標簽（空格分隔）： matlab 積分 數值 計算

文章目錄

• • 標簽（空格分隔）： matlab 積分 數值 計算
  • @[toc]
• matlab數值積分
• • 1 Gauss-Hermite積分
  • • 1.1 測試Gauss-Hermite積分函數gaussHermiteIntegral()
    • 1.2 gaussHermiteIntegral()
  • 2 Gauss-Laguerre積分
  • • 2.1 測試Gauss-Laguerre積分函數gaussLaguerreIntegral()
    • 2.2 gaussLaguerreIntegral()
  • 3 Gauss-Legendre積分W(x)=1
  • • 3.1 測試Gauss-Legendre積分W(x)=1函數gaussLegendreIntegral()
    • 3.2 gaussLegendreIntegral()
  • 4 Gauss積分的積分系數計算
  • • 4.1 測試Gauss積分的積分系數計算的函數
    • 4.2 Gauss-Hermite積分系數求解函數
    • 4.3 Gauss-Laguerre積分系數求解函數
    • 4.4 Gauss-Legendre積分系數求解函數
  • 5 NewtonCotes積分
  • • 5.1 測試NewtonCotes積分函數newtonCotesIntegral()
    • 5.2 newtonCotesIntegral()
  • 6 Romberg積分
  • • 6.1 測試Romberg積分函數rombergIntegral()
    • 6.2 rombergIntegral()
  • 7 復化積分
  • • 7.1 測試復化積分函數complexSimpson()
    • 7.2 complexSimpson()
    • 7.3 complexTrangleInt()
  • 8 自適應運算積分
  • • 8.1 測試自適應運算積分函數selfAdaptIntegral()
    • 8.2 selfAdaptIntegral()
  • 9 積分系數——二
  • • 9.1 Gauss-Legendre積分系數
    • 9.2 newton cotes積分系數
  • 聯系作者 definedone@163.com

matlab數值積分

1 Gauss-Hermite積分

1.1 測試Gauss-Hermite積分函數gaussHermiteIntegral()

clc,clear format long f=@(x)(exp(-x.^4)) % % % % f=@(x)((1/sqrt(2*pi))*exp(-x.^2/2)) %標準正太分布 % % area=[];nn=[]; % % for n=1:10 % % nn=[nn,n]; % % area=[area,gaussHermiteIntegral(f,-inf,inf,n)]; % % end,nn,area % % darea=abs(diff(area)); % % %% 系統計算的積分精確值 % % f_=@(x)(f(-x)) % % area_system=quadgk(f,0,inf,'RelTol',1e-8,'AbsTol',1e-12)+... % % quadgk(f_,0,inf,'RelTol',1e-8,'AbsTol',1e-12) % % d_my_system=abs(area-area_system); % % log_darea=round(log10(darea)*10) % % log_d_my_system=round(log10(d_my_system)*10) % 1.812804954110955 n=5 [area,gErrer]=gaussHermiteIntegral(f,-inf,inf,n,true) %% 從以下數據看（前向差商，系統誤差）：前向差商公式完全能夠作為事后誤差估計 % log_darea = -19 -28 -36 -45 -52 -60 -69 -90 -81 % log_d_my_system = -19 -29 -37 -45 -53 -61 -69 -82 -83 -86 %% 從以下數據看：Gauss Hermite積分的精確度e隨積分節點個數n呈現關系：log10(e)=n/10%因此采用10n個節點的gauss積分作為程序中的誤差階數 % f = @(x)(exp(-x.^4)) % n =100 % area = 1.812804956399165 % area_system = 1.812804954110955 % d_my_system = 2.288210509959754e-09 % n=41 % d_my_system = 2.403635556458283e-05 % n=20 % d_my_system = 0.001299981246872 % n=10 % d_my_system = 0.014102477783775

運行結果：

f = @(x)(exp(-x.^4)) n =5 area =1.812809811622279 gErrer =3.459687060525241e-05

1.2 gaussHermiteIntegral()

