标签（空格分隔）： matlab 积分数值计算

文章目录

- 标签（空格分隔）： matlab 积分数值计算
- @[toc]
matlab数值积分
- 1 Gauss-Hermite积分
- - 1.1 测试Gauss-Hermite积分函数gaussHermiteIntegral()
  - 1.2 gaussHermiteIntegral()
- 2 Gauss-Laguerre积分
- - 2.1 测试Gauss-Laguerre积分函数gaussLaguerreIntegral()
  - 2.2 gaussLaguerreIntegral()
- 3 Gauss-Legendre积分W(x)=1
- - 3.1 测试Gauss-Legendre积分W(x)=1函数gaussLegendreIntegral()
  - 3.2 gaussLegendreIntegral()
- 4 Gauss积分的积分系数计算
- - 4.1 测试Gauss积分的积分系数计算的函数
  - 4.2 Gauss-Hermite积分系数求解函数
  - 4.3 Gauss-Laguerre积分系数求解函数
  - 4.4 Gauss-Legendre积分系数求解函数
- 5 NewtonCotes积分
- - 5.1 测试NewtonCotes积分函数newtonCotesIntegral()
  - 5.2 newtonCotesIntegral()
- 6 Romberg积分
- - 6.1 测试Romberg积分函数rombergIntegral()
  - 6.2 rombergIntegral()
- 7 复化积分
- - 7.1 测试复化积分函数complexSimpson()
  - 7.2 complexSimpson()
  - 7.3 complexTrangleInt()
- 8 自适应运算积分
- - 8.1 测试自适应运算积分函数selfAdaptIntegral()
  - 8.2 selfAdaptIntegral()
- 9 积分系数——二
- - 9.1 Gauss-Legendre积分系数
  - 9.2 newton cotes积分系数
- 联系作者 definedone@163.com

matlab数值积分

1 Gauss-Hermite积分

1.1 测试Gauss-Hermite积分函数gaussHermiteIntegral()

clc,clear
format long
f=@(x)(exp(-x.^4))
% % % % f=@(x)((1/sqrt(2*pi))*exp(-x.^2/2)) %标准正太分布
% % area=[];nn=[];
% % for n=1:10
% %     nn=[nn,n];
% %     area=[area,gaussHermiteIntegral(f,-inf,inf,n)];
% % end,nn,area
% % darea=abs(diff(area));
% % %% 系统计算的积分精确值
% % 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)
%% 从以下数据看（前向差商，系统误差）：前向差商公式完全能够作为事后误差估计
% 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
%% 从以下数据看：Gauss Hermite积分的精确度e随积分节点个数n呈现关系：log10(e)=n/10%因此采用10n个节点的gauss积分作为程序中的误差阶数
% 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

运行结果：

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)
%高斯积分
%理论依据：数值分析方法 奚梅成
%func 被积函数；
%[a,b] 积分区间
%n 高斯积分的阶数
%%
%name：邓能财     Date: 2013/12/23
%% 默认参数
if nargin<5 withError=false; gErrer=0; end
% 查错
assert(a==-inf && b==inf,'积分区间必须为(-inf,inf)!');
assert(ismember(n,[1:10]),'积分精确度数量级必须在1~10之间!');
%% 函数变换
expfunc=@(x)(exp(x^2)*func(x));
%% 数据：10,20,30,...,maxn*10阶高斯积分的积分系数
maxn=10; %   (±积分节点node，积分系数coef)，‘±’被省略
data= struct('node',cell(1,maxn),'coef',cell(1,maxn));
%data()= struct('node',{[]},'coef',{[]});
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3.3241469234223e-01
1.1079587242244e-01   ]},'coef',{[4.1890734835153e-53
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1.2153798684410e-01
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1.9846285025419e-01
2.1889262958744e-01   ]});
%% 计算
area=0;
noden=data(n).node;
coefn=data(n).coef;
k=n;
n=10*n;  %n转换为表示Gauss积分的积分节点或积分系数的个数
if n/2-floor(n/2)==1/2 %当阶数为奇数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 %当阶数为偶数for i=1:n/2area=area+coefn(i)*(expfunc(noden(i))+expfunc(-noden(i)));end
end
%% 误差:前向（n-1阶）差商作为事后误差估计
if withErrorarea_=gaussHermiteIntegral(func,a,b,k-1,false);gErrer=abs(area-area_);
end
end

