定積分的近似計(jì)算

1、數(shù)學(xué)實(shí)驗(yàn)報(bào)告實(shí)驗(yàn)序號(hào)：03 日期：2014年 11 月14日班級(jí)一班姓名學(xué)號(hào)實(shí)驗(yàn)名稱定積分的近似運(yùn)算問(wèn)題背景描述： 利用牛頓萊布尼茲公式雖然可以精確地計(jì)算定積分的值，但它僅適用于被積函數(shù)的原函數(shù)能用初等函數(shù)表達(dá)出來(lái)的情形如果這點(diǎn)辦不到或者不容易辦到，這就有必要考慮近似計(jì)算的方法在定積分的很多應(yīng)用問(wèn)題中，被積函數(shù)甚至沒(méi)有解析表達(dá)式，可能只是一條實(shí)驗(yàn)記錄曲線，或者是一組離散的采樣值，這時(shí)只能應(yīng)用近似方法去計(jì)算相應(yīng)的定積分 本實(shí)驗(yàn)將主要研究定積分的三種近似計(jì)算算法：矩形法、梯形法、拋物線法對(duì)于定積分的近似數(shù)值計(jì)算。實(shí)驗(yàn)?zāi)康模?1加深理解積分理論中分割、近似、求和、取極限的思想方法； 2了解定積分近似

2、計(jì)算的矩形法、梯形法與拋物線法； 3會(huì)用MATLAB語(yǔ)言編寫(xiě)求定積分近似值的程序，會(huì)用MALAB中的命令求定積分實(shí)驗(yàn)原理與數(shù)學(xué)模型： 1 矩形法 根據(jù)定積分的定義，每一個(gè)積分和都可以看作是定積分的一個(gè)近似值，即 在幾何意義上，這是用一系列小矩形面積近似小曲邊梯形的結(jié)果，所以把這個(gè)近似計(jì)算方法稱為矩形法不過(guò)，只有當(dāng)積分區(qū)間被分割得很細(xì)時(shí)，矩形法才有一定的精確度 針對(duì)不同的取法，計(jì)算結(jié)果會(huì)有不同。 （1） 左點(diǎn)法：對(duì)等分區(qū)間 ， 在區(qū)間上取左端點(diǎn)，即取。 （2）右點(diǎn)法：同（1）中劃分區(qū)間，在區(qū)間上取右端點(diǎn)，即取。 （3）中點(diǎn)法：同（1）中劃分區(qū)間，在區(qū)間上取中點(diǎn)，即取。 2 梯形法等分區(qū)間，相應(yīng)函

3、數(shù)值為 （）曲線上相應(yīng)的點(diǎn)為 （）將曲線的每一段弧用過(guò)點(diǎn)，的弦（線性函數(shù)）來(lái)代替，這使得每個(gè)上的曲邊梯形成為真正的梯形，其面積為，于是各個(gè)小梯形面積之和就是曲邊梯形面積的近似值，即 ，稱此式為梯形公式。3 拋物線法將積分區(qū)間作等分，分點(diǎn)依次為，對(duì)應(yīng)函數(shù)值為（），曲線上相應(yīng)點(diǎn)為（）現(xiàn)把區(qū)間上的曲線段用通過(guò)三點(diǎn)，的拋物線來(lái)近似代替，然后求函數(shù)從到的定積分：由于，代入上式整理后得同樣也有將這個(gè)積分相加即得原來(lái)所要計(jì)算的定積分的近似值：，即這就是拋物線法公式.實(shí)驗(yàn)所用軟件及版本：Matlab2012b Wps2013主要內(nèi)容（要點(diǎn)）：1.實(shí)現(xiàn)實(shí)驗(yàn)內(nèi)容中的例子，即采用矩形法，梯形法，拋物線法計(jì)算，取n=

4、258,并比較三種方法的精確程度。2.分別用梯形法與拋物線法，計(jì)算，取并嘗試直接使用函數(shù)trapz()、quad()進(jìn)行計(jì)算求解，比較結(jié)果的差異3. 試計(jì)算定積分（注意：可以運(yùn)用trapz()、quad()或附錄程序求解嗎？為什么？）4.將的近似計(jì)算結(jié)果與matlab中各命令的計(jì)算結(jié)果相比較，使猜測(cè)matlab中的數(shù)值積分命令最可能采用了哪一種近似計(jì)算方法？并找出其他例子支持你的觀點(diǎn)。5.通過(guò)整個(gè)實(shí)驗(yàn)內(nèi)容和練習(xí)，你能否做出一些理論上的小結(jié)，即針對(duì)什么類型的函數(shù)（具有某種單調(diào)特性和凹凸特性），用某種近似計(jì)算方法所得結(jié)果更接近于實(shí)際值？6 學(xué)習(xí)fulu2sum.m的程序設(shè)計(jì)方法，嘗試用函數(shù) sum