function [area,gErrer]=gaussHermiteIntegral(func,a,b,n,withError) %高斯積分 %理論依據：數值分析方法 奚梅成 %func 被積函數； %[a,b] 積分區間 %n 高斯積分的階數 %% %name：鄧能財 Date: 2013/12/23 %% 默認參數 if nargin<5 withError=false; gErrer=0; end % 查錯 assert(a==-inf && b==inf,'積分區間必須為(-inf,inf)!'); assert(ismember(n,[1:10]),'積分精確度數量級必須在1~10之間!'); %% 函數變換 expfunc=@(x)(exp(x^2)*func(x)); %% 數據：10,20,30,...,maxn*10階高斯積分的積分系數 maxn=10; % (±積分節點node，積分系數coef)，‘±’被省略 data= struct('node',cell(1,maxn),'coef',cell(1,maxn)); %data()= struct('node',{[]},'coef',{[]}); data(1)= struct('node',{[3.4361591188377e+00 2.5327316742328e+00 1.7566836492999e+00 1.0366108297895e+00 3.4290132722370e-01 ]},'coef',{[7.6404328552326e-06 1.3436457467812e-03 3.3874394455481e-02 2.4013861108231e-01 6.1086263373533e-01 ]}); data(2)= struct('node',{[5.3874808900112e+00 4.6036824495507e+00 3.9447640401156e+00 3.3478545673832e+00 2.7888060584281e+00 2.2549740020893e+00 1.7385377121166e+00 1.2340762153953e+00 7.3747372854539e-01 2.4534070830090e-01 ]},'coef',{[2.2293936455341e-13 4.3993409922731e-10 1.0860693707693e-07 7.8025564785321e-06 2.2833863601635e-04 3.2437733422379e-03 2.4810520887464e-02 1.0901720602002e-01 2.8667550536283e-01 4.6224366960061e-01 ]}); data(3)= struct('node',{[6.8633452935299e+00 6.1382792201239e+00 5.5331471515675e+00 4.9889189685899e+00 4.4830553570925e+00 4.0039086038612e+00 3.5444438731553e+00 3.0999705295864e+00 2.6671321245356e+00 2.2433914677615e+00 1.8267411436037e+00 1.4155278001982e+00 1.0083382710467e+00 6.0392105862555e-01 2.0112857654887e-01 ]},'coef',{[2.9082547163104e-21 2.8103336037565e-17 2.8786070806795e-14 8.1061862974598e-12 9.1785804243669e-10 5.1085224507758e-08 1.5790948873248e-06 2.9387252289230e-05 3.4831012431868e-04 2.7379224730677e-03 1.4703829704827e-02 5.5144176870234e-02 1.4673584754089e-01 2.8013093083921e-01 3.8639488954181e-01 ]}); data(4)= struct('node',{[8.0987611392509e+00 7.4115825314855e+00 6.8402373052494e+00 6.3282553512201e+00 5.8540950560304e+00 5.4066542479701e+00 4.9792609785453e+00 4.5675020728444e+00 4.1682570668325e+00 3.7792067534352e+00 3.3985582658596e+00 3.0248798839013e+00 2.6569959984429e+00 2.2939171418751e+00 1.9347914722823e+00 1.5788698949316e+00 1.2254801090463e+00 8.7400661235709e-01 5.2387471383228e-01 1.7453721459758e-01 ]},'coef',{[2.5910578133142e-29 8.5440614848689e-25 2.5675935123395e-21 1.9891810183765e-18 6.0083587891597e-16 8.8057076445574e-14 7.1565280526724e-12 3.5256207913729e-10 1.1212360832276e-08 2.4111441636703e-07 3.6315761506931e-06 3.9369339810925e-05 3.1385359454133e-04 1.8714968295980e-03 8.4608880082581e-03 2.9312565536172e-02 7.8474605865404e-02 1.6337873271327e-01 2.6572825187738e-01 3.3864327742559e-01 ]}); data(5)= struct('node',{[9.1824069581293e+00 8.5227710309178e+00 7.9756223682056e+00 7.4864094298642e+00 7.0343235097706e+00 6.6086479738554e+00 6.2029525192747e+00 5.8129946754204e+00 5.4357860872249e+00 5.0691175849172e+00 4.7112936661690e+00 4.3609731604546e+00 4.0170681728581e+00 3.6786770625153e+00 3.3450383139379e+00 3.0154977695745e+00 2.6894847022677e+00 2.3664939042987e+00 2.0460719686864e+00 1.7278065475159e+00 1.4113177548983e+00 1.0962511289577e+00 7.8227172955461e-01 4.6905905667824e-01 1.5630254688947e-01 ]},'coef',{[1.8283576319454e-37 1.6732639420678e-32 1.2152047767042e-28 2.1376568843541e-25 1.4170933459735e-22 4.4709843726175e-20 7.7423829723092e-18 8.0942618938896e-16 5.4659440315878e-14 2.5066555238998e-12 8.1118773649361e-11 1.9090405438119e-09 3.3467934040213e-08 4.4570299668179e-07 4.5816827079555e-06 3.6840190537807e-05 2.3426989210925e-04 1.1890117817496e-03 4.8532638261719e-03 1.6031941068412e-02 4.3079159156766e-02 9.4548935477086e-02 1.7003245567716e-01 2.5113085633200e-01 3.0508512920440e-01 ]}); data(6)= struct('node',{[1.0159109246180e+01 9.5209036770133e+00 8.9923980014049e+00 8.5205692841176e+00 8.0851886542490e+00 7.6758399375049e+00 7.2862765943956e+00 6.9123815321893e+00 6.5512591670629e+00 6.2007735579934e+00 5.8592901963942e+00 5.5255210861387e+00 5.1984265345763e+00 4.8771500774732e+00 4.5609737579358e+00 4.2492864359560e+00 3.9415607339262e+00 3.6373358761707e+00 3.3362046535476e+00 3.0378033382307e+00 2.7418037480697e+00 2.4479069023077e+00 2.1558378712292e+00 1.8653415312330e+00 1.5761790119750e+00 1.2881246748689e+00 1.0009634995607e+00 7.1448878167258e-01 4.2850006422063e-01 1.4280123870344e-01 ]},'coef',{[3.7989555024173e-45 3.2023914909894e-40 3.9179142693654e-36 1.3335820885372e-32 1.7163780212922e-29 1.0294248187970e-26 3.3457568162026e-24 6.5125663921413e-22 8.1536403865806e-20 6.9232479198565e-18 4.1524441111527e-16 1.8166245761912e-14 5.9484305160636e-13 1.4889573490645e-11 2.8993590128044e-10 4.4568227752248e-09 5.4755546192770e-08 5.4335161342049e-07 4.3942869362670e-06 2.9187419041555e-05 1.6027733468185e-04 7.3177355696551e-04 2.7913248289531e-03 8.9321783603078e-03 2.4061272766109e-02 5.4718970932183e-02 1.0529876369779e-01 1.7177615691888e-01 2.3786890495866e-01 2.7985311752283e-01 ]}); data(7)= struct('node',{[1.1055240743138e+01 1.0434459776321e+01 9.9210064725726e+00 9.4631239608462e+00 9.0410623133341e+00 8.6446518655005e+00 8.2677951152318e+00 7.9064749419688e+00 7.5578680649896e+00 7.2198940023019e+00 6.8909628903904e+00 6.5698241787567e+00 6.2554707188783e+00 5.9470752244714e+00 5.6439466223452e+00 5.3454991415733e+00 5.0512298489716e+00 4.7607019531714e+00 4.4735321501382e+00 4.1893808634926e+00 3.9079445988681e+00 3.6289498686373e+00 3.3521483007724e+00 3.0773126524498e+00 2.8042335229303e+00 2.5327166122907e+00 2.2625804097899e+00 1.9936542226231e+00 1.7257764756010e+00 1.4587932269484e+00 1.1925568563654e+00 9.2692488970662e-01 6.6175893081257e-01 3.9692367564299e-01 1.3228598727032e-01 ]},'coef',{[5.9928891485445e-50 1.3088219454484e-44 1.9775381832792e-41 5.1459796204458e-40 1.2902253872744e-36 1.3859983185653e-33 7.5869522154891e-31 2.5172152573746e-28 5.3429813146915e-26 7.6615884430059e-24 7.7454741231880e-22 5.7095124891098e-20 3.1533210598112e-18 1.3342434683070e-16 4.4061175397096e-15 1.1534952560119e-13 2.4259623022220e-12 4.1457910332392e-11 5.8137344484274e-10 6.7473312881327e-09 6.5292934189487e-08 5.3024856335052e-07 3.6344904047183e-06 2.1131233267306e-05 1.0466994285007e-04 4.4339999663333e-04 1.6117353953559e-03 5.0416367476379e-03 1.3605327560619e-02 3.1741278190051e-02 6.4133640985369e-02 1.1238816501982e-01 1.7101015293604e-01 2.2612844383734e-01 2.5999310620316e-01 ]}); data(8)= struct('node',{[1.1887863560471e+01 1.1281694164989e+01 1.0780796472469e+01 1.0334491039538e+01 9.9234351145692e+00 9.5376679163021e+00 9.1712174787265e+00 8.8201501286611e+00 8.4817021873216e+00 8.1538381574635e+00 7.8350036783293e+00 7.5239772908069e+00 7.2197764697589e+00 6.9215953729258e+00 6.6287620808576e+00 6.3407083213300e+00 6.0569474734510e+00 5.7770582280616e+00 5.5006722123152e+00 5.2274644551007e+00 4.9571459285134e+00 4.6894576329528e+00 4.4241658477652e+00 4.1610582741258e+00 3.8999408693874e+00 3.6406352232272e+00 3.3829763625124e+00 3.1268108983731e+00 2.8719954485287e+00 2.6183952824718e+00 2.3658831480742e+00 2.1143382465070e+00 1.8636453287449e+00 1.6136938918511e+00 1.3643774570540e+00 1.1155929146033e+00 8.6723992270397e-01 6.1922034962757e-01 3.7143774948304e-01 1.2379686317313e-01 ]},'coef',{[3.8685383727931e-49 2.0252403350315e-45 5.6114941460751e-43 2.7312591323502e-40 8.0059947621974e-39 4.8548677202834e-38 1.6512006592794e-36 6.3922427517268e-35 1.8964852968049e-32 4.3114531467056e-30 6.8829012692744e-28 7.9853336688990e-26 6.9341381005532e-24 4.6134698831073e-22 2.3978402197188e-20 9.8965290426435e-19 3.2891620153310e-17 8.9094610374559e-16 1.9875208348038e-14 3.6848498201918e-13 5.7232797184518e-12 7.4997199146582e-11 8.3430559260916e-10 7.9229232891485e-09 6.4544629501197e-08 4.5305355658615e-07 2.7507246274012e-06 1.4496444191130e-05 6.6517412418152e-05 2.6647788669334e-04 9.3431768696117e-04 2.8731975330897e-03 7.7640394053997e-03 1.8465751154428e-02 3.8708599914838e-02 7.1600747691843e-02 1.1698073380430e-01 1.6893796927590e-01 2.1577494627433e-01 2.4383585380721e-01 ]}); data(9)= struct('node',{[1.2668764375434e+01 1.2075130816911e+01 1.1584960252805e+01 1.1148508777694e+01 1.0746787108514e+01 1.0370015493271e+01 1.0012330403392e+01 9.6698698743375e+00 9.3399210724884e+00 9.0204865095226e+00 8.7100414635652e+00 8.4073884130035e+00 8.1115647551405e+00 7.8217816701739e+00 7.5373821269924e+00 7.2578111510466e+00 6.9825942253569e+00 6.7113212484547e+00 6.4436343875168e+00 6.1792187234935e+00 5.9177949371442e+00 5.6591135131207e+00 5.4029500908349e+00 5.1491016937770e+00 4.8973836402140e+00 4.6476269884214e+00 4.3996764055654e+00 4.1533883754886e+00 3.9086296798929e+00 3.6652761017604e+00 3.4232113106648e+00 3.1823258978579e+00 2.9425165353484e+00 2.7036852380959e+00 2.4657387122777e+00 2.2285877756048e+00 1.9921468380504e+00 1.7563334332534e+00 1.5210677923813e+00 1.2862724534456e+00 1.0518719000410e+00 8.1779222425490e-01 5.8396080911088e-01 3.5030602639582e-01 1.1675694609164e-01 ]},'coef',{[1.3926602246895e-51 1.9568360107978e-47 1.0639144183847e-44 7.9089094484501e-43 7.1933465481042e-41 3.4996773126588e-39 9.2942987391649e-38 5.0159398402133e-38 3.1216254599340e-37 2.8172065374851e-36 3.9144464842405e-34 5.9935373474292e-32 7.7735858553266e-30 7.7182478105170e-28 5.9828467035793e-26 3.6826786261206e-24 1.8269175754861e-22 7.3963427902401e-21 2.4705506323895e-19 6.8737797964624e-18 1.6064742921236e-16 3.1773129595293e-15 5.3533724860157e-14 7.7293031086162e-13 9.6137828692418e-12 1.0350233726258e-10 9.6863849248741e-10 7.9104166149095e-09 5.6567852900933e-08 3.5533045829498e-07 1.9661408494659e-06 9.6077760741792e-06 4.1557921102543e-05 1.5944214368481e-04 5.4359224952963e-04 1.6496055361812e-03 4.4622998595035e-03 1.0773786624642e-02 2.3243228173755e-02 4.4849988132390e-02 7.7467627605513e-02 1.1985655590239e-01 1.6619497260996e-01 2.0661430370392e-01 2.3035797018195e-01 ]}); data(10)= struct('node',{[1.3406487338145e+01 1.2823799749488e+01 1.2342964222860e+01 1.1915061943114e+01 1.1521415400787e+01 1.1152404385585e+01 1.0802260753685e+01 1.0467185421343e+01 1.0144509941293e+01 9.8322698077780e+00 9.5289658233901e+00 9.2334208902192e+00 8.9446892173255e+00 8.6619961681345e+00 8.3846969404163e+00 8.1122473111628e+00 7.8441823844608e+00 7.5801008078575e+00 7.3196528223045e+00 7.0625310602489e+00 6.8084633528588e+00 6.5572070319215e+00 6.3085443611121e+00 6.0622788326143e+00 5.8182321352035e+00 5.5762416493299e+00 5.3361583601384e+00 5.0978451050891e+00 4.8611750917912e+00 4.6260306357872e+00 4.3923020786827e+00 4.1598868551310e+00 3.9286886834277e+00 3.6986168593185e+00 3.4695856364186e+00 3.2415136796310e+00 3.0143235803312e+00 2.7879414239820e+00 2.5622964023726e+00 2.3373204639069e+00 2.1129479963712e+00 1.8891155374270e+00 1.6657615087415e+00 1.4428259702159e+00 1.2202503912190e+00 9.9797743609810e-01 7.7595076154015e-01 5.5411482359162e-01 3.3241469234223e-01 1.1079587242244e-01 ]},'coef',{[4.1890734835153e-53 6.6685952840498e-49 1.7149047609884e-45 4.6329593470698e-43 2.3803897130070e-41 3.1234305446425e-39 2.4962135456919e-38 2.5988685504385e-37 1.3109926053898e-36 1.4501664855704e-36 5.1295692004930e-37 1.3477869587572e-35 2.8227154103471e-35 5.9582141086172e-34 7.7461504141821e-32 7.1102466014413e-30 5.0387971358429e-28 2.9171459549279e-26 1.3948232985713e-24 5.5610285249618e-23 1.8649975920043e-21 5.3023160356162e-20 1.2868329542238e-18 2.6824921883091e-17 4.8298353231531e-16 7.5488968761282e-15 1.0288749372616e-13 1.2278785143771e-12 1.2879038257424e-11 1.1913006349369e-10 9.7479212538641e-10 7.0758572838838e-09 4.5681275084851e-08 2.6290974837540e-07 1.3517971591104e-06 6.2215248177779e-06 2.5676159384549e-05 9.5171627785510e-05 3.1729197104330e-04 9.5269218854862e-04 2.5792732600591e-03 6.3030002856081e-03 1.3915665220232e-02 2.7779127385933e-02 5.0175812677429e-02 8.2051827391224e-02 1.2153798684410e-01 1.6313003050278e-01 1.9846285025419e-01 2.1889262958744e-01 ]}); %% 計算 area=0; noden=data(n).node; coefn=data(n).coef; k=n; n=10*n; %n轉換為表示Gauss積分的積分節點或積分系數的個數 if n/2-floor(n/2)==1/2 %當階數為奇數for i=1:floor(n/2)area=area+coefn(i)*(expfunc(noden(i))+expfunc(-noden(i)));endi=floor(n/2)+1;area=area+coefn(i)*expfunc(noden(i)); elseif n/2-floor(n/2)==0 %當階數為偶數for i=1:n/2area=area+coefn(i)*(expfunc(noden(i))+expfunc(-noden(i)));end end %% 誤差:前向（n-1階）差商作為事后誤差估計 if withErrorarea_=gaussHermiteIntegral(func,a,b,k-1,false);gErrer=abs(area-area_); end end

