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@ifhtml
@set htmltex
@end ifhtml
@iftex
@set htmltex
@end iftex

@node Sparse Matrices
@chapter Sparse Matrices

@menu
* Basics::                      Creation and Manipulation of Sparse Matrices
* Sparse Linear Algebra::       Linear Algebra on Sparse Matrices
* Iterative Techniques::        Iterative Techniques
* Real Life Example::           Using Sparse Matrices
@end menu

@node Basics
@section 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 of 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 nonzero elements of the matrix are
stored.  Not only does 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
nonzero elements to accelerate their calculations.

A matrix type that stores only the nonzero 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.

@menu
* Storage of Sparse Matrices::
* Creating Sparse Matrices::
* Information::
* Operators and Functions::
@end menu

@node Storage of Sparse Matrices
@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 complexity
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 @nospell{Saad} @footnote{Y. Saad "SPARSKIT: A basic
toolkit for sparse matrix computation", 1994,
@url{https://www-users.cs.umn.edu/~saad/software/SPARSKIT/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 nonzero 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 their 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 the compressed column
format.  It is similar to the Yale format.
@footnote{@url{https://en.wikipedia.org/wiki/Sparse_matrix#Yale_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 nonzero elements in each
column, rather than their positions.  Thus assuming that the matrix has
more nonzero 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 simplification in the code, in that there is no special case
for the first or last columns.  A short example, demonstrating this in
C is.

@example
@group
  for (j = 0; j < nc; j++)
    for (i = cidx(j); i < cidx(j+1); i++)
       printf ("nonzero element (%i,%i) is %d\n",
           ridx(i), j, data(i));
@end group
@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 nonzero elements of this matrix are

@example
@group
   (1, 1)  @xresult{} 1
   (1, 2)  @xresult{} 2
   (2, 4)  @xresult{} 3
   (3, 4)  @xresult{} 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, 1, 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 at zero, while in Octave itself the row and
column indexing starts at one.  Thus, the number of elements in the
@var{i}-th column is given by
@code{@var{cidx} (@var{i} + 1) - @var{cidx} (@var{i})}.

Although Octave uses a compressed column format, 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, which makes certain operations faster.  However, it imposes
the need to sort the elements on the creation of sparse matrices.  Having
disordered 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 Creating Sparse Matrices
@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{diag}, 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 normal random distributions of elements.  They have
exactly the same calling convention, where @code{sprand (@var{r}, @var{c},
@var{d})}, creates an @var{r}-by-@var{c} sparse matrix with a density of
filled elements of @var{d}.

Other functions of interest that directly create sparse matrices, are
@dfn{diag} 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,

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

@noindent
creates a sparse (@var{n}+1)-by-(@var{n}+1) sparse matrix with a single
diagonal defined.

@DOCSTRING(spdiags)

@DOCSTRING(speye)

@DOCSTRING(spones)

@DOCSTRING(sprand)

@DOCSTRING(sprandn)

@DOCSTRING(sprandsym)

The recommended way for the user to create a sparse matrix, is to create
two vectors containing 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
@group
  ri = ci = d = [];
  for j = 1:c
    ri = [ri; randperm(r,n)'];
    ci = [ci; j*ones(n,1)];
    d = [d; rand(n,1)];
  endfor
  s = sparse (ri, ci, d, r, c);
@end group
@end example

@noindent
creates an @var{r}-by-@var{c} sparse matrix with a random distribution
of @var{n} (<@var{r}) elements per column.  The elements of the vectors
do not need to be sorted in any particular order as Octave will sort
them prior to storing the data.  However, pre-sorting the data will
make the 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
@group
s = spconvert ([1 2 3 4; 1 3 4 4; 1 2 3 0]')
@xresult{} Compressed Column Sparse (rows=4, cols=4, nnz=3)
      (1 , 1) -> 1
      (2 , 3) -> 2
      (3 , 4) -> 3
@end group
@end example

An example of creating and filling a matrix might be

@example
@group
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 group
@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 of the above loop.
Therefore the @dfn{spalloc} function ignores the @var{nz} argument and
does not pre-assign the memory for the matrix.  Therefore, it is vitally
important that code using to above structure should be vectorized
as much as possible to minimize the number of assignments and reduce the
number of memory allocations.

