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| {{Redirect|Gaussian integration|the integral of a Gaussian function|Gaussian integral}}
| | == 「マグマ生き物はどのような... ' == |
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| In [[numerical analysis]], a '''quadrature rule''' is an approximation of the [[integral|definite integral]] of a [[function (mathematics)|function]], usually stated as a [[weighted sum]] of function values at specified points within the domain of integration.
| | これらのマグマの純蓮火悪魔パワーへの自動蒸発は、明らかにわずかなを妨害する資格はない [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-6.html casio 腕時計 説明書]。<br>シャオヤン速度は、前年に比べて、私は、何倍も速く知っていますが、時間の10分よりも弱いない<br>今日、彼は、同時に、彼の身長次第に深いマグマだが、ある周りを探したくさん顔「カラー」溶岩平原を遅く、彼が感じることができた、微妙な味をたくさん持っている、速い近い彼に来ている [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-13.html カシオ アナログ 腕時計]。<br><br>「マグマ生き物はどのような... [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-11.html カシオ腕時計 g-shock] '<br><br>シャオヤンかすかな笑顔、優しくYiduo足突然、突然、巨大な魂の変動は、眉毛からの高潮はオープン、瞬時に、周囲のマグマの爆発がシャドウマグマに隠された道を開いている間に噴火甲高い悲鳴 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-10.html casio電波腕時計]...<br><br>「チチ！ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-5.html カシオ 腕時計 スタンダード] '<br>突然マグマの周りに激しく投げ、これらの悲鳴の出現により<br> |
| (See [[numerical integration]] for more on [[quadrature (mathematics)|quadrature]] rules.)
| | 相关的主题文章： |
| An ''n''-point '''Gaussian quadrature rule''', named after [[Carl Friedrich Gauss]], is a quadrature rule constructed to yield an exact result for [[polynomial]]s of degree 2''n'' − 1 or less by a suitable choice of the points ''x''<sub>''i''</sub> and weights ''w''<sub>''i''</sub> for ''i'' = 1,...,''n''.
| | <ul> |
| The domain of integration for such a rule is conventionally taken as [−1, 1],
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| so the rule is stated as
| | <li>[http://www.glinbbs.com/home.php?mod=space&uid=16147 http://www.glinbbs.com/home.php?mod=space&uid=16147]</li> |
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| | <li>[http://www.alcoholfreesocial.com/blogs/post/80055 http://www.alcoholfreesocial.com/blogs/post/80055]</li> |
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| | <li>[http://www.thnw.gov.cn/bbs/ShowPost.asp?ThreadID=18513 http://www.thnw.gov.cn/bbs/ShowPost.asp?ThreadID=18513]</li> |
| | |
| | </ul> |
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| :<math>\int_{-1}^1 f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i).</math>
| | == 」リッピング」 == |
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| Gaussian quadrature as above will only produce accurate results if the function ''f''(''x'') is well approximated by a polynomial function within the range [-1,1]. The method is not, for example, suitable for functions with [[singularity (math)|singularities]]. However, if the integrated function can be written as
| | 目、銀のロール「色」リールの間では、彼の手のひらで登場しました [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-2.html カシオの時計]。<br>分よりも非常に寛大でリールに銀色「色」リールを<br>、全身が明るい銀色「色」だったが、それは、それを見つけることができる密度の高い小さな赤い「色」線が太いかどうかを確認するように注意です。それは一般的に血液と同様に、リールの各部分を広げている [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-9.html カシオ 時計 プロトレック]。<br><br>目は、このロールリールを見て、シャオヤンの心は知らず知らずのうちに鼓動を加速させていると、このリールは彼が神韻におけるいくつかの奇妙なものが満たさ見つけた唯一のものであるようだ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-3.html 電波時計 カシオ]。<br>'それは何ですか？'<br><br>一部の容疑者「混乱」は風変わりなので、このことにより、叫びをつぶやい<br>、シャオヤンを直接開いていないが、後者でスローリールから悪魔の人形を召喚するために手を伸ばした：」リッピング」<br><br>悪魔の人形リールキャッチは、少しもためらうことなく、手のひらをゆっくりと広がってスクロールします [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-7.html カシオ電波ソーラー腕時計レディース]。<br><br>'強打 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-0.html カシオ電波ソーラー腕時計]！ |
| <math>f(x) = \omega(x) g(x)\,</math>, <!--- Do not delete "\,": It improves the formula display in certain browsers. ---> | | 相关的主题文章： |
| where ''g''(''x'') is approximately polynomial, and ''ω''(''x'') is known{{how|date=January 2014}}, then there are alternative weights <math>w_i'</math> such that
| | <ul> |
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| | <li>[http://bbs.hongxi.com/forum.php?mod=viewthread&tid=1129032&fromuid=99021 http://bbs.hongxi.com/forum.php?mod=viewthread&tid=1129032&fromuid=99021]</li> |
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| | <li>[http://www.languiren.cc/forum.php?mod=viewthread&tid=255756 http://www.languiren.cc/forum.php?mod=viewthread&tid=255756]</li> |
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| | <li>[http://topart798.com/plus/feedback.php?aid=23781 http://topart798.com/plus/feedback.php?aid=23781]</li> |
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| | </ul> |
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| :<math>\int_{-1}^1 f(x)\,dx = \int_{-1}^1 \omega(x) g(x)\,dx \approx \sum_{i=1}^n w_i' g(x_i).</math>
| | == アオ上記の直面している == |
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| Common weighting functions include
| | アオ上記の直面している [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-9.html カシオ 時計 プロトレック]。<br><br>'ブーム [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-5.html gps 腕時計 カシオ]！'<br>そのはすぐに、さらにいくつかのタンブリングを再生した後、空気中の体は、一般的に短期的な凧のように、それは青を支払う見てちょうど重く、見物人の心のジャンプを中心に押し込まれたように<br>拳肉に触れ、低い声で遠く地面から数十メートルにわたってザラ、無謀に行動する<br><br>死ん静かに囲まYishaザラShimoji、という青ボディを払って [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-4.html カシオ ソーラー電波腕時計]！<br><br>第四百七十一章白ギャングの強さ<br><br>第四百七十一章白ギャングの強さ<br><br>広い敷地の上に、静かな雰囲気を凍結し、遠くの青十メートルの支払い動か横たわっ道路の光景を見て、恐怖を隠すことができない、目のタッチを持っています [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-8.html カシオ gショック 腕時計]。<br><br>これは、トップ70かそこらに入ることができる、それが全体の中庭に置かれても、青の強さを支払うために、ああ、サムスンの強い闘志ですが、今、彼はで敗北した |
| <math>\omega(x)=1/\sqrt{1-x^2}\,</math> <!-- Do not delete "\,": it improves the display of the formula on certain browsers. ---> | | 相关的主题文章： |
| ([[Chebyshev–Gauss quadrature|Chebyshev–Gauss]]) and <math>\omega(x)=e^{-x^2}</math> ([[Gauss–Hermite quadrature|Gauss–Hermite]]).
