lines 12-69 of file: xrst/algorithm.xrst

{xrst_begin det_by_minor}
{xrst_spell
   th
}

Determinant Using Expansion by Minors
#####################################

ell
***
We use :math:`\ell` to denote the row and column dimension
of the square matrix under consideration.

Algorithm
*********
This algorithm computes :math:`|A|` the determinant of a square matrix
:math:`A \in \B{R}^{\ell \times \ell}` .
In the special case where *\ell* is one, the determinant
is just the element of the matrix.
Otherwise, the determinant is computed using the formula

.. math::

   | A | = + | A(0,0) | - | A(0,1) |  + | A(0,2) | \cdots \pm  | A(0,\ell-1) |

where the last term is + (-) if *\ell* is odd (even) and
:math:`A(i,j) \in \B{R}^{(\ell-1) \times (\ell-1)}` is the
:math:`(i,j)` minor of *A*\ ; i.e., the square sub-matrix formed by deleting
the *i*\ -th row and *j*\ -th column of *A* .

option
******
The only option used by this algorithm is ``n_arg`` ; see below:

n_arg
=====
This is the number of arguments to the algorithm
which is the number of elements in the matrix; i.e., :math:`\ell^2` .
There is an assert checking that *n_arg* > 0.

n_other
=======
This option is not used except that it may be checked to make sure it is zero.

Implementation
==============
:ref:`cpp_det_by_minor-name` , :ref:`py_det_by_minor-name` .

Derivative
**********
The partial derivative of :math:`|A|` with respect to
the :math:`(i,j)`\ -th element :math:`A_{i,j}` is

.. math::

   \frac{ \partial |A| }{ \partial A_{i,j} } = (-1)^{i+j} | A(i,j) |

{xrst_end det_by_minor}