@DOCSTRING(full)

@DOCSTRING(spalloc)

@DOCSTRING(sparse)

@DOCSTRING(spconvert)

The above problem of memory reallocation can be avoided in
oct-files.  However, the construction of a sparse matrix from an oct-file
is more complex than can be discussed here.  @xref{External Code Interface},
for a full description of the techniques involved.

@node Information
@subsection Finding Information about Sparse Matrices

There are a number of functions that allow information concerning
sparse matrices to be obtained.  The most basic of these is
@dfn{issparse} that identifies whether a particular Octave object is
in fact a sparse matrix.

Another very basic function is @dfn{nnz} that returns the number of
nonzero entries there are in a sparse matrix, while the function
@dfn{nzmax} returns the amount of storage allocated to the sparse
matrix.  Note that Octave tends to crop unused memory at the first
opportunity for sparse objects.  There are some cases of user created
sparse objects where the value returned by @dfn{nzmax} will not be
the same as @dfn{nnz}, but in general they will give the same
result.  The function @dfn{spstats} returns some basic statistics on
the columns of a sparse matrix including the number of elements, the
mean and the variance of each column.

@DOCSTRING(issparse)

@DOCSTRING(nnz)

@DOCSTRING(nonzeros)

@DOCSTRING(nzmax)

@DOCSTRING(spstats)

When solving linear equations involving sparse matrices Octave
determines the means to solve the equation based on the type of the
matrix (@pxref{Sparse Linear Algebra}).  Octave probes the
matrix type when the div (/) or ldiv (\) operator is first used with
the matrix and then caches the type.  However the @dfn{matrix_type}
function can be used to determine the type of the sparse matrix prior
to use of the div or ldiv operators.  For example,

@example
@group
a = tril (sprandn (1024, 1024, 0.02), -1) ...
    + speye (1024);
matrix_type (a);
ans = Lower
@end group
@end example

@noindent
shows that Octave correctly determines the matrix type for lower
triangular matrices.  @dfn{matrix_type} can also be used to force
the type of a matrix to be a particular type.  For example:

@example
@group
a = matrix_type (tril (sprandn (1024, ...
   1024, 0.02), -1) + speye (1024), "Lower");
@end group
@end example

This allows the cost of determining the matrix type to be
avoided.  However, 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 identify the
matrix type

There are several graphical means of finding out information about
sparse matrices.  The first is the @dfn{spy} command, which displays
the structure of the nonzero elements of the
matrix.  @xref{fig:spmatrix}, for an example of the use of
@dfn{spy}.  More advanced graphical information can be obtained with the
@dfn{treeplot}, @dfn{etreeplot} and @dfn{gplot} commands.

@float Figure,fig:spmatrix
@center @image{spmatrix,4in}
@caption{Structure of simple sparse matrix.}
@end float

One use of sparse matrices is in graph theory, where the
interconnections between nodes are represented as an adjacency
matrix.  That is, if the i-th node in a graph is connected to the j-th
node.  Then the ij-th node (and in the case of undirected graphs the
@nospell{ji-th} node) of the sparse adjacency matrix is nonzero.  If each node
is then associated with a set of coordinates, then the @dfn{gplot}
command can be used to graphically display the interconnections
between nodes.

As a trivial example of the use of @dfn{gplot} consider the example,

@example
@group
A = sparse ([2,6,1,3,2,4,3,5,4,6,1,5],
    [1,1,2,2,3,3,4,4,5,5,6,6],1,6,6);
xy = [0,4,8,6,4,2;5,0,5,7,5,7]';
gplot (A,xy)
@end group
@end example

@noindent
which creates an adjacency matrix @code{A} where node 1 is connected
to nodes 2 and 6, node 2 with nodes 1 and 3, etc.  The coordinates of
the nodes are given in the n-by-2 matrix @code{xy}.
@ifset htmltex
@xref{fig:gplot}.

