# ApCoCoA-1:MatlabCoCoAPoly

# Introduction

The CoCoAPoly class represents a polynomial over a given polynomial ring (See CoCoARing).

It is based on a representation of the terms by using the logarithm of the term. For example the polynomial *x^2+2y+3xy+4* defined over the Ring Q[x,y] can be represented in a matrix of the following form:

coeff | power x | power y |

1 | 2 | 0 |

2 | 0 | 1 |

3 | 1 | 1 |

4 | 0 | 0 |

# Constructor

The constructor for a CoCoAPoly can be used in one of the three different forms:

- CoCoAPoly('keyword1',value1,'keyword2','value2',...).

Available keywords are listed below:

Keyword | Description | Default |

Ring | A CoCoARing object | Q[x[1]] |

Terms | A term representing matrix as described above. Can also be the string 'One' or 'Zero'. This will generate the constant 1 or 0 polynomial. | [0 0] |

Table 1: Keywords for CoCoAPoly

- A string can be passed as a single argument. For example: CoCoAPoly('x^2+2y'). The string must hold the following conventions:
- Indeterminates: a-z
- + or - seperates two terms
- Coefficients are placed left to the indeterminate and without a '*' character. Decimal delimiter is '.'. If coefficient=1 it is not needed to be typed.
- The ring is build upon the found indeterminates over Q with a DegRevLex ordering. If only a constant is given the constant polynomial over Q[x] is returned. To define a poly over an indeterminate which is not part of the current polynomial use coefficient = 0.

- A real numeric value can be passed as a single argument. For example: CoCoAPoly(2). This will return a constant polynomial over Q[x[1]].

# Attributes

ring | A CoCoARing |

terms | a polynomial representing matrix as described above. |

# Methods

Method | Operator | Description |

display | Is used by Matlab to print a CoCoAPoly to the command window. E.g. when a command is not terminated by a ';'. | |

end | returns the last coloumn or row of CoCoAPoly array | |

eq | == | Compares two polynomials. Returns 1 if all attributes are equal. Of polynomials are not equal 0 is returned. |

ge | >= | Compares two polynomials accordingto the given term ordering. Returns 1 if first polynomial is greater equal the second polynomial. |

get | Returns the attributes of a polynomial. Syntax: get(polyVar, 'Keyword'), where polyVar is a CoCoAPoly and 'Keyword' is one of the keywords listed in table 1. Additionaly the keyword 'String' can be used. This will return the polynomial as a string. The keyword 'Latex' will return a string using Latex syntax. | |

gt | > | Compares two polynomials accordingto the given term ordering. Returns 1 if first polynomial is greater than the second polynomial. |

le | <= | Compares two polynomials accordingto the given term ordering. Returns 1 if first polynomial is less equal the second polynomial. |

lt | < | Compares two polynomials accordingto the given term ordering. Returns 1 if first polynomial is less than the second polynomial. |

minus | - | Subtract one polynomial from another. |

mpower | ^ | Returns the i-th power of a polynomial. |

mtimes | * | Multiplies two polynomials. |

ne | ~= | returns NOT eq. |

plus | + | Adds two polynomials. |

subsasgn | () | Writes a polynomial to the (i,j) coordinate of a CoCoAPoly array. |

subsref | () | Returns the (i,j) element of a CoCoAPoly array, {i,j} returns the element as a string. |

times | .* | Multiplies arrays of polynomials elementwise. If both arrays have the same dimension than each element of poly1 is multiplied with the corresponding element in poly2. If one of the poly's is a single value, this value is multiplied to each element in the other poly. If both poly's are single values they are simply multiplied. |

uminus | - | support for leading minus symbol. |

uplus | + | support for leading plus symbol. |

# Additional methods

All coefficients are stored in Matlab as of typ double. Thus the conversion in the
Interface between Matlab and the lib's rounding error's are introduced.
This is in particular relevant when comparing polynomials. To overcome rounding issues
the CoCoAPoly class provides six approximate compare functions.

Method | Description |

approxeq | Approximative equal. Each term is compared. |

approxge | Approximative greater equal. For the greater part only leading terms are compared, for equal all terms are compared. |

approxgt | Approximative greater. Only leading terms are compared. |

approxle | Approximative less equal. For the greater part only leading terms are compared, for equal all terms are compared. |

approxlt | Approximative less. Only leading terms are compared. |

approxne | Approximative not equal. Each term is compared. |

The coefficients are compared within a range of precision: |coeff1 - coeff2| < precLevel => coeff1 = coeff2

Default precLevel = 10^-15 [Min. diff. between two doubles is: 2.22044604925031e-16]

The precision level is offered as a third optional argument.

# Examples

A collection of examples is provided in the file TestCoCoAPoly.m. This file defines the function TestCoCoAPoly which will run and display many examples. A few common examples are presented below.

- Default polynomial: Constant 0 over Q[x[1]]

<matlab> pDefault = CoCoAPoly(); </matlab>

- Create two polynomials over same ring

<matlab> r1 = CoCoARing('Typ','Q','Indets',{'x','y'},'Ordering','DegRevLex'); p1 = CoCoAPoly('Ring', r1, 'Terms', [1 1 2 ; 1 2 1 ; 1 1 1]); p2 = CoCoAPoly('Ring', r1, 'Terms', [1 1 2 ; 1 0 0 ; 1 0 1]); </matlab>

- Add the two polynomials

<matlab> pAdded = p1+p2; </matlab>

- To read the attributes of a CoCoARing use the following command (this examples reads all attributes):

<matlab> [a1,a2] = get(p1,'Ring','Terms'); </matlab>

- Create an array of CoCoAPolys and access an element

<matlab> pArray = [p1,p2 ; p2,p1]; pElement = pArray(1,2); </matlab>

- Passing a string to create a CoCoAPoly

<matlab> poly = CoCoAPoly('45.4x^2x-x^1y^2+0z+5'); </matlab>