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.

example

Properties

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`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

$lpoly (z) = \sum_{m = 1}^{k} C (m) z^{1 - m} .$

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

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

$lpoly (z) = \sum_{m = 1}^{k} C (m) z^{d - m + 1} .$

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

Data Types: double

Object Functions

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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

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Basic Mathematical Operations Applied to Laurent Polynomials

Open Live Script

Create three Laurent polynomials:

$a (z) = 1 + z^{- 1}$
$b (z) = z^{2} + 3 z + z^{- 1}$
$c (z) = z^{3} + 3 z^{2} + 5 z + 7$

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

Open Live Script

Create two Laurent polynomials:

$a (z) = z - 1$
$b (z) = - 2 z^{3} + 6 z^{2} - 7 z + 2$

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

laurentPolynomial

Description

Creation

Syntax

Description

Properties

Coefficients — Laurent polynomial coefficients 1 (default) | real-valued vector

MaxOrder — Maximum order 0 (default) | integer

Object Functions

Specific to laurentPolynomial

Common to laurentPolynomial and laurentMatrix

Examples

Basic Mathematical Operations Applied to Laurent Polynomials

Properties of Laurent Polynomials

Extended Capabilities

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

Version History

See Also

`Coefficients` — Laurent polynomial coefficients
`1` (default) | real-valued vector

`MaxOrder` — Maximum order
`0` (default) | integer

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