@float Figure,fig:gplot
@center @image{gplot,4in}
@caption{Simple use of the @dfn{gplot} command.}
@end float
@end ifset

The dependencies between the nodes of a Cholesky@tie{}factorization can be
calculated in linear time without explicitly needing to calculate the
Cholesky@tie{}factorization by the @code{etree} command.  This command
returns the elimination tree of the matrix and can be displayed
graphically by the command @code{treeplot (etree (A))} if @code{A} is
symmetric or @code{treeplot (etree (A+A'))} otherwise.

@DOCSTRING(spy)

@DOCSTRING(etree)

@DOCSTRING(etreeplot)

@DOCSTRING(gplot)

@DOCSTRING(treeplot)

@DOCSTRING(treelayout)

@node Operators and Functions
@subsection Basic Operators and Functions on Sparse Matrices

@menu
* Sparse Functions::
* Return Types of Operators and Functions::
* Mathematical Considerations::
@end menu

@node Sparse Functions
@subsubsection Sparse Functions

Many Octave functions have been overloaded to work with either sparse or full
matrices.  There is no difference in calling convention when using an
overloaded function with a sparse matrix, however, there is also no access to
potentially sparse-specific features.  At any time the sparse matrix specific
version of a function can be used by explicitly calling its function name.

The table below lists all of the sparse functions of Octave.  Note that the
names of the specific sparse forms of the functions are typically the same as
the general versions with a @dfn{sp} prefix.  In the table below, and in the
rest of this article, the specific sparse versions of functions are used.

@c Table includes in comments the missing sparse functions

@table @asis
@item Generate sparse matrices:
  @dfn{spalloc}, @dfn{spdiags}, @dfn{speye}, @dfn{sprand},
  @dfn{sprandn}, @dfn{sprandsym}

@item Sparse matrix conversion:
  @dfn{full}, @dfn{sparse}, @dfn{spconvert}

@item Manipulate sparse matrices
  @dfn{issparse}, @dfn{nnz}, @dfn{nonzeros}, @dfn{nzmax},
  @dfn{spfun}, @dfn{spones}, @dfn{spy}

@item Graph Theory:
  @dfn{etree}, @dfn{etreeplot}, @dfn{gplot},
  @dfn{treeplot}
@c @dfn{treelayout}

@item Sparse matrix reordering:
  @dfn{amd}, @dfn{ccolamd}, @dfn{colamd}, @dfn{colperm}, @dfn{csymamd},
  @dfn{dmperm}, @dfn{symamd}, @dfn{randperm}, @dfn{symrcm}

@item Linear algebra:
  @dfn{condest}, @dfn{eigs}, @dfn{matrix_type},
  @dfn{normest}, @dfn{normest1}, @dfn{sprank}, @dfn{spaugment}, @dfn{svds}

@item Iterative techniques:
  @dfn{ichol}, @dfn{ilu}, @dfn{pcg}, @dfn{pcr}
@c @dfn{bicg}, @dfn{bicgstab}, @dfn{cholinc}, @dfn{cgs}, @dfn{gmres},
@c @dfn{lsqr}, @dfn{minres}, @dfn{qmr}, @dfn{symmlq}

@item Miscellaneous:
  @dfn{spparms}, @dfn{symbfact}, @dfn{spstats}
@end table

In addition all of the standard Octave mapper functions (i.e., basic
math functions that take a single argument) such as @dfn{abs}, etc.@:
can accept sparse matrices.  The reader is referred to the documentation
supplied with these functions within Octave itself for further
details.

@node Return Types of Operators and Functions
@subsubsection Return Types of Operators and Functions

The two basic reasons 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 number of nonzero elements.

