常州大学数值分析作业 第四章
姓名：李俊乾专业：化学
用最小二乘法求形如y=axe的拟合函数。
答：Matlab程序
function [a,b]=ec(x,y)
Y=log(y)';
A=zeros(5,3);
for i=1:5
A(i,1)=1;
A(i,2)=log(x(i));
A(i,3)=i;
end
c=inv(A'*A)*(A'*Y);
a=exp(c(1));
b=c(3);
for i=1:5
y=a*x.*exp(b*x);
end
return
x=[1 2 3 4 5];
y=[1.222 2.984 5.466 8.902 13.592]; [a,b]=ec(x,y)
输出结果为：
a = 1.000202219673205
b = 0.200293860504786
plot(x,y,'b*',x,1*x.*exp(0.2*x),'r-')
7、已知人体表面积S和人体身高h，体重w有近似关系式S=α0hα1wα2。
试根据身高，体重及相应的人体表面积的一组观测值
工程（专）学号：14102932
（h i,w i,S i）(i=0,1,2….n)来估计参数α0α1
α2的大小
答：Matlab程序：
function [a0,a1,a2]=ec2(h,w)
S=log(s)';N=length(h);
A=zeros(N,3);
for i=1:5
A(i,1)=1;
A(i,2)=log(h(i));
A(i,3)=log(w(i));
end
c=inv(A'*A)*(A'*S);
a0=exp(c(1));
a1=c(2);
a2=c(3);
return
%给出数据
h=[175 172 183 164 156];
w=[80 90 80 70 65];
s=[1000 900 1200 750 800];
[a0,a1,a2]=ec2(h,w,s)
输出结果为：
a0 =1.614815742043648e-04
a1 =3.383163094165866
a2 =-0.419165011582663
8、学习Matlab内部的函数lsqcurvefit，并
设计数值实验使用lsqcurvefit。
答：Matlab内部函数lsqcurvefit是用来解决
非线性拟合的最小二乘问题的。
其调用格式为：
x= lsqcurvefit(fun,x0,xdata,ydata) x=lsqcurvefit(fun,x0,xdata,ydata
,lb,ub)
x=lsqcurvefit(fun,x0,xdata,ydata
,lb,ub,options)
[x,resnorm] = lsqcurvefit(…)
[x,resnorm,residual]=lsqcurvefit (…)
[x,resnorm,residual,exitflag]= lsqcurvefit(…)
[x,resnorm,residual,exitflag,out put] = lsqcurvefit(…)
[x,resnorm,residual,exitflag,out put,lambda] = lsqcurvefit(…) [x,resnorm,residual,exitflag,out put,lambda,jacobian]
=lsqcurvefit(…)
输入参数：
fun为待拟合函数，计算x处拟合函数值，其定义为function F=myfun(x,xdata)
x0为初始解向量，即拟合参数的初始解；xdata，ydata为满足关系ydata=F(x, xdata)的数据；
lb、ub为解向量的下界和上界lb≤x≤ub，若没有指定界，则lb=[ ]，ub=[ ]；
options为指定的优化参数；
输出参数：
x为迭代得出解向量，即拟合出的参数；resnorm=sum ((fun(x,xdata)-ydata).^2)，即x 处残差平方和,最小二乘式值;
residual=fun(x,xdata)-ydata，即在x处的残差；
exitflag为终止迭代的条件；
output为输出的优化信息；
lambda为解x处的Lagrange乘子；jacobian为解x处拟合函数fun的jacobian 矩阵。
function F = myfun(x,xdata)
F=(x(1).*xdata).*(exp(x(2).*xda ta));
xdata=[1,2,3,4,5];
ydata=[1.222,2.984,5.466,8.902,13.592];
x0=[0,0];
[x,resnorm]=lsqcurvefit(@myfun,x0,xdata,yda ta)
输出结果为：
Local minimum found.
