lmlib.polynomial#

Package Abstract :: This module provides a calculus for uni- and multivariate polynomials using the vector exponent notation [Wildhaber2019], Chapter 6. This calculus simplifies to use polynomials in (squared error) cost functions, e.g., as localized signal models.

Polynomial Classes#

`Poly`(coef, expo)	Univariate polyonimial in vector exponent notation: \(\alpha^\mathsf{T} x^q\)
`MPoly`(coefs, expos)	Multivariate polynomials \({\tilde{\alpha}}^\mathsf{T} (x^q \otimes y^r)\), or with factorized coefficient vector \((\alpha \otimes \beta )^\mathsf{T} (x^q \otimes y^r)\).

Polynomial Operators#

Operators for Univariate Polynomials#

Operators handling univariate polynomials. Return parameters are highlighted in \(\color{blue}{blue}\).

Sum of Polynomials \(\alpha^\mathsf{T} x^q + \beta^\mathsf{T} x^r\)#

`poly_sum`(polys)	\(\alpha^\mathsf{T} x^q + \dots + \beta^\mathsf{T} x^r = \color{blue}{\tilde{\alpha}^\mathsf{T} x^\tilde{q}}\)
`poly_sum_coef`(polys)	\(\alpha^\mathsf{T} x^q + \dots + \beta^\mathsf{T} x^r = \color{blue}{\tilde{\alpha}}^\mathsf{T} x^\tilde{q}\)
`poly_sum_coef_Ls`(expos)	\(\alpha^\mathsf{T} x^q + \dots + \beta^\mathsf{T} x^r = (\color{blue}{\Lambda_1} \alpha + \dots + \color{blue}{\Lambda_N}\beta)^\mathsf{T} x^{\tilde{q}}\)
`poly_sum_expo`(expos)	\(\alpha^\mathsf{T} x^q + \dots + \beta^\mathsf{T} x^r = \tilde{\alpha}^\mathsf{T} x^{\color{blue}{\tilde{q}}}\)
`poly_sum_expo_Ms`(expos)	\(\alpha^\mathsf{T} x^q + \dots + \beta^\mathsf{T} x^r = \tilde{\alpha}^\mathsf{T} x^{\color{blue}{M_1} q + \dots +\color{blue}{M_N} r}\)

>>> import lmlib as lm
>>>
>>> p1 = lm.Poly([1, 3, 5], [0, 1, 2])
>>> p2 = lm.Poly([2, -1], [0, 1])
>>>
>>> p_sum = lm.poly_sum((p1, p2))
>>> print(p_sum)
[ 1.  3.  5.  2. -1.], [0. 1. 2. 0. 1.]

Product of Polynomials \(\alpha^\mathsf{T} x^q \cdot \beta^\mathsf{T} x^r\)#

`poly_prod`(polys)	\(\alpha^\mathsf{T} x^q \cdot \beta^\mathsf{T} x^q = \color{blue}{\tilde{\alpha}^\mathsf{T} x^\tilde{q}}\)
`poly_prod_coef`(polys)	\(\alpha^\mathsf{T} x^q \cdot \beta^\mathsf{T} x^r = \color{blue}{\tilde{\alpha}}^\mathsf{T} x^\tilde{q}\)
`poly_prod_expo`(expos)	\(\alpha^\mathsf{T} x^q \cdot \beta^\mathsf{T} x^r = \tilde{\alpha}^\mathsf{T} x^{\color{blue}{\tilde{q}}}\)
`poly_prod_expo_Ms`(expos)	\(\alpha^\mathsf{T} x^q \cdot \beta^\mathsf{T} x^r = \tilde{\alpha}^\mathsf{T} x^{\color{blue}{M_1} q + \color{blue}{M_2} r}\)

>>> import lmlib as lm
>>>
>>> p1 = lm.Poly([1, 3, 5], [0, 1, 2])
>>> p2 = lm.Poly([2, -1], [0, 1])
>>>
>>> p_prod = lm.poly_prod((p1, p2))
>>> print(p_prod)
[ 2 -1  6 -3 10 -5], [0. 1. 1. 2. 2. 3.]

Square of a Polynomial \((\alpha^\mathsf{T} x^q)^2\)#

`poly_square`(poly)	\((\alpha^\mathsf{T} x^q)^2 = \color{blue}{\tilde{\alpha}^\mathsf{T} x^\tilde{q}}\)
`poly_square_coef`(poly)	\(\color{blue}{\tilde{\alpha}}^\mathsf{T} x^\tilde{q}\)
`poly_square_expo`(expo)	\(\tilde{\alpha}^\mathsf{T} x^{\color{blue}{\tilde{q}}}\)
`poly_square_expo_M`(expo)	\(\tilde{\alpha}^\mathsf{T} x^{\color{blue}{M} q}\)

>>> import lmlib as lm
>>>
>>> p1 = lm.Poly([1, 3, 5], [0, 1, 2])
>>>
>>> p_square = lm.poly_square(p1)
>>> print(p_square)
[ 1  3  5  3  9 15  5 15 25], [0. 1. 2. 1. 2. 3. 2. 3. 4.]

