A library to represent and manipulate multivariate polynomials
Project description
Multivariate Polynomial Library
PolyMat is a Python library designed for the representation and manipulation of multivariate polynomial matrices.
Features
- Expression Building: Create polynomial expressions using various operators provided by the library.
- Efficient Internal Representation: Uses a sparse internal structure to optimize intermediate computations.
- Deferred Computation: Polynomial matrices are evaluated lazily using the statemonad library, meaning expressions are computed only when needed.
- Performance Optimized: Designed for speed, the library outperforms other symbolic computation tools like
sympy
, making it ideal for large matrix expressions.
Installation
You can install PolyMat via pip:
pip install polymat
Basic Usage
In this example, we define a polynomial expressions using the +
and *
operators:
$f(x_1, x_2) = (x_1 + x_2) + (x_1 + x_1 x_2)$
Finally, different representations of the polynomial are printed.
import polymat
# Initialize state
state = polymat.init_state()
# Define polynomial variables and stack them into a vector
names = ('x1', 'x2')
x1, x2 = (polymat.define_variable(n) for n in names)
x = polymat.v_stack((x1, x2))
# Create a polynomial expression using arithmetic operations
f = (x1 + x2) + (x1 + x1*x2)
# Print a human-readable string representation
print(f'{f}')
# Print the internal Python representation of the expression
print(f'{f=}')
# sympy representation
state, sympy_repr = polymat.to_sympy(f).apply(state)
print(f'{sympy_repr}')
# array representation
state, array_repr = polymat.to_array(f, x).apply(state)
print(f'{array_repr.data[1]=}') # Dense numpy array
print(f'{array_repr.data[2].toarray()=}') # Sparse scipy array converted to numpy
Operations
Creating Polynomial Expressions
- Polynomial Variable: Define a polynomial variable.
x = polymat.define_variable('x')
- From Data: Create a polynomial expression from:
- Tuple of numbers and polynomial variables
j = polymat.from_(((0, -1), (1, 0))) # Matrix([[0, -1], [1, 0]])
numpy
arrays (possibly containing polynomial variables)i = polymat.from_(np.eye(2)) # Matrix([[1, 0], [0, 1]])
sympy
expressions (symbols are automatically converted to polynomial variables).
- Tuple of numbers and polynomial variables
Combining Polynomial Expressions
- Block Diagonal: Combine expression into block diagonal matrices.
xblk = polymat.block_diag((x, x)) # Matrix([[x, 0], [0, x]])
- Horizontal Stacking: Stack multiple polynomial expressions horizontally.
xhstack = polymat.h_stack((x, x)) # Matrix([[x, x]])
- Vertical Stacking: Stack multiple polynomial expressions vertically.
xvstack = polymat.v_stack((x, x)) # Matrix([[x], [x]])
Polynomial Expression Manipulation
- Arithmetic operations: Perform addition (
+
), subtraction (-
), scalar multiplication and division (*
,/
), matrix multiplication (@
), and exponentiation (**
).f = j * x**2 - i # Matrix([[-1.0, -1.0*x**2], [x**2, -1.0]])
- Caching: Cache the polynomial expression to store intermediate results and speed up computation.
x = x.cache()
- Combinations: Stack multiplied combinations of elements from polynomial vectors.
m = x.combinations((0, 1, 2)) # Matrix([[1], [x], [x**2]])
- Diagonalization: Extract a diagonal or construct diagonal matrix.
mdiag = m.diag() # Matrix([[1, 0, 0], [0, x, 0], [0, 0, x**2]])
- Differentiation: Compute the Jacobian matrix of a polynomial vector.
mdiff = m.diff(x) # Matrix([[0], [1], [2.0*x]])
- Evaluation: Replace variable symbols within tuple of floats.
meval = m.eval({x.symbol: (2,)}) # Matrix([[1], [2.0], [4.0]])
- Kronecker Product: Compute the Kronecker products.
fkron = f.kron(i) # Matrix([[-1.0, 0, -1.0*x**2, 0], [0, -1.0, 0, -1.0*x**2], [x**2, 0, -1.0, 0], [0, x**2, 0, -1.0]])
- Repmat: Repeat polynomial expressions.
xrepmat = x.repmat(3, 1) Matrix([[x], [x], [x]])
- Reshape: Modify the shape of polynomial matrices.
freshape = f.reshape(-1, 1) # Matrix([[-1.0], [x**2], [-1.0*x**2], [-1.0]])
- Summation: Sum the rows of the polynomial expression.
fsum = f.sum() # Matrix([[-1.0*x**2 - 1.0], [x**2 - 1.0]])
Specialized methods:
- Monomials Terms: Construct a monomial vector $Z(x)$ appearing in a polynomial expression.
p = x**3 - 2*x + 3 p_monom = p.to_linear_monomials(x) # Matrix([[1], [x], [x**3]])
- Linear Coefficient Vector: Compute a coefficient matrix $Q$ associated with a vector of monomials $Z(x)$ and a polynomial vector $p(x) = Q Z(x)$.
p_coeff = p.to_linear_coefficients(x, monomials=p_monom) # Matrix([[3, -2.0, 1]])
- Quadratic Monomial Terms: Construct a monomial vector $Z(x)$ for the quadratic form of the polynomial $p(x) = Z(x)^\top Q Z(x)$.
p_monom = p.to_quadratic_monomials(x) # Matrix([[1], [x], [x**2]])
- Quadratic Coefficient Matrix: Compute the symmetric coefficient matrix $Q$ appearing in the quadratic form of the polynomial $p(x) = Z(x)^\top Q Z(x)$.
p_coeff = p.to_gram_matrix(x, monomials=p_monom) # Matrix([[3.0, -1.0, 0], [-1.0, 0, 0.5], [0, 0.5, 0]])
Output
The output functions listed below perform stateful computations. As a result, they return a StateMonad object, which must be applied with a state object to generate the desired output value.
- Sympy Representation: Convert an experssion to a
sympy
representation.state, sympy_repr = polymat.to_sympy(f).apply(state) # Matrix([[-1.0, -1.0*x**2], [x**2, -1.0]])
- Array Representation: Convert polynomial expressions to an array representation (implemented through numpy and scipy array)..
state, farray = polymat.to_array(f, x).apply(state) # {0: array([[-1.], [ 0.], [ 0.], [-1.]]), 2: array([[ 0.], [ 1.], [-1.], [ 0.]])}
- Tuple Representation: Outputs constant parts as nested tuple.
# Setting assert_constant=False will prevent an exception form being raised, even if f is not a constant polynomial expression state, ftuple = polymat.to_tuple(f, assert_constant=False).apply(state) # ((-1.0,), (-1.0,))
- Polynomial Degrees: Obtain degrees of each polynomial matrix element.
state, fdegree = polymat.to_degree(f).apply(state) # ((0, 2), (2, 0))
- Shape of the Matrix: Retrieve the shape of the polynomial matrix.
state, fshape = polymat.to_shape(f).apply(state) # (2, 2)
References
Here are some references related to this probject:
- State-Monad is a Python library that encapsulates stateful computations into a monadic structure.
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