Optimization completed because the size of the gradient is less than the default value of the function tolerance. <stopping criteria details>
x =
0.999958348976391 0.200014132812834 resnorm = 8.067930437509675e-7。
数值分析第四章习题
数值分析第四章习题第四章习题1. 采用数值计算方法，画出dt t t x y x ?=0sin )(在]10 ,0[区间曲线，并计算)5.4(y 。
〖答案〗1.65412. 求函数x e x f 3sin )(=的数值积分?=π 0 )(dx x f s ，并请采用符号计算尝试复算。
〖答案〗s = 5.1354Warning: Explicit integral could not be found. > In sym.int at 58s =int(exp(sin(x)^3),x = 0 .. pi)3. 用quad 求取dx x e x sin 7.15?--ππ的数值积分，并保证积分的绝对精度为910-。
〖答案〗1.087849437547794. 求函数5.08.12cos 5.1)5(sin )(206.02++-=t t t et t f t 在区间]5,5[-中的最小值点。
〖答案〗最小值点是-1.28498111480531 相应目标值是-0.186048010065455. 设0)0(,1)0(,1)(2)(3)(22===+-dt dy y t y dt t dy dt t y d ，用数值法和符号法求5.0)(=t t y 。
〖答案〗数值解y_05 = 0.78958020790127符号解ys =1/2-1/2*exp(2*t)+exp(t)ys_05 =.789580356470605529168507052137806. 求矩阵b Ax =的解，A 为3阶魔方阵，b 是)13(?的全1列向量。
〖答案〗x =0.06670.06670.06677. 求矩阵b Ax =的解，A 为4阶魔方阵，b 是)14(?的全1列向量。
〖答案〗解不唯一x =-0.0074 -0.0809 0.1397 0.0662 0.0588 0.1176 -0.0588。
数值分析作业(3,4,5,6)
数值分析作业(3)思考题：1：（a）仅当系数矩阵是病态或奇异的时候，不选主元的Gauss消元法才会失败。
(b) 系数矩阵是对称正定的线性方程组总是良态的；(c) 两个对称矩阵的乘积依然是对称的；(d) 如果一个矩阵的行列式值很小，则它很接近奇异；(e) 两个上三角矩阵的乘积仍然是上三角矩阵；(f) 一个非奇异上三角矩阵的逆仍然是上三角矩阵；(g) 一个奇异矩阵不可能有LU分解；(h) 奇异矩阵的范数一定是零；(i) 范数为零的矩阵一定是零矩阵；（j）一个非奇异的对称阵，如果不是正定的则不能有Cholesky分解。
2: 全主元Gauss消元法与列主元Gauss消元法的基本区别是什么？它们各有什么优点？3：满足下面是我什么条件，可以判定矩阵接近奇异？（a）矩阵的行列式小；（b）矩阵的条件数小；（c）矩阵的范数小；（d）矩阵的条件数大；（e ）矩阵的范数大； （f ）矩阵的元素小4: Jacobi 迭代法和Gauss-Seidel 迭代法相比(a) 两种的基本差别是什么？(b) 哪种方法更适合并行计算？（c ）哪种方法更节省存储空间？（d ）Jacobi 迭代法是否运算速度更快？习题：1．对矩阵2112112112A -⎡⎤⎢⎥--⎢⎥=⎢⎥--⎢⎥-⎣⎦,试求A 的Cholesky 分解。
2. 对矩阵12122211111,222221112A A --⎡⎤⎡⎤⎢⎥⎢⎥==⎢⎥⎢⎥⎢⎥⎢⎥--⎣⎦⎣⎦证明：（1）求解以1A 为系数矩阵的线性方程组，Jacobi 迭代是收敛的，而Gauss-Seidel 迭代发散；（2）求解以2A 为系数矩阵的线性方程组，Jacobi 迭代是发散的，而Gauss-Seidel 迭代收敛。
3.对矩阵11,1a a A a a a a ⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦（1） 参数a 取什么值时，矩阵是正定的？（2） a 取什么值，求解以A 为系数矩阵的线性方程组，Jacobi 迭代是收敛的。
《数值分析》第四章答案
习题41. 给定x x f =)(在144,121,100=x 3点处的值，试以这3点建立)(x f 的2次(抛物)插值公式，利用插值公式115求的近似值并估计误差。
再给13169=建立3次插值公式，给出相应的结果。
解：x x f =)( 2121)(-='x x f ，2341)(--=''x x f ,2583)(-='''x x f ,27)4(1615)(--=x x f，72380529.10)115(=f1000=x ， 1211=x ， 1442=x ， 1693=x 100=y ， 111=y ， 122=y ， 133=y))(())(())(())(())(())(()(1202102210120*********x x x x x x x x y x x x x x x x x y x x x x x x x x y x L ----+----+----= )121144)(100144()121115)(100115(12)144121)(100121()144115)(100115(11)144100)(121100()144115)(121115(10)115(2----⨯+----⨯+----⨯=L=2344)6(1512)23(21)29(1511)44)(21()29)(6(10⨯-⨯⨯+-⨯-⨯⨯+----⨯72276.1006719.190683.988312.1=-+=))()((!3)()()(2102x x x x x x f x L x f ---'''=-ξ ，144100<<ξ )44115()121115()100115()(max 61)115()115(1441002-⨯-⨯-⋅'''≤-≤≤x f L f x 296151083615⨯⨯⨯⨯⨯≤-001631.0101631.02=⨯=- 实际误差 22101045.0)115()115(-⨯=-L f))()(())()(())()(())()(()(312101320130201032103x x x x x x x x x x x x y x x x x x x x x x x x x y x L ------+------= ))()(())()(())()(())()((23130321033212023102x x x x x x x x x x x x y x x x x x x x x x x x x y ------+------+ )169100()144100()121100()169115()144115()121115(10)115(3-⨯-⨯--⨯-⨯-⨯=L )169121()144121()100121()169115()144115()100115(11-⨯-⨯--⨯-⨯-⨯+)169144()121144()100144()169115()121115()100115(12-⨯-⨯--⨯-⨯-⨯+)144169()121169()100169()144115()121115()100115(13-⨯-⨯--⨯-⨯-⨯+)48()23(21)54()29(1511)69()44()21()54()29()6(10-⨯-⨯-⨯-⨯⨯+-⨯-⨯--⨯-⨯-⨯= 254869)29()6(1513)25(2344)54()6(1512⨯⨯-⨯-⨯⨯+-⨯⨯-⨯-⨯⨯+ 723571.10409783.0305138.2145186.11473744.1=+-+= ))()()((!4)()()(3210)4(3x x x x x x x x f x L x f ----=-ξ，169100<<ξ)169115)(144115)(121115)(10115(101615241)115()115(73----⨯⨯⨯≤--L f )54()29()6(151016152417-⨯-⨯-⨯⨯⨯⨯=- 0005505.0105505.03=⨯=-实际误差 321023429.0)115()115(-⨯=-L f 2. 设j x 为互异节点),,1,0(n j =求证： (1)k nj j k j x x l x =∑=)(0),,1,0(n k =;(2)0)()(0=-∑=x l x x j knj j ),,1(n k =。
数值分析(第四版)课后习题及答案
0.30
0.39
0.45
0.53
yj
0.5000
0.5477
0.6245
0.6708
0.7280
试求三次样条插值 S (x) 并满足条件
i) S(0.25) 1.0000, S(0.53) 0.6868; ii) S(0.25) S(0.53) 0.