2 Gauss-Laguerre積分

2.1 測試Gauss-Laguerre積分函數gaussLaguerreIntegral()

clc,clear format long % f=@(x)((x+1).^(-2)) % f=@(x)(exp(-x.^2)) % f=@(x)(exp(-x)) % f=@(x)(exp(-x.^4)) f=@(x)((1/sqrt(2*pi))*exp(-x.^2/2)) %標準正太分布 area=[];nn=[]; for n=1:10nn=[nn,n];area=[area,gaussLaguerreIntegral(f,0,inf,n)]; end,nn area_str=printArray(area) darea=abs(diff(area)); %% 系統計算的積分精確值 % area_system=quadgk(f,0,inf,'RelTol',1e-8,'AbsTol',1e-12) d_my_system=abs(area-.5); d_my_system_str=printArray(d_my_system) log_darea=round(log10(darea)*10); log_d_my_system=round(log10(d_my_system)*10); log_darea_str=printArray(log_darea) log_d_my_system_str=printArray(log_d_my_system) %% 測試誤差項 n=5 [area,gErrer]=gaussLaguerreIntegral(f,0,inf,n,true)

運行結果：

f = @(x)((1/sqrt(2*pi))*exp(-x.^2/2)) n =5 area =0.500000012302732 gErrer =9.329705445981773e-07

2.2 gaussLaguerreIntegral()

