lmlib.polynomial.poly.MPoly#
- class lmlib.polynomial.poly.MPoly(coefs, expos)#
Bases:
lmlib.polynomial.poly._PolyBaseMultivariate polynomials \({\tilde{\alpha}}^\mathsf{T} (x^q \otimes y^r)\), or with factorized coefficient vector \((\alpha \otimes \beta )^\mathsf{T} (x^q \otimes y^r)\).
This polynomial class is for multivariate polynomials in vector exponent notation, see [Wildhaber2019], Chapter 6.
Such a multivariate polynomial is in general given by
\[p(x) = \tilde{\alpha}^\mathsf{T}(x^q \otimes y^r) \ ,\]where \(\tilde{\alpha} \in \mathbb{R}^{Q \times R}\) is the coefficient vectors, \(q \in \mathbb{Z}_{\geq 0}^Q\) and \(r \in \mathbb{Z}_{\geq 0}^R\) the exponent vectors, and \(x \in \mathbb{R}\) and \(y \in \mathbb{R}\) the independent variables.
As a special case, if the coefficient vector is in the form of a Kronecker product, i.e.,
\[p(x) = (\alpha \otimes \beta)^\mathsf{T}(x^q \otimes y^r) \ ,\]where \(\alpha \in \mathbb{R}^Q\) and \(\beta \in \mathbb{R}^R\) are coefficient vectors, we denote a polynomial as factorized. This form often leads to algebraic simplifications (if it exists).
Examples
>>> # Bivariate (x,y) polynomial with factorized coefficients ([.2,.7],[-1.0,2.0,.1]) and terms x^1, x^2, y^1, y^2, y^3, and cross terms >>> l= MPoly(([.2,.7],[-1.0,2.0,.1]),([1,2],[1,2,3])) >>> l.coefs # gets coefficients (array([0.2, 0.7]), array([-1. , 2. , 0.1])) >>> l.coef_fac # gets factorized coefficients (if available) array([-0.2 , 0.4 , 0.02, -0.7 , 1.4 , 0.07]) >>> l.eval([.3,.7]) # evaluating polynomial for x=.3 and y=.7 array(0.0386589)
>>> # Bivariate (x,y) polynomial with non-factorized coefficients ([.2,.7,1.3,1.4,.2,-1.6]) and terms x^1, x^2, y^1, y^2, y^3, and cross terms >>> l= MPoly(([.2,.7,1.3,1.4,.2,-1.6],),([1,2],[1,2,3])) >>> l.eval([.3,.7]) # evaluating polynomial for x=.3 and y=.7 array(0.326298)
- Parameters
coefs (tuple of array_like) – Set of coefficient vector(s)
expos (tuple of array_like) – Set of exponent vector(s)
Methods
__init__(coefs, expos)Constructor method
eval(variables)Evaluates the polynomial for given values (variables)
Attributes
Coefficient vector (i.e., not factorized)
Factorized coefficient vectors
Exponent vectors
Number of dependent variables
- property coefs_fac#
Factorized coefficient vectors
- Type
ndarray, None
- eval(variables)#
Evaluates the polynomial for given values (variables)
- Parameters
variables (tuple) – Dependent variables of a polynomial. Each element in variables has the same shape which is also the output shape.
- Returns
out – Output of evaluated polynomial. Shape is identical as a dependent variable
- Return type
Example
>>> # evaluate bivariate polynomial at multiple positions (x=.1 ... .4, y=.5) >>> l = MPoly(([.2, .7], [1.3, 1.4, -.9],), ([0, 1], [0, 1, 2])) >>> l.eval(([.1, .2, .3, .4], [.5, .5, .5, .5])) array([0.47925, 0.6035, 0.72775, 0.852])