25. 若 f (x) C2 a,b, S (x) 是三次样条函数,证明
12. 在 1,1 上利用插值极小化求 1 f (x) tg 1x 的三次近似最佳逼近多项式.
13. 设 f (x) ex 在 1,1 上的插值极小化近似最佳逼近多项式为 Ln (x) ,若 f Ln 有界,
证明对任何 n 1,存在常数 n 、 n ,使
改用另一等价公式
ln(x x2 1) ln(x x2 1)
计算,求对数时误差有多大?
x1 1010 x2 1010 ; x1 x2 2.
14. 试用消元法解方程组
假定只用三位数计算,问结果是否可靠?
s 1 ab sin c,
0c
15. 已知三角形面积 2
n
x
k j
j1 f (xj )
0,0k n2; an1 ,k n1.
15. 证明 n 阶均差有下列性质:
i) 若 F (x) cf (x) ,则 F x0, x1,, xn cf x0, x1,, xn ;
ii) 若 F (x) f (x) g(x) ,则 F x0, x1,, xn f x0, x1,, xn g x0, x1,, xn .
5.
设 xk
x0
常州大学数值分析作业
常州大学数值分析作业1.解：（1）x = [ 3*π/8 π/2];Y = cos(x);x0 = π/3;[A,Y] = lagrange(x,y,x0);P1 = vpa(poly2sym(A),3)结果如下：P1 = 1.53*x - 0.974Y = 0.5102（2）x = [π/4 3*π/8π/2];Y = cos(x);[A,Y] = lagrange(x,y,x0);P2=vpa(poly2sym(A),3)结果如下：P2 = 1.18*x^2 - 0.455*x - 0.189Y = 0.4973（3）x = [0 π/8 π/4 3*π/8 π/2];Y = cos(x);[A,Y]=lagrange(x,y,x0);结果如下：P3 = x^4 + 0.00282*x^3 - 0.514*x^2 + 0.0232*x + 0.0287 Y = 0.50017.function [T]=aitken(x,y,x0,T0)If nargin == 3T0=[];endn0=size(T0,1);m=max(size(x));n=n0+m;T=zeros(n,n+1);T(1:n0,1:n0+1)=T0;T(n0+1:n,1)=x;T(n0+1:n, 2)=y; ifn0==0i0=2;elsei0=n0+1;Endx=[0 1];y=[0.5 1.25];x0=2.8;T0=aitken(x,y,x0);T=T0;x=[3.0,4.0]';y=[3.5,2.75]';x0=2.8;T=aitken(x,y,x0,T0);n=max(size(x))+size(T0,1);for i=1:nfor j=1:i+1fprintf('%10.4f',T(i,j));endfprintf('\n');EndReturn0.0000 0.5000 0 0 01.0000 1.25002.6000 0 03.0000 3.5000 3.3000 3.2300 04.0000 2.7500 2.0750 2.2850 3.419016.function [C,D,Y]=newpoly(x0,y0,x)if nargin < 2 | nargin> 3error( 'Incorrect Number of Inputs'); endif length(x0)~=length(y0)error('The length of x0 must be equal to it of y0');end n=length(x0); D=zeros(n,n); D(:,1)=y0'; for j=2:n%计算差商表for k=j:nIf abs(x0(k)-x0(k-j+1))<eps< bdsfid="127" p=""></eps<> error('DividedbyZero,therearetwonodesarethes ame');endD(k,j)=(D(k,j-1)-D(k-1,j-1))/(x0(k)-x0(k-j+1));EndEndC=D(n,n);For k=(n-1):-1:1C=conv(C,poly(x0(k)));m=length(C);C(m)=C(m)+D(k,k);endIf nargin==3Y=polyval(C,x);endC=fliplr(C);Returnx = [0 1 2 3 4 ];y = [0.5,1.25,2.75,3.5,2.75];[A,Y]=lagrange(x,y,x0)x0 = [0 1 2 3 4 ];y0 = [0.5,1.25,2.75,3.5,2.75];[C,D,X]=newpoly(x0,y0,x)plot(x,Y,'b-',x0,X,'r:')A = 0.5000 -0.3125 1.4687 -0.4375 0.0313Y = 0.5000 1.2500 2.75003.5000 2.7500C = 0.0313 -0.4375 1.4688 -0.3125 0.5000D = 0.5000 0 00 01.2500 0.7500 0 0 02.7500 1.5000 0.3750 0 03.5000 0.7500 -0.3750 -0.2500 02.7500 -0.7500 -0.7500 -0.1250 0.0313X = 0.5000 1.2500 2.7500 3.5000 2.7500fl(x)=0.5*x^4 - 0.312*x^3 + 1.47*x^2 - 0.438*x + 0.0312fn(x)=0.5*x^4 - 0.312*x^3 + 1.47*x^2 - 0.438*x + 0.03126. 解：对题中函数进行变形：原式→y/x = a* exp(b*x) →ln(y/x) = ln(a) + b*exp(x) 化为线性形式计算：>> a = [1 2 3 4 5];>> b = [1.222 2.984 5.466 8.902 13.592];>> x = exp(a);>> y = log(b)-log(a);>> n = 1; >> [C]=lspoly(x,y,n);>> y = vpa(poly2sym(C),3)结果如下：y = 0.00464*x + 0.384写成题中拟合函数的形式即为：y = 1.4679*x*exp(0.00464*x)7.function [a0,a1,a2]=ec2(h,w)S=log(s)';N=length(h);A=zeros(N,3);for i=1:5A(i,1)=1;A(i,2)=log(h(i));A(i,3)=log(w(i));endc=inv(A'*A)*(A'*S); a0=exp(c(1)); a1=c(2); a2=c(3);return%给出数据h=[175 172 183 164 156]; w=[80 90 80 70 65];s=[1000 900 1200 750 800]; [a0, a1,a2]=ec2(h,w,s)输出结果为：a0 =1.614815742043648e-04a1 =3.383163094165866a2 =-0.4191650115826638.x= lsqcurvefit(fun,x0,xdata,ydata)x=lsqcurvefit(fun,x0,xdata,ydata,lb,ub)x=lsqcurvefit(fun,x0,xdata,ydata,lb, ub,options)[x,resnorm] = lsqcurvefit(…)[x,resnorm,residual]=lsqcurvefit(…)[x,resnorm,residual,exitflag]= lsqcu rvefit(…)[x,resnorm,residual,exitflag,output] = lsqcurvefit(…)[x,resnorm,residual,exitflag,output,l ambda] = lsqcurvefit(…) [x,resnorm,residual,exitflag,output,l ambda,jacobian] =lsqcurvefit(…)function F = myfun(x,xdata)F=(x(1).