function [area,gErrer]=gaussLaguerreIntegral(func,a,b,n,withError) %高斯積分 %理論依據：數值分析方法 奚梅成 %func 被積函數； %[a,b] 積分區間 %n 高斯積分的階數 %% %name：鄧能財 Date: 2013/12/21 %% 默認參數 if nargin<5 withError=false; gErrer=0; end %% 查錯 maxn=10; % assert(a==0 && b==inf,'積分區間必須為[0,inf)!'); % assert(ismember(n,[1:maxn]),['積分精確度數量級必須在1~',num2str(maxn),'之間!']); %% 函數變換 expfunc=@(x)(exp(x)*func(x)); %% 數據：h,2h,3h,...,maxn*h階高斯積分的積分系數 h=5;% (積分節點node，積分系數coef) data= struct('node',cell(1,maxn),'coef',cell(1,maxn)); %data()= struct('node',{[]},'coef',{[]}); data(1)= struct('node',{[2.6356031971814e-011.4134030591065e+003.5964257710407e+007.0858100058588e+001.2640800844276e+01 ]},'coef',{[5.2175561058281e-013.9866681108317e-017.5942449681707e-023.6117586799220e-032.3369972385776e-05 ]}); data(2)= struct('node',{[1.3779347054049e-017.2945454950317e-011.8083429017403e+003.4014336978549e+005.5524961400638e+008.3301527467645e+001.1843785837900e+011.6279257831378e+012.1996585811981e+012.9920697012274e+01 ]},'coef',{[3.0844111576502e-014.0111992915527e-012.1806828761181e-016.2087456098678e-029.5015169751811e-037.5300838858754e-042.8259233495996e-054.2493139849627e-071.8395648239796e-099.9118272196090e-13 ]}); data(3)= struct('node',{[9.3307812017283e-024.9269174030189e-011.2155954120710e+002.2699495262037e+003.6676227217514e+005.4253366274136e+007.5659162266131e+001.0120228568019e+011.3130282482176e+011.6654407708330e+012.0776478899449e+012.5623894226729e+013.1407519169754e+013.8530683306486e+014.8026085572686e+01 ]},'coef',{[2.1823488594009e-013.4221017792288e-012.6302757794168e-011.2642581810593e-014.0206864921001e-028.5638778036118e-031.2124361472143e-031.1167439234425e-046.4599267620229e-062.2263169070963e-074.2274303849794e-093.9218972670411e-111.4565152640731e-131.4830270511133e-161.6005949062111e-20 ]}); data(4)= struct('node',{[7.0539889691990e-023.7212681800161e-019.1658210248327e-011.7073065310283e+002.7491992553094e+004.0489253138509e+005.6151749708616e+007.4590174536711e+009.5943928695811e+001.2038802546964e+011.4814293442631e+011.7948895520519e+012.1478788240285e+012.5451702793187e+012.9932554631701e+013.5013434240479e+014.0833057056729e+014.7619994047347e+015.5810795750064e+016.6524416525616e+01 ]},'coef',{[1.6874680185112e-012.9125436200607e-012.6668610286700e-011.6600245326951e-017.4826064668792e-022.4964417309283e-026.2025508445722e-031.1449623864769e-031.5574177302781e-041.5401440865225e-051.0864863665180e-065.3301209095567e-081.7579811790506e-093.7255024025123e-114.7675292515782e-133.3728442433624e-151.1550143395004e-171.5395221405823e-205.2864427255692e-241.6564566124990e-28 ]}); data(5)= struct('node',{[5.6704775452705e-022.9901089858699e-017.3590955543502e-011.3691831160352e+002.2013260537215e+003.2356758035580e+004.4764966150738e+005.9290837627004e+007.5998993099567e+009.4967492209324e+001.1629014911779e+011.4007957976545e+011.6647125597289e+011.9562898011469e+012.2775241986835e+012.6308772390969e+013.0194291163316e+013.4471097571922e+013.9190608803937e+014.4422349336162e+015.0264574993834e+015.6864967173940e+016.4466670615954e+017.3534234792100e+018.5260155562496e+01 ]},'coef',{[1.3752601422934e-012.5164527376491e-012.5617600280975e-011.8621549036244e-011.0319984810752e-014.4714161129934e-021.5305232886396e-024.1524146328771e-038.9209907325968e-041.5115601916424e-042.0065531801933e-052.0677743964319e-061.6346520222911e-079.7660150621244e-094.3277207941849e-101.3896009633895e-113.1389227925400e-134.8026148226043e-154.7358853648073e-172.8142053798431e-199.1649543959912e-221.4189400094973e-248.2736519440991e-281.1688817115428e-311.3158315000591e-36 ]});data(6)= struct('node',{[4.7407180540804e-02 2.4992391675316e-01 6.1483345439277e-01 1.1431958256661e+00 1.8364545546226e+00 2.6965218745572e+00 3.7258145077795e+00 4.9272937658499e+00 6.3045155909651e+00 7.8616932933703e+00 9.6037759854793e+00 1.1536546597956e+01 1.3666744693064e+01 1.6002221188981e+01 1.8552134840143e+01 2.1327204321783e+01 2.4340035764533e+01 2.7605554796781e+01 3.1141586701111e+01 3.4969652008249e+01 3.9116084949068e+01 4.3613652908485e+01 4.8503986163804e+01 5.3841385406508e+01 5.9699121859235e+01 6.6180617794438e+01 7.3441238595560e+01 8.1736810506728e+01 9.1556466522537e+01 1.0415752443106e+02 ]},'coef',{[1.1604408602039e-01 2.2085112475070e-01 2.4139982758787e-01 1.9463676844642e-01 1.2372841596688e-01 6.3678780368988e-02 2.6860475273381e-02 9.3380708816042e-03 2.6806968913369e-03 6.3512912194088e-04 1.2390745990689e-04 1.9828788438953e-05 2.5893509291317e-06 2.7409428405359e-07 2.3328311650249e-08 1.5807455747823e-09 8.4274791230481e-11 3.4851612347541e-12 1.0990180599189e-13 2.5883126679993e-15 4.4378380184999e-17 5.3659179405818e-19 4.3939492584790e-21 2.3114466007031e-23 7.2761720424752e-26 1.2409628392253e-28 9.9460139172742e-32 3.0921279934282e-35 3.1849751877368e-39 1.7664292620495e-43 ]}); data(7)= struct('node',{[4.0729209061714e-02 2.1468745273514e-01 5.2801038431934e-01 9.8138617345908e-01 1.5757259475775e+00 2.3122282965132e+00 3.1923979394931e+00 4.2180641250297e+00 5.3914027346738e+00 6.7149632799157e+00 8.1917017588376e+00 9.8250204970927e+00 1.1618816388907e+01 1.3577539357308e+01 1.5706263395763e+01 1.8010773287835e+01 2.0497671107147e+01 2.3174507997799e+01 2.6049948710370e+01 2.9133979209544e+01 3.2438171837678e+01 3.5976028769894e+01 3.9763434104800e+01 4.3819260118607e+01 4.8166197974019e+01 5.2831925058016e+01 5.7850795022134e+01 6.3266373675385e+01 6.9135413883648e+01 7.5534435134730e+01 8.2571405523583e+01 9.0408523864401e+01 9.9312973659441e+01 1.0979599094491e+02 1.2317325317538e+02 ]},'coef',{[1.0036223741924e-01 1.9648238336171e-01 2.2601456693434e-01 1.9627166789644e-01 1.3760074201313e-01 8.0030400464140e-02 3.9125771428883e-02 1.6186726526126e-02 5.6853255717743e-03 1.6972232991260e-03 4.3048506814168e-04 9.2634053063654e-05 1.6870264869314e-05 2.5916721474280e-06 3.3446109572665e-07 3.6077854386912e-08 3.2335690322103e-09 2.3913229815959e-10 1.4473323990302e-11 7.1013552776266e-13 2.7933695098016e-14 8.6948402151367e-16 2.1088459299447e-17 3.9129108212979e-19 5.4327218369250e-21 5.4936982290015e-23 3.9126007648144e-25 1.8797445108978e-27 5.7353329163766e-30 1.0118890306195e-32 6.4972453567684e-36 1.0249186153468e-41 1.0144829299011e-40 1.8914407768105e-43 2.6963083454085e-47 ]}); data(8)= struct('node',{[3.5700394308884e-02 1.8816228315870e-01 4.6269428131458e-01 8.5977296397294e-01 1.3800108205273e+00 2.0242091359228e+00 2.7933693535068e+00 3.6887026779083e+00 4.7116411465550e+00 5.8638508783437e+00 7.1472479081023e+00 8.5640170175862e+00 1.0116634048452e+01 1.1807892294005e+01 1.3640933712537e+01 1.5619285893339e+01 1.7746905950096e+01 2.0028232834575e+01 2.2468249983498e+01 2.5072560772426e+01 2.7847480009169e+01 3.0800145739445e+01 3.3938657084914e+01 3.7272245880476e+01 4.0811492823887e+01 4.4568603175334e+01 4.8557763533060e+01 5.2795611187217e+01 5.7301863323394e+01 6.2100179072775e+01 6.7219370927127e+01 7.2695158847612e+01 7.8572802911571e+01 8.4911231135705e+01 9.1789874671236e+01 9.9320808717447e+01 1.0767244063939e+02 1.1712230951269e+02 1.2820184198826e+02 1.4228004446916e+02 ]},'coef',{[8.8412106190338e-02 1.7681473909573e-01 2.1136311701596e-01 1.9408119531860e-01 1.4643428242412e-01 9.3326798435770e-02 5.0932204361044e-02 2.3976193015685e-02 9.7746252467145e-03 3.4579399930185e-03 1.0622468938969e-03 2.8327168532433e-04 6.5509405003248e-05 1.3116069073268e-05 2.2684528787793e-06 3.3796264822014e-07 4.3228213222805e-08 4.7284937709896e-09 4.4031741042515e-10 3.4724414847684e-11 2.3053815448611e-12 1.2797725979698e-13 5.8941771706254e-15 2.2322175699210e-16 6.8803366293934e-18 1.7056039068355e-19 3.3537088111640e-21 5.1461768068696e-23 6.0453446308727e-25 5.3139346735449e-27 3.3657690232258e-29 1.2178445191183e-31 4.2890086271870e-35 2.3067506710852e-34 2.3302702800589e-35 7.9126811843002e-37 1.4714096228146e-38 6.8575263257425e-41 1.0375271306839e-43 1.2955873193971e-47 ]}); data(9)= struct('node',{[3.1776956867006e-02 1.6747251657590e-01 4.1176935189346e-01 7.6501433412688e-01 1.2276394084198e+00 1.8002069428945e+00 2.4834175141916e+00 3.2781154527750e+00 4.1852949337855e+00 5.2061071301655e+00 6.3418686461714e+00 7.5940714118218e+00 8.9643942355385e+00 1.0454716247742e+01 1.2067132515935e+01 1.3803972172016e+01 1.5667819467697e+01 1.7661538267980e+01 1.9788300611284e+01 2.2051620115564e+01 2.4455391203194e+01 2.7003935367666e+01 2.9702056032584e+01 3.2555103986001e+01 3.5569055951601e+01 3.8750609641046e+01 4.2107299705812e+01 4.5647640501793e+01 4.9381303694811e+01 5.3319341780002e+01 5.7474473058814e+01 6.1861450326356e+01 6.6497545839445e+01 7.1403201447971e+01 7.6602919389220e+01 8.2126514283411e+01 8.8010926374694e+01 9.4302943609710e+01 1.0106347092232e+02 1.0837460196798e+02 1.1635218467520e+02 1.2517034745017e+02 1.3511619015017e+02 1.4673979649750e+02 1.6145944790909e+02 ]},'coef',{[7.9003863218578e-02 1.6064969814262e-01 1.9788792774995e-01 1.8978490012961e-01 1.5161633179829e-01 1.0374664065632e-01 6.1658521650479e-02 3.2073009065685e-02 1.4666691055305e-02 5.9109404105920e-03 2.1021480717925e-03 6.5997635763383e-04 1.8287173922646e-04 4.4687548597413e-05 9.6188526495207e-06 1.8207784506475e-06 3.0249820529759e-07 4.4003996132627e-08 5.5894745239333e-09 6.1800526258555e-10 5.9265751381681e-11 4.9096834353441e-12 3.4975836508939e-13 2.1317103205750e-14 1.1051884160140e-15 4.8425951880898e-17 1.7802221360264e-18 5.4452758047095e-20 1.3728113205342e-21 2.8219449500093e-23 4.6711191673266e-25 6.1366267898675e-27 6.3012553677705e-29 4.9199469546978e-31 2.8553923595618e-33 2.6513859772935e-35 1.7302431662816e-35 3.7028624824552e-36 1.3728993488078e-37 1.3069964719048e-38 3.7845759572530e-40 6.4931338105968e-44 1.0904747480551e-45 5.8188542963473e-48 7.4815330585462e-52 ]}); data(10)= struct('node',{[2.8630518339375e-02 1.5088293567693e-01 3.7094878153490e-01 6.8909069988105e-01 1.1056250235399e+00 1.6209617511025e+00 2.2356103759152e+00 2.9501833666418e+00 3.7653997744058e+00 4.6820893875593e+00 5.7011975747849e+00 6.8237909097945e+00 8.0510636693908e+00 9.3843453082584e+00 1.0825109031549e+01 1.2374981608757e+01 1.4035754599830e+01 1.5809397197845e+01 1.7698070933350e+01 1.9704146535462e+01 2.1830223306578e+01 2.4079151444412e+01 2.6454057841253e+01 2.8958376011937e+01 3.1595880956623e+01 3.4370729963090e+01 3.7287510610551e+01 4.0351297573586e+01 4.3567720269995e+01 4.6943043991603e+01 5.0484267963130e+01 5.4199244880169e+01 5.8096828017249e+01 6.2187054175689e+01 6.6481373878445e+01 7.0992944826619e+01 7.5737011547727e+01 8.0731404802478e+01 8.5997211136463e+01 9.1559690412534e+01 9.7449565614851e+01 1.0370489123669e+02 1.1037385880764e+02 1.1751919820311e+02 1.2522547013347e+02 1.3361202792273e+02 1.4285832548925e+02 1.5326037197260e+02 1.6538564331668e+02 1.8069834370921e+02 ]},'coef',{[7.1404726135184e-02 1.4714860696459e-01 1.8567162757483e-01 1.8438538252735e-01 1.5420116860636e-01 1.1168536990227e-01 7.1052885490196e-02 4.0020276911508e-02 2.0050623080072e-02 8.9608512036464e-03 3.5781124153157e-03 1.2776171567891e-03 4.0803024498372e-04 1.1652883223097e-04 2.9741704936942e-05 6.7778425265418e-06 1.3774795031715e-06 2.4928861817200e-07 4.0103543504264e-08 5.7233317481479e-09 7.2294342491785e-10 8.0617101422267e-11 7.9133931001156e-12 6.8157366176493e-13 5.1324267151836e-14 3.3656247656300e-15 1.9134763294156e-16 9.3855895940761e-18 3.9500701020349e-19 1.4177501120800e-20 4.3099605395263e-22 1.1012486685628e-23 2.3448759094638e-25 4.1195425783619e-27 5.8517301056039e-29 6.7131788910435e-31 1.2697958123263e-32 6.9071714060483e-34 3.9707498882293e-34 1.2868077986345e-33 1.1748415289134e-33 1.8729029481275e-34 2.9809110912453e-35 2.1474665129166e-36 5.6599382245170e-38 1.2115127694324e-39 8.4123926853140e-42 1.1227629730641e-44 9.4963283290577e-48 4.9608486792055e-52 ]}); %% 計算 area=0; noden=data(n).node; coefn=data(n).coef; k=h*n; %k表示Gauss積分的積分節點或積分系數的個數 for i=1:karea=area+coefn(i)*expfunc(noden(i)); end %% 誤差:前向（n-1階）差商作為事后誤差估計 if withErrorarea_=gaussLaguerreIntegral(func,a,b,n-1,false);gErrer=abs(area-area_); end end