| | <ul> |
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| | <li>[http://www.icxwl.com/plus/feedback.php?aid=12 http://www.icxwl.com/plus/feedback.php?aid=12]</li> |
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| | <li>[http://bbs.nandu.com/forum.php?mod=viewthread&tid=464287 http://bbs.nandu.com/forum.php?mod=viewthread&tid=464287]</li> |
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| | <li>[http://din-timelines.com/cgi-bin/forums/x-forum.cgi http://din-timelines.com/cgi-bin/forums/x-forum.cgi]</li> |
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| | </ul> |
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| It can be shown (see Press, et al., or Stoer and Bulirsch) that the evaluation points are just the [[root of a function|root]]s of a [[polynomial]] belonging to a class of [[orthogonal polynomials]].
| | == こんにちは下の女の子 == |
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| == Gauss–Legendre quadrature == <!-- The section "Other forms of Gaussian quadrature" below links to this section -->
| | 超越 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-2.html 腕時計 casio]。<br><br>群衆が最後に、シャオヤンの息をのむような、彼の唇を叩か、本当に、、まだステージのスターを改善するために戦っている人たちのレベル以上の戦いに一年以内にこのような行為をZheni紫ジンを期待していなかったスピード、ほとんどすべての[OK]を、彼の同等のベース精神「流体」を構築するために使用される。<br>こんにちは下の女の子<br>碑はその後、無力なしわ眉、そんなに気にするようで、戻って群衆に、最後にテレビドラマシャオヤン遊び心アリスアリスの口での驚異の外観になっていません [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-10.html casio 腕時計 データバンク]。<br><br>'、結果はしなかったことは驚くべきことは何もあなたが年以内のレベルに戦いを入力する必要はありません場合は、私は非常に驚かれることでしょう、あなたの才能と、誇りに思っありませんしないでください [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-12.html 腕時計 メンズ casio]。」肩をすくめたが、シャオヤン冗談言った [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-1.html カシオ 腕時計 バンド]。<br><br>は彼に、薫の子供が突然ダウンして小さな顔を渡り、白のいくつかの憤りを聞いた [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-9.html カシオ 腕時計 チタン]。<br><br>が引か薫の子供たちは再び床に座っ、頬の下シャオヤンウランバートルなし |
| [[File:Legendrepolynomials6.svg|thumb|Graphs of Legendre polynomials (up to ''n'' = 5)]]
| | 相关的主题文章： |
| For the simplest integration problem stated above, i.e. with <math>\omega(x)=1</math>,
| | <ul> |
| the associated polynomials are [[Legendre polynomials]], ''P''<sub>''n''</sub>(''x''), and the method is usually known as Gauss–Legendre quadrature. With the ''n''<sup>th</sup> polynomial normalized to give ''P''<sub>''n''</sub>(1) = 1, the ''i''<sup>''th''</sup> Gauss node, ''x''<sub>''i''</sub>, is the ''i''<sup>''th''</sup> root of ''P''<sub>''n''</sub>; its weight is given by {{Harv|Abramowitz|Stegun|1972|loc=p. 887}}
| | |
| :<math> w_i = \frac{2}{\left( 1-x_i^2 \right) [P'_n(x_i)]^2}. \,\!</math>
| | <li>[http://www.jorun.cn/plus/feedback.php?aid=17 http://www.jorun.cn/plus/feedback.php?aid=17]</li> |
| Some low-order rules for solving the integration problem are listed below.
