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Representing Polynomials with Classes

You can use classes to define new data types. This example implements a class that represents polynomials. The class stores the coefficients of the polynomial terms in a vector and overrides the default MATLAB® display to show the polynomials as powers of x. Using customized indexing, the class also enables you to evaluate the polynomials at one or more values of x using parentheses indexing syntax.

Class Requirements

The design requirements for the DocPolynom class are:

  • Value class behavior —Behave like MATLAB numeric variables when copied and passed to functions.

  • Scalar object behavior — Polynomial objects cannot be concatenated, and polynomial array size must always be (1,1).

  • Customized indexing behavior — Evaluate a polynomial using parentheses indexing syntax. p(x) evaluates the polynomial represented by object p at each value in x.

  • Specialized display — Use the coefficients stored in the polynomial object to display the polynomial as an algebraic expression.

  • Override addition, subtraction, and multiplication — Adding, subtracting, or multiplying polynomial objects returns the result of the corresponding algebraic operation on the two polynomials.

  • Double converter — Convert a polynomial object to a double array so it can be used with existing MATLAB functions that accept numeric inputs.

DocPolynom Class Members

The class defines the property coef for storage of the polynomial coefficients.

DocPolynom Class Properties

Name

Class

Default

Description

coef

double

[]

Vector of polynomial coefficients, in order of the highest exponent of x to lowest.

This table summarizes the methods for the DocPolynom class.

DocPolynom Class Methods

Name

Description

DocPolynom

Class constructor

double

Converts the DocPolynom object to a double. In other words, this method returns the coefficients in a vector of double values.

char

Creates a formatted display of the DocPolynom object as powers of x. This method is used by the disp method.

disp

Defines how MATLAB displays DocPolynom objects on the command line.

plus

Adds DocPolynom objects.

minus

Subtracts DocPolynom objects.

mtimes

Multiplies DocPolynom objects.

dispPoly

Evaluates the polynomial for one or more values and returns the results in an organized list instead of a vector of double values.

parenReference

Enables evaluation of a polynomial using parentheses indexing syntax. p(x) evaluates the polynomial object p at each value in x.

Using the DocPolynom Class

These examples show the basic use of the DocPolynom class. Create DocPolynom objects to represent f(x) = x3 − 2x − 5 and f(x) = 2x4 + 3x2 + 2x − 7.

p1 = DocPolynom([1 0 -2 -5])
p1 =
   x^3 - 2*x - 5

p2 = DocPolynom([2 0 3 2 -7])
p2 =
   2*x^4 + 3*x^2 + 2*x - 7

Find the roots of the polynomial p1. Use the double method of the object and pass the result to the roots function.

roots(double(p1))

ans =

   2.0946 + 0.0000i
  -1.0473 + 1.1359i
  -1.0473 - 1.1359i

Add the two polynomials p1 and p2. MATLAB calls the plus method defined for the DocPolynom class when you add two DocPolynom objects.

p1 + p2

ans = 

2*x^4 + x^3 + 3*x^2 - 12

DocPolynom Class Synopsis

Class CodeDescription
classdef DocPolynom < matlab.mixin.Scalar

Value class that implements a data type for polynomials. The class inherits from matlab.mixin.Scalar, which enforces scalar class behavior and also provides functionality to customize parentheses indexing. For more information on the superclass, see matlab.mixin.Scalar.

   properties
      coef
   end

Vector of polynomial coefficients.

   methods      
      function obj = DocPolynom(c)
         if nargin > 0
            if isa(c,'DocPolynom')
               obj.coef = c.coef;
            else
               obj.coef = c(:).';
            end
         end
      end

Class constructor that creates objects using either:

  • An existing DocPolynom object

  • A vector of doubles

For more information, see The DocPolynom Constructor.

      function obj = set.coef(obj,val)
         if ~isa(val,'double')
            error('Coefficients must be doubles.')
         end
         ind = find(val(:).'~=0);
         if isempty(ind)
            obj.coef = val;
         else
            obj.coef = val(ind(1):end);
         end
      end

Set method for coef property:

  • Restricts coefficients to type double

  • Removes leading zeros from the coefficient vector

For more information, see Remove Leading Zeros.

      function c = double(obj)
         c = obj.coef;
      end

Convert DocPolynom object to double by returning the coefficients.