*xdata).*(exp(x(2).*xdat a));xdata=[1,2,3,4,5];ydata=[1.222,2.984,5.466,8.902,13. 592]; x0=[0,0];[x,resnorm]=lsqcurvefit(@myfun,x0, xdata,ydata)输出结果为：Local minimum found.Optimization completed because t he size of the gradient is less than t he default value of the function toler ance.x =0.999958348976391 0.2000141328 12834aaresnorm = 8.067930437509675e -7。
《数值分析》杨大地 答案(第四章)
⑴ 2 3 1 ，求按模最大特征值和对应的特征向量，精确到小数三位。
解：由幂法公式有:
1i n
//P67 页
(1) | a r | max | a i | ,其中 ai 是uk −1 = (a1 , a2 , . . . , an )T 的各分量； (2) y k 1
∴ i t 为������ − ������������ 的特征值。 令 为 A 的特征向量，则有： A i 又∵ A tI A tI i t i t ∴ 也为������ − ������������ 的特征向量； ∴ i t 是 A tI 的特征值，且 A 和������ − ������������特征向量相同。
T
1
当计算到第 9 次时，λ 1 的小数点前三位精度开始稳定，满足题目要求，所以此时 A 矩阵的按模最大特 征值 1 =7.288，对应的特征向量为（1.000,0.523,0.242）T 5．若 A 的特征值为 1, 2 ,, n , t 是一实数，证明： i t 是 A tI 的特征值，且特征向量不变． 证明： ∵ A 的特征值为 ∴｜ i I A ｜=0 假设������是 A tI 的特征值，则有：
k 0 1 2 3 4 5 6 7 8 9 ukT 1.000,1.000,1.000 9.000,6.000,3.000 7.6667,4.3333,2.000 7.3913,3.9565,1.8261 7.3177,3.8530,1.7824 7.2966,3.8231,1.7701 7.2906,3.8146,1.7666 7.2887,3.8119,1.7655 7.2882,3.9112,1.7652 7.2880,3.8109,1.7651 ykT 1.000,1.000,1.000 1.000,0.6667,0.3333 1.000,0.5653,0.2609 1.000,0.5353,0.2471 1.000,0.5265,0.2436 1.000,0.5240,0.2426 1.000,0.5232,0.2423 1.000,0.5230,0.2422 1.000,0.5229,0.2422 9.0000 7.6667 7.5913 7.3177 7.2966 7.2916 7.2887 7.2882 7.2880
数值分析答案第四章
令
f (x) = x ，则
0 = −1 + 2 x1 + 3 x2
令 f ( x ) = x 2 ，则
2 2 = 1 + 2 x12 + 3 x2
从而解得
⎧ x1 = −0.2899 ⎧ x1 = 0.6899 或⎨ ⎨ ⎩ x2 = 0.5266 ⎩ x2 = 0.1266
令 f ( x ) = x 3 ，则
∫
1
−1
f ( x)dx = ∫ x3 dx = 0
−1
1
[ f ( −1) + 2 f ( x1 ) + 3 f ( x2 )] / 3 ≠ 0
故
∫
1
−1
f ( x)dx = [ f (− 1) + 2 f ( x1 ) + 3 f ( x2 )] / 3不成立。
h
因此，原求积公式具有 2 次代数精度。 （4）若
7 h T8 = [ f ( a) + 2∑ f ( xk ) + f ( b)] = 0.11140 2 k =1
复化辛普森公式为
7 7 h S8 = [ f ( a) + 4∑ f ( x 1 ) + 2∑ f ( xk ) + f ( b)] = 0.11157 k+ 6 k=0 k =1 2 1
令 f ( x ) = x 2 ，则
b 1 3 3 2 f ( x ) dx = ∫a ∫a x dx = 3 (b − a ) b −a 1 3 3 [7 f ( x0 ) + 32 f ( x1 ) + 12 f ( x2 )+ 32 f ( x (b − a ) 3 )+ 7 f ( x 4 )]= 90 3 b
数值分析大作业四
《数值分析》大作业四一、算法设计方案：复化梯形积分法，选取步长为1/500=0.002，迭代误差控制在E ≤1.0e-10①复化梯形积分法：11()[()()2()]2n bak hf x dx f a f b f a kh -=⎰≈+++∑，截断误差为：322()''()''(),[,]1212T b a b a R f h f a b n ηηη--=-=-∈其中。
复化Simpson 积分法，选取步长为1/50=0.02，迭代误差控制在E ≤1.0e-10②Simpson 积分法：121211()[()()4()2()]3m m bi i a i i hf x dx f a f b f x f x --==≈+++∑∑⎰， 截断误差为：4(4)(),[,]180s b a R h f a b ηη-=-∈。
③Guass积分法选用Gauss-Legendre 求积公式：111()()ni i i f x dx A f x -=≈∑⎰截断误差为：R= ()()n 2n 422n!2×(2[2!]2n 1f n n ⨯（2）η（））＋ η∈(1,1)。
选择9个节点：-0.9681602395,-0.8360311073,-0.6133714327,-0.3242534234,0,0.3242534234,0.6133714327,0.8360311073,0.9681602395， 对应的求积系数依次为：0.0812743884,0.1806481607,0.2606106964,0.3123470770,0.3302393550,0.3123470770,0.2606106964,0.1806481607,0.0812743884。
