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Functions |
| | DEFUN_DLD (chol, args, nargout,"-*- texinfo -*-\n\
@deftypefn {Loadable Function} {@var{r} =} chol (@var{a})\n\
@deftypefnx {Loadable Function} {[@var{r}, @var{p}] =} chol (@var{a})\n\
@deftypefnx {Loadable Function} {[@var{r}, @var{p}, @var{q}] =} chol (@var{s})\n\
@deftypefnx {Loadable Function} {[@var{r}, @var{p}, @var{q}] =} chol (@var{s}, 'vector')\n\
@deftypefnx {Loadable Function} {[@var{l}, @dots{}] =} chol (@dots{}, 'lower')\n\
@cindex Cholesky factorization\n\
Compute the Cholesky factor, @var{r}, of the symmetric positive definite\n\
matrix @var{a}, where\n\
@iftex\n\
@tex\n\
$ R^T R = A $.\n\
@end tex\n\
@end iftex\n\
@ifnottex\n\
\n\
@example\n\
@var{r}' * @var{r} = @var{a}.\n\
@end example\n\
@end ifnottex\n\
\n\
Called with one output argument @code{chol} fails if @var{a} or @var{s} is\n\
not positive definite. With two or more output arguments @var{p} flags\n\
whether the matrix was positive definite and @code{chol} does not fail. A\n\
zero value indicated that the matrix was positive definite and the @var{r}\n\
gives the factorization, and @var{p} will have a positive value otherwise.\n\
\n\
If called with 3 outputs then a sparsity preserving row/column permutation\n\
is applied to @var{a} prior to the factorization. That is @var{r}\n\
is the factorization of @code{@var{a}(@var{q},@var{q})} such that\n\
@iftex\n\
@tex\n\
$ R^T R = Q^T A Q$.\n\
@end tex\n\
@end iftex\n\
@ifnottex\n\
\n\
@example\n\
@var{r}' * @var{r} = @var{q}' * @var{a} * @var{q}.\n\
@end example\n\
@end ifnottex\n\
\n\
The sparsity preserving permutation is generally returned as a matrix.\n\
However, given the flag 'vector', @var{q} will be returned as a vector\n\
such that\n\
@iftex\n\
@tex\n\
$ R^T R = A (Q, Q)$.\n\
@end tex\n\
@end iftex\n\
@ifnottex\n\
\n\
@example\n\
@var{r}' * @var{r} = a (@var{q}, @var{q}).\n\
@end example\n\
@end ifnottex\n\
\n\
Called with either a sparse or full matrix and using the 'lower' flag,\n\
@code{chol} returns the lower triangular factorization such that\n\
@iftex\n\
@tex\n\
$ L L^T = A $.\n\
@end tex\n\
@end iftex\n\
@ifnottex\n\
\n\
@example\n\
@var{l} * @var{l}' = @var{a}.\n\
@end example\n\
@end ifnottex\n\
\n\
In general the lower triangular factorization is significantly faster for\n\
sparse matrices.\n\
@seealso{cholinv, chol2inv}\n\
@end deftypefn") |
| | DEFUN_DLD (cholinv, args,,"-*- texinfo -*-\n\
@deftypefn {Loadable Function} {} cholinv (@var{a})\n\
Use the Cholesky factorization to compute the inverse of the\n\
symmetric positive definite matrix @var{a}.\n\
@seealso{chol, chol2inv}\n\
@end deftypefn") |
| | DEFUN_DLD (chol2inv, args,,"-*- texinfo -*-\n\
@deftypefn {Loadable Function} {} chol2inv (@var{u})\n\
Invert a symmetric, positive definite square matrix from its Cholesky\n\
decomposition, @var{u}. Note that @var{u} should be an upper-triangular\n\
matrix with positive diagonal elements. @code{chol2inv (@var{u})}\n\