For more information, see Convert DocPolynom Objects to Other Classes.

      function str = char(obj)
         if all(obj.coef == 0)
            s = '0';
            str = s;
            return
         else
            d = length(obj.coef) - 1;
            s = cell(1,d);
            ind = 1;
            for a = obj.coef
               if a ~= 0
                  if ind ~= 1
                     if a > 0
                        s(ind) = {' + '};
                        ind = ind + 1;
                     else
                        s(ind) = {' - '};
                        a = -a;
                        ind = ind + 1;
                     end
                  end
                  if a ~= 1 || d == 0
                     if a == -1
                        s(ind) = {'-'};
                        ind = ind + 1;
                     else
                        s(ind) = {num2str(a)};
                        ind = ind + 1;
                        if d > 0
                           s(ind) = {'*'};
                           ind = ind + 1;
                        end
                     end
                  end
                  if d >= 2
                     s(ind) = {['x^' int2str(d)]};
                     ind = ind + 1;
                  elseif d == 1
                     s(ind) = {'x'};
                     ind = ind + 1;
                  end
               end
               d = d - 1;
            end
         end
         str = [s{:}];
      end

Convert DocPolynom object to char. For more information, see Convert DocPolynom Objects to Other Classes.

      function disp(obj)
         c = char(obj);
         if iscell(c)
            disp(['     ' c{:}])
         else
            disp(c)
         end
      end

Overload disp function. This method displays objects as output of the char method.

For more information, see Overload disp for DocPolynom.

      function dispPoly(obj,x)
         p = char(obj);
         y = zeros(length(x));
         disp(['f(x) = ',p])
         for k = 1:length(x)
            y(k) = polyval(obj.coef,x(k));
            disp(['   f(',num2str(x(k)),') = ',num2str(y(k))])
         end
      end

Return evaluated polynomial with formatted output. This method uses polyval in a loop to evaluate the polynomial at specified values of the independent variable.

For more information, see Display Evaluated Expression.

      function r = plus(obj1,obj2)
         obj1 = DocPolynom(obj1);
         obj2 = DocPolynom(obj2);
         k = length(obj2.coef) - length(obj1.coef);
         zp = zeros(1,k);
         zm = zeros(1,-k);
         r = DocPolynom([zp,obj1.coef] + [zm,obj2.coef]);
      end

      function r = minus(obj1,obj2)
         obj1 = DocPolynom(obj1);
         obj2 = DocPolynom(obj2);
         k = length(obj2.coef) - length(obj1.coef);
         zp = zeros(1,k);
         zm = zeros(1,-k);
         r = DocPolynom([zp,obj1.coef] - [zm,obj2.coef]);
      end

      function r = mtimes(obj1,obj2)
         obj1 = DocPolynom(obj1);
         obj2 = DocPolynom(obj2);
         r = DocPolynom(conv(obj1.coef,obj2.coef));
      end
   end

Define three arithmetic operators:

  • Polynomial addition

  • Polynomial subtraction

  • Polynomial multiplication

For information about this code, see Define Arithmetic Operators.

For general information about defining operators, see Operator Overloading.

   methods (Access = protected)
        function f = parenReference(obj,indexOp)
            n = cell2mat(indexOp(1).Indices);
            if numel(indexOp) == 1
                f = polyval(obj.coef,n);
            else
                f = polyval(obj.coef,n).(indexOp(2:end));
            end
        end
   end
end

Customize parentheses reference for DocPolynom objects. This implementation of parenReference enables users to evaluate a polynomial using parentheses indexing syntax. p(x) evaluates the polynomial represented by object p for each value in x.

For more information, see Redefine Parentheses Indexing.

 Expand for Class Code

The DocPolynom Constructor

This is the DocPolynom class constructor:

function obj = DocPolynom(c)
   if nargin > 0
      if isa(c,'DocPolynom')
         obj.coef = c.coef;
       else
          obj.coef = c(:).';
       end
    end
end

Constructor Calling Syntax

The DocPolynom constructor can accept two different input arguments:

  • An existing DocPolynom object — Calling the constructor with an existing DocPolynom object as an input argument returns a new DocPolynom object with the same coefficients as the input argument. The isa function checks for this input.

  • Coefficient vector — When the input argument is not a DocPolynom object, the constructor attempts to reshape the values into a row vector and assign them to the coef property.