Therefore, there is a certain density of nonzero 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
@group
speye (3) + 0
@xresult{}   1  0  0
  0  1  0
  0  0  1
@end group
@end example

@noindent
returns a full matrix as can be seen.

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 are 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 behavior 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

@noindent
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 Mathematical Considerations
@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.

First, the @qcode{"./"} and @qcode{".^"} operators must be used with care.
Consider what the examples

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

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

To correct this behavior 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
efficiency, and so the user is warned that calculations where the sign-bit
of zero is important must not be done using sparse matrices.

In general any function or operator used on a sparse matrix will
result in a sparse matrix with the same or a larger number of nonzero
elements than the original matrix.  This is particularly true for the
important case of sparse matrix factorizations.  The usual way to
address this is to reorder the matrix, such that its factorization is
sparser than the factorization of the original matrix.  That is the
factorization of @code{L * U = P * S * Q} has sparser terms @code{L}
and @code{U} than the equivalent factorization @code{L * U = S}.

Several functions are available to reorder depending on the type of the
matrix to be factorized.  If the matrix is symmetric positive-definite,
then @dfn{symamd} or @dfn{csymamd} should be used.  Otherwise
@dfn{amd}, @dfn{colamd} or @dfn{ccolamd} should be used.  For completeness
the reordering functions @dfn{colperm} and @dfn{randperm} are
also available.

@xref{fig:simplematrix}, for an example of the structure of a simple
positive definite matrix.

@float Figure,fig:simplematrix
@center @image{spmatrix,4in}
@caption{Structure of simple sparse matrix.}
@end float

The standard Cholesky@tie{}factorization of this matrix can be
obtained by the same command that would be used for a full
matrix.  This can be visualized with the command
@code{r = chol (A); spy (r);}.
@xref{fig:simplechol}.
The original matrix had
@ifinfo
@ifnothtml
43
@end ifnothtml
@end ifinfo
@ifset htmltex
598
@end ifset
nonzero terms, while this Cholesky@tie{}factorization has
@ifinfo
@ifnothtml
71,
@end ifnothtml
@end ifinfo
@ifset htmltex
10200,
@end ifset
with only half of the symmetric matrix being stored.  This
is a significant level of fill in, and although not an issue
for such a small test case, can represents a large overhead
in working with other sparse matrices.

The appropriate sparsity preserving permutation of the original
matrix is given by @dfn{symamd} and the factorization using this
reordering can be visualized using the command @code{q = symamd (A);
r = chol (A(q,q)); spy (r)}.  This gives
@ifinfo
@ifnothtml
29
@end ifnothtml
@end ifinfo
@ifset htmltex
399
@end ifset
nonzero terms which is a significant improvement.

The Cholesky@tie{}factorization itself can be used to determine the
appropriate sparsity preserving reordering of the matrix during the
factorization, In that case this might be obtained with three return
arguments as @code{[r, p, q] = chol (A); spy (r)}.

@float Figure,fig:simplechol
@center @image{spchol,4in}
@caption{Structure of the unpermuted Cholesky@tie{}factorization of the above matrix.}
@end float

@float Figure,fig:simplecholperm
@center @image{spcholperm,4in}
@caption{Structure of the permuted Cholesky@tie{}factorization of the above matrix.}
@end float

In the case of an asymmetric matrix, the appropriate sparsity
preserving permutation is @dfn{colamd} and the factorization using
this reordering can be visualized using the command
@code{q = colamd (A); [l, u, p] = lu (A(:,q)); spy (l+u)}.

Finally, Octave implicitly reorders the matrix when using the div (/)
and ldiv (\) operators, and so no the user does not need to explicitly
reorder the matrix to maximize performance.

