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@c Copyright (C) 1996, 1997, 2007 John W. Eaton
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@node Linear Algebra
@chapter Linear Algebra

This chapter documents the linear algebra functions of Octave.
Reference material for many of these functions may be found in
Golub and Van Loan, @cite{Matrix Computations, 2nd Ed.}, Johns Hopkins,
1989, and in @cite{@sc{Lapack} Users' Guide}, SIAM, 1992.

@menu
* Techniques used for Linear Algebra::
* Basic Matrix Functions::      
* Matrix Factorizations::       
* Functions of a Matrix::       
@end menu

@node Techniques used for Linear Algebra
@section Techniques used for Linear Algebra

Octave includes a poly-morphic solver, that selects an appropriate
matrix factorization depending on the properties of the 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 or a matrix
inverse is form is given by

@enumerate 1
@item If the matrix is upper or lower triangular sparse a forward or
backward substitution using the @sc{Lapack} xTRTRS function, and goto 4.

@c Permuted triangular matrices currently disabled in the code
@c
@c @item If the matrix is a upper triangular matrix with column permutations
@c or lower triangular matrix with row permutations, perform a forward or
@c backward substitution, and goto 5.

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

@item If the Cholesky factorization failed or the matrix is not
hermitian with a real positive diagonal, and the matrix is square, factorize 
using the @sc{Lapack} xGETRF function.

@item  If the matrix is not square, or any of the previous solvers flags
a singular or near singular matrix, find a least squares solution using
the @sc{Lapack} xGELSD function.
@end enumerate

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
used with care.

It should be noted that the test for whether a matrix is a candidate for
Cholesky factorization, performed above and by the @code{matrix_type}
function, does not give a certainty that the matrix is
Hermitian. However, the attempt to factorize the matrix will quickly
flag a non-Hermitian matrix.

@node Basic Matrix Functions
@section Basic Matrix Functions

@anchor{doc-balance}
@deftypefn {Loadable Function} {@var{aa} =} balance (@var{a}, @var{opt})
@deftypefnx {Loadable Function} {[@var{dd}, @var{aa}] =} balance (@var{a}, @var{opt})
@deftypefnx {Loadable Function} {[@var{cc}, @var{dd}, @var{aa}, @var{bb}] =} balance (@var{a}, @var{b}, @var{opt})

Compute @code{aa = dd \ a * dd} in which @code{aa} is a matrix whose
row and column norms are roughly equal in magnitude, and
@code{dd} = @code{p * d}, in which @code{p} is a permutation
matrix and @code{d} is a diagonal matrix of powers of two.  This allows
the equilibration to be computed without roundoff.  Results of
eigenvalue calculation are typically improved by balancing first.

If four output values are requested, compute @code{aa = cc*a*dd} and
@code{bb = cc*b*dd)}, in which @code{aa} and @code{bb} have non-zero
elements of approximately the same magnitude and @code{cc} and @code{dd}
are permuted diagonal matrices as in @code{dd} for the algebraic
eigenvalue problem.

The eigenvalue balancing option @code{opt} may be one of:

@table @asis
@item @code{"N"}, @code{"n"}
No balancing; arguments copied, transformation(s) set to identity.

@item @code{"P"}, @code{"p"}
Permute argument(s) to isolate eigenvalues where possible.

@item @code{"S"}, @code{"s"}
Scale to improve accuracy of computed eigenvalues.

@item @code{"B"}, @code{"b"}
Permute and scale, in that order. Rows/columns of a (and b)
that are isolated by permutation are not scaled.  This is the default
behavior.
@end table

Algebraic eigenvalue balancing uses standard @sc{Lapack} routines.

