perform right or left Euclidean algorithm

はじめる前に
新機能一覧
Maple ワークシートの作成
Mapleワークシートを共有
Maple ウィンドウのカスタマイズ
Maple システムのカスタマイズ
コネクティビティ
Mathematics
数学
- テンソル解析
- パッケージ
- ファンクションアドバイザ
- ベキ級数
- ベクトル解析
- 不活性関数
- 代数
  - Magma
  - 多項式
  - 式の処理
  - 有理式表現
  - 歪多項式
- 因数分解と方程式解法
- 基本数学
- 基本的な注意事項
- 変分法
- 変換
- 幾何
- 微分代数方程式
- 微分幾何学
- 微分方程式
- 微積分
- 数
- 数値計算
- 数学関数
- 数論
- 最適化
- 特殊関数
- 統計
- 線形代数
- 群論
- 評価
- 論理
- 過去のパッケージ
- 金融
- 離散数学
- piecewise examples
- グラフ電卓
- 区分関数
物理
プログラミング
グラフィックス
Student パッケージ
Science and Engineering
科学・エンジニアリング
アプリケーションと例題
リファレンス
システム
MapleSim
MapleSim ツールボックス
MapleSim ヘルプ
Tasks
Toolboxes
エラーメッセージガイド
マニュアル
数学アプリ

OreTools[Euclidean] - perform right or left Euclidean algorithm

	Calling Sequence
	Euclidean['right'](Poly1, Poly2, A, 'c1', 'c2') Euclidean(Poly1, Poly2, A, 'c1', 'c2') Euclidean['left'](Poly1, Poly2, A, 'c1', 'c2')

Parameters

Poly1, Poly2	-	nonzero Ore polynomials; to define an Ore polynomial, use the OrePoly structure.
A	-	Ore algebra; to define an Ore algebra, use the SetOreRing function.
'c1', 'c2'	-	(optional) unevaluated names

Description

•	The Euclidean['right'](Poly1, Poly2, A) or Euclidean(Poly1, Poly2, A) calling sequence returns a list [m, S] where m is a positive integer and S is an array with m nonzero Ore polynomials such that:

In addition, Remainder['right'](S[m-1], S[m], A) = 0. S is called the right Euclidean polynomial remainder sequence of Poly1 and Poly2.

•	If the fourth argument c1 of Euclidean['right'] or Euclidean is specified, it is assigned the first co-sequence of Poly1 and Poly2 so that:

and c1[m+1] Poly1 is a least common left multiple (LCLM) of Poly1 and Poly2.

•	If the fifth argument c2 of Euclidean['right'] or Euclidean is specified, it is assigned the second co-sequence of Poly1 and Poly2 so that:

and is an LCLM of Poly1 and Poly2.

•	The Euclidean['left'](Poly1, Poly2, A) calling sequence returns a list [m, S] where m is a positive integer and S is an array with m nonzero Ore polynomials such that:

In addition, . S is called the left Euclidean polynomial remainder sequence of Poly1 and Poly2.

•	If the fourth argument c1 of Euclidean['left'] is specified, it is assigned the first co-sequence of Poly1 and Poly2 so that:

and Poly1 c1[m+1] is a least common right multiple (LCRM) of Poly1 and Poly2.

•	If the fifth argument c2 of Euclidean['left'] is is specified, it is assigned the second co-sequence of Poly1 and Poly2 so that:

and Poly1 c1[m+1]= - Poly2 c2[m+1] is an LCRM of Poly1 and Poly2.

Examples

Perform the right Euclidean algorithm.

(1)

(2)

(3)

(4)

(5)

vector([OrePoly(1, 3*x, x^2-1-x, 23*x+1), OrePoly(x, x, x^2+x+1), OrePoly(-x*(-x^3-49*x^2-7*x+20+x^4)/(x^2+x+1)^2, (2*x^5-17*x^4+32*x^3-14*x^2-20*x-1)/(x^2+x+1)^2), OrePoly((-5*x+981*x^2+5*x^11+598*x^9+1451*x^8+2135*x^7-200*x^10+1781*x^6+1343*x^5+1850*x^4+1420*x^3-20+x^12)/(2*x^5-17*x^4+32*x^3-14*x^2-20*x-1)^2)])

(6)

Check the co-sequences.

for i to m do W1 := Multiply(c1[i], Ore1, A); W2 := Multiply(c2[i], Ore2, A); W := Add(W1, W2); C := Minus(W, S[i]); print(C) end do

(7)

Check the LCLM.

(8)

Perform the left Euclidean algorithm.

(9)

(10)

vector([OrePoly(1, 3*x, x^2-1-x, 23*x+1), OrePoly(x, x, x^2+x+1), OrePoly(-(18*x^5+99*x^4-12*x^3+x^7-1059*x^2-955*x-51)/(x^2+x+1)^3, (-21*x^4+20*x^3-17*x^2-289*x+2*x^5-136)/(x^2+x+1)^2), OrePoly((12308+39236*x+x^12+44*x^10+x^11+1037*x^8+1941*x^7+11909*x^6+556*x^9+36562*x^5+72548*x^4+88995*x^3+76169*x^2)/(-21*x^4+20*x^3-17*x^2-289*x+2*x^5-136)^2)])

(11)

Check the co-sequences.

for i to m do W1 := Multiply(Ore1, c1[i], A); W2 := Multiply(Ore2, c2[i], A); W := Add(W1, W2); C := Minus(W, S[i]); print(C) end do

(12)

Check the LCRM.

(13)

	See Also
	OreTools, OreTools/Add, OreTools/Minus, OreTools/Multiply, OreTools/OreAlgebra, OreTools/OrePoly, OreTools/Quotient, OreTools/Remainder, OreTools[SetOreRing]