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| | <li>[http://tonzawa.godream.ne.jp/cgi-bin/aska/aska.cgi http://tonzawa.godream.ne.jp/cgi-bin/aska/aska.cgi]</li> |
| | |
| | <li>[http://symbianity.com/viewtopic.php?f=5&t=703288 http://symbianity.com/viewtopic.php?f=5&t=703288]</li> |
| | |
| | </ul> |
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| {| class="wikitable" style="margin:auto; background:white;"
| | == 「家長は、隠れ家ですよろしければ == |
| ! Number of points, ''n'' !! Points, ''x''<sub>''i'' !! Weights, ''w''<sub>''i''</sub>
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| |- align="center"
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| | 1 || 0 || 2
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| |- align="center"
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| | 2 || <math>\pm \sqrt{1/3}</math> || 1
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| |- align="center"
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| | rowspan="2" | 3 || 0 || <sup>8</sup>⁄<sub>9</sub>
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| |- align="center"
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| | <math>\pm\sqrt{3/5}</math> || <sup>5</sup>⁄<sub>9</sub>
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| |- align="center"
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| | rowspan="2" | 4 || <math>\pm\sqrt{\Big( 3 - 2\sqrt{6/5} \Big)/7}</math> || <math>\tfrac{18+\sqrt{30}}{36}</math>
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| |- align="center"
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| | <math>\pm\sqrt{\Big( 3 + 2\sqrt{6/5} \Big)/7}</math> || <math>\tfrac{18-\sqrt{30}}{36}</math>
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| |- align="center"
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| | rowspan="3" | 5 || 0 || <sup>128</sup>⁄<sub>225</sub>
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| |- align="center"
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| | <math>\pm\tfrac13\sqrt{5-2\sqrt{10/7}}</math> || <math>\tfrac{322+13\sqrt{70}}{900}</math>
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| |- align="center"
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| | <math>\pm\tfrac13\sqrt{5+2\sqrt{10/7}}</math> || <math>\tfrac{322-13\sqrt{70}}{900}</math>
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| |}
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| == Change of interval ==
| | 限りある限り、ヘビテランの後、再び実行が次第に脇の意味のようなものを置くだけでなく、ガマの帝国になります [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-4.html カシオ 腕時計 バンド]。<br>かなり遠く離れて帝国から、2速、できるシャオヤンは、時間かそこらについては、それはゆっくりと、この広大な山の中で掃引Warcraftの目の端に山の中に直接あるが、地域<br>ヘビファミリは、常駐これらの森林シャトル蛇を見ることができる [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-5.html gps 腕時計 カシオ]。<br><br>ちょうどその時シャオヤン紫研究デュオと空のこの領域に滞在、誰かがそれを発見したと思われる広大な森が、現時点では、いくつかの図を迅速に、空を掃引する警察の「さらさ」メッシュが、ときにちょっとシャオヤン喬は、警察はすぐに多くの光を率いたとき [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-14.html カシオ腕時計 メンズ]。<br><br>「いくつかのヘビ家族の友人、私は女王メデューサ、お願いビーコンを探すために何かを持っている。 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-10.html カシオ 腕時計 スタンダード] 'このため、いくつかのヘビは、手、シャオヤンチェンシェンロードの強力なアーチをテラン。<br><br>「家長は、隠れ家ですよろしければ |
| An integral over [''a'', ''b''] must be changed into an integral over [−1, 1] before applying the Gaussian quadrature rule. This change of interval can be done in the following way:
| | 相关的主题文章： |
| | | <ul> |
| :<math>
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| \int_a^b f(x)\,dx = \frac{b-a}{2} \int_{-1}^1 f\left(\frac{b-a}{2}z
| | <li>[http://www.couromoda.com/cgi-bin/portalcm/listaexpositor.cgi http://www.couromoda.com/cgi-bin/portalcm/listaexpositor.cgi]</li> |
| + \frac{a+b}{2}\right)\,dz.
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| </math>
| | <li>[http://www.hnsxkyy.com/plus/feedback.php?aid=46 http://www.hnsxkyy.com/plus/feedback.php?aid=46]</li> |
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| Applying the Gaussian quadrature rule then results in the following approximation:
| | <li>[http://image.letuijian.com/home.php?mod=space&uid=4138 http://image.letuijian.com/home.php?mod=space&uid=4138]</li> |
| | | |
| :<math>
| | </ul> |
| \int_a^b f(x)\,dx \approx \frac{b-a}{2} \sum_{i=1}^n w_i f\left(\frac{b-a}{2}z_i + \frac{a+b}{2}\right).
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| </math>
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| == Other forms ==
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| The integration problem can be expressed in a slightly more general way by introducing a positive [[weight function]] ω into the integrand,
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| and allowing an interval other than [−1, 1].
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| That is, the problem is to calculate
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| :<math> \int_a^b \omega(x)\,f(x)\,dx </math>
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| for some choices of ''a'', ''b'', and ω.
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| For ''a'' = −1, ''b'' = 1, and ω(''x'') = 1,
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| the problem is the same as that considered above.
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| Other choices lead to other integration rules.
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| Some of these are tabulated below.
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| Equation numbers are given for [[Abramowitz and Stegun]] (A & S).
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| {| class="wikitable" style="margin:auto; background:white;"
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| ! Interval !! ω(''x'') !! Orthogonal polynomials !! A & S !! For more information, see ...
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| |- align="center"
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| | [−1, 1] || <math>1\,</math> || [[Legendre polynomials]] || 25.4.29 || Section ''[[#Gauss–Legendre quadrature|Gauss–Legendre quadrature]]'', above
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| |- align="center"
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| | (−1, 1) || <math>(1-x)^\alpha (1+x)^\beta,\quad \alpha, \beta > -1\,</math> || [[Jacobi polynomials]] || 25.4.33 (<math>\beta=0</math>) || [[Gauss–Jacobi quadrature]]
| |
| |- align="center"
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| | (−1, 1) || <math>\frac{1}{\sqrt{1 - x^2}}</math> || [[Chebyshev polynomials]] (first kind) || 25.4.38 || [[Chebyshev–Gauss quadrature]]
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| |- align="center"
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| | [−1, 1] || <math>\sqrt{1 - x^2}</math> || Chebyshev polynomials (second kind) || 25.4.40 || [[Chebyshev–Gauss quadrature]]
| |
| |- align="center"
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| | [0, ∞) || <math> e^{-x}\, </math> || [[Laguerre polynomials]] || 25.4.45 || [[Gauss–Laguerre quadrature]]
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| |- align="center"
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| | [0, ∞) || <math> x^\alpha e^{-x}\, </math> || Generalized [[Laguerre polynomials]] || || [[Gauss–Laguerre quadrature]]
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| |- align="center"
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| | (−∞, ∞) || <math> e^{-x^2} </math> || [[Hermite polynomials]] || 25.4.46 || [[Gauss–Hermite quadrature]]
| |
| |}
| |
| | |
| === Fundamental theorem ===
| |
| Let <math>p_n</math> be a nontrivial polynomial of degree ''n'' such that
| |
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| :<math> | |
| \int_a^b \omega(x) \, x^k p_n(x) \, dx = 0, \quad \text{for all }k=0,1,\ldots,n-1.