輸出函數：

function str=printArray(a) str=sprintf('%.13e\t',a);

3 Gauss-Legendre積分W(x)=1

3.1 測試Gauss-Legendre積分W(x)=1函數gaussLegendreIntegral()

clc,clear format long % f=@(x)(exp(x)) % a=1,b=10 f=@(x)(cos(x)) % a=0,b=pi/2 a=-pi/2,b=pi/2 % a=-pi,b=pi % a=-1,b=1 disp('各個階數(h*)的gaussLegendreIntegral積分') h=5 max=5 n_=[2:max];area_=[];gErrer_=[]; for n=2:max %% 適合區間長為數量級10左右的積分[area,gErrer]=gaussLegendreIntegral(f,a,b,n,true);area_=[area_,area];gErrer_=[gErrer_,gErrer]; end n_ area__str=printArray(area_) gErrer_str=printArray(gErrer_) area_system=quadl(f,a,b)

運行結果：

f = @(x)(cos(x)) a =-1.570796326794897 b =1.570796326794897 各個階數(h*)的gaussLegendreIntegral積分 h =5 max =5 n_ =2 3 4 5 area__str = 2.0000000000000e+00 2.0000000000000e+00 2.0000000000000e+00 2.0000000000000e+00 gErrer_str = 1.1028448909656e-07 3.0864200084579e-14 3.0864200084579e-14 2.7089441800854e-14 area_system =1.999999977471133 >>

3.2 gaussLegendreIntegral()

