最速下降法与共轭梯度法实验.doc

工程数值算法实验报告班级姓 名学 号课程名称时 间地 点一、实验名称： 实验一、最速下降法与共轭梯度法实验二、 实验目的：1理解最速下降法与共轭梯度法的基本原理；2最速下降法是一种局部极值搜寻方法，任一点的负梯度方向是函数值在该点下降最快的方向；3共轭梯度法适合求解二次函数最小值问题。三、 实验内容及要求：例1 求解，其中，需求选取初始点，终止误差为e = 0.1。采用最速下降法求解例2 用最速下降法求解无约束非线性规划问题： 其中,初始点，终止误差e = 0.1.采用共轭梯度法。四、插值法的实验步骤：步骤1：如何选择搜索方向步骤2：在确定搜索方向上，采用适合的方法进行一维搜索五、插值法的实验结果：例1结果：=1.9304 = 0.9744 例2结果： =-1 =1六、程序附录例1 定义函数function x=fsxsteep(f,e,a,b)% fsxsteep函数 最速下降法% x=fsxsteep(f,e,a,b)为输入函数 f为函数 e为允许误差 (a,b)为初始点;% fsx TJPU 2008.6.15x1=a;x2=b;Q=fsxhesse(f,x1,x2);x0=x1 x2;fx1=diff(f,x1); %对x1求偏导数fx2=diff(f,x2); %对x2求偏导数g=fx1 fx2; %梯度g1=subs(g); %把符号变量转为数值d=-g1;while (abs(norm(g1)=e)t=(-d)*d/(-d)*Q*d);t=(-d)*d/(-d)*Q*d); %求搜索方向x0=x0-t*g1; %搜索到的点v=x0;a=1 0*x0;b=0 1*x0;x1=a;x2=b;g1=subs(g);d=-g1;end;x=v;function x=fsxhesse(f,a,b)% fsxhesse函数 求函数的hesse矩阵；% 本程序仅是简单的求二次函数的hesse矩阵！；% x=fsxhesse(f)为输入函数 f为二次函数 x1,x2为自变量；% fsx TJPU 2008.6.15x1=a;x2=b;fx=diff(f,x1);%求f对x1偏导数fy=diff(f,x2);%求f对x2偏导数fxx=diff(fx,x1);%求二阶偏导数 对x1再对x1fxy=diff(fx,x2);%求二阶偏导数 对x1再对x2fyx=diff(fy,x1); %求二阶偏导数 对x2再对x1fyy=diff(fy,x2);%求二阶偏导数 对x2再对x2fxx=subs(fxx); %将符号变量转化为数值fxy=subs(fxy);fyx=subs(fyx);fyy=subs(fyy);x=fxx,fxy;fyx,fyy; %求hesse矩阵运行函数syms x1 x2;X=x1,x2;fx=(X(1)-2).4+(X(1)-2*X(2).2;z=fsxsteep(fx,0.1,0,3)例2 定义函数%conjugate gradient methods%method:FR,PRP,HS,DY,CD,WYL,LS%精确线搜索,梯度终止准则function m,k,d,a,X,g1,fv = conjgradme( G,b,c,X,e,method)if nargin=e k(i-1)=(m/m1)2; d(:,i)=-g1(:,i)+k(i-1)*d(:,i-1); a(i)=-(d(:,i)*g1(:,i)/(d(:,i)*G*d(:,i); %a1(i)=-(X(:,i)*G*d(:,i)+b*d(:,i)/(d(:,i)*G*d(:,i);a(i)=g1(:,i)*g1(:,i)/(d(:,i)*G*d(:,i); X(:,i+1)=X(:,i)+a(i)*d(:,i); g1=g1 subs(subs(g,x1,X(1,i+1),x2,X(2,i+1); m1=m; m=norm(g1(:,i+1); i=i+1; end case PRP while m=e k(i-1)=g1(:,i)*(g1(:,i)-g1(:,i-1)/(norm(g1(:,i-1)2; d(:,i)=-g1(:,i)+k(i-1)*d(:,i-1); a(i)=-(d(:,i)*g1(:,i)/(d(:,i)*G*d(:,i); X(:,i+1)=X(:,i)+a(i)*d(:,i); g1=g1 