference
@subsubsection symamd

@anchor{doc-symamd}
@deftypefn {Loadable Function} {@var{p} =} symamd (@var{s})
@deftypefnx {Loadable Function} {@var{p} =} symamd (@var{s}, @var{knobs})
@deftypefnx {Loadable Function} {[@var{p}, @var{stats}] =} symamd (@var{s})
@deftypefnx {Loadable Function} {[@var{p}, @var{stats}] =} symamd (@var{s}, @var{knobs})

For a symmetric positive definite matrix @var{s}, returns the permutation
vector p such that @code{@var{s} (@var{p}, @var{p})} tends to have a
sparser Cholesky factor than @var{s}. Sometimes SYMAMD works well for
symmetric indefinite matrices too. The matrix @var{s} is assumed to be
symmetric; only the strictly lower triangular part is referenced. @var{s}
must be square.

@var{knobs} is an optional input argument. If @var{s} is n-by-n, then
rows and columns with more than @code{@var{knobs} (1) * @var{n}} entries
are removed prior to ordering, and ordered last in the output permutation
@var{p}. If the @var{knobs} parameter is not present, then the default of
0.5 is used instead. @code{@var{knobs} (2)} controls the printing of
statistics and error messages.

@var{stats} is an optional 20-element output vector that provides data
about the ordering and the validity of the input matrix @var{s}. Ordering
statistics are in @code{@var{stats} (1:3)}. @code{@var{stats} (1) =
@var{stats} (2)} is the number of dense or empty rows and columns
ignored by SYMAMD and @code{@var{stats} (3)} is the number of garbage
collections performed on the internal data structure used by SYMAMD
(roughly of size @code{8.4 * nnz (tril (@var{s}, -1)) + 9 * @var{n}}
integers).

Octave built-in functions are intended to generate valid sparse matrices,
with no duplicate entries, with ascending row indices of the nonzeros
in each column, with a non-negative number of entries in each column (!)
and so on.  If a matrix is invalid, then SYMAMD may or may not be able
to continue.  If there are duplicate entries (a row index appears two or
more times in the same column) or if the row indices in a column are out
of order, then SYMAMD can correct these errors by ignoring the duplicate
entries and sorting each column of its internal copy of the matrix S (the
input matrix S is not repaired, however).  If a matrix is invalid in
other ways then SYMAMD cannot continue, an error message is printed, and
no output arguments (@var{p} or @var{stats}) are returned.  SYMAMD is
thus a simple way to check a sparse matrix to see if it's valid.

@code{@var{stats} (4:7)} provide information if SYMAMD was able to
continue. The matrix is OK if @code{@var{stats} (4)} is zero, or 1
if invalid. @code{@var{stats} (5)} is the rightmost column index that
is unsorted or contains duplicate entries, or zero if no such column
exists. @code{@var{stats} (6)} is the last seen duplicate or out-of-order
row index in the column index given by @code{@var{stats} (5)}, or zero
if no such row index exists. @code{@var{stats} (7)} is the number of
duplicate or out-of-order row indices. @code{@var{stats} (8:20)} is
always zero in the current version of SYMAMD (reserved for future use).

The ordering is followed by a column elimination tree post-ordering.


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. (see
@url{http://www.cise.ufl.edu/research/sparse/colamd})
@end deftypefn
@seealso{colperm, colamd}


@node symbfact, symrcm, symamd, Function Reference
@subsubsection symbfact

@anchor{doc-symbfact}
@deftypefn {Loadable Function} {[@var{count}, @var{h}, @var{parent}, @var{post}, @var{r}]} = symbfact (@var{s}, @var{typ})

Performs a symbolic factorization analysis on the sparse matrix @var{s}.
@end deftypefn


@node symrcm, , symbfact, Function Reference
@subsubsection symrcm



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