mystDoc: Update matrices

symengine · Mar 6, 2021 · 38bbaee · 38bbaee
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diff --git a/.github/workflows/test_tutorials.yml b/.github/workflows/test_tutorials.yml
@@ -9,7 +9,7 @@ jobs:
       - uses: actions/checkout@v2
       - name: Run tutorials
         run: |
-          export NB_PYTHON_PREFIX=$CONDA_PREFIX
+          export NB_PYTHON_PREFIX=$our_install_dir
           source bin/install_travis.sh
           ./binder/postBuild
           python bin/test_tutorials.py

diff --git a/binder/environment.yml b/binder/environment.yml
@@ -12,4 +12,3 @@ dependencies:
   - libflint
   - jupytext
   - papermill
-  - symengine
diff --git a/binder/postBuild b/binder/postBuild
@@ -1,10 +1,9 @@
 set -e
 mkdir build
 cd build
-export LDFLAGS="-Wl,-rpath,${NB_PYTHON_PREFIX}/lib -L${NB_PYTHON_PREFIX}/lib"
-cmake -DWITH_COTIRE=no -DBUILD_SHARED_LIBS=yes -DWITH_MPC=yes -DWITH_ARB=yes -DINTEGER_CLASS=flint -DWITH_LLVM=yes -DCMAKE_PREFIX_PATH=${NB_PYTHON_PREFIX} -DCMAKE_INSTALL_PREFIX=${NB_PYTHON_PREFIX} -DBUILD_TESTS=off -DBUILD_BENCHMARKS=off -DCMAKE_INSTALL_LIBDIR=lib ..
+export LDFLAGS="-Wl,-rpath"
+cmake -DWITH_COTIRE=no -DBUILD_SHARED_LIBS=yes -DWITH_MPC=yes -DWITH_ARB=yes -DINTEGER_CLASS=flint -DWITH_LLVM=yes -DBUILD_TESTS=off -DBUILD_BENCHMARKS=off ..
 make -j4
-make install
 sudo make install
 cd ..
 rm -rf build
diff --git a/docs/mystMD/Matrices.myst.md b/docs/mystMD/Matrices.myst.md
@@ -14,25 +14,178 @@ kernelspec:
 
 # Matrices
 
-```{code-cell}
-#include <chrono>
-#include <xcpp/xdisplay.hpp>
+```{note}
+As of now there is no pretty printer implemented for matrices ([issue](https://github.com/symengine/symengine/issues/1701))
+```
+
+
+The following header files will be necessary for this section.
 
+```{code-cell}
+#include <symengine/parser.h>
 #include <symengine/matrix.h>
 #include <symengine/add.h>
 #include <symengine/pow.h>
 #include <symengine/symengine_exception.h>
 #include <symengine/visitor.h>
+#include <symengine/printers/strprinter.h>
 ```
 
+## Base Elements
+
+We will need a set of basic elements to cover a reasonable set of operations. Namely we would like:
+
+- Two Matrices (A,B)
+- A Vector (X)
+
 ```{code-cell}
-SymEngine::vec_basic elems{SymEngine::integer(1),
+SymEngine::vec_basic elemsA{SymEngine::integer(1),
                            SymEngine::integer(0),
                            SymEngine::integer(-1),
                            SymEngine::integer(-2)};
-SymEngine::DenseMatrix A = SymEngine::DenseMatrix(2, 2, elems);
+SymEngine::DenseMatrix A = SymEngine::DenseMatrix(2, 2, elemsA);
 ```
 
