Low-level functions in the Modular package allow for working with or on part of a Matrix or Vector, or with the transpose of a Matrix or Vector. This is most useful in utilization of the package for algorithms that use some form of blocking, or when only a part of a calculation is needed.

Using standard Maple syntax to define sub-Matrices and sub-Vectors, for example, $M_{1..5,2..3}$) produces a copy of the data, which is inefficient when the copy is not required for some other purpose. This functionality is provided to avoid data duplication.

Examples of the available functionality include interpreting a row of a Matrix as a row Vector, interpreting a row Vector as a column Vector, interpreting some part of a Matrix as a smaller Matrix, etcetera.

The basic specification for a Matrix is of the form $\mathrm{Matrix},r$, $\mathrm{r1}..\mathrm{r2},c$, or $\mathrm{c1}..\mathrm{c2},'\mathrm{transpose}'$. Each of the three trailing entries is optional, but in order to provide a column-column range for a Matrix, the row-row range must be provided.

If the row is specified as a fixed value, for example, 2, but the column is not provided or specified as a range, the Matrix is interpreted as a row Vector with values corresponding to the specified row of the Matrix.

If the row is provided as a range and the column is a fixed value, the Matrix is interpreted as a column Vector with values corresponding to the specified column of the Matrix.

If both the row and column are fixed values, the selected entry of the Matrix is interpreted as a scalar (only valid for output).

If both the row and column are specified as ranges, the Matrix is interpreted as a sub-Matrix with the specified rows and columns.

Note: For all functions, it is invalid for a range to be empty. Ranges can include negative values, which are interpreted as offsets from the highest available value, similar to lists. For example, $1..\mathrm{-1}$ refers to the entire range.

The 'transpose' keyword is only available for input objects (not for output), and indicates that the specified sub-Matrix is transposed prior to performing the requested operation.

$\mathrm{with}\left(\mathrm{LinearAlgebra}\left[\mathrm{Modular}\right]\right)\:$

$M≔\mathrm{Mod}\left(13\,\mathrm{Matrix}\left(3\,4\,\left(i\,j\right)\mapsto \mathrm{rand}\left(\right)\right)\,\mathrm{integer}\left[\right]\right)$

$M≔\left[\begin{array}{cccc}10& 0& 8& 12\\ 2& 6& 0& 11\\ 9& 2& 11& 7\end{array}\right]$

$\mathrm{Copy}\left(13\,M\,2\right)$

$\left[\begin{array}{cccc}2& 6& 0& 11\end{array}\right]$

$\mathrm{Copy}\left(13\,M\,1..-1\,3\right)$

$\left[\begin{array}{c}8\\ 0\\ 11\end{array}\right]$

$\mathrm{Copy}\left(13\,M\,2..-1\,1..-2\,'\mathrm{transpose}'\right)$

$\left[\begin{array}{cc}2& 9\\ 6& 2\\ 0& 11\end{array}\right]$

$V≔\mathrm{Mod}\left(13\,\mathrm{Vector}\left[\mathrm{column}\right]\left(4\,\left(i\,j\right)\mapsto \mathrm{rand}\left(\right)\right)\,\mathrm{integer}\left[\right]\right)$

$V≔\left[\begin{array}{c}4\\ 11\\ 12\\ 11\end{array}\right]$

$\mathrm{Copy}\left(13\,V\,2..-2\right),\mathrm{Copy}\left(13\,V\,2..-2\,'\mathrm{transpose}'\right)$

$\left[\begin{array}{c}11\\ 12\end{array}\right],\left[\begin{array}{cc}11& 12\end{array}\right]$

$V≔\mathrm{Mod}\left(13\,\mathrm{Vector}\left[\mathrm{row}\right]\left(5\,\left(i\,j\right)\mapsto \mathrm{rand}\left(\right)\right)\,\mathrm{integer}\left[\right]\right)$

$V≔\left[\begin{array}{ccccc}8& 9& 4& 11& 10\end{array}\right]$

$\mathrm{Copy}\left(13\,V\,2..-2\right),\mathrm{Copy}\left(13\,V\,2..-2\,'\mathrm{transpose}'\right)$

$\left[\begin{array}{ccc}9& 4& 11\end{array}\right],\left[\begin{array}{c}9\\ 4\\ 11\end{array}\right]$

$M≔\mathrm{Mod}\left(13\,\mathrm{Matrix}\left(4\,3\,\left(i\,j\right)\mapsto \mathrm{rand}\left(\right)\right)\,\mathrm{integer}\left[\right]\right)$

$M≔\left[\begin{array}{ccc}3& 5& 5\\ 1& 6& 7\\ 3& 0& 8\\ 3& 11& 7\end{array}\right]$

$P≔\mathrm{Create}\left(13\,4\,4\,\mathrm{integer}\left[\right]\right)$

$P≔\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$

$$\begin{gathered} \mathbf{for}i\mathbf{to}4\mathbf{do} \\ \mathrm{Multiply}\left(13\,M\,i\,M\,i\,'\mathrm{transpose}'\,P\,i\,i\right) \\ \mathbf{end}\mathbf{do}\: \end{gathered}$$$P$

$\left[\begin{array}{cccc}7& 0& 0& 0\\ 0& 8& 0& 0\\ 0& 0& 8& 0\\ 0& 0& 0& 10\end{array}\right]$

$\mathrm{Multiply}\left(13\,M\,M\,'\mathrm{transpose}'\right)$

$\left[\begin{array}{cccc}7& 3& 10& 8\\ 3& 8& 7& 1\\ 10& 7& 8& 0\\ 8& 1& 0& 10\end{array}\right]$

$\mathrm{M1}≔\mathrm{Mod}\left(13\,\mathrm{Matrix}\left(3\,4\,\left(i\,j\right)\mapsto \mathrm{rand}\left(\right)\right)\,\mathrm{integer}\left[\right]\right)$

$\mathrm{M1}≔\left[\begin{array}{cccc}6& 4& 5& 3\\ 5& 4& 10& 4\\ 6& 9& 5& 9\end{array}\right]$

$\mathrm{M2}≔\mathrm{Mod}\left(13\,\mathrm{Matrix}\left(3\,4\,\left(i\,j\right)\mapsto \mathrm{rand}\left(\right)\right)\,\mathrm{integer}\left[\right]\right)$

$\mathrm{M2}≔\left[\begin{array}{cccc}10& 4& 6& 12\\ 6& 5& 11& 6\\ 4& 9& 7& 2\end{array}\right]$

$\mathrm{AddMultiple}\left(13\,1\,\mathrm{M1}\,2..3\,3..4\,\mathrm{M2}\,1..2\,2..3\right)$

$\left[\begin{array}{cc}1& 10\\ 10& 7\end{array}\right]$

$\mathrm{AddMultiple}\left(13\,1\,\mathrm{M1}\,2..3\,3..4\,\mathrm{M2}\,1\,2\right)$

$\left[\begin{array}{cc}1& 10\\ 10& 7\end{array}\right]$