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# Multivariate polynomial library
`polymatrix` is a library to represent and operate on multivariate polynomials.
It is currently mainly used to define Sum Of Squares optimization problems.
Some aspects of the library
* it has a lazy behavior:
* the library implements operators to build polynomial expression
* the actual polynomial representation is created by calling the `apply(state)` method on the polynomial expression
* a `sympy` expression can be converted to a polynomial expression using the `polymatrix.from_sympy` function
* multiple polynomial expressions are combined using functions like `polymatrix.v_stack` or `polymatrix.block_diag`.
* polynomial expressions are manipulated using bounded functions like `diff`, `reshape`, `substitute`, `sum` or `to_constant` (see `polymatrix/expression/mixins/expressionmixin.py`)
* an expression can be converted to matrix representation using `polymatrix.to_matrix_repr`. Again, to get the actual representation the `apply(state)` method needs to be called.
## Example
``` python
import sympy
import polymatrix
state = polymatrix.init_expression_state()
x1, x2 = sympy.symbols('x1, x2')
x = polymatrix.from_((x1, x2))
f1 = polymatrix.from_(x1 + x2)
f2 = polymatrix.from_(x1 + x1*x2)
expr = f1 + f2
# prints the data structure of the expression
# ExpressionImpl(underlying=AdditionExprImpl(left=FromSympyExprImpl(data=((x1 + x2,),)), right=FromSympyExprImpl(data=((x1*x2 + x1,),))))
print(expr)
state, poly_matrix = expr.apply(state)
# prints the data structure of the polynomial matrix
# PolyMatrixImpl(terms={(0, 0): {((0, 1), (1, 1)): 1, ((0, 1),): 2, ((1, 1),): 1}}, shape=(1, 1))
print(poly_matrix)
state, matrix_repr = polymatrix.to_matrix_repr((expr,), x).apply(state)
# prints the numpy matrix representations of the polynomial matrix
# array([[2., 1.]])
# array([[0. , 0.5, 0.5, 0. ]])
print(matrix_repr.data[0][1])
print(matrix_repr.data[0][2].toarray())
```
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