二、程序源代码：#include<stdio.h>#include<math.h>#include<stdlib.h>#define E 1.0e-10/****定义函数g和K*****/double g(double a){double b;b=exp(4*a)+(exp(a+4)-exp(-a-4))/(a+4);return b;}double K(double a,double b){double c;c=exp(a*b);return c;}/******复化梯形法******/void Tixing( ){double u[1001],x[1001],h,c[1001],e;int i,j,k;FILE *fp;fp=fopen("f:/result0. xls ","w");h=1.0/1500;for(i=0;i<3001;i++){x[i]=i*h-1;u[i]=g(x[i]);}for(k=0;k<100;k++){e=0;for(i=0;i<1001;i++){for(j=1,c[i]=0;j<N-1;j++)c[i]+=K(x[i],x[j])*u[j];u[i]=g(x[i])-h*c[i]-h/2*(K(x[i],x[0])*u[0]+K(x[i],x[N-1])*u[N-1]);e+=h*(exp(4*x[i])-u[i])*(exp(4*x[i])-u[i]);}if(e<=E) break;}for(i=0;i<1001;i++)fprintf(fp,"%.12lf,%.12lf\n",x[i],u[i]);fclose(fp);}/******复化Simpson法******/void simpson( ){double u[101],x[101],h,c[101],d[101],e;int i,j,k;FILE *fp;fp=fopen("f:/result1.xls","w");h=1.0/50;for(i=0;i<101;i++){x[i]=i*h-1;u[i]=g(x[i]);}for(k=0;k<50;k++){e=0;for(i=0;i<101;i++){for(j=1,c[i]=0,d[i]=0;j<51;j++){c[i]+=K(x[i],x[2*j-1])*u[2*j-1];if(j<50)d[i]+=K(x[i],x[2*j])*u[2*j];}u[i]=g(x[i])-4*h/3*c[i]-2*h/3*d[i]-h/3*(K(x[i],x[0])*u[0]+K(x[i],x[M-1])*u[M-1]);e+=h*(exp(4*x[i])-u[i])*(exp(4*x[i])-u[i]);}if(e<=E) break;}for(i=0;i<101;i++)fprintf(fp,"%.12lf,%.12lf\n",x[i],u[i]);fclose(fp);}/******Gauss积分法******/void gauss( ){double x[9]={-0.9681602395,-0.8360311073,-0.6133714327,-0.3242534234,0,\0.3242534234,0.6133714327,0.8360311073,0.9681602395},A[9]={0.0812743884,0.1806481607,0.2606106964,0.3123470770,0.3302393550,\0.3123470770,0.2606106964,0.1806481607,0.0812743884},u[9],c[9],e;int i,j,k;FILE *fp;fp=fopen("f:/result2. xls ","w");for(i=0;i<9;i++)u[i]=g(x[i]);for(k=0;k<50;k++){e=0;for(i=0;i<9;i++){for(j=0,c[i]=0;j<9;j++)c[i]+=A[j]*K(x[i],x[j])*u[j];u[i]=g(x[i])-c[i];e+=A[i]*(exp(4*x[i])-u[i])*(exp(4*x[i])-u[i]);}if(e<=E) break;}for(i=0;i<9;i++)fprintf(fp,"%.12lf,%.12lf\n",x[i],u[i]);fclose(fp);}/******主函数******/main(){Tixing ( );Simpson( );Gauss( );return 0;}三、运算结果复化梯形数据-10.018323-0.920.02523-0.9980.018471-0.9180.025433-0.9960.018619-0.9160.025637-0.9940.018768-0.9140.025843-0.9920.018919-0.9120.026051-0.990.019071-0.910.02626-0.9880.019224-0.9080.026471-0.9860.019378-0.9060.026683-0.9840.019534-0.9040.026897-0.9820.019691-0.9020.027113-0.980.019849-0.90.027331-0.9780.020008-0.8980.02755-0.9760.020169-0.8960.027772-0.9740.020331-0.8940.027995-0.9720.020494-0.8920.028219-0.970.020658-0.890.028446-0.9680.020824-0.8880.028674-0.9660.020992-0.8860.028905-0.9640.02116-0.8840.029137-0.9620.02133-0.8820.029371-0.960.021501-0.880.029607-0.9580.021674-0.8780.029844-0.9560.021848-0.8760.030084-0.9540.022023-0.8740.030326-0.9520.0222-0.8720.030569-0.950.022378-0.870.030815-0.9480.022558-0.8680.031062-0.9460.022739-0.8660.031311-0.9440.022922-0.8640.031563-0.9420.023106-0.8620.031816-0.940.023291-0.860.032072-0.9380.023478-0.8580.032329-0.9360.023667-0.8560.032589-0.9340.023857-0.8540.032851-0.9320.024048-0.8520.033114-0.930.024241-0.850.03338-0.9280.024436-0.8480.033648-0.9260.024632-0.8460.033918-0.9240.02483-0.8440.034191-0.9220.025029-0.8420.034465-0.840.034742-0.760.047841-0.8380.035021-0.7580.048225-0.8360.035302-0.7560.048613 