5、 改寫(xiě)附錄1和附錄3的程序，避免for 循環(huán)。實(shí)驗(yàn)過(guò)程記錄（含基本步驟、主要程序清單及異常情況記錄等）：一1.矩形法：format longn=258;a=0;b=1;syms x fxfx=1/(1+x2);i=1:n;xj=a+(i-1)*(b-a)/n; %左點(diǎn)xi=a+i*(b-a)/n; %右點(diǎn)xij=(xi+xj)/2;fxj=subs(fx,x,xj); %左點(diǎn)值fxi=subs(fx,x,xi); %右點(diǎn)值fxij=subs(fx,x,xij); %中點(diǎn)值f1=fxj*(b-a)/n;f2=fxi*(b-a)/n;f3=fxij*(b-a)/n;inum1=sum(f1)inu

6、m2=sum(f2)inum3=sum(f3)integrate=int(fx,0,1);integrate=double(integrate);fprintf(the relative error between inum1 and real-value is about: %gnn,. abs(inum1-integrate)/integrate)fprintf(the relative error between inum2 and real-value is about: %gnn,. abs(inum2-integrate)/integrate)fprintf(the relativ

7、e error between inum3 and real-value is about: %gnn,.abs(inum3-integrate)/integrate)調(diào)試結(jié)果：inum1 = 0.786366529681526inum2 = 0.784428545185402inum3 = 0.785398476379441the relative error between inum1 and real-value is about: 0.00123296the relative error between inum2 and real-value is about: 0.00123456

8、the relative error between inum3 and real-value is about: 3.98501e-07 2.梯形法format longn=258;a=0;b=1;syms x fxfx=1/1+x2;i=1:n;xj=a+(i-1)*(b-a)/n; %所有左點(diǎn)的數(shù)組xi=a+i*(b-a)/n; %所有右點(diǎn)的數(shù)組fxj=subs(fx,x,xj); %所有左點(diǎn)值fxi=subs(fx,x,xi); %所有右點(diǎn)值f=(fxi+fxj)/2*(b-a)/n; %梯形面積inum=sum(f) %加和梯形面積求解integrate=int(fx,1,2);in

9、tegrate=double(integrate)fprintf(The relative error between inum and real-value is about:%g/n/n,.abs(inum-integrate)/integrate)調(diào)試結(jié)果：inum = 1.333335837189272integrate = 3.333333333333334The relative error between inum and real-value is about:0.599999/n/n 3. .拋物線法：format longn=258;a=0;b=1;inum=0;syms

10、x fxfx=1/1+x2;for i=1:n xj=a+(i-1)*(b-a)/n; %左點(diǎn) xi=a+i*(b-a)/n; %右點(diǎn) xk=(xi+xj)/2; %中點(diǎn) fxj=subs(fx,x,xj); fxi=subs(fx,x,xi); fxk=subs(fx,x,xk);inum=inum+(fxj+4*fxk+fxi)*(b-a)/(6*n);endinumintegrate=int(fx,1,2);integrate=double(integrate);fprintf(The relative error between inum and real-value is about

11、:%g/n/n,.abs(inum-integrate)/integrate)調(diào)試結(jié)果：inum = 1.333333333333334The relative error between inum and real-value is about:0.6/n/n二：1.梯形法format longn=120;a=1;b=2;syms x fxfx=1/x;i=1:n;xj=a+(i-1)*(b-a)/n; %所有左點(diǎn)的數(shù)組xi=a+i*(b-a)/n; %所有右點(diǎn)的數(shù)組fxj=subs(fx,x,xj); %所有左點(diǎn)值fxi=subs(fx,x,xi); %所有右點(diǎn)值f=(fxi+fxj)/2

12、*(b-a)/n; %梯形面積inum=sum(f) %加和梯形面積求解integrate=int(fx,1,2);integrate=double(integrate)fprintf(The relative error between inum and real-value is about:%g/n/n,.abs(inum-integrate)/integrate)【調(diào)試結(jié)果】inum = 0.69315152080005integrate = 0.69314718055995The relative error between inum and real-value is about:

