.. highlight:: lua

.. include:: <isogrk1.txt>

Matrices
========

Overview
--------

GSL Shell main data types are *real* and *complex* matrices and many of the GSL routines works on this kind of data. GSL Shell does not have a separate type to rapresent a *vector* but we use column matrix for this purpose.

With matrix you can do basic algebraic operation just by using the '+' and '*' operators to perform element by element operations with the matrix operands. In order to perform real matrix multiplication you can use the functions :func:`mul` and :func:`prod`. The first one take an arbitrary number of arguments and perform the matrix products of its arguments. The :func:`prod` function instead takes exactly two arguments and perform the matrix product of the transpose of the first with the second argument. So, for example, if ``x`` and ``y`` are two column matrices the expression ``prod(x, y)`` gives their scalar product.

In order to create a new matrix you should use the :func:`new` function for a *real* matrix and :func:`cnew` for a *complex* matrix. Most of GSL Shell function comes in two variants for complex and real number. Generally the *complex* variant has the same name of the real one but with 'c' prefix.

The :func:`new` function takes two mandatory arguments, the number of rows and of columns and an optional third argument. If the third argument is not given all the elements of the matrix are initialised to zero. Otherwise, if you provide a function of two variables, lets say i and j, it will be used to initialise the element of the matrix. The provided function will be called for each element of the matrix with the index of row and column as an argument. This provides a very useful way to initialise matrices. As an example, let us suppose that we want to define a matrix ``m`` of dimension :math:`n \times n` whose elements are given by :math:`m_{jk} = \textrm{exp} (2 \pi i j k / n)`.

In GSL Shell we can define the matrix with the following command::

   -- we assume that n is a positive integer number
   m = cnew(n, n, |j,k| exp(2i*pi*(j-1)*(k-1)/n))

where you have previously defined n to be a small integer number.


Matrix methods
--------------

.. method:: get(i, j)
   
   This function returns the (i,j)-th element of a matrix. As a
   general convention in GSL Shell the index of the first element is 1.

.. method:: set(i, j, v)

   This function sets the value of the (i,j)-th element of the matrix to v.   

.. method:: dims()
   
   Returns two values, in the order, the number of rows and of columns of
   the matrix.

.. method:: copy()

   Returns a copy of the matrix.

.. method:: slice(k0, k1, n0, n1)

   Return a sub-matrix obtained from the original matrix by starting
   at the element (k0, k1) and taking n0 rows and n1 columns. The
   matrix that is returned is a "view" of the existing matrix in the
   sense that it make reference to the same underlying matrix. So if
   you modify the submatrix you will implicitely modify the original
   matrix and viceversa.

.. method:: norm()

   Return the Frobenius norm of the matrix. It is defined as:

   .. math::
      \|a\| = \sqrt{\sum_i \sum_j | a_{ij} |^2}

   where a\ :sub:`ij` are the elements of the matrix.

.. method:: row(i)

   Return the submatrix given by the i-th row of the matrix.

.. method:: col(j)

   Return the submatrix given by the j-th column of the matrix.

.. method:: rows()

   Return an iterator that gives all the rows of the matrix as a submatrix.

   Example to calculate the norm of each row of a matrix ``m``::
      
      for r in m:rows() do
         print(r:norm())
      end



Matrix Functions
----------------

All the functions described in this section have an equivalent function for complex valued matrix. The functions for complex matrix are obtained by adding the 'c' prefix to the name. So for example the function :func:`mul` has an equivalent for complex matrix whose name is :func:`cmul`.

.. function:: new(r, c[, finit])

   Returns a new matrix of ``r`` rows and ``c`` columns. If ``finit``
   is not given the matrix is initialized to 0. If ``finit`` is
   provided the function ``finit(i, j)`` is called for all the
   elements with the i arguments equal to the row index and j equal to
   the column index. Then the value returned by the function is assigned
   to the matrix elements.

.. function:: tr(m)

   Return the transpose of the matrix.

.. function:: hc(m)

   Return the hermitian conjugate of the matrix.

.. function:: tocomplex(m)

   Return a new complex matrix equal to m.

.. function:: diag(v)

   Given a column vector ``v`` of length ``n`` returns a diagonal
   matrix whose diagonal elements are equal to the elements of ``v``.

.. function:: unit(n)

   Return the unit matrix of dimension nxn.

.. function:: set(a, b)

   Set the matrix ``a`` to be equal to the matrix ``b``. It raise an
   error if the dimensions of the matrices are different. Please note
   that it is different than the statement::

     a = b

   because this latter simple make the variable ``a`` refer to the
   same matrix of ``b``. With the :func:`set` function you set each
   element of an existing matrix ``a`` to the same value of the
   corresponding element of ``b``.

.. function:: null(a)

   Set each element of the matrix ``a`` to zero.

.. function:: mul(a, b[, c, ...])

   Perform the matrix multiplication of the given arguments in the given order.

.. function:: prod(A, B)

   Return the matrix product of the transpose of A time B. It can be useful
   also to compute the scalar product of a column vector.
