Functions |
| | DEFUN_DLD (qr, args, nargout,"-*- texinfo -*-\n\
@deftypefn {Loadable Function} {[@var{Q}, @var{R}, @var{P}] =} qr (@var{A})\n\
@deftypefnx {Loadable Function} {[@var{Q}, @var{R}, @var{P}] =} qr (@var{A}, '0')\n\
@cindex QR factorization\n\
Compute the QR@tie{}factorization of @var{A}, using standard @sc{lapack}\n\
subroutines. For example, given the matrix @code{@var{A} = [1, 2; 3, 4]},\n\
\n\
@example\n\
[@var{Q}, @var{R}] = qr (@var{A})\n\
@end example\n\
\n\
@noindent\n\
returns\n\
\n\
@example\n\
@group\n\
@var{Q} =\n\
\n\
-0.31623 -0.94868\n\
-0.94868 0.31623\n\
\n\
@var{R} =\n\
\n\
-3.16228 -4.42719\n\
0.00000 -0.63246\n\
@end group\n\
@end example\n\
\n\
The @code{qr} factorization has applications in the solution of least\n\
squares problems\n\
@tex\n\
$$\n\
\\min_x \\left\\Vert A x - b \\right\\Vert_2\n\
$$\n\
@end tex\n\
@ifnottex\n\
\n\
@example\n\
@code{min norm(A x - b)}\n\
@end example\n\
\n\
@end ifnottex\n\
for overdetermined systems of equations (i.e.,\n\
@tex\n\
$A$\n\
@end tex\n\
@ifnottex\n\
@var{A}\n\
@end ifnottex\n\
is a tall, thin matrix). The QR@tie{}factorization is\n\
@tex\n\
$QR = A$ where $Q$ is an orthogonal matrix and $R$ is upper triangular.\n\
@end tex\n\
@ifnottex\n\
@code{@var{Q} * @var{Q} = @var{A}} where @var{Q} is an orthogonal matrix and\n\
@var{R} is upper triangular.\n\
@end ifnottex\n\
\n\
If given a second argument of '0', @code{qr} returns an economy-sized\n\
QR@tie{}factorization, omitting zero rows of @var{R} and the corresponding\n\
columns of @var{Q}.\n\
\n\
If the matrix @var{A} is full, the permuted QR@tie{}factorization\n\
@code{[@var{Q}, @var{R}, @var{P}] = qr (@var{A})} forms the\n\
QR@tie{}factorization such that the diagonal entries of @var{R} are\n\
decreasing in magnitude order. For example, given the matrix @code{a = [1,\n\
2; 3, 4]},\n\
\n\
@example\n\
[@var{Q}, @var{R}, @var{P}] = qr (@var{A})\n\
@end example\n\
\n\
@noindent\n\
returns\n\
\n\
@example\n\
@group\n\
@var{Q} = \n\
\n\
-0.44721 -0.89443\n\
-0.89443 0.44721\n\
\n\
@var{R} =\n\
\n\
-4.47214 -3.13050\n\
0.00000 0.44721\n\
\n\
@var{P} =\n\
\n\
0 1\n\
1 0\n\
@end group\n\
@end example\n\
\n\
The permuted @code{qr} factorization @code{[@var{Q}, @var{R}, @var{P}] = qr\n\
(@var{A})} factorization allows the construction of an orthogonal basis of\n\
@code{span (A)}.\n\
\n\
If the matrix @var{A} is sparse, then compute the sparse\n\
QR@tie{}factorization of @var{A}, using @sc{CSparse}. As the matrix @var{Q}\n\
is in general a full matrix, this function returns the @var{Q}-less\n\
factorization @var{R} of @var{A}, such that @code{@var{R} = chol (@var{A}' *\n\
@var{A})}.\n\
\n\
If the final argument is the scalar @code{0} and the number of rows is\n\
larger than the number of columns, then an economy factorization is\n\
returned. That is @var{R} will have only @code{size (@var{A},1)} rows.\n\
\n\
If an additional matrix @var{B} is supplied, then @code{qr} returns\n\
@var{C}, where @code{@var{C} = @var{Q}' * @var{B}}. This allows the\n\
least squares approximation of @code{@var{A} \\ @var{B}} to be calculated\n\
as\n\
\n\
@example\n\
@group\n\
[@var{C},@var{R}] = spqr (@var{A},@var{B})\n\
x = @var{R} \\ @var{C}\n\
@end group\n\
@end example\n\
@end deftypefn") |
| | DEFUN_DLD (qrupdate, args,,"-*- texinfo -*-\n\
@deftypefn {Loadable Function} {[@var{Q1}, @var{R1}] =} qrupdate (@var{Q}, @var{R}, @var{u}, @var{v})\n\
Given a QR@tie{}factorization of a real or complex matrix\n\
@w{@var{A} = @var{Q}*@var{R}}, @var{Q}@tie{}unitary and\n\
@var{R}@tie{}upper trapezoidal, return the QR@tie{}factorization\n\
of @w{@var{A} + @var{u}*@var{v}'}, where @var{u} and @var{v} are\n\
