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# Multivariate polynomial library
`polymatrix` is a library to represent and operate on multivariate polynomials.
It is used to define Sum Of Squares optimization problems.
## Key aspects
Some aspects of the library:
* it has a lazy behavior
* the library implements operators to build polynomial expression
* the actual polynomial is created by calling the [`apply(state)`](https://gitlab.nccr-automation.ch/michael.schneeberger/polymatrix/-/blob/main/polymatrix/expression/mixins/expressionbasemixin.py) method on the polynomial expression
* a `sympy` expression is converted to a polynomial expression using the [`polymatrix.from_sympy`](https://gitlab.nccr-automation.ch/michael.schneeberger/polymatrix/-/blob/main/polymatrix/__init__.py) function
* multiple polynomial expressions are combined using functions like
* [`polymatrix.v_stack`](https://gitlab.nccr-automation.ch/michael.schneeberger/polymatrix/-/blob/main/polymatrix/__init__.py)
* [`polymatrix.block_diag`](https://gitlab.nccr-automation.ch/michael.schneeberger/polymatrix/-/blob/main/polymatrix/__init__.py).
* polynomial expressions are manipulated using methods like
* [`diff`](https://gitlab.nccr-automation.ch/michael.schneeberger/polymatrix/-/blob/main/polymatrix/expression/mixins/expressionmixin.py),
* [`reshape`](https://gitlab.nccr-automation.ch/michael.schneeberger/polymatrix/-/blob/main/polymatrix/expression/mixins/expressionmixin.py)
* [`substitute`](https://gitlab.nccr-automation.ch/michael.schneeberger/polymatrix/-/blob/main/polymatrix/expression/mixins/expressionmixin.py)
* [`sum`](https://gitlab.nccr-automation.ch/michael.schneeberger/polymatrix/-/blob/main/polymatrix/expression/mixins/expressionmixin.py)
* [`to_constant`](https://gitlab.nccr-automation.ch/michael.schneeberger/polymatrix/-/blob/main/polymatrix/expression/mixins/expressionmixin.py)
* an expression is converted to matrix representation using [`polymatrix.to_matrix_repr`](https://gitlab.nccr-automation.ch/michael.schneeberger/polymatrix/-/blob/main/polymatrix/__init__.py). Again, to get the actual representation the [`apply(state)`](https://gitlab.nccr-automation.ch/michael.schneeberger/polymatrix/-/blob/main/polymatrix/statemonad/mixins/statemonadmixin.py) method needs to be called.
## Example
In this example, two polynomial expressions are defined using sympy expressions
$\quad f_1(x_1, x_2) = x_1 + x_2$
$\quad f_2(x_1, x_2) = x_1 + x_1 x_2$
Then, the two expression are combined using the [`__add__`](https://gitlab.nccr-automation.ch/michael.schneeberger/polymatrix/-/blob/main/polymatrix/expression/mixins/expressionmixin.py) (or equivalently `+`) function
$\quad f_3(x_1, x_2) = f_1(x_1, x_2) + f_2(x_1, x_2) = 2 x_1 + x_2 + x_1 x_2$
Finally, different representations of the polynomial are printed.
``` 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)
f3 = f1 + f2
# prints the data structure of the expression
# ExpressionImpl(underlying=AdditionExprImpl(left=FromSympyExprImpl(data=((x1 + x2,),)), right=FromSympyExprImpl(data=((x1*x2 + x1,),))))
print(f3)
state, poly_matrix = f3.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((f3,), 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]) # numpy array
print(matrix_repr.data[0][2].toarray()) # sparse scipy array
```
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