@DOCSTRING(amd)

@DOCSTRING(ccolamd)

@DOCSTRING(colamd)

@DOCSTRING(colperm)

@DOCSTRING(csymamd)

@DOCSTRING(dmperm)

@DOCSTRING(symamd)

@DOCSTRING(symrcm)

@node Sparse Linear Algebra
@section Linear Algebra on Sparse Matrices

Octave includes a polymorphic solver for sparse matrices, where
the exact solver used to factorize the matrix, depends on the properties
of the sparse matrix itself.  Generally, 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 diagonal, solve directly and goto 8

@item If the matrix is a permuted diagonal, solve directly taking into
account the permutations.  Goto 8

@item If the matrix is square, banded and if the band density is less
than that given by @code{spparms ("bandden")} continue, else goto 4.

@enumerate a
@item If the matrix is tridiagonal and the right-hand side is not sparse
continue, else goto 3b.

@enumerate
@item If the matrix is Hermitian, with a positive real diagonal, attempt
      Cholesky@tie{}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 8.
@end enumerate

@item If the matrix is Hermitian with a positive real diagonal, attempt
      Cholesky@tie{}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 8.
@end enumerate

@item If the matrix is upper or lower triangular perform a sparse forward
or backward substitution, and goto 8

@item If the matrix is an upper triangular matrix with column permutations
or lower triangular matrix with row permutations, perform a sparse forward
or backward substitution, and goto 8

@item If the matrix is square, Hermitian with a real positive diagonal, attempt
sparse Cholesky@tie{}factorization using @sc{cholmod}.

@item If the sparse Cholesky@tie{}factorization failed or the matrix is not
Hermitian with a real positive diagonal, and the matrix is square, factorize,
solve, and perform one refinement iteration using @sc{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 using
@sc{cxsparse}@footnote{The @sc{cholmod}, @sc{umfpack} and @sc{cxsparse}
packages were written by Tim Davis and are available at
@url{http://faculty.cse.tamu.edu/davis/suitesparse.html}}.
@end enumerate

The band density is defined as the number of nonzero values in the band
divided by the total number of values in the full band.  The banded
matrix solvers can be entirely disabled by using @dfn{spparms} to set
@code{bandden} to 1 (i.e., @code{spparms ("bandden", 1)}).

The QR@tie{}solver factorizes the problem with a @nospell{Dulmage-Mendelsohn}
decomposition, to separate the problem into blocks that can be treated
as over-determined, multiple well determined blocks, and a final
over-determined block.  For matrices with blocks of strongly connected
nodes this is a big win as LU@tie{}decomposition can be used for many
blocks.  It also significantly improves the chance of finding a solution
to over-determined problems rather than just returning a vector of
@dfn{NaN}'s.

All of the solvers above, can 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, triangular or diagonal 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 in these cases, and Octave relies on simpler
techniques to detect singular matrices or the underlying @sc{lapack} code in
the case of banded matrices.

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 that identifying the type of the matrix incorrectly
will lead to unpredictable results, and so @code{matrix_type} should be
used with care.

@DOCSTRING(normest)

@DOCSTRING(normest1)

@DOCSTRING(condest)

@DOCSTRING(spparms)

@DOCSTRING(sprank)

@DOCSTRING(symbfact)

For non square matrices, the user can also utilize the @code{spaugment}
function to find a least squares solution to a linear equation.

@DOCSTRING(spaugment)

Finally, the function @code{eigs} can be used to calculate a limited
number of eigenvalues and eigenvectors based on a selection criteria
and likewise for @code{svds} which calculates a limited number of
singular values and vectors.

@DOCSTRING(eigs)

@DOCSTRING(svds)

@node Iterative Techniques
@section Iterative Techniques Applied to Sparse Matrices

The left division @code{\} and right division @code{/} operators,
discussed in the previous section, use direct solvers to resolve a
linear equation of the form @code{@var{x} = @var{A} \ @var{b}} or
@code{@var{x} = @var{b} / @var{A}}.  Octave also includes a number of
functions to solve sparse linear equations using iterative techniques.

