laurentPolynomial

Create Laurent polynomial

Since R2021b

Description

Use the laurentPolynomial object to create a Laurent polynomial with real-valued polynomial coefficients. You can specify the maximum order of the polynomial. You can perform mathematical and logical operations on Laurent polynomials. You can also create a lifting scheme associated with a pair of Laurent polynomials.

Creation

Syntax

lpoly = laurentPolynomial

lpoly = laurentPolynomial(Name=Value)

Description

lpoly = laurentPolynomial creates the constant Laurent polynomial, where the constant is equal to 1 and the maximum order is equal to 0.

lpoly = laurentPolynomial(Name=Value) creates a Laurent polynomial with Properties specified by name-value arguments. For example, laurentPolynomial(MaxOrder=2) creates a Laurent polynomial with maximum order equal to 2. You can specify multiple name-value arguments.

Properties

Coefficients — Laurent polynomial coefficients

1 (default) | real-valued vector

Laurent polynomial coefficients, specified as a real-valued vector. If k is the length of the vector C, then lpoly = laurentPolynomial(Coefficients=C) represents the Laurent polynomial

$$\text{lpoly}(z)={\displaystyle \sum _{m=1}^{k}C}(m)z^{1-m}.$$

Example: If C = [4 3 2 1], then P = laurentPolynomial(Coefficients=C) represents the Laurent polynomial $$\text{P}(z)=4+3z^{-1}+2z^{-2}+z^{-3}.$$

Data Types: double

MaxOrder — Maximum order

0 (default) | integer

Maximum order of the Laurent polynomial, specified as an integer. If k is the length of the vector C and d is an integer, then lpoly = laurentPolynomial(Coefficients=C,MaxOrder=d) represents the Laurent polynomial

$$\text{lpoly}(z)={\displaystyle \sum _{m=1}^{k}C}(m)z^{d-m+1}.$$

Example: If C = [2 4 6 8], then P = laurentPolynomial(Coefficients=C,MaxOrder=1) represents the Laurent polynomial $$\text{P}(z)=2z+4+6z^{-1}+8z^{-2}.$$

Data Types: double

Object Functions

Specific to laurentPolynomial

degree	Degree of Laurent polynomial
euclid	Euclidean algorithm for Laurent polynomials
polyphase	Polyphase components of Laurent polynomial
mpower	Laurent polynomial exponentiation
horzcat	Horizontal concatenation of Laurent polynomials
vertcat	Vertical concatenation of Laurent polynomials
lp2filters	Laurent polynomials to filters
lp2LS	Laurent polynomials to lifting steps and normalization factors
ne	Laurent polynomials inequality test
rescale	Rescale Laurent polynomial

Common to laurentPolynomial and laurentMatrix

dyaddown	Dyadic downsampling of Laurent polynomial or Laurent matrix
dyadup	Dyadic upsampling of Laurent polynomial or Laurent matrix
eq	Laurent polynomials or Laurent matrices equality test
plus	Laurent polynomial or Laurent matrix addition
minus	Laurent polynomial or Laurent matrix subtraction
mtimes	Laurent polynomial or Laurent matrix multiplication
reflect	Laurent polynomial or Laurent matrix reflection
uminus	Unary minus for Laurent polynomial or Laurent matrix

Examples

Basic Mathematical Operations Applied to Laurent Polynomials

Create three Laurent polynomials:

• $$a(z)=1+z^{-1}$$

• $$b(z)=z^2+3z+z^{-1}$$

• $$c(z)=z^3+3z^2+5z+7$$

a = laurentPolynomial(Coefficients=[1 1])

a = 
  laurentPolynomial with properties:

    Coefficients: [1 1]
        MaxOrder: 0

b = laurentPolynomial(Coefficients=[1 3 0 1],MaxOrder=2)

b = 
  laurentPolynomial with properties:

    Coefficients: [1 3 0 1]
        MaxOrder: 2

c = laurentPolynomial(Coefficients=[1 3 5 7],MaxOrder=3)

c = 
  laurentPolynomial with properties:

    Coefficients: [1 3 5 7]
        MaxOrder: 3

Addition

Add the two polynomials $$a(z)$$ and $$b(z)$$. Use the helper function helperPrintLaurent to print the result in algebraic form.

polySum = plus(a,b)

polySum = 
  laurentPolynomial with properties:

    Coefficients: [1 3 1 2]
        MaxOrder: 2

res = helperPrintLaurent(polySum);
disp(res)

z^(2) + 3*z + 1 + 2*z^(-1)