    The coef property set method restricts property values to doubles. See Remove Leading Zeros for a description of the property set method.

This example uses a vector as the input argument to the DocPolynom constructor:

p = DocPolynom([1 0 -2 -5])
p = 
   x^3 - 2*x -5

This statement creates an instance of the DocPolynom class with the specified coefficients. The display of the object shows the equivalent polynomial using MATLAB language syntax. The DocPolynom class implements this display using the disp and char class methods.

Remove Leading Zeros

The DocPolynom class represents polynomials as row vectors containing coefficients ordered by descending powers. Zeros in the coefficient vector represent terms that are not in the polynomial. Leading zeros, therefore, can be ignored when forming the polynomial. In fact, some DocPolynom class methods use the length of the coefficient vector to determine the degree of the polynomials, so removing leading zeros from the coefficient vector ensures that the vector length represents the correct polynomial degree.

The DocPolynom class stores the coefficient vector in a property that uses a set method to remove leading zeros from the specified coefficients before setting the property value. 

function obj = set.coef(obj,val)
   if ~isa(val,'double')
     error('Coefficients must be doubles.')
   end
   ind = find(val(:).'~=0);
   if isempty(ind)
     obj.coef = val;
   else
      obj.coef = val(ind(1):end);
   end
end

Convert DocPolynom Objects to Other Classes

The DocPolynom class defines two methods to convert DocPolynom objects to other classes:

  • double — Converts to the double numeric type so functions can perform mathematical operations on the coefficients.

  • char — Converts to characters used to format output for display in the Command Window.

The Double Converter

The double converter method for the DocPolynom class returns the coefficient vector:

function c = double(obj)
    c = obj.coef;
end

For the DocPolynom object p, double returns a vector of class double.

p = DocPolynom([1 0 -2 -5]);
c = double(p)

c =

     1     0    -2    -5

The Character Converter

The char method returns a char vector that represents the polynomial displayed as powers of x. The char vector returned is a syntactically correct MATLAB expression.

The char method uses a cell array to collect the char vector components that make up the displayed polynomial. The disp method uses the char method to format the DocPolynom object for display. Users of DocPolynom objects are not likely to call the char or disp methods directly, but these methods enable the DocPolynom class to behave like other data classes in MATLAB.

Overload disp for DocPolynom

To provide a more useful display of DocPolynom objects, this class overloads disp in the class definition. This disp method relies on the char method to produce a text representation of the polynomial, which it then displays.

The char method returns a cell array or the character '0' if the coefficients are all zero. 

function disp(obj)
   c = char(obj);
   if iscell(c)
      disp(['     ' c{:}])
   else
      disp(c)
   end
end

When MATLAB Calls the disp Method

This statement creates a DocPolynom object. Because the statement is not terminated with a semicolon, the resulting output is displayed on the command line using the overloaded disp method.

p = DocPolynom([1 0 -2 -5])

p =
    x^3 - 2*x - 5

Display Evaluated Expression

The dispPoly method evaluates the polynomial for one or more values of x. The method loops through the input values of x and uses the polyval function with the coef property to evaluate the polynomial.

function dispPoly(obj,x)
   p = char(obj);
   y = zeros(length(x));
   disp(['f(x) = ',p,])
   for k = 1:length(x)
      y(k) = polyval(obj.coef,x(k));
      disp(['   f(',num2str(x(k)),') = ',num2str(y(k))])
    end
end

Create a DocPolynom object p:

p = DocPolynom([1 0 -2 -5])

p = 
   x^3 - 2*x - 5

Evaluate the polynomial at three values of x, [3 5 9]. Instead of returning a vector of values, the method uses function notation to present the results in an organized list.

dispPoly(p,[3 5 9])
f(x) = x^3 - 2*x - 5
   f(3) = 16
   f(5) = 110
   f(9) = 706

Define Arithmetic Operators

The DocPolynom class implements methods for three arithmetic operations.

Method and Syntax

Operation

plus(a,b)

Addition

minus(a,b)

Subtraction

mtimes(a,b)

Matrix multiplication

The overloaded plus, minus, and mtimes methods accept argument pairs that include at least one DocPolynom object.