Generalized eigenvalue problem balancing uses Ward's algorithm
(SIAM Journal on Scientific and Statistical Computing, 1981).
@end deftypefn


@anchor{doc-cond}
@deftypefn {Function File} {} cond (@var{a},@var{p})
Compute the @var{p}-norm condition number of a matrix. @code{cond (@var{a})} is
defined as @code{norm (@var{a}, @var{p}) * norm (inv (@var{a}), @var{p})}.
By default @code{@var{p}=2} is used which implies a (relatively slow)
singular value decomposition. Other possible selections are 
@code{@var{p}= 1, Inf, inf, 'Inf', 'fro'} which are generally faster.
@seealso{norm, inv, det, svd, rank}
@end deftypefn


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


@anchor{doc-dmult}
@deftypefn {Function File} {} dmult (@var{a}, @var{b})
If @var{a} is a vector of length @code{rows (@var{b})}, return
@code{diag (@var{a}) * @var{b}} (but computed much more efficiently).
@end deftypefn


@anchor{doc-dot}
@deftypefn {Function File} {} dot (@var{x}, @var{y}, @var{dim})
Computes the dot product of two vectors. If @var{x} and @var{y}
are matrices, calculate the dot-product along the first 
non-singleton dimension. If the optional argument @var{dim} is
given, calculate the dot-product along this dimension.
@end deftypefn


@anchor{doc-eig}
@deftypefn {Loadable Function} {@var{lambda} =} eig (@var{a})
@deftypefnx {Loadable Function} {[@var{v}, @var{lambda}] =} eig (@var{a})
The eigenvalues (and eigenvectors) of a matrix are computed in a several
step process which begins with a Hessenberg decomposition, followed by a
Schur decomposition, from which the eigenvalues are apparent.  The
eigenvectors, when desired, are computed by further manipulations of the
Schur decomposition.

The eigenvalues returned by @code{eig} are not ordered.
@end deftypefn


@anchor{doc-givens}
@deftypefn {Loadable Function} {@var{g} =} givens (@var{x}, @var{y})
@deftypefnx {Loadable Function} {[@var{c}, @var{s}] =} givens (@var{x}, @var{y})
@iftex
@tex
Return a $2\times 2$ orthogonal matrix
$$
 G = \left[\matrix{c & s\cr -s'& c\cr}\right]
$$
such that
$$
 G \left[\matrix{x\cr y}\right] = \left[\matrix{\ast\cr 0}\right]
$$
with $x$ and $y$ scalars.
@end tex
@end iftex
@ifinfo
Return a 2 by 2 orthogonal matrix
@code{@var{g} = [@var{c} @var{s}; -@var{s}' @var{c}]} such that
@code{@var{g} [@var{x}; @var{y}] = [*; 0]} with @var{x} and @var{y} scalars.
@end ifinfo

For example,

@example
@group
givens (1, 1)
     @result{}   0.70711   0.70711
         -0.70711   0.70711
@end group
@end example
@end deftypefn


@anchor{doc-inv}
@deftypefn {Loadable Function} {[@var{x}, @var{rcond}] = } inv (@var{a})
@deftypefnx {Loadable Function} {[@var{x}, @var{rcond}] = } inverse (@var{a})
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.
@end deftypefn


@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'
Probable 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 probably a
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 identify the
matrix type.

Also the test for positive definiteness is a low cost test for a hermitian
matrix with a real positive diagonal. This does not guarantee that the matrix
is positive definite, but only that it is a probable candidate. When such a
matrix is factorized, a Cholesky factorization is first attempted, and if
that fails the matrix is then treated with an LU factorization. Once the
matrix has been factorized, @code{matrix_type} will return the correct
classification of the matrix.
@end deftypefn


@anchor{doc-norm}
@deftypefn {Function File} {} norm (@var{a}, @var{p})
Compute the p-norm of the matrix @var{a}.  If the second argument is
missing, @code{p = 2} is assumed.

If @var{a} is a matrix:

@table @asis
@item @var{p} = @code{1}
1-norm, the largest column sum of the absolute values of @var{a}.

@item @var{p} = @code{2}
Largest singular value of @var{a}.