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1.6955160.212 2.334945 0.134 1.7091350.214 2.3537 0.136 1.7228630.216 2.372605 0.138 1.7367010.218 2.391662 0.14 1.750650.22 2.410872 0.142 1.7647120.222 2.430236 0.144 1.7788860.224 2.449756 0.146 1.7931740.226 2.469433 0.148 1.8075770.228 2.489268 0.15 1.8220960.23 2.509262 0.152 1.8367310.232 2.529417 0.154 1.8514840.234 2.549733 0.156 1.8663550.236 2.570213 0.158 1.8813460.238 2.590857 0.16 1.8964570.24 2.611667 0.162 1.911690.242 2.632645 0.164 1.9270450.244 2.65379 0.166 1.9425230.246 2.675106 0.168 1.9581260.248 2.6965930.25 2.7182520.33 3.743385 0.252 2.7400850.332 3.7734530.254 2.7620940.334 3.803761 0.256 2.7842790.336 3.834314 0.258 2.8066430.338 3.865111 0.26 2.8291860.34 3.896156 0.262 2.8519110.342 3.927451 0.264 2.8748180.344 3.958996 0.266 2.8979090.346 3.990796 0.268 2.9211850.348 4.02285 0.27 2.9446480.35 4.055162 0.272 2.96830.352 4.087734 0.274 2.9921420.354 4.120567 0.276 3.0161750.356 4.153664 0.278 3.0404010.358 4.187026 0.28 3.0648220.36 4.220657 0.282 3.0894390.362 4.254558 0.284 3.1142540.364 4.288731 0.286 3.1392680.366 4.323179 0.288 3.1644830.368 4.357903 0.29 3.18990.37 4.392906 0.292 3.2155220.372 4.42819 0.294 3.2413490.374 4.463758 0.296 3.2673840.376 4.499612 0.298 3.2936280.378 4.535753 0.3 3.3200830.38 4.572185 0.302 3.346750.382 4.608909 0.304 3.3736320.384 4.645928 0.306 3.4007290.386 4.683245 0.308 3.4280440.388 4.720861 0.31 3.4555790.39 4.75878 0.312 3.4833350.392 4.797003 0.314 3.5113130.394 4.835533 0.316 3.5395160.396 4.874373 0.318 3.5679460.398 4.913524 0.32 3.5966040.4 4.95299 0.322 3.6254930.402 4.992773 0.324 3.6546130.404 5.032876 0.326 3.6839670.406 5.0733 0.328 3.7135570.408 5.114050.41 5.1551260.497.099276 0.412 5.1965330.4927.1562980.414 5.2382720.4947.213778 0.416 5.2803460.4967.27172 0.418 5.3227590.4987.330127 0.42 5.3655120.57.389004 0.422 5.4086080.5027.448353 0.424 5.4520510.5047.508179 0.426 5.4958420.5067.568486 0.428 5.5399850.5087.629277 0.43 5.5844830.517.690556 0.432 5.6293380.5127.752327 0.434 5.6745540.5147.814595 0.436 5.7201330.5167.877362 0.438 5.7660770.5187.940634 0.44 5.8123910.528.004414 0.442 5.8590770.5228.068707 0.444 5.9061380.5248.133516 0.446 5.9535770.5268.198845 0.448 6.0013960.5288.264699 0.45 6.04960.538.331082 0.452 6.0981910.5328.397998 0.454 6.1471730.5348.465452 0.456 6.1965480.5368.533447 0.458 6.2463190.5388.601989 0.46 6.296490.548.671081 0.462 6.3470640.5428.740728 0.464 6.3980450.5448.810935 0.466 6.4494340.5468.881705 0.468 6.5012370.5488.953044 0.47 6.5534560.559.024956 0.472 6.6060940.5529.097445 0.474 6.6591550.5549.170517 0.476 6.7126420.5569.244175 0.478 6.7665580.5589.318426 0.48 6.8209080.569.393272 0.482 6.8756950.5629.46872 0.484 6.9309210.5649.544774 0.486 6.9865910.5669.621439 0.4887.0427080.5689.6987190.579.776620.6513.46367 0.5729.8551470.65213.571810.5749.9343050.65413.68082 0.57610.01410.65613.79071 0.57810.094530.65813.90147 0.5810.175610.6614.01313 0.58210.257340.66214.12569 0.58410.339730.66414.23915 0.58610.422780.66614.35352 0.58810.50650.66814.46881 0.5910.590890.6714.58502 