Define the + Operator

If either p or q is a DocPolynom object, this expression generates a call to the plus method overload defined by DocPolynom unless the other object is of higher precedence.

p + q

This method overloads the plus (+) operator for the DocPolynom class.

function r = plus(obj1,obj2)
   obj1 = DocPolynom(obj1);
   obj2 = DocPolynom(obj2);
   k = length(obj2.coef) - length(obj1.coef);
   zp = zeros(1,k);
   zm = zeros(1,-k);
   r = DocPolynom([zp,obj1.coef] + [zm,obj2.coef]);
end

The plus method performs these actions:

  • Ensure that both input arguments are DocPolynom objects so that expressions that involve a DocPolynom and a double work correctly.

  • Access the two coefficient vectors and, if necessary, pad one of them with zeros to make both the same length. The actual addition is simply the vector sum of the two coefficient vectors.

  • Call the DocPolynom constructor to create a properly typed object that is the result of adding the polynomials.

Define the - Operator

The minus operator (-) uses the same approach as the plus (+) operator. The minus method computes p - q. The dominant argument must be a DocPolynom object.

function r = minus(obj1,obj2)
   obj1 = DocPolynom(obj1);
   obj2 = DocPolynom(obj2);
   k = length(obj2.coef) - length(obj1.coef);
   zp = zeros(1,k);
   zm = zeros(1,-k);
   r = DocPolynom([zp,obj1.coef] - [zm,obj2.coef]);
end

Define the * Operator

The mtimes method computes the product p*q. The mtimes method implements matrix multiplication because the multiplication of two polynomials is the convolution (conv) of their coefficient vectors:

methods
   function r = mtimes(obj1,obj2)
      obj1 = DocPolynom(obj1);
      obj2 = DocPolynom(obj2);
      r = DocPolynom(conv(obj1.coef,obj2.coef));
   end
end

Using the Arithmetic Operators

Create a DocPolynom object.

p = DocPolynom([1 0 -2 -5]);

These two arithmetic operations call the DocPolynom plus and mtimes methods.

q = p + 1
r = p*q

q = 
    x^3 - 2*x - 4

r =
    x^6 - 4*x^4 - 9*x^3 + 4*x^2 + 18*x + 20

Redefine Parentheses Indexing

The DocPolynom class inherits from matlab.mixin.Scalar, which in turn inherits from the modular indexing class matlab.mixin.indexing.RedefinesParen. Overloading the parenReference method of RedefinesParen enables users to evaluate a polynomial represented by a DocPolynom object using parentheses indexing syntax.

For example, create a DocPolynom object p.

p = DocPolynom([1 0 -2 -5])

p =
    x^3 - 2*x - 5

The overloaded parenReference method evaluates the value of the polynomial at x = 3 and at x = 4 using this command.

p([3 4])
ans =
    16   51

Modular Indexing Implementation Details

The parenReference method handles expressions of the form p(x), where p is a DocPolynom object and x contains numeric inputs. Instead of a traditional MATLAB indexing operation, however, parenReference uses polyval to evaluate the polynomial using the coefficients stored in the coef property.

methods (Access = protected)
    function f = parenReference(obj,indexOp)
       n = cell2mat(indexOp(1).Indices);
       if numel(indexOp) == 1
          f = polyval(obj.coef,n);
       else
          f = polyval(obj.coef,n).(indexOp(2:end));
       end
    end
end

The method performs these steps:

  1. Extract the indexing values from indexOp, which is an instance of the matlab.indexing.IndexingOperation class. The indexOp object stores them as a cell array, and the method converts them to a numeric array and stores them in n.

  2. Calculate the number of indexing operations in the expression.

  3. Evaluate the polynomial at the values in n. The intended use of this syntax includes one parentheses indexing operation.

  4. If the method finds more than one indexing operation, it uses polyval to evaluate the polynomial and then forwards the rest of the indexing operations to MATLAB using the forwarding syntax, .(indexOp(2:end)). The class does not support any additional customized indexing operations, so MATLAB returns an error.

    For example, attempting to evaluate a polynomial object while also using dot indexing to try to access its coef property at the same time errors.

    p(5).coef(1)
    Dot indexing is not supported for variables of this type.
    For more information on forwarding operations with modular indexing, see Forward Indexing Operations.

Using the modular indexing class in this way means that only parentheses reference operations are customized. Dot access to properties and methods are unaffected and are handled by MATLAB as expected.

Note

The matlab.mixin.Scalar and matlab.mixin.indexing.RedefinesParen functionality was introduced in R2021b.