@item @var{p} = @code{Inf} or @code{"inf"}
@cindex infinity norm
Infinity norm, the largest row sum of the absolute values of @var{a}.

@item @var{p} = @code{"fro"}
@cindex Frobenius norm
Frobenius norm of @var{a}, @code{sqrt (sum (diag (@var{a}' * @var{a})))}.
@end table

If @var{a} is a vector or a scalar:

@table @asis
@item @var{p} = @code{Inf} or @code{"inf"}
@code{max (abs (@var{a}))}.

@item @var{p} = @code{-Inf}
@code{min (abs (@var{a}))}.

@item @var{p} = @code{"fro"}
Frobenius norm of @var{a}, @code{sqrt (sumsq (abs (a)))}.

@item other
p-norm of @var{a}, @code{(sum (abs (@var{a}) .^ @var{p})) ^ (1/@var{p})}.
@end table
@seealso{cond, svd}
@end deftypefn


@anchor{doc-null}
@deftypefn {Function File} {} null (@var{a}, @var{tol})
Return an orthonormal basis of the null space of @var{a}.

The dimension of the null space is taken as the number of singular
values of @var{a} not greater than @var{tol}.  If the argument @var{tol}
is missing, it is computed as

@example
max (size (@var{a})) * max (svd (@var{a})) * eps
@end example
@end deftypefn


@anchor{doc-orth}
@deftypefn {Function File} {} orth (@var{a}, @var{tol})
Return an orthonormal basis of the range space of @var{a}.

The dimension of the range space is taken as the number of singular
values of @var{a} greater than @var{tol}.  If the argument @var{tol} is
missing, it is computed as

@example
max (size (@var{a})) * max (svd (@var{a})) * eps
@end example
@end deftypefn


@anchor{doc-pinv}
@deftypefn {Loadable Function} {} pinv (@var{x}, @var{tol})
Return the pseudoinverse of @var{x}.  Singular values less than
@var{tol} are ignored. 

If the second argument is omitted, it is assumed that

@example
tol = max (size (@var{x})) * sigma_max (@var{x}) * eps,
@end example

@noindent
where @code{sigma_max (@var{x})} is the maximal singular value of @var{x}.
@end deftypefn


@anchor{doc-rank}
@deftypefn {Function File} {} rank (@var{a}, @var{tol})
Compute the rank of @var{a}, using the singular value decomposition.
The rank is taken to be the number  of singular values of @var{a} that
are greater than the specified tolerance @var{tol}.  If the second
argument is omitted, it is taken to be

@example
tol = max (size (@var{a})) * sigma(1) * eps;
@end example

@noindent
where @code{eps} is machine precision and @code{sigma(1)} is the largest
singular value of @var{a}.
@end deftypefn


@anchor{doc-trace}
@deftypefn {Function File} {} trace (@var{a})
Compute the trace of @var{a}, @code{sum (diag (@var{a}))}.
@end deftypefn


@anchor{doc-rref}
@deftypefn {Function File} {[@var{r}, @var{k}] =} rref (@var{a}, @var{tol})

Returns the reduced row echelon form of @var{a}. @var{tol} defaults
to @code{eps * max (size (@var{a})) * norm (@var{a}, inf)}.

Called with two return arguments, @var{k} returns the vector of
"bound variables", which are those columns on which elimination 
has been performed.

