lines 8-142 of file: cpp/include/cmpad/algo/det_of_minor.hpp

{xrst_begin_parent cpp_det_of_minor}
{xrst_spell
   obj
}

C++ Determinant of a Minor
##########################

Syntax
******
| |tab| ``# include <cmpad/algo/det_of_minor.hpp>``
| |tab| *d* = ``cmpad::det_of_minor`` ( *a* , *n* , *m* , *r* , *c* )

Prototype
*********
{xrst_literal
   // BEGIN PROTOTYPE
   // END PROTOTYPE
}

Purpose
*******
This template function
computes the determinant of a minor of the matrix :math:`A`
using expansion by minors.
It is for example and testing purposes only.
Expansion by minors is chosen as an example because it uses
a lot of floating point operations yet does not require much source code.
This is not an efficient method for computing a determinant;
for example, using an LU factorization would be faster.

Minor
*****
The elements of the :math:`m \times m` minor :math:`M`
of the matrix :math:`A` are defined,
for :math:`i = 0 , \ldots , m-1` and :math:`j = 0 , \ldots , m-1`, by

.. math::

   M_{i,j} = A_{R(i), C(j)}

where the function
:math:`R(i)` is defined by the :ref:`argument r<cpp_det_of_minor@r>` and
:math:`C(j)` is defined by the :ref:`argument c<cpp_det_of_minor@c>` .


Determinant of A
****************
If the following conditions hold, the minor is the
entire matrix :math:`A` and hence ``det_of_minor``
will return the determinant of :math:`A`:

#. :math:`m = n`.
#. for :math:`i = 0 , \ldots , n-1`, :math:`r[i] = i+1`,
   and :math:`r[n] = 0`.
#. for :math:`j = 0 , \ldots , n-1`, :math:`c[j] = j+1`,
   and :math:`c[n] = 0`.

Vector
******
This type satisfies the conditions for a
fun_obj :ref:`cpp_fun_obj@Vector` .

a
*
The elements of the :math:`n \times n` matrix :math:`A` are defined,
for :math:`i = 0 , \ldots , n-1` and :math:`j = 0 , \ldots , n-1`, by

.. math::

   A_{i,j} = a[ i * n + j]

n
*
This is the number of rows (and columns) in the square matrix :math:`A`.

m
*
This is the number of rows (and columns) in the square minor :math:`M`.

r
*
This defines the function :math:`R(i)`
which specifies the rows of the minor :math:`M`.
To be specific, the function :math:`R(i)`
for :math:`i = 1, \ldots , m-1` is defined by

.. math::

   \begin{eqnarray}
      R(0)   & = & r[n]
      \\
      R(i) & = & r[ R(i-1) ]
   \end{eqnarray}

All the elements of *r* have value less than or equal *n* ;
:math:`R(i) < n` and :math:`r[ R(m-1) ] = n` .
The elements of vector *r* are modified during the computation,
and restored to their original value before the return from ``det_of_minor`` .

c
*
This defines the function :math:`C(i)`
which specifies the columns of the minor :math:`M`.
To be specific, the function :math:`C(i)`
for :math:`j = 1, \ldots , m-1` is defined by

.. math::

   \begin{eqnarray}
      C(0)   & = & c[n]
      \\
      C(j) & = & c[ C(j-1) ]
   \end{eqnarray}

All the elements of *c* must have value less than or equal *n* ;
:math:`C(j) < n` and :math:`c[ C(m-1) ] = n` .
The elements of vector *c* are modified during the computation,
and restored to their original value before the return from ``det_of_minor`` .

d
*
The return value *d* is equal to the determinant of the minor :math:`M`
and has the same type as the elements of *a*.

{xrst_toc_table after
   cpp/xam/det_of_minor.cpp
}

Example
*******
:ref:`xam_det_of_minor.cpp-name`
contains an example and test of ``det_of_minor`` .

{xrst_end cpp_det_of_minor}