Numerical Summation - Maple Help

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
- 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
  - Maple の数値表現
  - MATLAB リンクパッケージ
  - 区間
  - 多項式近似代数（SNAP）パッケージ
  - 整数用関数
  - 補間とカーブフィッティング
  - 近似
    - evalf
    - IntegerRelations パッケージ
    - numapprox パッケージ
    - 連分数
    - fdiff
    - float
    - identify コマンド
    - testfloat
    - Tolerance
    - 桁
  - 離散変換
  - Numerical Analysis パッケージ
- 数学関数
- 数論
- 最適化
- 特殊関数
- 統計
- 線形代数
- 群論
- 評価
  - assume
  - evala
  - evalc
  - evalf
    - 概要
    - 詳細情報
    - int
    - sum コマンド
  - evalhf
  - allvalues
  - cost
  - Eval
  - evalb
  - evalindets
  - evaln
  - evaln example
  - evalpow
  - evalr
  - evalrC
  - fdiff
  - hfarray オブジェクト
  - option hfloat
  - value コマンド
  - 最後の名前の評価
  - 特殊な評価
- 論理
- 過去のパッケージ
- 金融
- 離散数学
  - グラフ理論
  - 組合せ論
  - 総和と差分方程式
    - evalf
      - sum コマンド
    - LREtools
    - QDifferenceEquations
    - sum コマンド
    - SumTools
    - hypergeom
    - RESol
    - rsolve
    - 総和
  - 離散変換
  - OrderBasis コマンド
  - 積
- 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

Numerical Summation

Calling Sequence

evalf(Sum(f, x=a..b))
evalf(sum(f, x=a..b))

evalf(Sum(f, x=RootOf(g(_Z))))

evalf(sum(f, x=RootOf(g(_Z))))

Parameters

f	-	algebraic expression; the summand
x	-	name; the variable of summation
a, b	-	endpoints of the interval of summation (can be infinite)
g(_Z)	-	algebraic expression with a finite number of roots in Z

Description

•

The most common command for numerical summation is evalf(Sum(f, x=a..b)) where the summation command is expressed in inert form to avoid first invoking the symbolic summation routines. It is also possible to invoke evalf on an unevaluated sum returned by the symbolic sum command, as in evalf(sum(f, x=a..b)), if it happens that symbolic sum fails (returns an unevaluated sum).

•	You can enter the command evalf/Sum using either the 1-D or 2-D calling sequence. For example, evalf(Sum(1/(x^2+1), x=-5..5)) is equivalent to

•	The evalf(Sum()) command attempts to compute a result that is accurate to within 1 ulp at the current setting of Digits, so increasing Digits increases the requested accuracy of the result.

•	The summand f may be another unevaluated sum or integral, that is, nested forms are supported.

Method

•	In the case of finite sums, or sums over RootOfs, the sum is computed directly through numerical evaluation of the summand.

•	In the case of infinite sums, Levin's u-transform is used, which has the additional effect that sums that formally diverge may return a result that can be interpreted as evaluation of the analytic extension of the series for the sum (see the examples below).

•	This behavior can be controlled through the use of the _EnvFormal environment variable. If this variable is assigned the value false, then this command will apply some (numeric) convergence tests to determine if the infinite sum in question is convergent, divergent, or otherwise non-convergent.

Examples

Below are evaluation of sums over RootOf.

(1)

(2)

Here are some examples of finite sums approaching infinity, and an infinite sum.

evalf(Sum(1/(x^2+1), x = -5 .. 5))

(3)

evalf(Sum(1/(x^2+1), x = -20 .. 20))

(4)

evalf(Sum(1/(x^2+1), x = -100 .. 100))

(5)

evalf(Sum(1/(x^2+1), x = -infinity .. infinity))

(6)

An example with nesting.

xpr := Int(BesselK(0, sqrt((2*n+1)^2+x^2))^2, x = -infinity .. infinity)

xpr := Int(BesselK(0, (4*n^2+4*n+1+x^2)^(1/2))^2, x = -infinity .. infinity)

(7)

evalf(Sum(xpr, n = 0 .. infinity))

(8)

The following is a case where Levin's u-transform computes the values for the analytic extension of the sum.

evalf(Sum(1/sqrt(x), x = 1 .. 10))

(9)

evalf(Sum(1/sqrt(x), x = 1 .. 100))

(10)

evalf(Sum(1/sqrt(x), x = 1 .. 1000))

(11)

evalf(Sum(1/sqrt(x), x = 1 .. infinity))

(12)

The infinite sum diverges.

sum(1/sqrt(x), x = 1 .. infinity)

(13)

This can be seen as a sum from the analytic extension of Zeta:

Zeta(z) = Sum(1/i^z, i = 1 .. infinity)

(14)

evalf(eval(Zeta(z) = Sum(1/i^z, i = 1 .. infinity), z = 1/2))

(15)

or using 'sum':

sum(1/sqrt(x), x = 1 .. infinity)

(16)

evalf(Sum(1/sqrt(x), x = 1 .. infinity))

(17)

evalf(Sum(1/sqrt(x), x = 1 .. infinity))

(18)

	See Also
	Digits, evalf, Sum, sum

References

Fessler, T.; Ford, W.F.; and Smith, D.A. "HURRY: An acceleration algorithm for scalar sequences and series." ACM Trans. Math. Software, Vol. 9, (1983): 346-354.