@end deftypefn


@node Matrix Factorizations
@section Matrix Factorizations

@anchor{doc-chol}
@deftypefn {Loadable Function} {} chol (@var{a})
@cindex Cholesky factorization
Compute the Cholesky factor, @var{r}, of the symmetric positive definite
matrix @var{a}, where
@iftex
@tex
$ R^T R = A $.
@end tex
@end iftex
@ifinfo

@example
r' * r = a.
@end example
@end ifinfo
@seealso{cholinv, chol2inv}
@end deftypefn


@anchor{doc-cholinv}
@deftypefn {Loadable Function} {} cholinv (@var{a})
Use the Cholesky factorization to compute the inverse of the
symmetric positive definite matrix @var{a}.
@seealso{chol, chol2inv}
@end deftypefn


@anchor{doc-chol2inv}
@deftypefn {Loadable Function} {} chol2inv (@var{u})
Invert a symmetric, positive definite square matrix from its Cholesky
decomposition, @var{u}.  Note that @var{u} should be an upper-triangular
matrix with positive diagonal elements.  @code{chol2inv (@var{u})}
provides @code{inv (@var{u}'*@var{u})} but it is much faster than
using @code{inv}.
@seealso{chol, cholinv}
@end deftypefn


@anchor{doc-hess}
@deftypefn {Loadable Function} {@var{h} =} hess (@var{a})
@deftypefnx {Loadable Function} {[@var{p}, @var{h}] =} hess (@var{a})
@cindex Hessenberg decomposition
Compute the Hessenberg decomposition of the matrix @var{a}.

The Hessenberg decomposition is usually used as the first step in an
eigenvalue computation, but has other applications as well (see Golub,
Nash, and Van Loan, IEEE Transactions on Automatic Control, 1979).  The
Hessenberg decomposition is
@iftex
@tex
$$
A = PHP^T
$$
where $P$ is a square unitary matrix ($P^HP = I$), and $H$
is upper Hessenberg ($H_{i,j} = 0, \forall i \ge j+1$).
@end tex
@end iftex
@ifinfo
@code{p * h * p' = a} where @code{p} is a square unitary matrix
(@code{p' * p = I}, using complex-conjugate transposition) and @code{h}
is upper Hessenberg (@code{i >= j+1 => h (i, j) = 0}).
@end ifinfo
@end deftypefn


@anchor{doc-lu}
@deftypefn {Loadable Function} {[@var{l}, @var{u}, @var{p}] =} lu (@var{a})
@cindex LU decomposition
Compute the LU decomposition of @var{a}, using subroutines from
@sc{Lapack}.  The result is returned in a permuted form, according to
the optional return value @var{p}.  For example, given the matrix
@code{a = [1, 2; 3, 4]},

@example
[l, u, p] = lu (a)
@end example

@noindent
returns

@example
l =

  1.00000  0.00000
  0.33333  1.00000

u =

  3.00000  4.00000
  0.00000  0.66667

p =

  0  1
  1  0
@end example

The matrix is not required to be square.
@end deftypefn


@anchor{doc-qr}
@deftypefn {Loadable Function} {[@var{q}, @var{r}, @var{p}] =} qr (@var{a})
@cindex QR factorization
Compute the QR factorization of @var{a}, using standard @sc{Lapack}
subroutines.  For example, given the matrix @code{a = [1, 2; 3, 4]},

@example
[q, r] = qr (a)
@end example

@noindent
returns

@example
q =

  -0.31623  -0.94868
  -0.94868   0.31623

r =

  -3.16228  -4.42719
   0.00000  -0.63246
@end example

The @code{qr} factorization has applications in the solution of least
squares problems
@iftex
@tex
$$
\min_x \left\Vert A x - b \right\Vert_2
$$
@end tex
@end iftex
@ifinfo

@example
@code{min norm(A x - b)}
@end example

@end ifinfo
for overdetermined systems of equations (i.e.,
@iftex
@tex
$A$
@end tex
@end iftex
@ifinfo
@code{a}
@end ifinfo
 is a tall, thin matrix).  The QR factorization is
@iftex
@tex
$QR = A$ where $Q$ is an orthogonal matrix and $R$ is upper triangular.
@end tex
@end iftex
@ifinfo
@code{q * r = a} where @code{q} is an orthogonal matrix and @code{r} is
upper triangular.
@end ifinfo