@DOCSTRING(pcg)

@DOCSTRING(pcr)

The speed with which an iterative solver converges to a solution can be
accelerated with the use of a pre-conditioning matrix @var{M}.  In this
case the linear equation @code{@var{M}^-1 * @var{x} = @var{M}^-1 *
@var{A} \ @var{b}} is solved instead.  Typical pre-conditioning matrices
are partial factorizations of the original matrix.

@DOCSTRING(ichol)

@DOCSTRING(ilu)

@node Real Life Example
@section Real Life Example using Sparse Matrices

A common application for sparse matrices is in the solution of Finite
Element Models.  Finite element models allow numerical solution of
partial differential equations that do not have closed form solutions,
typically because of the complex shape of the domain.

In order to motivate this application, we consider the boundary value
Laplace equation.  This system can model scalar potential fields, such
as heat or electrical potential.  Given a medium
@tex
$\Omega$ with boundary $\partial\Omega$.  At all points on the $\partial\Omega$
the boundary conditions are known, and we wish to calculate the potential in
$\Omega$.
@end tex
@ifnottex
Omega with boundary dOmega.  At all points on the dOmega
the boundary conditions are known, and we wish to calculate the potential in
Omega.
@end ifnottex
Boundary conditions may specify the potential (Dirichlet
boundary condition), its normal derivative across the boundary
(@nospell{Neumann} boundary condition), or a weighted sum of the potential and
its derivative (Cauchy boundary condition).

In a thermal model, we want to calculate the temperature in
@tex
$\Omega$
@end tex
@ifnottex
Omega
@end ifnottex
and know the boundary temperature (Dirichlet condition)
or heat flux (from which we can calculate the @nospell{Neumann} condition
by dividing by the thermal conductivity at the boundary).  Similarly,
in an electrical model, we want to calculate the voltage in
@tex
$\Omega$
@end tex
@ifnottex
Omega
@end ifnottex
and know the boundary voltage (Dirichlet) or current
(@nospell{Neumann} condition after diving by the electrical conductivity).
In an electrical model, it is common for much of the boundary
to be electrically isolated; this is a @nospell{Neumann} boundary condition
with the current equal to zero.

The simplest finite element models will divide
@tex
$\Omega$
@end tex
@ifnottex
Omega
@end ifnottex
into simplexes (triangles in 2-D, pyramids in 3-D).
@ifset htmltex
We take as a 3-D example a cylindrical liquid filled tank with a small
non-conductive ball from the EIDORS project@footnote{EIDORS - Electrical
Impedance Tomography and Diffuse optical Tomography Reconstruction Software
@url{http://eidors3d.sourceforge.net}}.  This is model is designed to reflect
an application of electrical impedance tomography, where current patterns
are applied to such a tank in order to image the internal conductivity
distribution.  In order to describe the FEM geometry, we have a matrix of
vertices @code{nodes} and simplices @code{elems}.
@end ifset

The following example creates a simple rectangular 2-D electrically
conductive medium with 10 V and 20 V imposed on opposite sides
(Dirichlet boundary conditions).  All other edges are electrically
isolated.

@example
@group
   node_y = [1;1.2;1.5;1.8;2]*ones(1,11);
   node_x = ones(5,1)*[1,1.05,1.1,1.2, ...
             1.3,1.5,1.7,1.8,1.9,1.95,2];
   nodes = [node_x(:), node_y(:)];

   [h,w] = size (node_x);
   elems = [];
   for idx = 1:w-1
     widx = (idx-1)*h;
     elems = [elems; ...
       widx+[(1:h-1);(2:h);h+(1:h-1)]'; ...
       widx+[(2:h);h+(2:h);h+(1:h-1)]' ];
   endfor

   E = size (elems,1); # No. of simplices
   N = size (nodes,1); # No. of vertices
   D = size (elems,2); # dimensions+1
@end group
@end example

This creates a N-by-2 matrix @code{nodes} and a E-by-3 matrix
@code{elems} with values, which define finite element triangles:

@example
@group
  nodes(1:7,:)'
    1.00 1.00 1.00 1.00 1.00 1.05 1.05 @dots{}
    1.00 1.20 1.50 1.80 2.00 1.00 1.20 @dots{}

  elems(1:7,:)'
    1    2    3    4    2    3    4 @dots{}
    2    3    4    5    7    8    9 @dots{}
    6    7    8    9    6    7    8 @dots{}
@end group
@end example

Using a first order FEM, we approximate the electrical conductivity
distribution in
@tex
$\Omega$
@end tex
@ifnottex
Omega
@end ifnottex
as constant on each simplex (represented by the vector @code{conductivity}).
Based on the finite element geometry, we first calculate a system (or
stiffness) matrix for each simplex (represented as 3-by-3 elements on the
diagonal of the element-wise system matrix @code{SE}).  Based on @code{SE}
and a N-by-DE connectivity matrix @code{C}, representing the connections
between simplices and vertices, the global connectivity matrix @code{S} is
calculated.

@example
  ## Element conductivity
  conductivity = [1*ones(1,16), ...
         2*ones(1,48), 1*ones(1,16)];

  ## Connectivity matrix
  C = sparse ((1:D*E), reshape (elems', ...
         D*E, 1), 1, D*E, N);

  ## Calculate system matrix
  Siidx = floor ([0:D*E-1]'/D) * D * ...
         ones(1,D) + ones(D*E,1)*(1:D) ;
  Sjidx = [1:D*E]'*ones (1,D);
  Sdata = zeros (D*E,D);
  dfact = factorial (D-1);
  for j = 1:E
     a = inv ([ones(D,1), ...
         nodes(elems(j,:), :)]);
     const = conductivity(j) * 2 / ...
         dfact / abs (det (a));
     Sdata(D*(j-1)+(1:D),:) = const * ...
         a(2:D,:)' * a(2:D,:);
  endfor
  ## Element-wise system matrix
  SE = sparse(Siidx,Sjidx,Sdata);
  ## Global system matrix
  S = C'* SE *C;
@end example

The system matrix acts like the conductivity
@tex
$S$
@end tex
@ifnottex
@code{S}
@end ifnottex
in Ohm's law
@tex
$SV = I$.
@end tex
@ifnottex
@code{S * V = I}.
@end ifnottex
Based on the Dirichlet and @nospell{Neumann} boundary conditions, we are able
to solve for the voltages at each vertex @code{V}.

@example
  ## Dirichlet boundary conditions
  D_nodes = [1:5, 51:55];
  D_value = [10*ones(1,5), 20*ones(1,5)];

  V = zeros (N,1);
  V(D_nodes) = D_value;
  idx = 1:N; # vertices without Dirichlet
             # boundary condns
  idx(D_nodes) = [];

  ## Neumann boundary conditions.  Note that
  ## N_value must be normalized by the
  ## boundary length and element conductivity
  N_nodes = [];
  N_value = [];

  Q = zeros (N,1);
  Q(N_nodes) = N_value;

  V(idx) = S(idx,idx) \ ( Q(idx) - ...
            S(idx,D_nodes) * V(D_nodes));
@end example

Finally, in order to display the solution, we show each solved voltage
value in the z-axis for each simplex vertex.
@ifset htmltex
@xref{fig:femmodel}.
@end ifset

@example
@group
  elemx = elems(:,[1,2,3,1])';
  xelems = reshape (nodes(elemx, 1), 4, E);
  yelems = reshape (nodes(elemx, 2), 4, E);
  velems = reshape (V(elemx), 4, E);
  plot3 (xelems,yelems,velems,"k");
  print "grid.eps";
@end group
@end example


@ifset htmltex
@float Figure,fig:femmodel
@center @image{grid,4in}
@caption{Example finite element model the showing triangular elements.
The height of each vertex corresponds to the solution value.}
@end float
@end ifset