| |
| </math>
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| If we pick the ''n'' nodes ''x''<sub>''i''</sub> to be the zeros of ''p''<sub>''n''</sub>, then there exist ''n'' weights ''w''<sub>''i''</sub> which make the Gauss-quadrature computed integral exact for all polynomials <math>h(x)</math> of degree 2''n'' − 1 or less. Furthermore, all these nodes ''x''<sub>''i''</sub> will lie in the open interval (''a'', ''b'') {{harv|Stoer|Bulirsch|2002|pp=172–175}}.
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| The polynomial <math>p_n</math> is said to be an orthogonal polynomial of degree ''n'' associated to the weight function <math>\omega (x)</math>. It is unique up to a constant normalization factor. The idea underlying the proof is that, because of its sufficiently low degree, <math>h(x)</math> can be divided by <math>p_n(x)</math> to produce a quotient <math>q(x)</math> of degree strictly lower than ''n'', and a remainder <math>r(x)</math> of still lower degree, so that both will be orthogonal to <math>p_n(x)</math>, by the defining property of <math>p_n(x)</math>. Thus
| |
| :<math> \int_a^b \omega(x)\,h(x)\,dx = \int_a^b \omega(x)\,r(x)\,dx. </math>
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| Because of the choice of nodes ''x''<sub>''i''</sub>, the corresponding relation
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| :<math>\sum_{i=1}^n w_i h(x_i) = \sum_{i=1}^n w_i r(x_i)</math>
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| holds also. The exactness of the computed integral for <math>h(x)</math> then follows from corresponding exactness for polynomials of degree only ''n'' or less (as is <math>r(x)</math>).
| |
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| ====General formula for the weights ====
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| The weights can be expressed as
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| :<math>w_{i} = \frac{a_{n}}{a_{n-1}}\frac{\int_{a}^{b}\omega(x)p_{n-1}(x)^{2}dx}{p'_{n}(x_{i})p_{n-1}(x_{i})}</math> (1)
| |
| where <math>a_{k}</math> is the coefficient of <math>x^{k}</math> in <math>p_{k}(x)</math>. To prove this, note that using [[Lagrange interpolation]] one can express <math>r(x)</math> in terms of <math>r(x_{i})</math> as
| |
| :<math>r(x) = \sum_{i=1}^{n}r(x_{i})\prod_{\begin{smallmatrix}1\leq j\leq n\\j\neq i\end{smallmatrix}}\frac{x-x_{j}}{x_{i}-x_{j}}</math>
| |
| because r(x) has degree less than n and is thus fixed by the values it attains at n different points. Multiplying both sides by <math>\omega(x)</math> and integrating from a to b yields
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| :<math>\int_{a}^{b}\omega(x)r(x)dx= \sum_{i=1}^{n}r(x_{i})\int_{a}^{b}\omega(x)\prod_{\begin{smallmatrix}1\leq j\leq n\\j\neq i\end{smallmatrix}}\frac{x-x_{j}}{x_{i}-x_{j}}dx</math> | |
| The weights <math>w_{i}</math> are thus given by
| |
| :<math>w_{i} = \int_{a}^{b}\omega(x)\prod_{\begin{smallmatrix}1\leq j\leq n\\j\neq i\end{smallmatrix}}\frac{x-x_{j}}{x_{i}-x_{j}}dx</math>
| |
| This integral expression for <math>w_{i}</math> can be expressed in terms of the orthogonal polynomials <math>p_{n}(x)</math> and <math>p_{n+1}(x)</math> as follows.
| |
| | |
| We can write
| |
| :<math>\prod_{\begin{smallmatrix}1\leq j\leq n\\j\neq i\end{smallmatrix}}\left(x-x_{j}\right) = \frac{\prod_{1\leq j\leq n} \left(x - x_{j}\right)}{x-x_{i}} = \frac{p_{n}(x)}{a_{n}\left(x-x_{i}\right)}</math>
| |
| where <math>a_{n}</math> is the coefficient of <math>x^{n}</math> in <math>p_{n}(x)</math>. Taking the limit of x to <math>x_{i}</math> yields using L'Hôpital's rule
| |
| :<math>\prod_{\begin{smallmatrix}1\leq j\leq n\\j\neq i\end{smallmatrix}}\left(x_{i}-x_{j}\right) = \frac{p'_{n}(x_{i})}{a_{n}}</math>
| |
| We can thus write the integral expression for the weights as
| |
| :<math>w_{i} = \frac{1}{p'_{n}(x_{i})}\int_{a}^{b}\omega(x)\frac{p_{n}(x)}{x-x_{i}}dx</math> (2)
| |
| | |
| In the integrand, writing
| |
| :<math>\frac{1}{x-x_{i}} = \frac{1-\left(\frac{x}{x_{i}}\right)^{k}}{x-x_{i}} + \left(\frac{x}{x_{i}}\right)^{k} \frac{1}{x-x_{i}}</math>
| |
| yields
| |
| :<math>\int_{a}^{b}\omega(x)\frac{x^{k}p_{n}(x)}{x-x_{i}}dx= x_{i}^{k}\int_{a}^{b}\omega(x)\frac{p_{n}(x)}{x-x_{i}}dx</math>
| |
| provided <math>k\leq n</math>, because <math>\frac{1-\left(\frac{x}{x_{i}}\right)^{k}}{x-x_{i}} </math> is a polynomial of degree k-1 which is then orthogonal to <math>p_{n}(x)</math>. So, if q(x) is a polynomial of at most nth degree we have
| |
| :<math>\int_{a}^{b}\omega(x)\frac{p_{n}(x)}{x-x_{i}}dx=\frac{1}{q(x_{i})}\int_{a}^{b}\omega(x)\frac{q(x)p_{n}(x)}{x-x_{i}}dx </math>
| |
| We can evaluate the integral on the right hand side for <math>q(x) = p_{n-1}(x)</math> as follows. Because <math>\frac{p_{n}(x)}{x-x_{i}}</math> is a polynomial of degree n-1, we have