The permuted QR factorization @code{[@var{q}, @var{r}, @var{p}] =
qr (@var{a})} forms the QR factorization such that the diagonal
entries of @code{r} are decreasing in magnitude order.  For example,
given the matrix @code{a = [1, 2; 3, 4]},

@example
[q, r, p] = qr(a)
@end example

@noindent
returns

@example
q = 

  -0.44721  -0.89443
  -0.89443   0.44721

r =

  -4.47214  -3.13050
   0.00000   0.44721

p =

   0  1
   1  0
@end example

The permuted @code{qr} factorization @code{[q, r, p] = qr (a)}
factorization allows the construction of an orthogonal basis of
@code{span (a)}.
@end deftypefn


@anchor{doc-qz}
@deftypefn {Loadable Function} {@var{lambda} =} qz (@var{a}, @var{b})
Generalized eigenvalue problem @math{A x = s B x},
@var{QZ} decomposition. There are three ways to call this function:
@enumerate
@item @code{lambda = qz(A,B)}

Computes the generalized eigenvalues
@iftex
@tex
$\lambda$
@end tex
@end iftex
@ifinfo
@var{lambda}
@end ifinfo
of @math{(A - s B)}.
@item @code{[AA, BB, Q, Z, V, W, lambda] = qz (A, B)}

Computes qz decomposition, generalized eigenvectors, and 
generalized eigenvalues of @math{(A - sB)}
@iftex
@tex
$$ AV = BV{ \rm diag }(\lambda) $$
$$ W^T A = { \rm diag }(\lambda)W^T B $$
$$ AA = Q^T AZ, BB = Q^T BZ $$
@end tex
@end iftex
@ifinfo
@example
@group

    A*V = B*V*diag(lambda)
    W'*A = diag(lambda)*W'*B
    AA = Q'*A*Z, BB = Q'*B*Z

@end group
@end example
@end ifinfo
with @var{Q} and @var{Z} orthogonal (unitary)= @var{I}

@item @code{[AA,BB,Z@{, lambda@}] = qz(A,B,opt)}

As in form [2], but allows ordering of generalized eigenpairs
for (e.g.) solution of discrete time algebraic Riccati equations.
Form 3 is not available for complex matrices, and does not compute
the generalized eigenvectors @var{V}, @var{W}, nor the orthogonal matrix @var{Q}.
@table @var
@item opt
for ordering eigenvalues of the GEP pencil.  The leading  block
of the revised pencil contains all eigenvalues that satisfy:
@table @code
@item "N"
= unordered (default) 

@item "S"
= small: leading block has all |lambda| <=1 

@item "B"
= big: leading block has all |lambda| >= 1 

@item "-"
= negative real part: leading block has all eigenvalues
in the open left half-plane

@item "+"
= nonnegative real part: leading block has all eigenvalues
in the closed right half-plane
@end  table
@end table
@end enumerate

Note: qz performs permutation balancing, but not scaling (see balance).
Order of output arguments was selected for compatibility with MATLAB

@seealso{balance, dare, eig, schur}
@end deftypefn


@anchor{doc-qzhess}
@deftypefn {Function File} {[@var{aa}, @var{bb}, @var{q}, @var{z}] =} qzhess (@var{a}, @var{b})
Compute the Hessenberg-triangular decomposition of the matrix pencil
@code{(@var{a}, @var{b})}, returning
@code{@var{aa} = @var{q} * @var{a} * @var{z}},
@code{@var{bb} = @var{q} * @var{b} * @var{z}}, with @var{q} and @var{z}
orthogonal.  For example,

@example
@group
[aa, bb, q, z] = qzhess ([1, 2; 3, 4], [5, 6; 7, 8])
@result{} aa = [ -3.02244, -4.41741;  0.92998,  0.69749 ]
@result{} bb = [ -8.60233, -9.99730;  0.00000, -0.23250 ]
@result{}  q = [ -0.58124, -0.81373; -0.81373,  0.58124 ]
@result{}  z = [ 1, 0; 0, 1 ]
@end group
@end example

The Hessenberg-triangular decomposition is the first step in
Moler and Stewart's QZ decomposition algorithm.

