Newton Raphson gives answers divided by 0

I'm trying to estimate 4 angles and 2 distances using the newton raphson method with phi4 and theta as the input. I have 6 functions that equal to 0 and when i try substituting to calculate dx (the array of all my values) i get a division by 0. I transposed some arrays to columns, but id be surprised if thats the issue. The functions themselves check out and im mostly using matlab functions to make this work so I don't really know what could be wrong

Angle=3;
phi4=pi/180*Angle;

theta=linspace(0,2*pi, 72);

%First guesses
phi1=ones(1, 72);
phi2=ones(1, 72);
phi3=ones(1, 72);
phi5=ones(1, 72);
d=ones(1,72);
s=ones(1, 72);


syms phi1V phi2V phi3V phi5V dV sV;

xV=[sV, dV, phi1V, phi2V, phi3V, phi5V];


ds=0;
dd=0;
dphi1=0;
dphi2=0;
dphi3=0;
dphi4=0;

dx=[ds, dd, dphi1, dphi2, dphi3, phi5]';

lim=0.01;

i=1;

while(i<73)

%redefine functions for changes in theta
f1=25-sV+dV*cos(phi2V)+3.1*cos(phi3V);
f2=-5+dV*sin(phi2V)+3.1*sin(phi3V);
f3=-25+6.3*cos(theta(i))+27*cos(phi1V)-19.2*cos(phi2V);
f4=11+6.3*sin(theta(i))+27*sin(phi1V)-19.2*sin(phi2V);
f5=-40+6.3*cos(theta(i))+(dV-19.2)*cos(phi2V)+29.1*cos(phi5V)+16*cos(phi4);
f6=-24+6.3*sin(theta(i))+(dV-19.2)*sin(phi2V)+29.1*sin(phi5V)+16*sin(phi4);

f=[f1, f2, f3, f4, f5, f6]';

J=jacobian(f, xV);

M=-inv(J)*f;


x=[s(i), d(i), phi1(i), phi2(i), phi3(i), phi5(i)];

    firstloop=1;

    while(ds > lim | dd > lim | dphi1 > lim | dphi2 > lim | dphi3 > lim | dphi4 > lim | firstloop==1)
        
        dx=subs(M, xV, x);
        x=x+dx;
        firstloop=0;
        
    end
    
    i=i+1;
end

Error using sym/subs
Division by zero.

Answer 1

[Sam Chak]

Hi @Jean-Paul

During the computation of M, certain elements involve divisions by . In the line 'dx = subs(M, xV, x)', an attempt is made to substitute the variables in 'xV = [sV, dV, phi1V, phi2V, phi3V, phi5V]' with 'x = [1, 1, 1, 1, 1, 1]', which leads to . Consequently, it triggers the 'Division-by-zero' error message.

Answer 2

[Torsten]

Your linear systems in the course of the Newton iterations cannot be solved since the Jacobian is rank-deficient (it has only rank 5 instead of rank 6 as needed). This usually indicates that the equations you try to solve are not independent or even contradictory.

Angle=3;
phi4=pi/180*Angle;

theta=linspace(0,2*pi, 72);

%First guesses
phi1=ones(1, 72);
phi2=ones(1, 72);
phi3=ones(1, 72);
phi5=ones(1, 72);
d=ones(1,72);
s=ones(1, 72);


syms phi1V phi2V phi3V phi5V dV sV;

xV=[sV, dV, phi1V, phi2V, phi3V, phi5V];


ds=0;
dd=0;
dphi1=0;
dphi2=0;
dphi3=0;
dphi4=0;

%dx=[ds, dd, dphi1, dphi2, dphi3, phi5]';

lim=0.01;

i=1;

while(i<73)

%redefine functions for changes in theta
f1=25-sV+dV*cos(phi2V)+3.1*cos(phi3V);
f2=-5+dV*sin(phi2V)+3.1*sin(phi3V);
f3=-25+6.3*cos(theta(i))+27*cos(phi1V)-19.2*cos(phi2V);
f4=11+6.3*sin(theta(i))+27*sin(phi1V)-19.2*sin(phi2V);
f5=-40+6.3*cos(theta(i))+(dV-19.2)*cos(phi2V)+29.1*cos(phi5V)+16*cos(phi4);
f6=-24+6.3*sin(theta(i))+(dV-19.2)*sin(phi2V)+29.1*sin(phi5V)+16*sin(phi4);

f=[f1, f2, f3, f4, f5, f6]';

J=jacobian(f, xV);