Add 2 to $$b(z)$$.

consSum = b+2;
res = helperPrintLaurent(consSum);
disp(res)

z^(2) + 3*z + 2 + z^(-1)

Subtraction

Subtract $$a(z)$$ from $$b(z)$$.

polyDiff = minus(b,a);
res = helperPrintLaurent(polyDiff);
disp(res)

z^(2) + 3*z - 1

Subtract $$a(z)$$ from 1.

consDiff = 1-a;
res = helperPrintLaurent(consDiff);
disp(res)

- z^(-1)

Multiplication

Multiply $$a(z)$$ and $$b(z)$$.

polyProd = mtimes(a,b);
res = helperPrintLaurent(polyProd);
disp(res)

z^(2) + 4*z + 3 + z^(-1) + z^(-2)

Compute $$a(z)c(z)-b(z)$$.

polyProd2 = a*c-b;
res = helperPrintLaurent(polyProd2);
disp(res)

z^(3) + 3*z^(2) + 5*z + 12 + 6*z^(-1)

To multiply a Laurent polynomial by a constant, use the rescale function.

consProd = rescale(b,7);
res = helperPrintLaurent(consProd);
disp(res)

7*z^(2) + 21*z + 7*z^(-1)

Exponentiation

Raise $$a(z)$$ to the fourth power.

polyPow = mpower(a,4);
res = helperPrintLaurent(polyPow);
disp(res)

1 + 4*z^(-1) + 6*z^(-2) + 4*z^(-3) + z^(-4)

Compute $$b^2(z)-c(z)$$.

polyPow2 = b^2-c;
res = helperPrintLaurent(polyPow2);
disp(res)

z^(4) + 5*z^(3) + 6*z^(2) - 3*z - 1 + z^(-2)

Properties of Laurent Polynomials

Create two Laurent polynomials:

• $$a(z)=z-1$$

• $$b(z)=-2z^3+6z^2-7z+2$$

a = laurentPolynomial(Coefficients=[1 -1],MaxOrder=1);
b = laurentPolynomial(Coefficients=[-2 6 -7 2],MaxOrder=3);

Reflection

Obtain the reflection of $$b(z)$$.

br = reflect(b);
res = helperPrintLaurent(br);
disp(res)

2 - 7*z^(-1) + 6*z^(-2) - 2*z^(-3)

Unary Minus

Confirm the sum of $$b(z)$$ and its unary negation is equal to 0.

b+uminus(b)

ans = 
  laurentPolynomial with properties:

    Coefficients: 0
        MaxOrder: 0

Degree

Multiply $$a(z)$$ and $$b(z)$$. Confirm the degree of the product is equal to the sum of the degrees of $$a(z)$$ and $$b(z)$$.

ab = a*b;
degree(ab)

ans = 4

degree(a)+degree(b)

ans = 4

Exponentiation

Raise $$a(z)$$ to the third power. Confirm the result is not equal to $$b(z)$$.

a3 = a^3;
a3 ~= b

ans = logical
   1

Rescale

Confirm $$a(z)$$ raised to the third power is equal to $$-b(z)/2-z/2$$.

zt = laurentPolynomial(Coefficients=[-1/2],MaxOrder=1);
b2 = rescale(b,-1/2)+zt;
eq(a3,b2)

ans = logical
   1

Dyadic Operations

Create the Laurent polynomial $$c(z)=\sum _{k=-3}^4(-1{)}^{k}k{z}^k$$. Obtain the degree of $$c(z)$$.

cfs = (-1).^(-3:4).*(-3:4);
c = laurentPolynomial(Coefficients=fliplr(cfs),MaxOrder=4);
res = helperPrintLaurent(c);
disp(res)

4*z^(4) - 3*z^(3) + 2*z^(2) - z + z^(-1) - 2*z^(-2) + 3*z^(-3)

degree(c)

ans = 7

Obtain the dyadic upsampling and downsampling of $$c(z)$$. Obtain the degree of both polynomials.

dUp = dyadup(c)

dUp = 
  laurentPolynomial with properties:

    Coefficients: [4 0 -3 0 2 0 -1 0 0 0 1 0 -2 0 3]
        MaxOrder: 8

degree(dUp)

ans = 14

dDown = dyaddown(c)

dDown = 
  laurentPolynomial with properties:

    Coefficients: [4 2 0 -2]
        MaxOrder: 2

degree(dDown)

ans = 3

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Version History

Introduced in R2021b