function [area,gErrer]=gaussLegendreIntegral(func,a,b,n,withError) %高斯積分 %理論依據：數值分析方法 奚梅成 %func 被積函數； %[a,b] 積分區間 %n 高斯積分的階數 %% %name：鄧能財 Date: 2013/12/21 %% 默認參數 if nargin<5 withError=false; gErrer=0; end %% 數據：1-8階高斯積分的積分系數 h=5; maxn=10; % (±積分節點node，積分系數coef)，‘±’被省略 data= struct('node',cell(1,maxn),'coef',cell(1,maxn)); % data(1)= struct('node',{[0 ]},'coef',{[2]}); % data(2)= struct('node',{[0.5773502692]},'coef',{[1]}); % data(3)= struct('node',{[0.7745966692 0 ]},'coef',{[0.5555555556 0.8888888889]}); % data(4)= struct('node',{[0.8611363116 0.3399810436]},'coef',{[0.3478548451 0.6521451549]}); % data(5)= struct('node',{[0.9061798459 0.5384693101 0 ]},'coef',{[0.2369268851 0.4786286705 0.5688888889]}); % data(6)= struct('node',{[0.9324695142 0.6612093865 0.2386191861]},'coef',{[0.1713244924 0.3607615730 0.4679139346]}); % data(7)= struct('node',{[0.9491079123 0.7415311856 0.4058451514 0 ]},'coef',{[0.1294849662 0.2797053915 0.3818300505 0.4179591837]}); % data(8)= struct('node',{[0.9602898565 0.7966664774 0.5255324099 0.1834346425]},'coef',{[0.1012285363 0.2223810345 0.3137066459 0.3626837834]}); data(1)= struct('node',{[9.0617984593866e-01 5.3846931010568e-01 -4.7505928254367e-17 ]},'coef',{[2.3692688505619e-01 4.7862867049937e-01 5.6888888888889e-01 ]}); data(2)= struct('node',{[9.7390652851717e-01 8.6506336668898e-01 6.7940956829902e-01 4.3339539412925e-01 1.4887433898163e-01 ]},'coef',{[6.6671344308688e-02 1.4945134915058e-01 2.1908636251598e-01 2.6926671931000e-01 2.9552422471475e-01 ]}); data(3)= struct('node',{[9.8799251802049e-01 9.3727339240071e-01 8.4820658341043e-01 7.2441773136017e-01 5.7097217260854e-01 3.9415134707756e-01 2.0119409399743e-01 1.0426645532607e-16 ]},'coef',{[3.0753241996117e-02 7.0366047488109e-02 1.0715922046717e-01 1.3957067792615e-01 1.6626920581699e-01 1.8616100001556e-01 1.9843148532711e-01 2.0257824192556e-01 ]}); data(4)= struct('node',{[9.9312859918510e-01 9.6397192727791e-01 9.1223442825133e-01 8.3911697182222e-01 7.4633190646015e-01 6.3605368072652e-01 5.1086700195083e-01 3.7370608871542e-01 2.2778585114164e-01 7.6526521133497e-02 ]},'coef',{[1.7614007139152e-02 4.0601429800386e-02 6.2672048334110e-02 8.3276741576705e-02 1.0193011981724e-01 1.1819453196152e-01 1.3168863844918e-01 1.4209610931838e-01 1.4917298647260e-01 1.5275338713073e-01 ]}); data(5)= struct('node',{[9.9555696979050e-01 9.7666392145952e-01 9.4297457122897e-01 8.9499199787828e-01 8.3344262876083e-01 7.5925926303736e-01 6.7356636847347e-01 5.7766293024122e-01 4.7300273144572e-01 3.6117230580939e-01 2.4386688372099e-01 1.2286469261071e-01 2.3477116549018e-17 ]},'coef',{[1.1393798501026e-02 2.6354986615032e-02 4.0939156701307e-02 5.4904695975835e-02 6.8038333812357e-02 8.0140700335000e-02 9.1028261982964e-02 1.0053594906705e-01 1.0851962447426e-01 1.1485825914571e-01 1.1945576353578e-01 1.2224244299031e-01 1.2317605372672e-01 ]}); data(6)= struct('node',{[9.9689348407465e-01 9.8366812327975e-01 9.6002186496831e-01 9.2620004742927e-01 8.8256053579205e-01 8.2956576238277e-01 7.6777743210483e-01 6.9785049479332e-01 6.2052618298924e-01 5.3662414814202e-01 4.4703376953809e-01 3.5270472553088e-01 2.5463692616789e-01 1.5386991360858e-01 5.1471842555318e-02 ]},'coef',{[7.9681924961671e-03 1.8466468311090e-02 2.8784707883324e-02 3.8799192569627e-02 4.8402672830593e-02 5.7493156217619e-02 6.5974229882181e-02 7.3755974737705e-02 8.0755895229420e-02 8.6899787201083e-02 9.2122522237786e-02 9.6368737174644e-02 9.9593420586795e-02 1.0176238974840e-01 1.0285265289356e-01 ]}); data(7)= struct('node',{[9.9770656909960e-01 9.8793576444385e-01 9.7043761603923e-01 9.4534514820783e-01 9.1285426135932e-01 8.7321912502522e-01 8.2674989909223e-01 7.7381025228691e-01 7.1481450155663e-01 6.5022436466589e-01 5.8054534474976e-01 5.0632277324149e-01 4.2813754151781e-01 3.4660155443081e-01 2.6235294120930e-01 1.7605106116599e-01 8.8371343275659e-02 -8.0123445265982e-18 ]},'coef',{[5.8834334204433e-03 1.3650828348362e-02 2.1322979911483e-02 2.8829260108895e-02 3.6110115863463e-02 4.3108422326170e-02 4.9769370401354e-02 5.6040816212370e-02 6.1873671966080e-02 6.7222285269087e-02 7.2044794772560e-02 7.6303457155442e-02 7.9964942242325e-02 8.3000593728857e-02 8.5386653392099e-02 8.7104446997184e-02 8.8140530430276e-02 8.8486794907104e-02 ]}); data(8)= struct('node',{[9.9823770971056e-01 9.9072623869946e-01 9.7725994998377e-01 9.5791681921379e-01 9.3281280827868e-01 9.0209880696887e-01 8.6595950321226e-01 8.2461223083331e-01 7.7830565142652e-01 7.2731825518993e-01 6.7195668461418e-01 6.1255388966798e-01 5.4946712509513e-01 4.8307580168618e-01 4.1377920437160e-01 3.4199409082576e-01 2.6815218500725e-01 1.9269758070137e-01 1.1608407067526e-01 3.8772417506051e-02 ]},'coef',{[4.5212770985336e-03 1.0498284531152e-02 1.6421058381909e-02 2.2245849194167e-02 2.7937006980023e-02 3.3460195282548e-02 3.8782167974472e-02 4.3870908185673e-02 4.8695807635072e-02 5.3227846983937e-02 5.7439769099392e-02 6.1306242492929e-02 6.4804013456601e-02 6.7912045815234e-02 7.0611647391287e-02 7.2886582395805e-02 7.4723169057968e-02 7.6110361900626e-02 7.7039818164248e-02 7.7505947978425e-02 ]}); data(9)= struct('node',{[9.9860364518194e-01 9.9264999844720e-01 9.8196871503454e-01 9.6660831039689e-01 9.4664169099563e-01 9.2216393671900e-01 8.9329167175324e-01 8.6016247596066e-01 8.2293422050209e-01 7.8178431259391e-01 7.3690884894549e-01 6.8852168077120e-01 6.3685339445322e-01 5.8215021256935e-01 5.2467282046292e-01 4.6469512391964e-01 4.0250294385854e-01 3.3839265425060e-01 2.7266976975238e-01 2.0564748978326e-01 1.3764520598325e-01 6.8986980163144e-02 1.0816665110908e-16 ]},'coef',{[3.5826631552839e-03 8.3231892962177e-03 1.3031104991583e-02 1.7677535257937e-02 2.2239847550579e-02 2.6696213967578e-02 3.1025374934516e-02 3.5206692201609e-02 3.9220236729302e-02 4.3046880709165e-02 4.6668387718374e-02 5.0067499237952e-02 5.3228016731269e-02 5.6134878759786e-02 5.8774232718842e-02 6.1133500831067e-02 6.3201440073820e-02 6.4968195750723e-02 6.6425348449842e-02 6.7565954163608e-02 6.8384577378670e-02 6.8877316977661e-02 6.9041824829232e-02 ]}); data(10)= struct('node',{[9.9886640442007e-01 9.9403196943209e-01 9.8535408404801e-01 9.7286438510669e-01 9.5661095524281e-01 9.3665661894488e-01 9.1307855665579e-01 8.8596797952361e-01 8.5542976942995e-01 8.2158207085934e-01 7.8455583290040e-01 7.4449430222607e-01 7.0155246870682e-01 6.5589646568544e-01 6.0770292718495e-01 5.5715830451465e-01 5.0445814490746e-01 4.4980633497404e-01 3.9341431189757e-01 3.3550024541944e-01 2.7628819377953e-01 2.1600723687604e-01 1.5489058999815e-01 9.3174701560086e-02 3.1098338327189e-02 ]},'coef',{[2.9086225531552e-03 6.7597991957456e-03 1.0590548383651e-02 1.4380822761485e-02 1.8115560713489e-02 2.1780243170125e-02 2.5360673570013e-02 2.8842993580535e-02 3.2213728223579e-02 3.5459835615146e-02 3.8568756612588e-02 4.1528463090147e-02 4.4327504338803e-02 4.6955051303949e-02 4.9400938449466e-02 5.1655703069581e-02 5.3710621888997e-02 5.5557744806213e-02 5.7189925647728e-02 5.8600849813222e-02 5.9785058704266e-02 6.0737970841770e-02 6.1455899590316e-02 6.1936067420683e-02 6.2176616655347e-02 ]}); %% 積分變量的線性變換 y=@(x)(((b+a)+(b-a)*x)/2); %% 計算 area=0; noden=data(n).node; coefn=data(n).coef; k=n; n=n*h; if n/2-floor(n/2)==1/2 %當階數為奇數 for i=1:floor(n/2)area=area+coefn(i)*(func(y(noden(i)))+func(y(-1*noden(i))));endi=floor(n/2)+1;area=area+coefn(i)*func(y(noden(i))); elseif n/2-floor(n/2)==0 %當階數為偶數 for i=1:n/2area=area+coefn(i)*(func(y(noden(i)))+func(y(-1*noden(i))));end end area=area*(b-a)/2; %% 誤差:前向（n-1階）差商作為事后誤差估計 if withErrorarea_=gaussLegendreIntegral(func,a,b,k-1,false);gErrer=abs(area-area_); end end

輸出函數：

function str=printArray(a) str=sprintf('%.13e\t',a);