| |
| :<math>\frac{p_{n}(x)}{x-x_{i}} = a_{n}x^{n-1} + s(x)</math>
| |
| where s(x) is a polynomial of degree <math>n-2</math>. Since <math>s(x)</math> is orthogonal to <math>p_{n-1}(x)</math> we have
| |
| :<math>\int_{a}^{b}\omega(x)\frac{p_{n}(x)}{x-x_{i}}dx=\frac{a_{n}}{p_{n-1}(x_{i})}\int_{a}^{b}\omega(x)p_{n-1}(x)x^{n-1}dx </math> | |
| We can then write
| |
| :<math>x^{n-1} = \left(x^{n-1} - \frac{p_{n-1}(x)}{a_{n-1}}\right) + \frac{p_{n-1}(x)}{a_{n-1}}</math>
| |
| The term in the brackets is a polynomial of degree <math>n-2</math>, which is therefore orthogonal to <math>p_{n-1}(x)</math>. The integral can thus be written as
| |
| :<math>\int_{a}^{b}\omega(x)\frac{p_{n}(x)}{x-x_{i}}dx=\frac{a_{n}}{a_{n-1}p_{n-1}(x_{i})}\int_{a}^{b}\omega(x)p_{n-1}(x)^{2}dx </math>
| |
| According to Eq. (2), the weights are obtained by dividing this by <math>p'_{n}(x_{i})</math> and that yields the expression in Eq. (1).
| |
| | |
| ====Proof that the weights are positive====
| |
| Consider the following polynomial of degree 2n-2
| |
| :<math>f(x) = \prod_{\begin{smallmatrix}1\leq j\leq n\\j\neq i\end{smallmatrix}}(x-x_{j})^{2}</math>
| |
| where as above the <math>x_{j}</math> are the roots of the polynomial <math>p_{n}(x)</math>. Since the degree of f(x) is less than 2n-1, the Gaussian quadrature formula involving the weights and nodes obtained from <math>p_{n}(x)</math> applies. Since <math>f(x_{j})=0</math> for j not equal to i, we have
| |
| :<math>\int_{a}^{b}\omega(x)f(x)=\sum_{j=1}^{N}w_{j}f(x_{j}) = w_{i} f(x_{i}).</math>
| |
| Since both <math>\omega(x)</math> and f(x) are non-negative functions, it follows that <math>w_{i}>0</math>.
| |
| | |
| === Computation of Gaussian quadrature rules ===
| |
| For computing the nodes <math>x_i</math> and weights <math>w_i</math> of Gaussian quadrature rules, the fundamental tool is the three-term recurrence relation satisfied by the set of orthogonal polynomials associated to the corresponding weight function. For ''n'' points, these nodes and weights can be computed in ''O''(''n''<sup>2</sup>) operations by an algorithm derived by Gautschi (1968).
| |
| | |
| ==== Gautschi's theorem ====
| |
| Gautschi's theorem (Gautschi, 1968) states that orthogonal polynomials <math>p_r</math> with <math>(p_r,p_s)=0</math> for <math>r\ne s</math> for a scalar product <math>( , )</math> to be specified later, degree<math>(p_r)=r</math> and leading coefficient one (i.e. [[monic polynomial|monic]] orthogonal polynomials) satisfy the recurrence relation
| |
| :<math>p_{r+1}(x)=(x-a_{r,r})p_r(x)-a_{r,r-1}p_{r-1}(x)\ldots-a_{r,0}p_0(x)</math> | |
| for <math>r=0,1,\ldots,n-1</math> where <math>n</math> is the maximal degree which can be taken to
| |
| be infinity, and where <math>a_{r,s}=(xp_r,p_s)/(p_s,p_s)</math>. First of all, it is obvious that the polynomials defined by the recurrence relation starting with <math>p_0(x)=1</math> have leading coefficient one and correct degree. Given the starting point by <math>p_0</math>, the orthogonality of <math>p_r</math> can be shown by induction. For <math>r=s=0</math> one has
| |
| :<math>(p_1,p_0)=((x-a_{0,0}p_0,p_0)=(xp_0,p_0)-a_{0,0}(p_0,p_0)=(xp_0,p_0)-(xp_0,p_0)=0.</math>
| |
| Now if <math>p_0,p_1,\ldots,p_r</math> are orthogonal, then also <math>p_{r+1}</math>, because in
| |
| :<math>(p_{r+1},p_s)=(xp_r,p_s)-a_{r,r}(p_r,p_s)-a_{r,r-1}(p_{r-1},p_s)\ldots-a_{r,0}(p_0,p_s)</math>
| |
| all scalar products vanish except for the first one and the one where <math>p_s</math> meets the same
| |
| orthogonal polynom. Therefore,
| |
| :<math>(p_{r+1},p_s)=(xp_r,p_s)-a_{r,s}(p_s,p_s)=(xp_r,p_s)-(xp_r,p_s)=0.</math> | |
| However, if the scalar product satisfies <math>(xf,g)=(f,xg)</math> (which is the case for Gaussian quadrature), the recurrence relation reduces to a three-term recurrence relation: For <math>s\le r-1</math>, <math>xp_s</math> is a polynomial of degree less or equal to <math>r-1</math>. On the other hand, <math>p_r</math> is orthogonal to every polynomial of degree less or equal to <math>r-1</math>. Therefore, one has <math>(xp_r,p_s)=(p_r,xp_s)=0</math> and <math>a_{r,s}=0</math> for <math>s<r-1</math>. The recurrence relation then simplifies to <math>p_{r+1}(x)=(x-a_{r,r})p_r(x)-a_{r,r-1}p_{r-1}(x)</math> or
| |
| :<math>p_{r+1}(x)=(x-a_r)p_r(x)-b_rp_{r-1}(x)</math>
| |
| (with the convention <math>p_{-1}(x)\equiv 0</math>) where
| |
| :<math>a_r:=\frac{(xp_r,p_r)}{(p_r,p_r)},\qquad
| |
| b_r:=\frac{(xp_r,p_{r-1})}{(p_{r-1},p_{r-1})}=\frac{(p_r,p_r)}{(p_{r-1},p_{r-1})}</math>
| |
| (the last because of <math>(xp_r,p_{r-1})=(p_r,xp_{r-1})=(p_r,p_r)</math>, since <math>xp_{r-1}</math> differs from <math>p_r</math> by a degree less than <math>r</math>).