Algorithm taken from Golub and Van Loan, @cite{Matrix Computations, 2nd
edition}.
@end deftypefn


@anchor{doc-schur}
@deftypefn {Loadable Function} {@var{s} =} schur (@var{a})
@deftypefnx {Loadable Function} {[@var{u}, @var{s}] =} schur (@var{a}, @var{opt})
@cindex Schur decomposition
The Schur decomposition is used to compute eigenvalues of a
square matrix, and has applications in the solution of algebraic
Riccati equations in control (see @code{are} and @code{dare}).
@code{schur} always returns
@iftex
@tex
$S = U^T A U$
@end tex
@end iftex
@ifinfo
@code{s = u' * a * u}
@end ifinfo
where
@iftex
@tex
$U$
@end tex
@end iftex
@ifinfo
@code{u}
@end ifinfo
 is a unitary matrix
@iftex
@tex
($U^T U$ is identity)
@end tex
@end iftex
@ifinfo
(@code{u'* u} is identity)
@end ifinfo
and
@iftex
@tex
$S$
@end tex
@end iftex
@ifinfo
@code{s}
@end ifinfo
is upper triangular.  The eigenvalues of
@iftex
@tex
$A$ (and $S$)
@end tex
@end iftex
@ifinfo
@code{a} (and @code{s})
@end ifinfo
are the diagonal elements of
@iftex
@tex
$S$.
@end tex
@end iftex
@ifinfo
@code{s}.
@end ifinfo
If the matrix
@iftex
@tex
$A$
@end tex
@end iftex
@ifinfo
@code{a}
@end ifinfo
is real, then the real Schur decomposition is computed, in which the
matrix
@iftex
@tex
$U$
@end tex
@end iftex
@ifinfo
@code{u}
@end ifinfo
is orthogonal and
@iftex
@tex
$S$
@end tex
@end iftex
@ifinfo
@code{s}
@end ifinfo
is block upper triangular
with blocks of size at most
@iftex
@tex
$2\times 2$
@end tex
@end iftex
@ifinfo
@code{2 x 2}
@end ifinfo
along the diagonal.  The diagonal elements of
@iftex
@tex
$S$
@end tex
@end iftex
@ifinfo
@code{s}
@end ifinfo
(or the eigenvalues of the
@iftex
@tex
$2\times 2$
@end tex
@end iftex
@ifinfo
@code{2 x 2}
@end ifinfo
blocks, when
appropriate) are the eigenvalues of
@iftex
@tex
$A$
@end tex
@end iftex
@ifinfo
@code{a}
@end ifinfo
and
@iftex
@tex
$S$.
@end tex
@end iftex
@ifinfo
@code{s}.
@end ifinfo

The eigenvalues are optionally ordered along the diagonal according to
the value of @code{opt}.  @code{opt = "a"} indicates that all
eigenvalues with negative real parts should be moved to the leading
block of
@iftex
@tex
$S$
@end tex
@end iftex
@ifinfo
@code{s}
@end ifinfo
(used in @code{are}), @code{opt = "d"} indicates that all eigenvalues
with magnitude less than one should be moved to the leading block of
@iftex
@tex
$S$
@end tex
@end iftex
@ifinfo
@code{s}
@end ifinfo
(used in @code{dare}), and @code{opt = "u"}, the default, indicates that
no ordering of eigenvalues should occur.  The leading
@iftex
@tex
$k$
@end tex
@end iftex
@ifinfo
@code{k}
@end ifinfo
columns of
@iftex
@tex
$U$
@end tex
@end iftex
@ifinfo
@code{u}
@end ifinfo
always span the
@iftex
@tex
$A$-invariant
@end tex
@end iftex
@ifinfo
@code{a}-invariant
@end ifinfo
subspace corresponding to the
@iftex
@tex
$k$
@end tex
@end iftex
@ifinfo
@code{k}
@end ifinfo
leading eigenvalues of
@iftex
@tex
$S$.
@end tex
@end iftex
@ifinfo
@code{s}.
@end ifinfo
@end deftypefn