4 Gauss積分的積分系數計算

4.1 測試Gauss積分的積分系數計算的函數

clc,clear % GaussLegendre積分：(積分節點 積分系數) % disp('GaussLegendre積分：(積分節點+-node 積分系數coef)') % for n=1:16 % sprintf('************階數 %d',n) % [x, w] = GaussLegendre_2(n); % k=floor(n/2); % if k~=n/2, spac=1:k+1;else, spac=1:n/2; end % node=-x(spac),coef=w(spac) % end %% GaussLaguerre積分：(積分節點 積分系數) % disp('GaussLaguerre積分：(積分節點 積分系數)') % alpha=1e-30 % for n=1:16 % sprintf('************階數 %d',n) % [node,coef]=GaussLaguerre_2(n,alpha) % end %% GaussHermite 積分：(積分節點 積分系數) % disp('GaussHermite積分：(積分節點 積分系數)') % for n=1:20 % sprintf('************階數 %d',n) % [x, w] =GaussHermite_2(n); % k=floor(n/2); % if k~=n/2, spac=1:k+1;else, spac=1:n/2; end % node=-x(spac),coef=w(spac) % end %% printGaussIntData(1,5);

printGaussIntData()函數（打印Gauss積分數據表）：

function printGaussIntData(type,h) %% 打印Gauss積分數據表 str=''; for n=h:h:h*10if type==2[x, w] =GaussLaguerre_2(n,1e-100);elseif type==1[x, w] =GaussLegendre_2(n);elseif type==3[x, w] =GaussHermite_2(n);endk=floor(n/2);if k~=n/2, spac=1:k+1;else, spac=1:n/2; endx=-x(spac); w=w(spac);end%data()= struct('node',{[]},'coef',{[]});sprintf('%d',n)str=[str,'data(',num2str((n/h)),...')= struct(#&#node#&#,{[',printMatrix(x,'%.13e',false),...']},#&#coef#&#,{[',printMatrix(w,'%.13e',false),']});',sprintf('\n')]; end,str end

輸出矩陣的函數：

function s=printMatrix(m,style,with_endl) if nargin<2, style='%.13e'; end if nargin<3, with_endl=true; end [dim1,dim2]=size(m); s=''; for i=1:dim1s=[s,sprintf([style,' '],m(i,:)),sprintf('\n')]; end if ~with_endls(end)=''; end end

4.2 Gauss-Hermite積分系數求解函數

function [x, w] = GaussHermite_2(n) %name：鄧能財 Date: 2013/12/23 i = 1:n-1; a = sqrt(i/2); CM = diag(a,1) + diag(a,-1); [V L] = eig(CM); [x ind] = sort(diag(L)); V = V(:,ind)'; w = sqrt(pi) * V(:,1).^2; end

4.3 Gauss-Laguerre積分系數求解函數

function [x, w] = GaussLaguerre_2(n, alpha) %name：鄧能財 Date: 2013/12/23 i = 1:n; a = (2*i-1) + alpha; b = sqrt( i(1:n-1) .* ((1:n-1) + alpha) ); CM = diag(a) + diag(b,1) + diag(b,-1); [V L] = eig(CM); [x ind] = sort(diag(L)); V = V(:,ind)'; w = gamma(alpha+1) .* V(:,1).^2; end

4.4 Gauss-Legendre積分系數求解函數

function [x, w] = GaussLegendre_2(n) % http://www.mathworks.cn/matlabcentral/fileexchange/8067-gauss-laguerre %對應《數值分析方法 奚梅成的Gauss-Legendre積分》 %name：鄧能財 Date: 2013/12/23 i = 1:n-1; a = i./sqrt(4*i.^2-1); CM = diag(a,1) + diag(a,-1); [V L] = eig(CM); [x ind] = sort(diag(L)); V = V(:,ind)'; w = 2 * V(:,1).^2; end

5 NewtonCotes積分

5.1 測試NewtonCotes積分函數newtonCotesIntegral()

clc,clear format long disp('各個階數的Newton-Cote`s積分') % f=@(x)((x+1).^(-2)) f = @(x)(cos(x)) a = -1.57079632679490 b = 1.57079632679490 % f=@(x)(exp(-x.^2)) % f=@(x)(exp(-x)) % f=@(x)(exp(-x.^4)) % f=@(x)((1/sqrt(2*pi))*exp(-x.^2/2)) %標準正太分布 % f=@(x)(sin(x)./x) % f=@(x)(x) % a=0,b=1 area=[];nn=[];gErrer=[]; for n=2:8nn=[nn,n];[arean,gErrern]=newtonCotesIntegral(f,a,b,n,true);area=[area,arean];gErrer=[gErrer,gErrern]; end,nn, area_str=printArray(area) gErrer_str=printArray(gErrer) %% 系統計算的積分精確值 area_system=quad(f,a,b,1e-13) d_my_system=abs(area-area_system); log_darea=round(log10(gErrer)*10) log_d_my_system=round(log10(d_my_system)*10) % %% 測試誤差項 % a=0,b=.01 % n=3 % [area,gErrer]=newtonCotesIntegral(f,a,b,n,true) % area_system=quad(f,a,b,1e-13)

輸出數組的函數：

function str=printArray(a) str=sprintf('%.13e\t',a);

運行結果：

各個階數的Newton-Cote`s積分 f = @(x)(cos(x)) a =-1.570796326794900 b =1.570796326794900 nn =2 3 4 5 6 7 8 area_str = 1.5707963267949e+00 2.2672492052928e+00 2.2888179796358e+00 2.2609034070862e+00 2.2389648201115e+00 2.2178021611519e+00 2.1987231887715e+00 gErrer_str = 3.1341507153454e+00 1.7494105356846e+00 6.8206453262057e-02 1.1112991494925e-01 1.0995339715508e-01 1.3352735099341e-01 1.5154966451227e-01 area_system =2

5.2 newtonCotesIntegral()

function [area,gErrer]=newtonCotesIntegral(func,a,b,n,withError) %Newton-Cotes積分 %理論依據：數值分析方法 奚梅成 %func 被積函數； %[a,b] 積分區間 %n Newton-Cotes積分的階數 %% %name：鄧能財 Date: 2013/12/21 %% 默認參數 if nargin<5 withError=false; gErrer=0; end %% 查錯 maxn=8; % assert(a~=inf && b~=inf,'積分區間不能包含inf!'); % assert(ismember(n,[1:maxn]),['積分精確度數量級必須在1~',num2str(maxn),'之間!']); %% 數據：1,2,...,maxn階Newton-Cotes積分的積分系數% (積分系數coef) data= struct('coef',cell(1,maxn)); %data()= struct('coef',{[]}); data(1)= struct('coef',{[1]}); data(2)= struct('coef',{[ 1/2 1/2 ]}); data(3)= struct('coef',{[ 1/6 4/6 1/6 ]}); data(4)= struct('coef',{[ 1/8 3/8 3/8 1/8 ]}); data(5)= struct('coef',{[ 7/90 16/45 2/15 16/45 7/90 ]}); data(6)= struct('coef',{[ 19/288 25/96 25/144 25/144 25/96 19/288 ]}); data(7)= struct('coef',{[ 41/840 9/35 9/280 34/105 9/280 9/35 41/840 ]}); data(8)= struct('coef',{[ 751/17280 3577/17280 1323/17280 2989/17280 2989/17280 1323/17280 3577/17280 751/17280 ]}); %% 計算 area=0; coefn=data(n).coef; xi=a; h=(b-a)/n; for i=1:nxi=xi+h;area=area+coefn(i)*func(xi); end area=area*(b-a); %% 誤差:前向（n-1階）差商作為事后誤差估計 if withErrorarea_=newtonCotesIntegral(func,a,b,n-1,false);gErrer=abs(area-area_);gErrer=gErrer*(10^(.1*(n+1))); %根據試驗值調整誤差估計 end % %% 誤差:兩分段（n-1階）積分和與原積分之差作為事后誤差估計 % if withError % area_=(newtonCotesIntegral(func,a,(a+b)/2,n,false)+... % newtonCotesIntegral(func,(a+b)/2,b,n,false)); % gErrer=abs(area-area_); % end end

6 Romberg積分

6.1 測試Romberg積分函數rombergIntegral()

clc,clear format long f = @(x)(cos(x)) a = -1.57079632679490 b = 1.57079632679490 '各種精度的Romberg積分' % f=@(x)(exp(x)) % a=1,b=2 area_=[];pre_=[];k_=[]; for i=2:5pre=10^(-2*i)[area,k]=rombergIntegral(f,a,b,pre);pre_=[pre_,pre];area_=[area_,area];k_=[k_,k]; end, k_ area_str=printArray(area_) pre_str=printArray(pre_) area_system=quadl(f,a,b)

輸出數組的函數：

function str=printArray(a) str=sprintf('%.13e\t',a);

運行結果：

f = @(x)(cos(x)) a =-1.570796326794900 b =1.570796326794900 ans = 各種精度的Romberg積分 pre =1.000000000000000e-04 pre =1.000000000000000e-06 pre =1.000000000000000e-08 pre =1.000000000000000e-10 k_ =4 5 6 6 area_str = 2.0000055499797e+00 1.9999999945873e+00 2.0000000000013e+00 2.0000000000013e+00 pre_str = 1.0000000000000e-04 1.0000000000000e-06 1.0000000000000e-08 1.0000000000000e-10 area_system =1.999999977471133