 ```{code-cell}
 A.__str__()
 ```
+
+```{code-cell}
+SymEngine::vec_basic elemsB{SymEngine::integer(5),
+                           SymEngine::integer(2),
+                           SymEngine::integer(-7),
+                           SymEngine::integer(-3)};
+SymEngine::DenseMatrix B = SymEngine::DenseMatrix(2, 2, elemsB);
+B.__str__()
+```
+
+Note that the real utility of working with SymEngine is the ability to include free symbols as first class members of the matrix.
+
++++
+
+## Basic Operations
+
+The key thing to remember that as a `C++` library, we need to pre-allocated variable sizes and types. Furthermore, the unary operators are **not** overloaded, so we will call functions for each of the standard operations. The general form of each of these is:
+
+**operation**(_term1_,_term2_,**output**)
+
++++
+
+### Matrix-Matrix
+
+#### Addition
+The addition of two dense matrices is carried out as expected.
+
+\begin{align*}
+\mathbf{A}+\mathbf{B} & = \begin{bmatrix}
+ a_{11} & a_{12} & \cdots & a_{1n} \\
+ a_{21} & a_{22} & \cdots & a_{2n} \\
+ \vdots & \vdots & \ddots & \vdots \\
+ a_{m1} & a_{m2} & \cdots & a_{mn} \\
+\end{bmatrix} +
+
+\begin{bmatrix}
+ b_{11} & b_{12} & \cdots & b_{1n} \\
+ b_{21} & b_{22} & \cdots & b_{2n} \\
+ \vdots & \vdots & \ddots & \vdots \\
+ b_{m1} & b_{m2} & \cdots & b_{mn} \\
+\end{bmatrix} \\
+& = \begin{bmatrix}
+ a_{11} + b_{11} & a_{12} + b_{12} & \cdots & a_{1n} + b_{1n} \\
+ a_{21} + b_{21} & a_{22} + b_{22} & \cdots & a_{2n} + b_{2n} \\
+ \vdots & \vdots & \ddots & \vdots \\
+ a_{m1} + b_{m1} & a_{m2} + b_{m2} & \cdots & a_{mn} + b_{mn} \\
+\end{bmatrix} \\
+\end{align*}
+
+```{code-cell}
+SymEngine::DenseMatrix C = SymEngine::DenseMatrix(2, 2);
+add_dense_dense(A, B, C);
+C.__str__()
+```
+
+#### Multiplication
+
++++
+
+### Matrix-Scalar
+#### Addition
+
+## Gaussian Elimination
+
+One of the main use-cases for any library with matrices is the ability to perform Gaussian elimination. We will consider an example from the literature {cite}`nakosFractionfreeAlgorithmsLinear1997`.
+
+```{code-cell}
+    // Fraction-Free Algorithms for Linear and Polynomial Equations, George C
+    // Nakos,
+    // Peter R Turner et. al.
+   SymEngine::DenseMatrix A = SymEngine::DenseMatrix(4, 4, {SymEngine::integer(1), SymEngine::integer(2), SymEngine::integer(3), SymEngine::integer(4),
+                           SymEngine::integer(2), SymEngine::integer(2), SymEngine::integer(3), SymEngine::integer(4),
+                           SymEngine::integer(3), SymEngine::integer(3), SymEngine::integer(3), SymEngine::integer(4),
+                           SymEngine::integer(9), SymEngine::integer(8), SymEngine::integer(7), SymEngine::integer(6)});
+  SymEngine::DenseMatrix  B = SymEngine::DenseMatrix(4, 4);
+fraction_free_gaussian_elimination(A, B);
+B.__str__()
+```
+
+## Hadamard Product
+
+```{warning}
+This will be part of the next release
+```
+
+
+This is essentially the element wise multiplication of two matrices, so for two matrices of the same dimension we have $(A \circ B)_{ij} = (A \odot B)_{ij} = (A)_{ij} (B)_{ij}$.
+
+$$
+\begin{bmatrix}
+    a_{11} & a_{12} & a_{13}\\
+    a_{21} & a_{22} & a_{23}\\
+    a_{31} & a_{32} & a_{33}
+  \end{bmatrix} \circ \begin{bmatrix}
+    b_{11} & b_{12} & b_{13}\\
+    b_{21} & b_{22} & b_{23}\\
+    b_{31} & b_{32} & b_{33}
+  \end{bmatrix} = \begin{bmatrix}
+    a_{11}\, b_{11} & a_{12}\, b_{12} & a_{13}\, b_{13}\\
+    a_{21}\, b_{21} & a_{22}\, b_{22} & a_{23}\, b_{23}\\
+    a_{31}\, b_{31} & a_{32}\, b_{32} & a_{33}\, b_{33}
+\end{bmatrix}
+$$
+
+For `SymEngine` this has been implemented recently as the `element_mul_matrix` function.
+
+```{code-cell}
+SymEngine::DenseMatrix D = SymEngine::DenseMatrix(2, 2);
+B.elementwise_mul_matrix(A, D);
+D.__str__()
+```
+
+# NumPY Convenience Functions
+A set of functions are provided to allow for `numpy`-esque generation of matrices.
+- `eye` for generating a 2-D matrix with ones on the diagonal and zeros elsewhere
+- `diag` for generating diagonal matrices
+- `ones` for generating a dense matrix of ones
+- `zeros` for generating a dense matrix of zeros
+
++++
+
+## Identity Matrices and Offsets
+The `eye` function is used to create 2D matrices with ones on the diagonal and zeros elsewhere; with an optional parameter to offset the position of the ones.
+
+```{code-cell}
+SymEngine::DenseMatrix npA = SymEngine::DenseMatrix(3, 3);
+SymEngine::eye(npA);
+npA.__str__()
+```
+
+```{code-cell}
+SymEngine::eye(npA,1);
+npA.__str__()
+```
+
+## Diagonal Matrix Helpers
+The `diag` function
+
+```{bibliography} references.bib
+```
diff --git a/docs/mystMD/index.md b/docs/mystMD/index.md
@@ -4,4 +4,5 @@ This part of the documentation deals with the use of basic C++ structures.
 ```{toctree}
 :maxdepth: 2
 Expressions <firststeps.myst.md>
+Matrices <Matrices.myst.md>
 ```
diff --git a/docs/mystMD/references.bib b/docs/mystMD/references.bib
@@ -0,0 +1,13 @@
+@article{nakosFractionfreeAlgorithmsLinear1997,
+  title = {Fraction-Free Algorithms for Linear and Polynomial Equations},
+  author = {Nakos, George C. and Turner, Peter R. and Williams, Robert M.},
+  year = {1997},
+  month = sep,
+  volume = {31},
+  pages = {11--19},
+  issn = {0163-5824},
+  doi = {10.1145/271130.271133},
+  abstract = {This paper extends the ideas behind Bareiss's fraction-free Gauss elimination algorithm in a number of directions. First, in the realm of linear algebra, algorithms are presented for fraction-free LU "factorization" of a matrix and for fraction-free algorithms for both forward and back substitution. These algorithms are valid not just for integer computation but also for any matrix system where the entries are taken from a unique factorization domain such as a polynomial ring. The second part of the paper introduces the application of the fraction-free formulation to resultant algorithms for solving systems of polynomial equations. In particular, the use of fraction-free polynomial arithmetic and triangularization algorithms in computing the Dixon resultant of a polynomial system is discussed.},
+  journal = {ACM SIGSAM Bulletin},
+  number = {3}
+}