| |
| | |
| ==== The Golub-Welsch algorithm ====
| |
| The three-term recurrence relation can be written in the matrix form <math>J\tilde{P}=x\tilde{P}</math>
| |
| where <math>\tilde{P}=[p_0(x),p_1(x),...,p_{n-1}(x)]^{T}</math> and <math>J</math> is the so-called Jacobi matrix:
| |
| :<math>
| |
| \mathbf{J}=\left(
| |
| \begin{array}{llllll}
| |
| a_0 & 1 & 0 & \ldots & \ldots & \ldots\\
| |
| b_1 & a_1 & 1 & 0 & \ldots & \ldots \\
| |
| 0 & b_2 & a_2 & 1 & 0 & \ldots \\
| |
| 0 & \ldots & \ldots & \ldots & \ldots & 0 \\
| |
| \ldots & \ldots & 0 & b_{n-2} & a_{n-2} & 1 \\
| |
| \ldots & \ldots & \ldots & 0 & b_{n-1} & a_{n-1}
| |
| \end{array}
| |
| \right).
| |
| </math>
| |
| The zeros <math>x_j</math> of the polynomials up to degree <math>n</math> which are used as nodes for the Gaussian quadrature can be found by computing the eigenvalues of this [[tridiagonal matrix]]. This procedure is known as ''Golub–Welsch algorithm''.
| |
| | |
| For computing the weights and nodes, it is preferable to consider the symmetric tridiagonal matrix <math>\mathcal{J}</math> with elements <math>\mathcal{J}_{i,i}=J_{i,i}=a_{i-1},\, i=1,\ldots,n</math> and <math>\mathcal{J}_{i-1,i}=\mathcal{J}_{i,i-1}=\sqrt{J_{i,i-1}J_{i-1,i}}=\sqrt{b_{i-1}},\, i=2,\ldots,n.</math>
| |
| <math>\mathbf{J}</math> and <math>\mathcal{J}</math> are [[similar matrices]] and therefore have the same eigenvalues (the nodes). The weights can be computed from the corresponding eigenvectors: If <math>\phi^{(j)}</math> is a normalized eigenvector (i.e., an eigenvector with euclidean norm equal to one) associated to the eigenvalue <math>x_j</math>, the corresponding weight can be computed from
| |
| the first component of this eigenvector, namely:
| |
| | |
| :<math>
| |
| w_j=\mu_0 \left(\phi_1^{(j)}\right)^2
| |
| </math>
| |
| | |
| where <math>\mu_0</math> is the integral of the weight function
| |
| | |
| :<math>
| |
| \mu_0=\int_a^b \omega(x) dx.
| |
| </math>
| |
| | |
| See, for instance, {{harv|Gil|Segura|Temme|2007}} for further details.
| |
| | |
| There are alternative methods for obtaining the same weights and nodes in ''O''(''n'') operations using the Prüfer Transform.
| |
| | |
| === Error estimates ===
| |
| | |
| The error of a Gaussian quadrature rule can be stated as follows {{harv|Stoer|Bulirsch|2002|loc=Thm 3.6.24}}.
| |
| For an integrand which has 2''n'' continuous derivatives,
| |
| | |
| :<math> \int_a^b \omega(x)\,f(x)\,dx - \sum_{i=1}^n w_i\,f(x_i)
| |
| = \frac{f^{(2n)}(\xi)}{(2n)!} \, (p_n,p_n) </math>
| |
| | |
| for some ξ in (''a'', ''b''), where ''p''<sub>''n''</sub> is the monic (i.e. the leading coefficient is 1) orthogonal polynomial of degree ''n'' and where
| |
| | |
| :<math> (f,g) = \int_a^b \omega(x) f(x) g(x) \, dx . \,\!</math>
| |
| | |
| In the important special case of ω(''x'') = 1, we have the error estimate {{Harv|Kahaner|Moler|Nash|1989|loc=§5.2}}
| |
| | |
| :<math> \frac{(b-a)^{2n+1} (n!)^4}{(2n+1)[(2n)!]^3} f^{(2n)} (\xi) , \qquad a < \xi < b . \,\!</math> | |
| | |
| Stoer and Bulirsch remark that this error estimate is inconvenient in practice,
| |
| since it may be difficult to estimate the order 2''n'' derivative,
| |
| and furthermore the actual error may be much less than a bound established by the derivative.