6.2 rombergIntegral()

function [area,k]=rombergIntegral(func,a,b,prec) %Romberg積分 %理論依據：數值分析方法 奚梅成 %func 函數； %[a,b] 積分區間 %pre 精度 %程序特點：密切結合理論，將內存和運算量達到極小化%只記錄最后一排需用的數字 %% %name：鄧能財 Date: 2013/12/20 %這幾乎是我編寫的第一個【調試只出一個錯誤】的程序 %% 計算 %取初始分段數為1 h=b-a; t=[];%記錄最后一排需用的數字 %Romberg積分的階數：k%即所儲存的T的長度 k=1; %計算梯形積分T0(h) t=(func(a)+func(b))/2; while truet=[0,t];k=k+1;%h1用于T0(h)與T0(h/2)之間的遞推計算h1=0;for xi=a+h/2:h:b-h/2h1=h1+func(xi);end, h1=h1*h;%計算梯形積分T0(h/2)t(1)=(t(2)+h1)/2;h=h/2;%計算所得積分最精確值bit=1;%記錄2^(2k)for i=2:kbit=bit*4;t(i)=t(i-1)+(t(i-1)-t(i))/(bit-1);end%判斷是否達到精度要求if abs(t(end)-t(end-1))<precbreak;end end area=t(end); end

7 復化積分

7.1 測試復化積分函數complexSimpson()

clc,clear % f=@(x)(exp(x)) % a=1,b=2 % m2=f(2) % m4=m2 f = @(x)(cos(x)) a = -1.57079632679490 b = 1.57079632679490 '各種精度的復化梯形積分積分' % m2=1 m4=1 area_=[];pre_=[];n_=[]; for i=2:5pre=10^(-2*i);[area,n]=complexSimpson(f,a,b,pre,m4);pre_=[pre_,pre];area_=[area_,area];n_=[n_,n]; end, n_ area_str=printArray(area_) pre_str=printArray(pre_) area_system=quadl(f,a,b) % pre=1e-6 % ss=complexTrangleInt(f,a,b,pre,m2) % ss=complexSimpson(f,a,b,pre,m4)

輸出數組的函數：

function str=printArray(a) str=sprintf('%.13e\t',a);

運行結果：

f = @(x)(cos(x)) a =-1.570796326794900 b =1.570796326794900 ans = 各種精度的復化梯形積分積分 m4 =1 n_ =3 9 28 89 area_str = 2.0008631896735e+00 2.0000103477058e+00 2.0000001100950e+00 2.0000000010782e+00 pre_str = 1.0000000000000e-04 1.0000000000000e-06 1.0000000000000e-08 1.0000000000000e-10 area_system =1.999999977471133

7.2 complexSimpson()

function [ss,n]=complexSimpson(func,a,b,pre,m4) %func函數 %a,b積分區間 %pre精度 %m4區間上4階導的最大值 %name：鄧能財 Date: 2013/12/23 %%%%%%%%%%%%%%%%%%%%%%：%【表需在尾部添加分號‘；’ n=floor((sqrt((b-a)^5)*m4/(2880*pre))^(1/4))+1%【 h=(b-a)/(2*n); ss=func(a)+4*func(a+h)+func(b); ss1=0; %n為奇數的項 ss2=0; %n為偶數的項 for xi=a+2*h:2*h:a+2*(n-1)*hss1=ss1+func(xi+h);ss2=ss2+func(xi); end ss1,ss2 ss=ss+4*ss1+2*ss2; ss=ss*(h/3); end

7.3 complexTrangleInt()

function [ss,n]=complexTrangleInt(func,a,b,pre,m2) %func函數 %a,b積分區間 %pre精度 %m2區間上二階導的最大值 %name：鄧能財 Date: 2013/12/23 %%%%%%%%%%%%%%%%%%%%%%：%【表需在尾部添加分號‘；’ n=floor((sqrt((b-a)^3)*m2/(12*pre))^.5)+1;%【 h=(b-a)/n; ss=.5*(func(a)+func(b)); for xi=a+h:h:a+(n-1)*hss=ss+func(xi); end ss=ss*h; end

8 自適應運算積分

8.1 測試自適應運算積分函數selfAdaptIntegral()

clc,clear format long % f=@(x)(exp(x)) % a=1,b=10 % pre=1e-13 f = @(x)(cos(x)) a = -1.57079632679490 b = 1.57079632679490 % ss=complexTrangleInt(f,a,b,pre,m2) % [area,deep]=selfAdaptIntegral(f,a,b,pre) '各種精度的自適應Simpson積分' area_=[];pre_=[];deep_=[]; for i=2:5pre=10^(-2*i);[area,deep]=selfAdaptIntegral(f,a,b,pre);pre_=[pre_,pre];area_=[area_,area];deep_=[deep_,deep]; end, area_str=printArray(area_) pre_str=printArray(pre_) deep_ area_system=quadl(f,a,b)

輸出數組的函數：

function str=printArray(a) str=sprintf('%.13e\t',a);

運行結果：

f = @(x)(cos(x)) a =-1.570796326794900 b =1.570796326794900 ans = 各種精度的自適應Simpson積分 area_str = 2.0000526243412e+00 2.0000002039922e+00 2.0000000017060e+00 2.0000000000498e+00 pre_str = 1.0000000000000e-04 1.0000000000000e-06 1.0000000000000e-08 1.0000000000000e-10 deep_ =1 3 5 6 area_system =1.999999977471133

8.2 selfAdaptIntegral()

function [area,deep]=selfAdaptIntegral(func,a,b,prec) %Simpson公式的自適應積分 %理論依據：數值分析方法 奚梅成 %name：鄧能財 Date: 2013/12/23 %func 函數； %[a,b] 積分區間 %pre 精度 %算法：遞歸 %程序特點：密切結合理論，將內存和運算量達到極小化 %% 計算 %%disp('##Start') %取分段次數為n %n=3時，Simpson積分系數序列為: 1424241%，已經能夠發揮Simpson公式的作用，且分段數最少 n=3; h=(b-a)/n; %計算梯形積分T_n s1=(func(a)+func(b))/2; for xi=a+h:h:b-hs1=s1+func(xi); end, s1=s1*h; %H_n(f) h1=0; for xi=a+h/2:h:b-h/2h1=h1+func(xi); end, h1=h1*h; %計算梯形積分T_2n s2=(s1+h1)/2; %計算Simpson積分S_n s1=(4*s2-s1)/3; %H_2n(f) h=h/2; h2=0; for xi=a+h/2:h:b-h/2h2=h2+func(xi); end, h2=h2*h; %計算Simpson積分S_2n s2=s1/2+(4*h2-h1)/6; %% 判斷是否達到要求精度 deep=1; if(abs(s1-s2)<15*prec)area=s2;%disp('##End')return; else %不滿足精度，進一步細分[area1,deep1]=selfAdaptIntegral(func,a,(a+b)/2,prec/2);[area2,deep2]=selfAdaptIntegral(func,(a+b)/2,b,prec/2);area=area1+area2;deep=deep+max(deep1,deep2);%disp('##End')return; end end

9 積分系數——二

9.1 Gauss-Legendre積分系數

格式：序號【 積分節點 ******積分系數 1【0 ********* 2 2【±0.5773502692 ********* 1 3【±0.7745966692 0********* 0.5555555556 0.8888888889 4【 ±0.8611363116 ±0.3399810436 ********* 0.3478548451 0.6521451549 5【±0.9061798459 ±0.5384693101 0********* 0.2369268851 0.4786286705 0.5688888889 6【 ±0.9324695142 ±0.6612093865 ±0.2386191861********* 0.1713244924 0.3607615730 0.4679139346 7【 ±0.9491079123 ±0.7415311856 ±0.4058451514 0********* 0.1294849662 0.2797053915 0.3818300505 0.4179591837 8【 ±0.9602898565 ±0.7966664774 ±0.5255324099 ±0.1834346425********* 0.1012285363 0.2223810345 0.3137066459 0.3626837834

9.2 newton cotes積分系數

表6—1 【】表錯誤 n Ci(n-1)2 1/2 1/2 3 1/6 4/6 1/6 4 1/8 3/8 3/8 1/8 5 7/90 16/45 2/15 16/45 7/90 6 19/288 25/96 25/144 25/144 25/96 19/288 7 41/840 9/35 9/280 34/105 9/280 9/35 41/840 8 751/17280 3577/17280 1323/17280 2989/17280 2989/17280 1323/17280 3577/17280 751/17280 9 989/28350 5888/28350 -928/28350 10496/28350 -4540/28350 10496/28350 -928/28350 【5888/28350】 989/28350

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