Newton Raphson gives answers divided by 0

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Jean-Paul
Jean-Paul 2024-3-10
编辑： Matt J 2024-3-11
Ran in:
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.

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采纳的回答

Sam Chak
Sam Chak 2024-3-10
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.

更多回答（2 个）

Torsten
Torsten 2024-3-10
编辑：Torsten 2024-3-10
Ran in:
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
Matt J
Matt J 2024-3-11
编辑：Matt J 2024-3-11
Ran in:
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

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