2 Gauss-Laguerre积分

2.1 测试Gauss-Laguerre积分函数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)) %标准正太分布
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));
%% 系统计算的积分精确值
% 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)
%% 测试误差项
n=5
[area,gErrer]=gaussLaguerreIntegral(f,0,inf,n,true)

运行结果：

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)
%高斯积分
%理论依据：数值分析方法 奚梅成
%func 被积函数；
%[a,b] 积分区间
%n 高斯积分的阶数
%%
%name：邓能财     Date: 2013/12/21
%% 默认参数
if nargin<5 withError=false; gErrer=0; end
%% 查错
maxn=10;
% assert(a==0 && b==inf,'积分区间必须为[0,inf)!');
% assert(ismember(n,[1:maxn]),['积分精确度数量级必须在1~',num2str(maxn),'之间!']);
%% 函数变换
expfunc=@(x)(exp(x)*func(x));
%% 数据：h,2h,3h,...,maxn*h阶高斯积分的积分系数
h=5;%   (积分节点node，积分系数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   ]});
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1.0415752443106e+02   ]},'coef',{[1.1604408602039e-01
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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   ]});
%% 计算
area=0;
noden=data(n).node;
coefn=data(n).coef;
k=h*n;  %k表示Gauss积分的积分节点或积分系数的个数
for i=1:karea=area+coefn(i)*expfunc(noden(i));
end
%% 误差:前向（n-1阶）差商作为事后误差估计
if withErrorarea_=gaussLaguerreIntegral(func,a,b,n-1,false);gErrer=abs(area-area_);
end
end

输出函数：

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

3 Gauss-Legendre积分W(x)=1

3.1 测试Gauss-Legendre积分W(x)=1函数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('各个阶数(h*)的gaussLegendreIntegral积分')
h=5
max=5
n_=[2:max];area_=[];gErrer_=[];
for n=2:max
%% 适合区间长为数量级10左右的积分[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)

运行结果：

f = @(x)(cos(x))
a =-1.570796326794897
b =1.570796326794897
各个阶数(h*)的gaussLegendreIntegral积分
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)
%高斯积分
%理论依据：数值分析方法 奚梅成
%func 被积函数；
%[a,b] 积分区间
%n 高斯积分的阶数
%%
%name：邓能财     Date: 2013/12/21
%% 默认参数
if nargin<5 withError=false; gErrer=0; end
%% 数据：1-8阶高斯积分的积分系数
h=5;
maxn=10; %   (±积分节点node，积分系数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   ]});
%% 积分变量的线性变换
y=@(x)(((b+a)+(b-a)*x)/2);
%% 计算
area=0;
noden=data(n).node;
coefn=data(n).coef;
k=n;
n=n*h;
if n/2-floor(n/2)==1/2 %当阶数为奇数   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 %当阶数为偶数  for i=1:n/2area=area+coefn(i)*(func(y(noden(i)))+func(y(-1*noden(i))));end
end
area=area*(b-a)/2;
%% 误差:前向（n-1阶）差商作为事后误差估计
if withErrorarea_=gaussLegendreIntegral(func,a,b,k-1,false);gErrer=abs(area-area_);
end
end

输出函数：

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

4 Gauss积分的积分系数计算

4.1 测试Gauss积分的积分系数计算的函数

clc,clear
% GaussLegendre积分：(积分节点    积分系数)
% disp('GaussLegendre积分：(积分节点+-node    积分系数coef)')
% for n=1:16
%     sprintf('************阶数 %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积分：(积分节点    积分系数)
% disp('GaussLaguerre积分：(积分节点    积分系数)')
% alpha=1e-30
% for n=1:16
%     sprintf('************阶数 %d',n)
%     [node,coef]=GaussLaguerre_2(n,alpha)
% end
%% GaussHermite 积分：(积分节点    积分系数)
% disp('GaussHermite积分：(积分节点    积分系数)')
% for n=1:20
%     sprintf('************阶数 %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()函数（打印Gauss积分数据表）：

function printGaussIntData(type,h)
%% 打印Gauss积分数据表
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