subs(subs(g,x1,X(1,i+1),x2,X(2,i+1); m=norm(g1(:,i+1); i=i+1; end case HS while m=e k(i-1)=g1(:,i)*(g1(:,i)-g1(:,i-1)/(d(:,i-1)*(g1(:,i)-g1(:,i-1); d(:,i)=-g1(:,i)+k(i-1)*d(:,i-1); a(i)=-(d(:,i)*g1(:,i)/(d(:,i)*G*d(:,i); X(:,i+1)=X(:,i)+a(i)*d(:,i); g1=g1 subs(subs(g,x1,X(1,i+1),x2,X(2,i+1); m=norm(g1(:,i+1); i=i+1; end case DY while m=e k(i-1)=g1(:,i)*g1(:,i)/(d(:,i-1)*(g1(:,i)-g1(:,i-1); d(:,i)=-g1(:,i)+k(i-1)*d(:,i-1); a(i)=-(d(:,i)*g1(:,i)/(d(:,i)*G*d(:,i); X(:,i+1)=X(:,i)+a(i)*d(:,i); g1=g1 subs(subs(g,x1,X(1,i+1),x2,X(2,i+1); m=norm(g1(:,i+1); i=i+1; end case LS while m=e k(i-1)=g1(:,i)*(g1(:,i)-g1(:,i-1)/(d(:,i-1)*(-g1(:,i-1); d(:,i)=-g1(:,i)+k(i-1)*d(:,i-1); a(i)=-(d(:,i)*g1(:,i)/(d(:,i)*G*d(:,i); %a(i)=-(X(:,i)*G*d(:,i)+b*d(:,i)/(d(:,i)*G*d(:,i); X(:,i+1)=X(:,i)+a(i)*d(:,i); g1=g1 subs(subs(g,x1,X(1,i+1),x2,X(2,i+1); m=norm(g1(:,i+1); i=i+1; end case CD while m=e k(i-1)=g1(:,i)*g1(:,i)/(d(:,i-1)*(-g1(:,i-1); d(:,i)=-g1(:,i)+k(i-1)*d(:,i-1); a(i)=-(d(:,i)*g1(:,i)/(d(:,i)*G*d(:,i); X(:,i+1)=X(:,i)+a(i)*d(:,i); g1=g1 subs(subs(g,x1,X(1,i+1),x2,X(2,i+1); m=norm(g1(:,i+1); i=i+1; end case WYL while m=e k(i-1)=g1(:,i)*(g1(:,i)-(m/m1)*g1(:,i-1)/(m12); d(:,i)=-g1(:,i)+k(i-1)*d(:,i-1); a(i)=-(d(:,i)*g1(:,i)/(d(:,i)*G*d(:,i); %a(i)=-(X(:,i)*G*d(:,i)+b*d(:,i)/(d(:,i)*G*d(:,i); X(:,i+1)=X(:,i)+a(i)*d(:,i); g1=g1 subs(subs(g,x1,X(1,i+1),x2,X(2,i+1); m1=m; m=norm(g1(:,i+1); i=i+1; endendfv=subs(subs(f,x1,X(1,i),x2,X(2,i); endl1=X(1,i);l2=X(2,i);w1=X(1,1);w2=X(2,1);v1=min(l1,w1)-abs(l1-w1)/10:abs(l1-w1)/10:max(l1,w1)+abs(l1-w1)/10;v2=min(l2,w2)-abs(l2-w2)/10:abs(l2-w2)/10:max(l2,w2)+abs(l2-w2)/10;x,y=meshgrid(v1,v2);s=size(x);z=zeros(size(x);for i=1:s(1) for j=1:s(2) z(i,j)=1/2*x(i,j),y(i,j)*G*x(i,j);y(i,j)+b*x(i,j);y(i,j)+c; endendpx,py = gradient(z,.2,.2);contour(v1,v2,z), hold on, quiver(v1,v2,px,py)C,h = cont