CONVOLUTION POLYNOMIAL DIVISION TEMPLATE

Research Paper

E-ISSN NO : 2455-295X | VOLUME : 3 | ISSUE : 9 | SEP 2017

CONVOLUTION POLYNOMIAL DIVISION TEMPLATE

FENG CHENG CHANG 1 1 ALLWAVE

CORPORATION, 3860 DEL AMO BLVD, SUITE 404, TORRANCE, CALIFORNIA 90503.

ABSTRACT The division of a pair of giving polynomials to find its quotient and remainder is derived by applying the convolution matrix. The process of matrix formulation with compact template is simple, efficient and direct, comparing to the familiar classical longhand division and synthetic polynomial division.

Typical numerical examples are provided to show the merit of the approaches.

Keywords: Polynomial division; Longhand division; Synthetic division; Convolution matrix; Matrix formulation.

Introduction and Formulation The division of two univariate polynomials to find the quotient and remainder is expressed as

or

where b(x) and a(x) are the given dividend and divisor of degrees n and m, n ³ m , and q(x) and r(x) are the resulted quotient and remainder of degrees n – m and m – 1 or less, respectively,

In practical calculation, we can omit the x factor in this linear algebraic equation. Also if its square matrix is found to be non-singular, the desired coefficients of q(x) and r(x) can thus be uniquely obtained.

This algebraic matrix equation may also be further separated into two sets of equations for computing our desired entries:



Then the coefficients of x in both side of polynomial division equation after substituting of the expansion forms of b(x), a(x) and q(x), r(x) will give the following relation: and where it is understood that The polynomial division is then written in the convolution matrix form [1,2] as:

Here, from the given entries

and

the desired

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Research Paper

E-ISSN NO : 2455-295X | VOLUME : 3 | ISSUE : 9 | SEP 2017

entrie must be computed consecutively in the entry order, and the entries

in

b( x ) r( x)  q( x )  a( x) a( x)

may then be found accordingly.

Applying the relationship among all of these entries, the polynomial division manipulation may further be caste into either of the following templates:

we shall find the desired results as

q( x ) 

4 3 11 2 64 176 x  x  x  3 9 27 81

r( x) 

619 4 533 3 148 2 77 215 x  x  x  x  81 81 81 81 81

by applying either one of the following approaches:

(1) Longhand polynomial division

 43  119  2764  17681 3  1  7  5  4  2 ) 4 5 1 7 6 1  2  3  7 4  43  283  203  163  83  113  253  13  32  53 

The desired entries of quotient and reminder can then be directed determined without writing down any intermediate data as in the familiar longhand polynomial division and synthetic polynomial division.

11 3



11 9



77 9



55 9



44 9

2  3  7  229

 649  809  619  299  94 3  7  649  2764  44827  32027  25627  12827  17627  26527  23327  24427  20927 7  17627  17681  1232  88081  70481  35281 81

It follows that our desired coefficients may also be recursively computed by the following formulas:

 61981  53381  14881  7781  21581 (2) Synthetic polynomial division

4 A simple computer routine in MATLAB is presented. Inputs b and a, and outputs q and r are the coefficient vectors of given dividend b(x) and divisor a(x), and resulted quotient q(x) and reminder r(x), respectively.

 13  73  53  43

5 1 7 6 1  43  119  6427  176 81 28 77 448  3  9  27  1232 81  203

2

3

7

 559  320  880 27 81 16 256 44  3  9  27  704 81

 23

 83  4  113  649  176  619 27 81

 229  128  352 27 81

 533  148 81 81

 7781  215 81

(3) Convolution polynomial division

Typical Example with Remarks For given

b( x )  4 x 8  5 x 7  x 6  7 x 5  6 x 4  x 3  2 x 2  3 x  7 a ( x )  3x 5  x 4  7 x 3  5 x 2  4 x  2

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 4   3  5   1     1   7     7   5  6    4     1   2  2       3    7     

E-ISSN NO : 2455-295X | VOLUME : 3 | ISSUE : 9 | SEP 2017

3 1 3 7 1 5 7 4 5 2 4 2

     3   1 1  7 1   5 1  4 1  2 1

  43   11   9    6427   176    81    619  81  533    81    148   81    7781    215   81 

(4) Convolution polynomial division (Separated form)  4   5     1     7   6   1     2     3   7 





     

3 1 7 5

 4  2     

  3   1 3  7 1 3 

  43    11   9  64    27  176    81 

5 4 2

  619   43  81    533    11   81   9  64     27    148 81  77   176    81    81    215  81 

7 5 4 2

1  7  5   4  2 

REFERENCES 1. B. Dayton, “Polynomial GCDs by linear algebra,” Theory of Equations, Northeastern Illinois University, www.neiu.edu/~bhdayton, 2004.

2. F. C. Chang, “Polynomial division by convolution,” Applied Mathematics E-Notes, vol. 11, pp. 249-254, 2011.

3. L. Zhou, “Short division of polynomials,” The college Mathematics Journal, 40 (2009), pp. 44-46.

4. D. Bini and V. Pan, “Polynomial division and its computational complexity,” Journal of Complexity, 2 (1986), pp. 179-203.

(5) Convolution polynomial division (Template form)

Comparing of those approaches in this typical example, we found that the convolution polynomial division is much simple and effective. The desired quotient and remainder are readily determined without computing any intermediate data as in the familiar classical longhand polynomial division and synthetic polynomial division [3,4].

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