column vectors (rank-1 update) or matrices with equal number of columns\n\
(rank-k update). Notice that the latter case is done as a sequence of rank-1\n\
updates; thus, for k large enough, it will be both faster and more accurate\n\
to recompute the factorization from scratch.\n\
\n\
The QR@tie{}factorization supplied may be either full\n\
(Q is square) or economized (R is square).\n\
\n\
@seealso{qr, qrinsert, qrdelete}\n\
@end deftypefn") |
| | DEFUN_DLD (qrinsert, args,,"-*- texinfo -*-\n\
@deftypefn {Loadable Function} {[@var{Q1}, @var{R1}] =} qrinsert (@var{Q}, @var{R}, @var{j}, @var{x}, @var{orient})\n\
Given a QR@tie{}factorization of a real or complex matrix\n\
@w{@var{A} = @var{Q}*@var{R}}, @var{Q}@tie{}unitary and\n\
@var{R}@tie{}upper trapezoidal, return the QR@tie{}factorization of\n\
@w{[A(:,1:j-1) x A(:,j:n)]}, where @var{u} is a column vector to be\n\
inserted into @var{A} (if @var{orient} is @code{\"col\"}), or the\n\
QR@tie{}factorization of @w{[A(1:j-1,:);x;A(:,j:n)]}, where @var{x}\n\
is a row vector to be inserted into @var{A} (if @var{orient} is\n\
@code{\"row\"}).\n\
\n\
The default value of @var{orient} is @code{\"col\"}.\n\
If @var{orient} is @code{\"col\"},\n\
@var{u} may be a matrix and @var{j} an index vector\n\
resulting in the QR@tie{}factorization of a matrix @var{B} such that\n\
@w{B(:,@var{j})} gives @var{u} and @w{B(:,@var{j}) = []} gives @var{A}.\n\
Notice that the latter case is done as a sequence of k insertions;\n\
thus, for k large enough, it will be both faster and more accurate to\n\
recompute the factorization from scratch.\n\
\n\
If @var{orient} is @code{\"col\"},\n\
the QR@tie{}factorization supplied may be either full\n\
(Q is square) or economized (R is square).\n\
\n\
If @var{orient} is @code{\"row\"}, full factorization is needed.\n\
@seealso{qr, qrupdate, qrdelete}\n\
@end deftypefn") |
| | DEFUN_DLD (qrdelete, args,,"-*- texinfo -*-\n\
@deftypefn {Loadable Function} {[@var{Q1}, @var{R1}] =} qrdelete (@var{Q}, @var{R}, @var{j}, @var{orient})\n\
Given a QR@tie{}factorization of a real or complex matrix\n\
@w{@var{A} = @var{Q}*@var{R}}, @var{Q}@tie{}unitary and\n\
@var{R}@tie{}upper trapezoidal, return the QR@tie{}factorization of\n\
@w{[A(:,1:j-1) A(:,j+1:n)]}, i.e., @var{A} with one column deleted\n\
(if @var{orient} is \"col\"), or the QR@tie{}factorization of\n\
@w{[A(1:j-1,:);A(:,j+1:n)]}, i.e., @var{A} with one row deleted (if\n\
@var{orient} is \"row\").\n\
\n\
The default value of @var{orient} is \"col\".\n\
\n\
If @var{orient} is @code{\"col\"},\n\
@var{j} may be an index vector\n\
resulting in the QR@tie{}factorization of a matrix @var{B} such that\n\
@w{A(:,@var{j}) = []} gives @var{B}.\n\
Notice that the latter case is done as a sequence of k deletions;\n\
thus, for k large enough, it will be both faster and more accurate to\n\
recompute the factorization from scratch.\n\
\n\
If @var{orient} is @code{\"col\"},\n\
the QR@tie{}factorization supplied may be either full\n\
(Q is square) or economized (R is square).\n\
\n\
If @var{orient} is @code{\"row\"}, full factorization is needed.\n\
@seealso{qr, qrinsert, qrupdate}\n\
@end deftypefn") |
| | DEFUN_DLD (qrshift, args,,"-*- texinfo -*-\n\
@deftypefn {Loadable Function} {[@var{Q1}, @var{R1}] =} qrshift (@var{Q}, @var{R}, @var{i}, @var{j})\n\
Given a QR@tie{}factorization of a real or complex matrix\n\
@w{@var{A} = @var{Q}*@var{R}}, @var{Q}@tie{}unitary and\n\
@var{R}@tie{}upper trapezoidal, return the QR@tie{}factorization\n\
of @w{@var{A}(:,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{qr, qrinsert, qrdelete}\n\
@end deftypefn") |
