Question

Compute the first 8 terms x(1), x(2), …, x(8) with newton’s method sequence{x(k)} for minimizing the function f(x1, x2)=2x1^4+x2^2-4x1x2+5x2 with initial point x(0)=(0,0).

please give values of eight terms (eager for...)

please give codes of Matlab （please give the code)

thanks!

Answer 1

%Matlab code for Newton method clear all close all syms x1 x2 %functions for which intersection have to find f (x1,x2) 2 (x1) 4+ (x2) 2-4*x1*x2+5*x2; Displaying the equation fprintf( Function for which optimization have to do In) disp(f) creating gradient and Hessian matrix f xl(xl, x2)-diff (f,xl); f x2(x1, x2)-diff (f, x2); hx1 (x1, x2)-diff (f xl,xl); h x2 (x1, x2)-diff(f_x2, x2); hx12 (xl,x2)-diff(f_xl, x2)i h x21 (x1,x2)-diff (f_x2, x1) %initial guess x11-0; x21-0; %loop for Newton iterations fprint f ( "\nFor initial condition x1-%f and x2-%f \n',x11 , x21) cnt-0 %Loop for 10 iterations for i=1:10 cnt-cnt+l; gradf double ([f_xl (x11, x21); f x2 (x11, x21)1); Hf-double( [h_xl(x11, x21) h _x12 (x11,x21); h_x21(x11, x21) h x2(x11,x21)]) x11-y (1); x21-y (2); fprint f ("\nAfter %d iteration \n..cnt) fprint f ("\tX1-8.8f and X2_%·8f \n',x11,x21) end ,,88응움88응움8888888888움움end of code 88888888888888888888888888 Function for which optimization have to do 2*x1 4 - 4*x1*x2 + x2*2 + 5*x2 symbolic function inputs: x1, x2 For initial condition xl-0.000000 and x2-0.000000 After 1 iteration x1-1.25000000 and X2=0.00000000 After 2 iteration

x1-0.72033898 and x 2--1.05932203 After 3 iteration x1--0.90260633 and X2--4.30521267 After 4 iteration x1=-1.88402029 and x2=-6.26804059 After 5 iteration x1-1.51574180 and X2--5.53148360 After 6 iteration x1-1.39412209 and X2-5.28824418 After 7 iteration Xi--1.38057119 and X2--5.26114239 After 8 iteration x1-1.38040894 and X2--5.26081788 After 9 iteration XI.-1.38040892 and X2=-5.26081783 After 10 iteration xi=-1.38040892 and X2=-5.26081783 Published with MATLAB® R2018a

%Matlab code for Newton method
clear all
close all

syms x1 x2
%functions for which intersection have to find
f(x1,x2)=2*(x1)^4+(x2)^2-4*x1*x2+5*x2;

%Displaying the equation
fprintf('Function for which optimization have to do \n')
disp(f)

%creating gradient and Hessian matrix
f_x1(x1,x2)=diff(f,x1);
f_x2(x1,x2)=diff(f,x2);
h_x1(x1,x2)=diff(f_x1,x1);
h_x2(x1,x2)=diff(f_x2,x2);
h_x12(x1,x2)=diff(f_x1,x2);
h_x21(x1,x2)=diff(f_x2,x1);

%Initial guess
x11=0;x21=0;

    %loop for Newton iterations
    fprintf('\nFor initial condition x1=%f and x2=%f \n',x11,x21)
    cnt=0;
    %Loop for 10 iterations
    for i=1:10
        cnt=cnt+1;
        gradf=double([f_x1(x11,x21);f_x2(x11,x21)]);
        Hf=double([h_x1(x11,x21) h_x12(x11,x21);h_x21(x11,x21) h_x2(x11,x21)]);
      
        y=[x11;x21]-Hf\gradf;
        x11=y(1); x21=y(2);
        fprintf('\nAfter %d iteration \n',cnt)
        fprintf('\tX1=%.8f and X2=%.8f \n',x11,x21)
    end

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