%M=-inv(J)*f


x=[s(i), d(i), phi1(i), phi2(i), phi3(i), phi5(i)]

    firstloop=1;

    while(ds > lim | dd > lim | dphi1 > lim | dphi2 > lim | dphi3 > lim | dphi4 > lim | firstloop==1)
        
        dx=-subs(J,xV,x)\subs(f,xV,x);%subs(M, xV, x)
        rank(subs(J,xV,x))
        rank([subs(J,xV,x),subs(f,xV,x)])
        x=x+dx;
        firstloop=0;
        
    end
    
    i=i+1;
end

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

x = 1×6

1 1 1 1 1 1

Warning: Solution does not exist because the system is inconsistent.

ans = 5

ans = 6

Answer 3

[Matt J]

Raw Newton-Raphson is a pretty terrible method, but I assume you'll tell us you have no choice...

[f,M]=preanalysis(); %Symbolic pre-analysis


%%%%Numeric stuff -- Newton-Raphson

Angle=3;
phi4=pi/180*Angle;

theta=linspace(0,2*pi, 72);

lim=1e-6;

i=1;

clear solutions

while(i<73)
 
    F=@(x) f(x, theta(i),phi4);
    Update=@(x) M(x, theta(i),phi4);

    x=F(zeros(6,1))/1000; %Initial guess - needs some experimentation
    
    dx=inf;
    firstloop=1;

    while( all(abs(dx)>lim) | firstloop==1)
        
        dx=Update(x);
        if any(~isfinite(dx)),
            warning 'Bad update. Aborting...'
            break; 
        end
        x=x+dx;
        
        x(1:4)=atan2(  sin(x(1:4))  , cos(x(1:4))); %normalize the 2*pi periodicity
        
        firstloop=0;
        
    end
    
    solutions{i}=x;

    i=i+1;
end

solutions=cell2mat(solutions);

solutions(:,1:20)

ans = 6×20

0.2505 0.2273 0.2054 0.1852 0.1665 0.1495 0.1342 0.1205 -1.3914 -1.3886 -1.3832 -1.3755 -1.3657 -1.3539 -1.3404 0.0657 0.0650 -1.2913 -1.2727 -1.2534 1.1719 1.1649 1.1619 1.1628 1.1675 1.1756 1.1870 1.2013 -2.5013 -2.5284 -2.5536 -2.5767 -2.5979 -2.6170 -2.6342 1.3896 1.4184 -2.6733 -2.6823 -2.6893 -1.1402 0.8784 0.7734 2.8885 -0.6993 -2.6619 -0.4288 1.4046 2.4737 -1.4770 1.7734 0.2960 -0.9963 -0.8606 1.4129 -0.8061 -2.7819 -2.7265 -1.8485 0.0196 0.9171 0.9085 0.8983 0.8866 0.8737 0.8598 0.8450 0.8294 0.7709 -2.6653 -2.6950 -2.7217 0.6895 2.9977 0.6579 -0.1126 -0.9554 3.0019 0.6113 0.6041 19.2667 18.8273 18.4260 18.0661 17.7507 17.4828 17.2651 17.1002 21.3498 -36.4085 -36.4210 -36.5029 21.1271 -62.8851 20.4115 119.0125 60.3767 -62.0026 18.8311 18.3472 6.4689 27.3019 31.5546 8.4895 3.5565 10.3017 1.9542 45.0152 31.9151 32.4906 51.2072 24.6759 6.1926 138.3313 -16.2680 8.5029 -95.8681 524.2912 -2.9438 89.7521

function [F,M]=preanalysis()
%%%Symbolic stuff
    
    syms phi1 phi2 phi3 phi5 d s theta phi4  real 
    
    f1=25-s+d*cos(phi2)+3.1*cos(phi3);
    f2=-5+d*sin(phi2)+3.1*sin(phi3);
    f3=-25+6.3*cos(theta)+27*cos(phi1)-19.2*cos(phi2);
    f4=11+6.3*sin(theta)+27*sin(phi1)-19.2*sin(phi2);
    f5=-40+6.3*cos(theta)+(d-19.2)*cos(phi2)+29.1*cos(phi5)+16*cos(phi4);
    f6=-24+6.3*sin(theta)+(d-19.2)*sin(phi2)+29.1*sin(phi5)+16*sin(phi4);
    
    f=[f1, f2, f3, f4, f5, f6]';
    
    x=[phi1 phi2 phi3 phi5 d s]';
    
    J=jacobian(f, x);
    
    M=matlabFunction(-J\f,'Vars',{x,theta,phi4});

    F=matlabFunction(f,'Vars',{x,theta,phi4});

end