数值分析-上机作业.doc

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《数值分析》上机作业

姓名：
学号：
学院：
专业：

第一章
4题
设计程序如下：
clear;
%定义x，y的区间
x=-10:0.2:10; %步长为100
y=-10:0.2:10;
%在XOY平面生成二维网格数据
[X,Y]=meshgrid(x,y);
%对二元函数进行表达
a=-abs(X);b=X+Y;c=X.^2+Y.^2+1;
Z=exp(a)+cos(b)+1./c;
%绘制三维图形
mesh(X,Y,Z);

图1. 区间等分100份 图2.区间等分200份

图3.区间等分400份

第二章
2题
设计程序如下：
Clear
n=8; V=220; R=27;
a=[0 -2 -2 -2 -2 -2 -2 -2]; b=[2 5 5 5 5 5 5 5]; c=[-2 -2 -2 -2 -2 -2 -2];
I=[V/R 0 0 0 0 0 0 0];
for i=2:n
a(i)=a(i)/b(i-1);
b(i)=b(i)-c(i-1)*a(i);
I(i)=I(i)-a(i)*I(i-1);
end;
I(n)=I(n)/b(n);
for i=n-1:-1:1
I(i)=(I(i)-c(i)*I(i+1))/b(i);
end;
I;
fprintf(“各电路的电流量I为”);
运行结果如下：
各电路的电流量I为
I=8.1478 4.0737 2.0365 1.0175 0.5073 0.2506 0.1194 0.0477

第三章
2题
高斯赛德尔迭代
设计程序如下：
：A=[10,1,2,3,4;1,9,-1,2,-3;2,-1,7,3,-5;3,2,3,12,-1;4,-3,-5,-1,15;]
x=[0;0;0;0;0];
b=[12;-27;14;-17;12]
c=0.000001
L=-tril(A,-1)
U=-triu(A,1)
D=(diag(diag(A)))
X=inv(D-L)*U*x+inv(D-L)*b;
k=1;
while norm(X-x,inf)>= c
x=X;
X=inv(D-L)*U*x+inv(D-L)*b;
k=k+1;
end
X
k
计算结果：X =
1.0000
-2.0000
3.0000
-2.0000
1.0000
k =37

Jacobi迭代法：
设计程序如下：
Jacobi：b=[12;-27;14;-17;12]
x = [0;0;0;0;0;]
k = 0;
r = 1;
e=0.000001
A=[10,1,2,3,4;1,9,-1,2,-3;2,-1,7,3,-5;3,2,3,12,-1;4,-3,-5,-1,15;]
D = diag(diag(A));
B = inv(D)*(D-A);
f = inv(D)*b;
p = max(abs(eig(B)));
if p >= 1
'迭代法不收敛'
return
end

while r >e
x0 = x;
x = B*x0 + f;
k = k + 1;
r = norm (x-x0,inf);
end
x
k
计算结果：x =
1.0000
-2.0000
3.0000
-2.0000
1.0000
k =65
SOR：A=[10,1,2,3,4;1,9,-1,2,-3;2,-1,7,3,-5;3,2,3,12,-1;4,-3,-5,-1,15]
x=[0;0;0;0;0];
b=[12;-27;14;-17;12]
e=0.000001
w=1.44;
L=-tril(A,-1)
U=-triu(A,1)
D=(diag(diag(A)))
X=inv(D-w*L)*((1-w)*D+w*U)*x+w*inv(D-w*L)*b
n=1;
while norm(X-x,inf)>=e
x=X;
X=inv(D-w*L)*((1-w)*D+w*U)*x+w*inv(D-w*L)*b;
n=n+1;
end
X
n
计算结果：X =
1.0000
-2.0000
3.0000
-2.0000
1.0000
n =22
结果分析：由迭代次数的比较可知该情况下Jacobi迭代法比Gauss-Seidel迭代法收敛的慢

第四章
2题
设计程序如下
%幂法和反幂法
A=[12 6 -6;6 16 2;-6 2 16];
x0=[1;1;1];
%% 幂法求按模最大特征值
x=x0;
N = 3;
lamada = max(x0);
for k =1:N
y = x/lamada;
x = A*y;
lamada = max(x);
end
lamada
%% 反幂法求精确值
lamda0 = floor(lamada);
I = [1 0 0;0 1 0;0 0 1];
[L,U,P] = lu(A-lamda0*I);

x=x0;
k=0;
error = 1;
maxe = 1e-10;
alpha = max(x0);
beta = alpha;
while error>maxe
alpha = beta;
y = x/alpha;
z = L\y;
x = U\z;
beta = max(abs(x));
error = abs(1/beta-1/alpha);
k=k+1;
if(k==100)
break;
end
end
lamda = lamda0+1/beta
k