0.59210.675960.67214.70217 0.59410.761710.67414.82026 0.59610.848150.67614.9393 0.59810.935280.67815.05929 0.611.023110.6815.18025 0.60211.111650.68215.30218 0.60411.20090.68415.42509 0.60611.290870.68615.54898 0.60811.381560.68815.67387 0.6111.472980.6915.79977 0.61211.565130.69215.92667 0.61411.658020.69416.0546 0.61611.751660.69616.18355 0.61811.846050.69816.31354 0.6211.94120.716.44457 0.62212.037110.70216.57665 0.62412.133790.70416.7098 0.62612.231250.70616.84401 0.62812.32950.70816.97931 0.6312.428530.7117.11569 0.63212.528360.71217.25316 0.63412.628990.71417.39174 0.63612.730420.71617.53143 0.63812.832680.71817.67225 0.6412.935750.7217.81419 0.64213.039650.72217.95728 0.64413.144390.72418.10151 0.64613.249960.72618.24691 0.64813.356390.72818.393470.7318.541210.8125.53363 0.73218.690130.81225.738720.73418.840250.81425.94545 0.73618.991580.81626.15385 0.73819.144120.81826.36392 0.7419.297890.8226.57568 0.74219.452890.82226.78914 0.74419.609140.82427.00431 0.74619.766640.82627.22121 0.74819.925410.82827.43985 0.7520.085450.8327.66025 0.75220.246780.83227.88242 0.75420.409410.83428.10638 0.75620.573340.83628.33213 0.75820.738580.83828.5597 0.7620.905160.8428.78909 0.76221.073070.84229.02033 0.76421.242330.84429.25342 0.76621.412950.84629.48839 0.76821.584940.84829.72524 0.7721.758310.8529.964 0.77221.933080.85230.20467 0.77422.109250.85430.44728 0.77622.286830.85630.69184 0.77822.465840.85830.93836 0.7822.646290.8631.18686 0.78222.828190.86231.43735 0.78423.011550.86431.68986 0.78623.196380.86631.9444 0.78823.382690.86832.20098 0.7923.570510.8732.45962 0.79223.759830.87232.72034 0.79423.950670.87432.98315 0.79624.143040.87633.24807 0.79824.336960.87833.51513 0.824.532440.8833.78432 0.80224.729490.88234.05568 0.80424.928110.88434.32922 0.80625.128340.88634.60496 0.80825.330170.88834.882910.8935.163090.94643.99154 0.89235.445520.94844.344880.89435.730220.9544.701070.89636.017210.95245.060110.89836.306510.95445.422040.936.598120.95645.786870.90236.892080.95846.154630.90437.188410.9646.525350.90637.487110.96246.899050.90837.788210.96447.275750.9138.091730.96647.655470.91238.397680.96848.038240.91438.70610.9748.424090.91639.016990.97248.813040.91839.330380.97449.205110.9239.646280.97649.600330.92239.964720.97849.998720.92440.285720.9850.400320.92640.60930.98250.805140.92840.935480.98451.213210.9341.264280.98651.624560.93241.595720.98852.039210.93441.929820.9952.45720.93642.26660.99252.878540.93842.606090.99453.303270.9442.948310.99653.73140.94243.293270.99854.162980.94443.64101154.59802复化Simpson数据：-1 0.018319929 -0.34 0.256658088 0.32 3.596641805 -0.98 0.0198445 -0.32 0.278035042 0.34 3.896195298-0.96 0.021494322 -0.3 0.301192133 0.36 4.220697765-0.94 0.023283225 -0.28 0.326278124 0.38 4.572227037-0.92 0.025220379 -0.26 0.353453177 0.4 4.95303418-0.9 0.027320224 -0.24 0.382891765 0.42 5.365557596-0.88 0.029594431 -0.22 0.41478194 0.44 5.812438891-0.86 0.032059069 -0.16 0.527292277 0.54 8.671138204-0.84 0.034728638 -0.14 0.571209036 0.56 9.39333156-0.82 0.037621263 -0.12 0.61878367 0.58 10.17567433-0.8 0.040754615 -0.1 0.670320427 0.6 11.02317608-0.78 0.044149394 -0.08 0.726149698 0.62 