provides @code{inv (@var{u}'*@var{u})} but it is much faster than\n\
using @code{inv}.\n\
@seealso{chol, cholinv}\n\
@end deftypefn") |
| | DEFUN_DLD (cholupdate, args, nargout,"-*- texinfo -*-\n\
@deftypefn {Loadable Function} {[@var{R1}, @var{info}] =} cholupdate (@var{R}, @var{u}, @var{op})\n\
Update or downdate a Cholesky factorization. Given an upper triangular\n\
matrix @var{R} and a column vector @var{u}, attempt to determine another\n\
upper triangular matrix @var{R1} such that\n\
@itemize @bullet\n\
@item\n\
@var{R1}'*@var{R1} = @var{R}'*@var{R} + @var{u}*@var{u}'\n\
if @var{op} is \"+\"\n\
@item\n\
@var{R1}'*@var{R1} = @var{R}'*@var{R} - @var{u}*@var{u}'\n\
if @var{op} is \"-\"\n\
@end itemize\n\
\n\
If @var{op} is \"-\", @var{info} is set to\n\
@itemize\n\
@item 0 if the downdate was successful,\n\
@item 1 if @var{R}'*@var{R} - @var{u}*@var{u}' is not positive definite,\n\
@item 2 if @var{R} is singular.\n\
@end itemize\n\
\n\
If @var{info} is not present, an error message is printed in cases 1 and 2.\n\
@seealso{chol, qrupdate}\n\
@end deftypefn") |
| | DEFUN_DLD (cholinsert, args, nargout,"-*- texinfo -*-\n\
@deftypefn {Loadable Function} {[@var{R1}, @var{info}] =} cholinsert (@var{R}, @var{j}, @var{u})\n\
Given a Cholesky@tie{}factorization of a real symmetric or complex hermitian\n\
positive definite matrix @w{@var{A} = @var{R}'*@var{R}}, @var{R}@tie{}upper triangular,\n\
return the Cholesky@tie{}factorization of\n\
@var{A1}, where @w{A1(p,p) = A}, @w{A1(:,j) = A1(j,:)' = u} and\n\
@w{p = [1:j-1,j+1:n+1]}. @w{u(j)} should be positive.\n\
On return, @var{info} is set to\n\
@itemize\n\
@item 0 if the insertion was successful,\n\
@item 1 if @var{A1} is not positive definite,\n\
@item 2 if @var{R} is singular.\n\
@end itemize\n\
\n\
If @var{info} is not present, an error message is printed in cases 1 and 2.\n\
@seealso{chol, cholupdate, choldelete}\n\
@end deftypefn") |
| | DEFUN_DLD (choldelete, args,,"-*- texinfo -*-\n\
@deftypefn {Loadable Function} {@var{R1} =} choldelete (@var{R}, @var{j})\n\
Given a Cholesky@tie{}factorization of a real symmetric or complex hermitian\n\
positive definite matrix @w{@var{A} = @var{R}'*@var{R}}, @var{R}@tie{}upper triangular,\n\
return the Cholesky@tie{}factorization of @w{A(p,p)}, where @w{p = [1:j-1,j+1:n+1]}.\n\
@seealso{chol, cholupdate, cholinsert}\n\
@end deftypefn") |
| | DEFUN_DLD (cholshift, args,,"-*- texinfo -*-\n\
@deftypefn {Loadable Function} {@var{R1} =} cholshift (@var{R}, @var{i}, @var{j})\n\
Given a Cholesky@tie{}factorization of a real symmetric or complex hermitian\n\
positive definite matrix @w{@var{A} = @var{R}'*@var{R}}, @var{R}@tie{}upper triangular,\n\
return the Cholesky@tie{}factorization of\n\
@w{@var{A}(p,p)}, where @w{p} is the permutation @*\n\
@code{p = [1:i-1, shift(i:j, 1), j+1:n]} if @w{@var{i} < @var{j}} @*\n\
or @*\n\
@code{p = [1:j-1, shift(j:i,-1), i+1:n]} if @w{@var{j} < @var{i}}. @*\n\
\n\
@seealso{chol, cholinsert, choldelete}\n\
@end deftypefn") |