13、6.26164e-006/n/n2.拋物線法：format longn=120;a=1;b=2;inum=0;syms x fxfx=1/x;for i=1:n xj=a+(i-1)*(b-a)/n; %左點(diǎn) xi=a+i*(b-a)/n; %右點(diǎn) xk=(xi+xj)/2; %中點(diǎn) fxj=subs(fx,x,xj); fxi=subs(fx,x,xi); fxk=subs(fx,x,xk); inum=inum+(fxj+4*fxk+fxi)*(b-a)/(6*n);endinumintegrate=int(fx,1,2);integrate=double(integrate);fprin

14、tf(The relative error between inum and real-value is about:%g/n/n,.abs(inum-integrate)/integrate)【調(diào)試結(jié)果】inum = 0.69314718056936The relative error between inum and real-value is about:1.35886e-011/n/n3.使用函數(shù)trapz()x=1:1/120:2;y=1./x;trapz(x,y)【調(diào)試結(jié)果】ans =0.693151520800054.使用函數(shù)quad()quad(1./x,1,2)【調(diào)試結(jié)果】a

15、ns = 0.69314719986297四1.梯形法：format long n=100;a=0;b=1;inum=0;syms x fx fx=1/(1+x2); for i=1:n xj=a+(i-1)*(b-a)/n; xi=a+i*(b-a)/n; fxj=subs(fx,x,xj); fxi=subs(fx,x,xi); inum=inum+(fxj+fxi)*(b-a)/(2*n); end inum integrate=int(fx,0,1) integrate=double(integrate) fprintf(The relative error between inum

16、and real-value is about: %gnn,. abs(inum-integrate)/integrate)【調(diào)試結(jié)果】:inum = 0.785393996730783integrate = pi/4 integrate = 0.785398163397448The relative error between inum and real-value is about: 5.30516e-062.矩形法：format long n=100;a=0;b=1;inum1=0;inum2=0;inum3=0; syms x fx fx=1/(1+x2); for i=1:n xj=

17、a+(i-1)*(b-a)/n; xi=a+i*(b-a)/n; fxj=subs(fx,x,xj); fxi=subs(fx,x,xi); fxij=subs(fx,x,(xi+xj)/2); inum1=inum1+fxj*(b-a)/n; inum2=inum2+fxi*(b-a)/n; inum3=inum3+fxij*(b-a)/n; end inum1 inum2 inum3 integrate=int(fx,0,1) integrate=double(integrate) fprintf(The relative error between inum1 and real-valu

18、e is about: %gnn,. abs(inum1-integrate)/integrate) fprintf(The relative error between inum2 and real-value is about: %gnn,. abs(inum2-integrate)/integrate) fprintf(The relative error between inum3 and real-value is about: %gnn,. abs(inum3-integrate)/integrate) 【調(diào)試結(jié)果】: inum1 = 0.787893996730782 inum2

19、 = 0.782893996730782 inum3 = 0.785400246730781 integrate = pi/4 integrate = 0.785398163397448The relative error between inum1 and real-value is about: 0.00317779The relative error between inum2 and real-value is about: 0.0031884nThe relative error between inum3 and real-value is about: 2.65258e-063.

20、梯形法 求和:format long n=100;a=0;b=1; syms x fx fx=1/(1+x2); i=1:n; xj=a+(i-1)*(b-a)/n; %所有左點(diǎn)的數(shù)組 xi=a+i*(b-a)/n; %所有右點(diǎn)的數(shù)組 fxj=subs(fx,x,xj); %所有左點(diǎn)值 fxi=subs(fx,x,xi); %所有右點(diǎn)值 f=(fxi+fxj)/2*(b-a)/n; %梯形面積 inum=sum(f) %加和梯形面積求解 integrate=int(fx,0,1) integrate=double(integrate) fprintf(The relative error be

21、tween inum and real-value is about: %gnn,. abs(inum-integrate)/integrate) 【調(diào)試結(jié)果】: inum = 0.785393996730783 integrate = pi/4 integrate = 0.785398163397448 The relative error between inum and real-value is about: 5.30516e-064.拋物線法：format long %2*n=200; n=100;a=0;b=1;inum=0; syms x fx fx=1/(1+x2); for