Shift of a Polynomial \(\alpha^\mathsf{T} (x+ \gamma)^q\)#

`poly_shift`(poly, gamma)	\(\alpha^\mathsf{T} (x+ \gamma)^q = \color{blue}{\tilde{\alpha}^\mathsf{T} x^\tilde{q}}\)
`poly_shift_coef`(poly, gamma)	\(\color{blue}{\tilde{\alpha}}^\mathsf{T} x^\tilde{q}\)
`poly_shift_coef_L`(expo, gamma)	\(\color{blue}{\Lambda} \alpha^\mathsf{T} x^{\tilde{q}}\)
`poly_shift_expo`(expo)	\(\tilde{\alpha}^\mathsf{T} x^{\color{blue}{\tilde{q}}}\)

>>> import lmlib as lm
>>>
>>> p1 = lm.Poly([1, 3, 5], [0, 1, 2])
>>> gamma = 2
>>> p_shift = lm.poly_shift(p1, gamma)
>>> print(p_shift)
[27. 23.  5.], [0 1 2]

Dilation of a Polynomial \(\alpha^\mathsf{T} (\eta x)^q\)#

`poly_dilation`(poly, eta)	\(\alpha^\mathsf{T} (\eta x)^q = \color{blue}{\tilde{\alpha}^\mathsf{T} x^q}\)
`poly_dilation_coef`(poly, eta)	\(\alpha^\mathsf{T} (\eta x)^q =\color{blue}{\tilde{\alpha}}^\mathsf{T} x^q\)
`poly_dilation_coef_L`(expo, eta)	\(\alpha^\mathsf{T} (\eta x)^q =\color{blue}{\Lambda} \alpha^\mathsf{T} x^{q}\)

>>> import lmlib as lm
>>>
>>> p1 = lm.Poly([1, 3, 5], [0, 1, 2])
>>> eta = -5
>>> p_dilation = lm.poly_dilation (p1, eta)
>>> print(p_dilation)
[  1 -15 125], [0 1 2]

Integral of a Polynomial \(\int (\alpha^\mathsf{T} x^q) dx\)#

`poly_int`(poly)	\(\int \big(\alpha^{\mathsf{T}}x^q\big) dx = \color{blue}{\tilde{\alpha}^\mathsf{T} x^\tilde{q}}\)
`poly_int_coef`(poly)	\(\int \big(\alpha^{\mathsf{T}}x^q\big) dx = \color{blue}{\tilde{\alpha}}^\mathsf{T} x^\tilde{q}\)
`poly_int_coef_L`(expo)	\(\int \big(\alpha^{\mathsf{T}}x^q\big) dx = \color{blue}{\Lambda} \alpha^\mathsf{T} x^{\tilde{q}}\)
`poly_int_expo`(expo)	\(\int \big(\alpha^{\mathsf{T}}x^q\big) dx = \tilde{\alpha}^\mathsf{T} x^{\color{blue}{\tilde{q}}}\)

>>> import lmlib as lm
>>>
>>> p1 = lm.Poly([1, 3, 5], [0, 1, 2])
>>>
>>> p_int = lm.poly_int(p1)
>>> print(p_int)
[1.         1.5        1.66666667], [1 2 3]

Derivative of a Polynomial \(\frac{d}{dx} (\alpha^\mathsf{T} x^q)\)#

`poly_diff`(poly)	\(\frac{d}{dx} \big(\alpha^{\mathsf{T}}x^q\big) = \color{blue}{\tilde{\alpha}^\mathsf{T} x^\tilde{q}}\)
`poly_diff_coef`(poly)	\(\frac{d}{dx} \big(\alpha^{\mathsf{T}}x^q\big) =\color{blue}{\tilde{\alpha}}^\mathsf{T} x^\tilde{q}\)
`poly_diff_coef_L`(expo)	\(\frac{d}{dx} \big(\alpha^{\mathsf{T}}x^q\big) =\color{blue}{\Lambda} \alpha^\mathsf{T} x^{\tilde{q}}\)
`poly_diff_expo`(expo)	\(\frac{d}{dx} \big(\alpha^{\mathsf{T}}x^q\big) =\tilde{\alpha}^\mathsf{T} x^{\color{blue}{\tilde{q}}}\)

>>> import lmlib as lm
>>>
>>> p1 = lm.Poly([1, 3, 5], [0, 1, 2])
>>>
>>> p_diff = lm.poly_diff(p1)
>>> print(p_diff)
[ 0  3 10], [0 0 1]

Operators for Multivariate Polynomials#

Operators handling multi variate polynomials. Return parameters are highlighted in \(\color{blue}{blue}\).

lmlib.statespace.applications.TSLM

lmlib.polynomial.poly.Poly