@anchor{doc-svd}
@deftypefn {Loadable Function} {@var{s} =} svd (@var{a})
@deftypefnx {Loadable Function} {[@var{u}, @var{s}, @var{v}] =} svd (@var{a})
@cindex singular value decomposition
Compute the singular value decomposition of @var{a}
@iftex
@tex
$$
 A = U S V^H
$$
@end tex
@end iftex
@ifinfo

@example
A = U*S*V'
@end example
@end ifinfo

The function @code{svd} normally returns the vector of singular values.
If asked for three return values, it computes
@iftex
@tex
$U$, $S$, and $V$.
@end tex
@end iftex
@ifinfo
U, S, and V.
@end ifinfo
For example,

@example
svd (hilb (3))
@end example

@noindent
returns

@example
ans =

  1.4083189
  0.1223271
  0.0026873
@end example

@noindent
and

@example
[u, s, v] = svd (hilb (3))
@end example

@noindent
returns

@example
u =

  -0.82704   0.54745   0.12766
  -0.45986  -0.52829  -0.71375
  -0.32330  -0.64901   0.68867

s =

  1.40832  0.00000  0.00000
  0.00000  0.12233  0.00000
  0.00000  0.00000  0.00269

v =

  -0.82704   0.54745   0.12766
  -0.45986  -0.52829  -0.71375
  -0.32330  -0.64901   0.68867
@end example

If given a second argument, @code{svd} returns an economy-sized
decomposition, eliminating the unnecessary rows or columns of @var{u} or
@var{v}.
@end deftypefn


@c FIXME -- should there be a new section here?

@anchor{doc-housh}
@deftypefn {Function File} {[@var{housv}, @var{beta}, @var{zer}] =} housh (@var{x}, @var{j}, @var{z})
Compute Householder reflection vector @var{housv} to reflect @var{x}
to be the jth column of identity, i.e.,

@example
@group
(I - beta*housv*housv')x =  norm(x)*e(j) if x(1) < 0,
(I - beta*housv*housv')x = -norm(x)*e(j) if x(1) >= 0
@end group
@end example

@noindent
Inputs

@table @var
@item x
vector
@item j
index into vector
@item z
threshold for zero  (usually should be the number 0)
@end table

@noindent
Outputs (see Golub and Van Loan):

@table @var
@item beta
If beta = 0, then no reflection need be applied (zer set to 0)
@item housv
householder vector
@end table
@end deftypefn


@anchor{doc-krylov}
@deftypefn {Function File} {[@var{u}, @var{h}, @var{nu}] =} krylov (@var{a}, @var{v}, @var{k}, @var{eps1}, @var{pflg})
Construct an orthogonal basis @var{u} of block Krylov subspace

@example
[v a*v a^2*v ... a^(k+1)*v]
@end example

@noindent
Using Householder reflections to guard against loss of orthogonality.

If @var{v} is a vector, then @var{h} contains the Hessenberg matrix
such that @code{a*u == u*h+rk*ek'}, in which @code{rk =
a*u(:,k)-u*h(:,k)}, and @code{ek'} is the vector
@code{[0, 0, @dots{}, 1]} of length @code{k}.  Otherwise, @var{h} is
meaningless.

If @var{v} is a vector and @var{k} is greater than
@code{length(A)-1}, then @var{h} contains the Hessenberg matrix such
that @code{a*u == u*h}.

The value of @var{nu} is the dimension of the span of the krylov
subspace (based on @var{eps1}).

If @var{b} is a vector and @var{k} is greater than @var{m-1}, then
@var{h} contains the Hessenberg decomposition of @var{a}.