输出矩阵的函数：

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积分系数求解函数

function [x, w] = GaussHermite_2(n)
%name：邓能财     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积分系数求解函数

function [x, w] = GaussLaguerre_2(n, alpha)
%name：邓能财     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积分系数求解函数

function [x, w] = GaussLegendre_2(n)
% http://www.mathworks.cn/matlabcentral/fileexchange/8067-gauss-laguerre
%对应《数值分析方法 奚梅成的Gauss-Legendre积分》
%name：邓能财     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积分

5.1 测试NewtonCotes积分函数newtonCotesIntegral()

clc,clear
format long
disp('各个阶数的Newton-Cote`s积分')
% 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)) %标准正太分布
% 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)
%% 系统计算的积分精确值
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)
% %% 测试误差项
% a=0,b=.01
% n=3
% [area,gErrer]=newtonCotesIntegral(f,a,b,n,true)
% area_system=quad(f,a,b,1e-13)

输出数组的函数：

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

运行结果：

各个阶数的Newton-Cote`s积分
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积分
%理论依据：数值分析方法 奚梅成
%func 被积函数；
%[a,b] 积分区间
%n Newton-Cotes积分的阶数
%%
%name：邓能财     Date: 2013/12/21
%% 默认参数
if nargin<5 withError=false; gErrer=0; end
%% 查错
maxn=8;
% assert(a~=inf && b~=inf,'积分区间不能包含inf!');
% assert(ismember(n,[1:maxn]),['积分精确度数量级必须在1~',num2str(maxn),'之间!']);
%% 数据：1,2,...,maxn阶Newton-Cotes积分的积分系数%   (积分系数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 ]});
%% 计算
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);
%% 误差:前向（n-1阶）差商作为事后误差估计
if withErrorarea_=newtonCotesIntegral(func,a,b,n-1,false);gErrer=abs(area-area_);gErrer=gErrer*(10^(.1*(n+1))); %根据试验值调整误差估计
end
% %% 误差:两分段（n-1阶）积分和与原积分之差作为事后误差估计
% 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积分

6.1 测试Romberg积分函数rombergIntegral()

clc,clear
format long
f =  @(x)(cos(x))
a = -1.57079632679490
b = 1.57079632679490
'各种精度的Romberg积分'
% 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)

输出数组的函数：

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

运行结果：

f = @(x)(cos(x))
a =-1.570796326794900
b =1.570796326794900
ans =
各种精度的Romberg积分
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积分
%理论依据：数值分析方法 奚梅成
%func 函数；
%[a,b] 积分区间
%pre 精度
%程序特点：密切结合理论，将内存和运算量达到极小化%只记录最后一排需用的数字
%%
%name：邓能财     Date: 2013/12/20
%这几乎是我编写的第一个【调试只出一个错误】的程序
%% 计算
%取初始分段数为1
h=b-a;
t=[];%记录最后一排需用的数字
%Romberg积分的阶数：k%即所储存的T的长度
k=1;
%计算梯形积分T0(h)
t=(func(a)+func(b))/2;
while truet=[0,t];k=k+1;%h1用于T0(h)与T0(h/2)之间的递推计算h1=0;for xi=a+h/2:h:b-h/2h1=h1+func(xi);end, h1=h1*h;%计算梯形积分T0(h/2)t(1)=(t(2)+h1)/2;h=h/2;%计算所得积分最精确值bit=1;%记录2^(2k)for i=2:kbit=bit*4;t(i)=t(i-1)+(t(i-1)-t(i))/(bit-1);end%判断是否达到精度要求if abs(t(end)-t(end-1))<precbreak;end
end
area=t(end);
end

7 复化积分

7.1 测试复化积分函数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
'各种精度的复化梯形积分积分'
% 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)

输出数组的函数：

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

运行结果：

f = @(x)(cos(x))
a =-1.570796326794900
b =1.570796326794900
ans =
各种精度的复化梯形积分积分
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函数
%a,b积分区间
%pre精度
%m4区间上4阶导的最大值
%name：邓能财     Date: 2013/12/23
%%%%%%%%%%%%%%%%%%%%%%：%【表需在尾部添加分号‘；’
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为奇数的项
ss2=0; %n为偶数的项
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函数
%a,b积分区间
%pre精度
%m2区间上二阶导的最大值
%name：邓能财     Date: 2013/12/23
%%%%%%%%%%%%%%%%%%%%%%：%【表需在尾部添加分号‘；’
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 自适应运算积分

8.1 测试自适应运算积分函数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)
'各种精度的自适应Simpson积分'
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)

