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 mul() and prod(). The first one take an arbitrary number of arguments and perform the matrix products of its arguments. The 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 new() function for a real matrix and cnew() for a complex matrix. The basic function and operations that operates on matrix can accept both real or complex matrix. If one of the operand is complex all the other operands are implicitly converted to a complex matrix and a complex matrix is returned.
The 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 whose elements are given by
.
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.
To access an element of a matrix placed at the i-th row and the j-th column you can use the syntax m:get(i, j) where m is the matrix. You can also change the value with the following syntax m:set(i, j, x) to set the element to the value x.
In GSL shell the matrix with only one column are considered like vectors and you can reference their elements just by indexing the vector like in v[i]. This syntax can be used also to assign a new value to the given element like we illustrate in the following example:
-- we define a vector, it is just a matrix with one column
v = vector {0, 0.2, 0.4, 7.6, 0.8, 1.0}
-- we can obtain the value of the 4th element with the syntax
v[4]
-- but we can change also its value
v[4] = 0.6
For the other side simple indexing will not work for matrix with more than one column and in this latter case you shound use the methods get() and set().
To obtain the dimension of a matrix you should use the dim() function that takes the matrix as argument and returns its number of rows and columns. Here an example:
-- we define a matrix
m = matrix {{0.2, 4, -0.5}, {2, 0, 1.1}}
-- we obtain the number of rows and columns, r and c respectively
r, c = dim(m)
Note
You can also use the length operator # with a matrix to obtain just the number of rows.
Return the Frobenius norm of the matrix. It is defined as:
where aij are the elements of the matrix.
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
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 mul() has an equivalent for complex matrix whose name is cmul().
Convert the table t into a matrix. The table should be in the form {{row1_v1, row1_v2, ...}, {row2_v1, row2_v2, ...}, ...} where each term is a number. You should also ensure that all the lines contains the same number of elements. Example:
th = pi/6
-- create a 2x2 rotation matrix using the angle th
m = matrix {{cos(th), sin(th)}, {-sin(th), cos(th)}}
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 set() function you set each element of an existing matrix a to the same value of the corresponding element of b.