# 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) $$(\alpha^\mathsf{T} x^q)^2 = \color{blue}{\tilde{\alpha}^\mathsf{T} x^\tilde{q}}$$ $$\color{blue}{\tilde{\alpha}}^\mathsf{T} x^\tilde{q}$$ $$\tilde{\alpha}^\mathsf{T} x^{\color{blue}{\tilde{q}}}$$ $$\tilde{\alpha}^\mathsf{T} x^{\color{blue}{M} q}$$
>>> 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) $$\color{blue}{\Lambda} \alpha^\mathsf{T} x^{\tilde{q}}$$ $$\tilde{\alpha}^\mathsf{T} x^{\color{blue}{\tilde{q}}}$$
>>> 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) $$\int \big(\alpha^{\mathsf{T}}x^q\big) dx = \color{blue}{\tilde{\alpha}}^\mathsf{T} x^\tilde{q}$$ $$\int \big(\alpha^{\mathsf{T}}x^q\big) dx = \color{blue}{\Lambda} \alpha^\mathsf{T} x^{\tilde{q}}$$ 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) $$\frac{d}{dx} \big(\alpha^{\mathsf{T}}x^q\big) =\color{blue}{\tilde{\alpha}}^\mathsf{T} x^\tilde{q}$$ $$\frac{d}{dx} \big(\alpha^{\mathsf{T}}x^q\big) =\color{blue}{\Lambda} \alpha^\mathsf{T} x^{\tilde{q}}$$ 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 $$\color{blue}{blue}$$.