输出数组的函数：

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

运行结果：

f = @(x)(cos(x))
a =-1.570796326794900
b =1.570796326794900
ans =
各种精度的自适应Simpson积分
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公式的自适应积分
%理论依据：数值分析方法 奚梅成
%name：邓能财     Date: 2013/12/23
%func 函数；
%[a,b] 积分区间
%pre 精度
%算法：递归
%程序特点：密切结合理论，将内存和运算量达到极小化
%% 计算
%%disp('##Start')
%取分段次数为n
%n=3时，Simpson积分系数序列为: 1424241%，已经能够发挥Simpson公式的作用，且分段数最少
n=3;
h=(b-a)/n;
%计算梯形积分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;
%计算梯形积分T_2n
s2=(s1+h1)/2;
%计算Simpson积分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;
%计算Simpson积分S_2n
s2=s1/2+(4*h2-h1)/6;
%% 判断是否达到要求精度
deep=1;
if(abs(s1-s2)<15*prec)area=s2;%disp('##End')return;
else %不满足精度，进一步细分[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 积分系数——二

9.1 Gauss-Legendre积分系数

格式：序号【 积分节点 ******积分系数
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积分系数

表6—1
【】表错误
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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end

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matlab数值积分的计算

标签（空格分隔）： matlab 积分数值计算

文章目录

matlab数值积分

1 Gauss-Hermite积分

1.1 测试Gauss-Hermite积分函数gaussHermiteIntegral()

1.2 gaussHermiteIntegral()

2 Gauss-Laguerre积分

2.1 测试Gauss-Laguerre积分函数gaussLaguerreIntegral()

2.2 gaussLaguerreIntegral()

3 Gauss-Legendre积分W(x)=1

3.1 测试Gauss-Legendre积分W(x)=1函数gaussLegendreIntegral()

3.2 gaussLegendreIntegral()

4 Gauss积分的积分系数计算

4.1 测试Gauss积分的积分系数计算的函数

4.2 Gauss-Hermite积分系数求解函数

4.3 Gauss-Laguerre积分系数求解函数

4.4 Gauss-Legendre积分系数求解函数

5 NewtonCotes积分

5.1 测试NewtonCotes积分函数newtonCotesIntegral()

5.2 newtonCotesIntegral()

6 Romberg积分

6.1 测试Romberg积分函数rombergIntegral()

6.2 rombergIntegral()

7 复化积分

7.1 测试复化积分函数complexSimpson()

7.2 complexSimpson()

7.3 complexTrangleInt()

8 自适应运算积分

8.1 测试自适应运算积分函数selfAdaptIntegral()

8.2 selfAdaptIntegral()

9 积分系数——二

9.1 Gauss-Legendre积分系数

9.2 newton cotes积分系数

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matlab数值积分的计算

标签（空格分隔）： matlab 积分 数值 计算

文章目录

matlab数值积分

1 Gauss-Hermite积分

1.1 测试Gauss-Hermite积分函数gaussHermiteIntegral()

1.2 gaussHermiteIntegral()

2 Gauss-Laguerre积分

2.1 测试Gauss-Laguerre积分函数gaussLaguerreIntegral()

2.2 gaussLaguerreIntegral()

3 Gauss-Legendre积分W(x)=1

3.1 测试Gauss-Legendre积分W(x)=1函数gaussLegendreIntegral()

3.2 gaussLegendreIntegral()

4 Gauss积分的积分系数计算

4.1 测试Gauss积分的积分系数计算的函数

4.2 Gauss-Hermite积分系数求解函数

4.3 Gauss-Laguerre积分系数求解函数

4.4 Gauss-Legendre积分系数求解函数

5 NewtonCotes积分

5.1 测试NewtonCotes积分函数newtonCotesIntegral()

5.2 newtonCotesIntegral()

6 Romberg积分

6.1 测试Romberg积分函数rombergIntegral()

6.2 rombergIntegral()

7 复化积分

7.1 测试复化积分函数complexSimpson()

7.2 complexSimpson()

7.3 complexTrangleInt()

8 自适应运算积分

8.1 测试自适应运算积分函数selfAdaptIntegral()

8.2 selfAdaptIntegral()

9 积分系数——二

9.1 Gauss-Legendre积分系数

9.2 newton cotes积分系数

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标签（空格分隔）： matlab 积分数值计算