第五章
1题
设计程序如下：
subplot(2,1,1)
syms x
for i=-5:5
f(i+6)=1/(1+4*(i)^2);
end

for i=1:10
for k=11:-1:i+1
f(k)=(f(k)-f(k-1))/(i);
end
end

a=-5:0.1:5;
b=1./(1+4.*(a).^2);
plot(a,b,'+');
title('\fontname{隶书} 原函数和插值多项式')
hold on

p=0;

for i=1:11
q=1;
for j=1:1:i-1
q=q*(x-j+6);
end;
p=p+f(i)*q;
end;
p

axis([-5,5,-1,5]);

x=-5:0.1:5;
e=subs(p)
plot(x,e);
hold on
text(2,3,'+++插值函数')
text(2,2.2,'-----原函数')

subplot(2,1,2)

e-b
plot(x,e-b);
hold on;
title('\fontname{隶书} 误差图')

p =

(36*x)/6565 + (3550298616520539*(x + 4)*(x + 5))/1152921504606846976 + (2689247898264063*(x + 3)*(x + 4)*(x + 5))/1152921504606846976 + (1806978031308661*(x + 2)*(x + 3)*(x + 4)*(x + 5))/576460752303423488 + (462354082176629*(x + 1)*(x + 2)*(x + 3)*(x + 4)*(x + 5))/144115188075855872 - (5850230976024283*x*(x + 1)*(x + 2)*(x + 3)*(x + 4)*(x + 5))/1152921504606846976 + (6518522480310501*x*(x - 1)*(x + 1)*(x + 2)*(x + 3)*(x + 4)*(x + 5))/2305843009213693952 - (2258610859405831*x*(x - 1)*(x + 1)*(x - 2)*(x + 2)*(x + 3)*(x + 4)*(x + 5))/2305843009213693952 + (1143600435142193*x*(x - 1)*(x + 1)*(x - 2)*(x + 2)*(x - 3)*(x + 3)*(x + 4)*(x + 5))/4611686018427387904 - (7319042784910035*x*(x - 1)*(x + 1)*(x - 2)*(x + 2)*(x - 3)*(x + 3)*(x - 4)*(x + 4)*(x + 5))/147573952589676412928 + 49/1313

第六章
2题
设计的程序如下：
Clear
X=[2 3 5 6 7 9 10 11 12 14 16 17 19 20];
Y=[106.42 108.26 109.58 109.50 109.86 110.00 109.93 110.59 110.60 110.72 110.90 110.76 111.10 111.30];
D=1./X;
S=1./Y;
S=S';
A=ones(14,2);
for i=1:14
A(i,2)=D(i);
end;
c=A\S;
a=c(1);
b=c(2);
syms x;
y=x/(a*x+b);
disp('y=x/(a*x+b)');
a
b x1=[1:0.1:21];
y1=x1./(a*x1+b);
plot(X,Y,'*',x1,y1,'k');
title('最小二乘拟合函数');
legend(' 原始点','y=x/(a*x+b)');

运行结果如下：
y=x/(a*x+b)
a =0.0090
b = 8.4169e-004

第七章
1题
设计的程序如下：
a=0;
X=[];
for b=-5:0.05:5
w=1.0e-3;
F1=cos(a)+cos(b^2/2);
F2=cos((a+b)^2/8);
S0=((b-a)/6)*(F1+4*F2);
m=2;
h=(b-a)/4;
F3=0;
for k=1:2^(m-1)
F3=F3+cos((a+(2*k-1)*h)^2/2);
end
S=(h/3)*(F1+2*F2+4*F3);
while abs(S-S0)>15*w
m=m+1;
h=h/2;
F2=F2+F3;
S0=S;
F3=0;
for k=1:2^(m-1)
F3=F3+cos((a+(2*k-1)*h)^2/2);
end
S=(h/3)*(F1+2*F2+4*F3);
end
X=[X S];
end
Y=[];
for b=-5:0.05:5
w=1.0e-7;
F1=sin(a)+sin(b^2/2);
F2=sin((a+b)^2/2);
S0=((b-a)/6)*(F1+4*F2);
m=2;
h=(b-a)/4;
F3=0;
for k=1:2^(m-1)
F3=F3+sin(a+(2*k-1)*h)^2/2;
end
S=(h/3)*(F1+2*F2+4*F3);
while abs(S-S0)>15*w
m=m+1;
h=h/2;
F2=F2+F3;
S0=S;
F3=0;
for k=1:2^(m-1)
F3=F3+sin((a+(2*k-1)*h)^2/2);
end
S=(h/3)*(F1+2*F2+4*F3);
end
Y=[Y S];
end
plot(X,Y,'r.-')