| |
| Another approach is to use two Gaussian quadrature rules of different orders,
| |
| and to estimate the error as the difference between the two results.
| |
| For this purpose, Gauss–Kronrod quadrature rules can be useful.
| |
| | |
| Important consequence of the above equation is that Gaussian quadrature of order ''n'' is accurate for all polynomials up to degree 2''n''–1.
| |
| | |
| === Gauss–Kronrod rules ===
| |
| {{main|Gauss–Kronrod quadrature formula}}
| |
| | |
| If the interval [''a'', ''b''] is subdivided,
| |
| the Gauss evaluation points of the new subintervals never coincide with the previous evaluation points (except at zero for odd numbers),
| |
| and thus the integrand must be evaluated at every point.
| |
| ''Gauss–Kronrod rules'' are extensions of Gauss quadrature rules generated by adding <math>n+1</math> points to an <math>n</math>-point rule in such a way that the resulting rule is of order <math>3n+1</math>.
| |
| This allows for computing higher-order estimates while re-using the function values of a lower-order estimate.
| |
| The difference between a Gauss quadrature rule and its Kronrod extension are often used as an estimate of the approximation error.
| |
| | |
| === Gauss–Lobatto rules ===
| |
| Also known as '''Lobatto quadrature''' {{Harv|Abramowitz|Stegun|1972|loc=p. 888}}, named after Dutch mathematician [[Rehuel Lobatto]].
| |
| | |
| It is similar to Gaussian quadrature with the following differences:
| |
| # The integration points include the end points of the integration interval.
| |
| # It is accurate for polynomials up to degree 2''n''–3, where ''n'' is the number of integration points.
| |
| | |
| Lobatto quadrature of function ''f''(''x'') on interval [–1, +1]:
| |
| :<math>
| |
| \int_{-1}^1 {f(x) \, dx} =
| |
| \frac {2} {n(n-1)}[f(1) + f(-1)] +
| |
| \sum_{i = 2} ^{n-1} {w_i f(x_i)} + R_n.
| |
| </math>
| |
| | |
| Abscissas: <math>x_i</math> is the <math>(i-1)</math><sup>st</sup> zero of <math>P'_{n-1}(x)</math>.
| |
| | |
| Weights:
| |
| :<math>
| |
| w_i = \frac{2}{n(n-1)[P_{n-1}(x_i)]^2} \quad (x_i \ne \pm 1).
| |
| </math> | |
| | |
| Remainder:
| |
| <math> | |
| R_n = \frac
| |
| {- n (n-1)^3 2^{2n-1} [(n-2)!]^4}
| |
| {(2n-1) [(2n-2)!]^3}
| |
| f^{(2n-2)}(\xi), \quad (-1 < \xi < 1)
| |
| </math>
| |
| | |
| Some of the weights are:
| |
| | |
| {| class="wikitable" style="margin:auto; background:white;"
| |
| ! Number of points, ''n'' !! Points, ''x''<sub>''i'' !! Weights, ''w''<sub>''i''</sub>
| |
| |- align="center"
| |
| | rowspan="2" | <math>3</math> || <math>0</math> || <math>\frac{4}{3}</math>
| |
| |- align="center"
| |
| | <math>\pm 1</math> || <math>\frac{1}{3}</math>
| |
| |- align="center"
| |
| | rowspan="2" | <math>4</math> || <math>\pm \sqrt{\frac {1} {5}}</math> || <math>\frac{5}{6}</math>
| |
| |- align="center"
| |
| | <math>\pm 1</math> || <math>\frac{1}{6}</math>
| |
| |- align="center"
| |
| | rowspan="3" | <math>5</math> || <math>0</math> || <math>\frac{32}{45}</math>
| |
| |- align="center"
| |
| | <math>\pm\sqrt{\frac {3} {7}}</math> || <math>\frac{49}{90}</math>
| |
| |- align="center"
| |
| | <math>\pm 1</math> || <math>\frac{1}{10}</math>
| |
| |}
| |
| | |
| ==See also==
| |
| *[[Euler–Maclaurin formula]]
| |
| *[[Clenshaw–Curtis quadrature]]
| |
| | |
| == References ==
| |
| * {{AS ref| 25.4, Integration }}
| |
| *{{cite journal|first1=Donald G. |last1=Anderson
| |
| |title=Gaussian quadrature formulae for int_0^1 -ln(x)f(x) dx
| |
| |year=1965 | volume=19 | number=91 | pages=477–481 | journal=Math. Comp.