11.94126383-0.76 0.047826844 -0.06 0.78662861 0.64 12.93581634-0.74 0.051810827 -0.04 0.85214479 0.66 14.01320231-0.72 0.056126648 -0.02 0.92311742 0.68 15.1803205-0.7 0.060802006 0 1.0000013 0.7 16.44464467 -0.68 0.065866854 0.02 1.083288424 0.72 17.81427057 -0.66 0.071353499 0.04 1.173512427 0.74 19.29796874 -0.64 0.077297255 0.06 1.271250748 0.76 20.90523965 -0.62 0.083735917 0.08 1.377129533 0.78 22.64637562 -0.6 0.090711017 0.1 1.491826493 0.8 24.53252554 -0.58 0.098266855 0.12 1.616076341 0.82 26.57576756 -0.56 0.106452202 0.14 1.750674449 0.84 28.78918506 -0.54 0.11531904 0.16 1.896482943 0.86 31.18695183 -0.52 0.12492459 0.18 2.054435268 0.88 33.78442141 -0.5 0.135329888 0.2 2.225543071 0.9 36.59822683 -0.48 0.14660204 0.22 2.410901825 0.92 39.64638571 -0.46 0.158812728 0.24 2.611698647 0.94 42.94841704 -0.44 0.17204064 0.26 2.829219145 0.96 46.52546475 -0.42 0.18636997 0.28 3.064856356 0.98 50.40043451 -0.4 0.201892977 0.3 3.320119013 1 54.59813904 -0.38 0.218708553 0.46 6.296539601-0.36 0.236924875 0.48 6.820959636-0.2 0.449328351 0.5 7.389057081-0.18 0.486751777 0.52 8.0044696750102030405060四、讨论①在满足相同精度要求的情况下复化梯形积分法比复化Simpson 积分法计算所需节点数多，计算量大。
数值分析-第四章学习小结
数值分析-第四章学习小结-CAL-FENGHAI-(2020YEAR-YICAI)_JINGBIAN第4章非线性方程与非线性方程组的迭代解法--------学习小结一、本章学习体会通过本章的学习，我了解了怎么求出非线性方程和非线性方程组的根，只是有很少类型的非线性方程能解出根的解析表达式，对于大多数非线性方程，只能用数值方法求出它的根的近似值。
我学习了非线性方程与非线性方程组的迭代解法。
我感到要想求非线性方程组的精确解是不容易的，困难程度远远超过线性方程组的求解。
首先要了解迭代公式的基本思想，迭代法是一种逐次逼近法,用某个固定公式反复校正根的近似值,使之逐步精确化,最后得到满足精度要求的解，实质上是一个逐步显示化的过程。
最基本的就是在高中学过的二分法，需要在给定的区域选择根，然后在二分，在从中舍弃一个，再选，直到所选的根符合题目所给的条件，但是二分法只能求实根，并且只能求单根和奇数重根，不能求偶数重根和复数根，所以又有它的缺陷，后面又学了斯蒂芬森加速法和牛顿法。
算法都是离不开模型的，我们在学习某种算法时，一定要结合数学模型才能把知识理解到位，比如本章结合几何思想能够很好的理解算法公式的推导说明。
运用这么多的算法去求解非线性方程组，只是能最大程度的求解线性方程组的精确解，但不是精确解。
我们在今后的学习工作中，也可以自己去创造一种算法，使求解更加精确容易。
在求解非线性方程的解的时候，我们要有如下思路：1.如何选取迭代公式；2.如何判断迭代公式的收敛速度；3.如何进行迭代公式的修正，以加速收敛；4.如何选取最适合的迭代方法二、 本章知识梳理1、非线性方程的迭代解法简单迭代法及其收敛性简单迭代法的基本思想)(0)(x x x f ϕ=⇔=迭代法的基本思想是将隐式方程)(x x ϕ=的求根问题归结为计算一组显式公式)(1k k x x ϕ=+一般形式： ,2,1,0),(1==+k x x k k ϕ收敛条件：a 、非局部收敛定理b 、局部收敛定理简单迭代法的收敛速度线性收敛的条件m 阶收敛的条件迭代过程的加速加权法 迭代：)(1k k x x ϕ=+ 改进：k k k x LL x L x ---=++11111 埃特金(Aitken)加速法设序列}{k x 线性收敛到s112212)(++++=+---≈k kk k k k k x x x x x x x s Newton 法（切线法）基本思想:(1)构造法:0)(='s ϕ(2)几何上:逐步线性化方法(3)Taylor 展开 ))((')()(k k k x x x f x f x f -+≈迭代函数：)(')()(x f x f x x -=ϕ 迭代公式： ,2,1,0,)(')(1=-=+k x f x f x x k k k k 几何意义收敛性（1）局部收敛定理（2）非局部收敛定理牛顿下山法)(')(1k k k k x f x f x x -=+)(')(1k k k k x f x f x x λ-=+ k k k x x x )1(11λλ-+=++ 其中10≤<λ称为下山因子通过适当选取下山因子保证函数值)(k x f 能单调下降。
数值分析第四章数值积分与数值微分习题答案汇编
(1)采用龙贝格方法可得
k
0
1.333333
1
1.166667
1.099259
2
1.116667
1.100000
1.099259
3
1.103211
1.098726
1.098641
1.098613
4
1.099768
1.098620
1.098613
1.098613
1.098613
故有
(2)采用高斯公式时
从而解得
令 ，则
故 成立。
令 ，则
故此时，
故
具有3次代数精度。
（2）若
令 ，则
令 ，则
令 ，则
从而解得
令 ，则
故 成立。
令 ，则
故此时，
因此，
具有3次代数精度。
（3）若
令 ，则
令 ，则
令 ，则
从而解得
或
令 ，则
故 不成立。
因此，原求积公式具有2次代数精度。
（4）若
令 ，则
令 ，则
令 ，则
故有
令 ，则
10.2075712
10.2075943
10.2075939
10.2075936
9、如果你亲戚朋友送你一件DIY手工艺制品你是否会喜欢？5
10.2112607
10.2075909
因此不难看出，自制饰品在校园里也大有市场所在。对于那些走在流行前端的女生来说，〝捕捉〞新事物便〝捕捉〞到了时尚与个性。10.2075922
用 的高斯—勒让德公式计算积分
他们的成功秘诀在于“连锁”二字。凭借“连锁”，他们在女孩们所喜欢的小玩意上玩出了大名堂。小店连锁，优势明显，主要有：