第八章
设计的程序如下：
a=0.5;
b=128.52*log((513+0.6651*a)/(513-0.6651*a));
while(abs(a-b)>1.0e-8)
a=b;
b=128.52*log((513+0.6651*a)/(513-0.6651*a));
end

第九章
1题
Adams预测-校正
设计的程序如下：
a=0;
b=3.14;
N=30;
y(1)=1;
h=(b-a)/3;
x(1)=a;
n=2;
while n<5
f(n-1)=-y(n-1)+2*cos(x(n-1));
K1=h*f(n-1);
K2=h*(-(y(n-1)+K1/2)+2*cos(x(n-1)+h/2));
K3=h*(-(y(n-1)+K2/2)+2*cos(x(n-1)+h/2));
K4=h*(-(y(n-1)+K3)+2*cos(x(n-1)+h));
y(n)=y(n-1)+(1/6)*(K1+2*K2+2*K3+K4);
x(n)=a+(n-1)*h;
f(n)=-y(n)+2*cos(x(n));
n=n+1;
end
p(4)=0;
c(4)=0;
while n<30
x(n)=x(n-1)+h;
f(n-1)=-y(n-1)+2*cos(x(n-1));
p(n)=y(n-1)+h/24*(55*f(n-1)-59*f(n-2)+37*f(n-3)-9*f(n-4));
m(n)=p(n)+251/270*(c(n-1)-p(n-1));
c(n)=y(n-1)+h/24*(9*(-m(n)+2*cos(x(n)))+19*f(n-1)-5*f(n-2)+f(n-3));
y(n)=c(n)-19/270*(c(n)-p(n));
n=n+1;

end
for i=1:1:29
err(i)=sin(x(i))+cos(x(i))-y(i);
end
x,y,err
x =

Columns 1 through 11

0 1.0467 2.0933 3.1400 4.1867 5.2333 6.2800 7.3267 8.3733 9.4200 10.4667

Columns 12 through 22

11.5133 12.5600 13.6067 14.6533 15.7000 16.7467 17.7933 18.8400 19.8867 20.9333 21.9800

Columns 23 through 29

23.0267 24.0733 25.1200 26.1667 27.2133 28.2600 29.3067

y =

Columns 1 through 11

1.0000 1.3511 0.3504 -1.0011 -1.5466 -0.3516 1.4053 1.3272 0.0146 -0.7104 -1.4114

Columns 12 through 22

-1.0258 1.7493 2.0501 -1.1763 -0.7545 0.5015 -2.6858 0.2454 5.9972 -2.3873 -5.8269

Columns 23 through 29

6.9872 -0.6083 -11.7898 13.6088 9.5706 -28.5289 8.7914

err =

Columns 1 through 11

0 0.0151 0.0171 0.0027 0.1798 -0.0180 -0.4085 0.0402 0.3572 -0.2848 0.0435

Columns 12 through 22

0.6518 -0.7557 -0.6816 1.5524 -0.2375 -1.8706 2.3075 0.7450 -4.6275 2.7678 4.8381

Columns 23 through 29

-8.3574 0.2256 12.7770 -12.2380 -9.1857 27.5433 -10.1628

RK法：
设计的程序如下：
a=0;
b=3.14;
N=30;
Y0=1;
h=(b-a)/(100);
n=1;
x0=a;
while n<N
x=x0+h;
K1=-Y0+2*cos(x0);
K2=-(Y0+h/2*K1)+2*cos(x0+h/2);
K3=-(Y0+h*K2/2)+2*cos(x0+h/2);
K4=-(Y0+h*K3)+2*cos(x0+h);
Y=Y0+(h/6)*(K1+2*(K2+K3)+K4);
n=n+1;
e=cos(x)+sin(x)-Y;
x0=x;
Y0=Y;
end
x,Y, e
x = 0.9106

Y = 1.4031

e =7.4211e-009

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