22、i=1:n x0=a+(2*i-2)*(b-a)/(2*n);x1=a+(2*i-1)*(b-a)/(2*n);x2=a+(2*i-0)*(b-a)/(2*n);fx0=subs(fx,x,x0); fx1=subs(fx,x,x1); fx2=subs(fx,x,x2); si=(fx0+4*fx1+fx2)*(b-a)/(6*n);inum=inum+si; end inum integrate=int(fx,0,1) integrate=double(integrate) fprintf(拋物線的相對(duì)誤差為：%en, abs(inum-integrate)/integrate) 【調(diào)試結(jié)

23、果】:inum = 0.785398163397448integrate = pi/4 integrate = 0.785398163397448數(shù)值計(jì)算：1.(符號(hào)求積分) ：int(1/(1+x2),x,0,1) int(sym(1/(1+x2),x,0,1) 【調(diào)試結(jié)果】: ans =pi/42.（拋物線法求數(shù)值積分）：quad(1./(1+x.2),0,1) 【調(diào)試結(jié)果】： ans = 0.785398149243260 3.（梯形法求數(shù)值積分）： x=0:0.001:1; y=1./(1+x.2); trapz(x,y)【調(diào)試結(jié)果】：ans = 0.785398121730782六.

24、1.矩形法：利用求和函數(shù)%矩陣法format longn=100;a=0;b=1;syms x fxfx=1/(1+x2);i=1:n;xj=a+(i-1)*(b-a)/n; %左點(diǎn)xi=a+i*(b-a)/n; %右點(diǎn)xij=(xi+xj)/2;fxj=subs(fx,x,xj); %左點(diǎn)值fxi=subs(fx,x,xi); %右點(diǎn)值fxij=subs(fx,x,xij); %中點(diǎn)值f1=fxj*(b-a)/n;f2=fxi*(b-a)/n;f3=fxij*(b-a)/n;inum1=sum(f1)inum2=sum(f2)inum3=sum(f3)integrate=int(fx,0,1

25、);integrate=double(integrate);fprintf(the relative error between inum1 and real-value is about: %gnn,. abs(inum1-integrate)/integrate)fprintf(the relative error between inum2 and real-value is about: %gnn,. abs(inum2-integrate)/integrate)fprintf(the relative error between inum3 and real-value is abo

26、ut: %gnn,.abs(inum3-integrate)/integrate)【調(diào)試結(jié)果】inum1 = 0.78789399673078 inum2 = 0.78289399673078 inum3 = 0.78540024673078the relative error between inum1 and real-value is about: 0.00317779the relative error between inum2 and real-value is about: 0.0031884the relative error between inum3 and real-va

27、lue is about: 2.65258e-0062.拋物線法：使用求和函數(shù)%拋物線format longn=100;a=0;b=1;syms x fxfx=1/(1+x2);i=1:n;xj=a+(i-1)*(b-a)/n; %左點(diǎn)xi=a+i*(b-a)/n; %右點(diǎn)xij=(xi+xj)/2;fxj=subs(fx,x,xj); %左點(diǎn)值fxi=subs(fx,x,xi); %右點(diǎn)值fxij=subs(fx,x,xij); %中點(diǎn)值f=(fxj+4*fxij+fxi)*(b-a)/(6*n);inum=sum(f)integrate=int(fx,0,1);integrate=doub

28、le(integrate);fprintf(the relative error between inum and real-value is about: %gnn,. abs(inum-integrate)/integrate)【調(diào)試結(jié)果】 inum = 0.78539816339745the relative error between inum and real-value is about: 2.82716e-016實(shí)驗(yàn)結(jié)果報(bào)告與實(shí)驗(yàn)總結(jié)：1 1.矩形法調(diào)試結(jié)果：inum1 = 0.786366529681526inum2 = 0.784428545185402inum3 = 0.7

29、85398476379441the relative error between inum1 and real-value is about: 0.00123296the relative error between inum2 and real-value is about: 0.00123456the relative error between inum3 and real-value is about: 3.98501e-072. 梯形法：調(diào)試結(jié)果：inum = 1.333335837189272integrate = 3.333333333333334The relative error between inum and real-value is about:0.599999/n/n 3. 拋物線法：調(diào)試結(jié)果：inum = 1.333333333333334The relative error between inum and real-value is about:0.6/n/n二：1.梯形法：調(diào)試結(jié)果：inum = 0.69315152080005integrate = 0.69314718055995The relative error between inum and real-value is about:6.26164e-006/n/n2