The optional parameter @var{eps1} is the threshold for zero.  The
default value is 1e-12.

If the optional parameter @var{pflg} is nonzero, row pivoting is used
to improve numerical behavior.  The default value is 0.

Reference: Hodel and Misra, "Partial Pivoting in the Computation of
Krylov Subspaces", to be submitted to Linear Algebra and its
Applications
@end deftypefn


@node Functions of a Matrix
@section Functions of a Matrix

@anchor{doc-expm}
@deftypefn {Loadable Function} {} expm (@var{a})
Return the exponential of a matrix, defined as the infinite Taylor
series
@iftex
@tex
$$
 \exp (A) = I + A + {A^2 \over 2!} + {A^3 \over 3!} + \cdots
$$
@end tex
@end iftex
@ifinfo

@example
expm(a) = I + a + a^2/2! + a^3/3! + ...
@end example

@end ifinfo
The Taylor series is @emph{not} the way to compute the matrix
exponential; see Moler and Van Loan, @cite{Nineteen Dubious Ways to
Compute the Exponential of a Matrix}, SIAM Review, 1978.  This routine
uses Ward's diagonal
@iftex
@tex
Pad\'e
@end tex
@end iftex
@ifinfo
Pade'
@end ifinfo
approximation method with three step preconditioning (SIAM Journal on
Numerical Analysis, 1977).  Diagonal
@iftex
@tex
Pad\'e
@end tex
@end iftex
@ifinfo
Pade'
@end ifinfo
 approximations are rational polynomials of matrices
@iftex
@tex
$D_q(a)^{-1}N_q(a)$
@end tex
@end iftex
@ifinfo

@example
     -1
D (a)   N (a)
@end example

@end ifinfo
 whose Taylor series matches the first
@iftex
@tex
$2 q + 1 $
@end tex
@end iftex
@ifinfo
@code{2q+1}
@end ifinfo
terms of the Taylor series above; direct evaluation of the Taylor series
(with the same preconditioning steps) may be desirable in lieu of the
@iftex
@tex
Pad\'e
@end tex
@end iftex
@ifinfo
Pade'
@end ifinfo
approximation when
@iftex
@tex
$D_q(a)$
@end tex
@end iftex
@ifinfo
@code{Dq(a)}
@end ifinfo
is ill-conditioned.
@end deftypefn


@anchor{doc-logm}
@deftypefn {Function File} {} logm (@var{a})
Compute the matrix logarithm of the square matrix @var{a}.  Note that
this is currently implemented in terms of an eigenvalue expansion and
needs to be improved to be more robust.
@end deftypefn


@anchor{doc-sqrtm}
@deftypefn {Loadable Function} {[@var{result}, @var{error_estimate}] =} sqrtm (@var{a})
Compute the matrix square root of the square matrix @var{a}.

Ref: Nicholas J. Higham. A new sqrtm for MATLAB. Numerical Analysis
Report No. 336, Manchester Centre for Computational Mathematics,
Manchester, England, January 1999.
@seealso{expm, logm, funm}
@end deftypefn


@anchor{doc-kron}
@deftypefn {Loadable Function} {} kron (@var{a}, @var{b})
Form the kronecker product of two matrices, 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


@anchor{doc-syl}
@deftypefn {Loadable Function} {@var{x} =} syl (@var{a}, @var{b}, @var{c})
Solve the Sylvester equation
@iftex
@tex
$$
 A X + X B + C = 0
$$
@end tex
@end iftex
@ifinfo

@example
A X + X B + C = 0
@end example
@end ifinfo
using standard @sc{Lapack} subroutines.  For example,

@example
@group
syl ([1, 2; 3, 4], [5, 6; 7, 8], [9, 10; 11, 12])
     @result{} [ -0.50000, -0.66667; -0.66667, -0.50000 ]
@end group
@end example
@end deftypefn

