octave-nkf: doc/interpreter/linalg.txi comparison

comparison doc/interpreter/linalg.txi @ 3372:f16c2ce14886

[project @ 1999-11-23 19:07:09 by jwe]

author	jwe
date	Tue, 23 Nov 1999 19:07:18 +0000
parents	bfe1573bd2ae
children	36405da8e173

comparison

equal deleted inserted replaced

-:86873384cd10
+:f16c2ce14886
 @end menu
 @node Basic Matrix Functions, Matrix Factorizations, Linear Algebra, Linear Algebra
 @section Basic Matrix Functions
-@deftypefn {Loadable Function} {@var{aa} =} balance (@var{a}, @var{opt})
+DOCSTRING(balance)
-@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})
-@code{[dd, aa] = balance (a)} returns @code{aa = dd \ a * dd}.
+DOCSTRING(cond)
-@code{aa} is a matrix whose row and column norms are roughly equal in
-magnitude, and @code{dd} = @code{p * d}, where @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.
-@code{[cc, dd, aa, bb] = balance (a, b)} returns @code{aa = cc*a*dd} and
+DOCSTRING(det)
-@code{bb = cc*b*dd)}, where @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} is selected as follows:
+DOCSTRING(eig)
-@table @asis
+DOCSTRING(givens)
-@item @code{"N"}, @code{"n"}
-No balancing; arguments copied, transformation(s) set to identity.
-@item @code{"P"}, @code{"p"}
+DOCSTRING(inv)
-Permute argument(s) to isolate eigenvalues where possible.
-@item @code{"S"}, @code{"s"}
+DOCSTRING(norm)
-Scale to improve accuracy of computed eigenvalues.
-@item @code{"B"}, @code{"b"}
+DOCSTRING(null)
-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.
+DOCSTRING(orth)
-Generalized eigenvalue problem balancing uses Ward's algorithm
+DOCSTRING(pinv)
-(SIAM Journal on Scientific and Statistical Computing, 1981).
-@end deftypefn
-@deftypefn {} {} cond (@var{a})
+DOCSTRING(rank)
-Compute the (two-norm) condition number of a matrix. @code{cond (a)} is
-defined as @code{norm (a) * norm (inv (a))}, and is computed via a
-singular value decomposition.
-@end deftypefn
-@deftypefn {Loadable Function} {} det (@var{a})
+DOCSTRING(trace)
-Compute the determinant of @var{a} using @sc{Linpack}.
-@end deftypefn
-@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.
-@end deftypefn
-@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
-@deftypefn {Loadable Function} {} inv (@var{a})
-@deftypefnx {Loadable Function} {} inverse (@var{a})
-Compute the inverse of the square matrix @var{a}.
-@end deftypefn
-@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 @var{a}.
-@item @var{p} = @code{2}
-Largest singular value of @var{a}.
-@item @var{p} = @code{Inf}
-@cindex infinity norm
-Infinity norm, the largest row sum 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}
-@code{max (abs (@var{a}))}.
-@item @var{p} = @code{-Inf}
-@code{min (abs (@var{a}))}.
-@item other
-p-norm of @var{a}, @code{(sum (abs (@var{a}) .^ @var{p})) ^ (1/@var{p})}.
-@end table
-@end deftypefn
-@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
-@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
-@deftypefn {Function File} {} 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
-@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} is the largest
-singular value of @var{a}.
-@end deftypefn
-@deftypefn {Function File} {} trace (@var{a})
-Compute the trace of @var{a}, @code{sum (diag (@var{a}))}.
-@end deftypefn
 @node Matrix Factorizations, Functions of a Matrix, Basic Matrix Functions, Linear Algebra
 @section Matrix Factorizations
-@deftypefn {Loadable Function} {} chol (@var{a})
+DOCSTRING(chol)
-@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
+DOCSTRING(hess)
-r' * r = a.
-@end example
-@end ifinfo
-@end deftypefn
-@deftypefn {Loadable Function} {@var{h} =} hess (@var{a})
+DOCSTRING(lu)
-@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
+DOCSTRING(qr)
-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
-@deftypefn {Loadable Function} {[@var{l}, @var{u}, @var{p}] =} lu (@var{a})
+DOCSTRING(qz)
-@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
+DOCSTRING(qzhess)
-[l, u, p] = lu (a)
-@end example
-@noindent
+DOCSTRING(schur)
-returns
-@example
+DOCSTRING(svd)
-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
-@end deftypefn
-@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, pi] = 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
-@deftypefn {Function File} {@var{lambda} =} qz (@var{a}, @var{b})
-Generalized eigenvalue problem @math{A x = s B x},
-@var{QZ} decomposition.  Three ways to call:
-@enumerate
-@item @code{lambda = qz(A,B)}
-Computes the generalized eigenvalues @var{lambda} of @math{(A - sB)}.
-@item @code{[AA, BB, Q, Z @{, V, W, lambda@}] = qz (A, B)}
-Computes qz decomposition, generalized eigenvectors, and
-generalized eigenvalues of @math{(A - sB)}
-@example
-@group
-A V = B V diag(lambda)
-W' A = diag(lambda) W' B
-AA = Q'*A*Z, BB = Q'*B*Z  with Q, Z orthogonal (unitary)= I
-@end group
-@end example
-@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 V, W, nor the orthogonal matrix 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-plant
-@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
-See also: balance, dare, eig, schur
-@end deftypefn
-@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
-@deftypefn {Loadable Function} {} qzval (@var{a}, @var{b})
-Compute generalized eigenvalues of the matrix pencil
-@iftex
-@tex
-$a - \lambda b$.
-@end tex
-@end iftex
-@ifinfo
-@code{@var{a} - lambda @var{b}}.
-@end ifinfo
-The arguments @var{a} and @var{b} must be real matrices.
-@end deftypefn
-@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
-blocks 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
-@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
-$$
-A = U\Sigma V^H
-$$
-@end tex
-@end iftex
-@ifinfo
-@example
-a = u * sigma * 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
 @node Functions of a Matrix,  , Matrix Factorizations, Linear Algebra
 @section Functions of a Matrix
-@deftypefn {Loadable Function} {} expm (@var{a})
+DOCSTRING(expm)
-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
+DOCSTRING(logm)
-expm(a) = I + a + a^2/2! + a^3/3! + ...
-@end example
-@end ifinfo
+DOCSTRING(sqrtm)
-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
+DOCSTRING(kron)
--1
-D (a)   N (a)
-@end example
-@end ifinfo
+DOCSTRING(syl)
-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
-@deftypefn {Loadable Function} {} 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
-@deftypefn {Loadable Function} {} sqrtm (@var{a})
-Compute the matrix square root 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
-@deftypefn {Function File} {} 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
-@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

Mercurial > hg > octave-nkf

comparison doc/interpreter/linalg.txi @ 3372:f16c2ce14886