| |
| }}
| |
| * {{citation| title = Calculation of Gauss Quadrature Rules | journal = Mathematics of Computation |first=Gene H. |last=Golub |authorlink=Gene Golub| first2= John H. |last2= Welsch|volume= 23| issue= 106 |year= 1969|pages=221–230| jstor = 2004418| doi = 10.1090/S0025-5718-69-99647-1
| |
| }}
| |
| *{{cite article|first1=Walter | last1=Gautschi
| |
| |title=Construction of Gauss–Christoffel Quadrature Formulas
| |
| |journal= Math. Comp. | year=1968
| |
| | volume=22 | issue= 102
| |
| | pages=251–270
| |
| |doi=10.1090/S0025-5718-1968-0228171-0 | mr=0228171
| |
| }}
| |
| *{{cite article|first1=Walter | last1=Gautschi
| |
| |title=On the construction of Gaussian quadrature rules from modified moments
| |
| |journal= Math. Comp. | year=1970 | volume=24 | pages=245–260
| |
| |doi=10.1090/S0025-5718-1970-0285117-6 | mr=0285177
| |
| }}
| |
| * {{ cite article | first1=R. | last1=Piessens | title=Gaussian quadrature formulas for the numerical integration of Bromwich's integral and the inversion of the laplace transform
| |
| |year=1971 | volume=5 | number=1 | journal= J. Eng. Math. | pages=1–9
| |
| |doi=10.1007/BF01535429}}
| |
| * {{cite article|first1=Bernard | last1=Danloy | title=Numerical construction of Gaussian quadrature formulas for int_0^1 (-log x) x^alpha f(x) dx and int_0^infty E_m(x) f(x) dx | journal = Math. Comp. | year=1973
| |
| |volume=27 | number=124 | pages=861–869 | doi= 10.1090/S0025-5718-1973-0331730-X |mr=0331730}}
| |
| * {{citation | last1=Kahaner | first1=David | last2=Moler | first2=Cleve | author2-link=Cleve Moler | last3=Nash | first3=Stephen | title=Numerical Methods and Software | year=1989 | publisher=[[Prentice-Hall]] | isbn=978-0-13-627258-8 }}
| |
| *{{cite journal | first1=Robin P. | last1=Sagar
| |
| |title=A Gaussian quadrature for the calculation of generalized Fermi-Dirac integrals
| |
| |year=1991 | journal=Comp. Phys. Commun. | volume=66 | pages=271–275 |number=2-3
| |
| |doi=10.1016/0010-4655(91)90076-W }}
| |
| *{{cite journal | first1=E. | last1=Yakimiw| title=Accurate computation of weights in classical Gauss-Christoffel quadrature rules
| |
| |year=1996 | journal=J. Comp. Phys. | volume=129 | pages=406–430| bibcode=1996JCoPh.129..406Y |doi=10.1006/jcph.1996.0258}}
| |
| * {{citation | first1=Dirk P. |last1=Laurie
| |
| |title=Accurate recovery of recursion coefficients from Gaussian quadrature formulas
| |
| |year=1999 | volume=112 | number=1-2 | pages=165–180
| |
| |journal=J. Comp. Appl. Math. | doi=10.1016/S0377-0427(99)00228-9 }}
| |
| * {{ cite journal| first1=Dirk P. | last1=Laurie
| |
| |title=Computation of Gauss-type quadrature formulas
| |
| |year=2001 | pages=201–217 | volume=127 | number=1-2 | journal=J. Comp. Appl. Math. | doi=10.1016/S0377-0427(00)00506-9 }}
| |
| * {{citation | last1=Stoer | first1=Josef | last2=Bulirsch | first2=Roland | year=2002 | title=Introduction to Numerical Analysis | edition=3rd | publisher=[[Springer-Verlag|Springer]] | isbn=978-0-387-95452-3<!-- 0-387-95452-X --> }}.
| |
| * {{dlmf | title=§3.5(v): Gauss Quadrature | id=3.5.v | last=Temme | first=Nico M.}}
| |
| * {{Citation |last1=Press|first1=WH|last2=Teukolsky|first2=SA|last3=Vetterling|first3=WT|last4=Flannery|first4=BP|year=2007|title=Numerical Recipes: The Art of Scientific Computing|edition=3rd|publisher=Cambridge University Press| publication-place=New York|isbn=978-0-521-88068-8|chapter=Section 4.6. Gaussian Quadratures and Orthogonal Polynomials|chapter-url=http://apps.nrbook.com/empanel/index.html?pg=179}}
| |
| * {{citation | last1=Gil | first1=Amparo | last2=Segura | first2=Javier |last3=Temme | first3=Nico M. | chapter=§5.3: Gauss quadrature| title=Numerical Methods for Special Functions | year=2007 | publisher=SIAM | isbn=978-0-89871-634-4 }}
| |
| * {{cite web | first1=R. J. |last1= Mathar | title=Gaussian quadrature of the integrals int_{-infty}^infty f(x) dx / cosh(x)
| |
| |url=http://vixra.org/abs/1303.0038 |year =2013}}
| |
| | |
| == External links ==
| |
| * {{springer|title=Gauss quadrature formula|id=p/g043510}}
| |
| * [http://www.alglib.net/integral/gq/ ALGLIB] contains a collection of algorithms for numerical integration (in C# / C++ / Delphi / Visual Basic / etc.)
| |
| * [http://www.gnu.org/software/gsl/ GNU Scientific Library] — includes [[C (programming language)|C]] version of [[QUADPACK]] algorithms (see also [[GNU Scientific Library]])
| |
| *[http://ans.hsh.no/home/skk/Publications/Lobatto/PRIMUS_KHATTRI.pdf From Lobatto Quadrature to the Euler constant e]
| |
| * [http://numericalmethods.eng.usf.edu/topics/gauss_quadrature.html Gaussian Quadrature Rule of Integration – Notes, PPT, Matlab, Mathematica, Maple, Mathcad] at ''Holistic Numerical Methods Institute''
| |
| * {{MathWorld|id=Legendre-GaussQuadrature|title=Legendre-Gauss Quadrature}}
| |
| * [http://demonstrations.wolfram.com/GaussianQuadrature/ Gaussian Quadrature] by Chris Maes and Anton Antonov, [[Wolfram Demonstrations Project]].
| |
| * [http://pomax.github.io/bezierinfo/legendre-gauss.html Tabulated weights and abscissae with Mathematica source code], high precision (16 and 256 decimal places) Legendre-Gaussian quadrature weights and abscissas, for ''n''=2 through ''n''=64, with Mathematica source code.
| |
| * [http://people.sc.fsu.edu/~jburkardt/math_src/arbitrary_weight_rule/arbitrary_weight_rule.html Mathematica source code distributed under the GNU LGPL] for abscissas and weights generation for arbitrary weighting functions W(x), integration domains and precisions.
| |
| | |
| [[Category:Numerical integration (quadrature)]]
| |