Newtons Method but error message "Singular Matrix"

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When I tried to solve theta and V with Newton for systems I got this error message
  1. Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 1.765187e-24.
Previously I had no problems just solving theta with Newtons for systems, but adding V got me an ill conditioned matrix. We tried different initial values to see if they were triggering, but all of them lead to the same error message. We would be very grateful for any help we could get!
function [theta,V]= newtonfsys1(theta1,V1)
t=1; it=0; maxit=100;theta0=0;V0=200/3.6;
while norm(t)>1e-9 & it<maxit
%Creates a list of x and y values
[v1,v2,v3,v4,v5]=varden(theta0,V0);
[x0,y0]=rungekutta(v1,v2,v3,v4,v5);
[v1,v2,v3,v4,v5]=varden(theta1,V1);
[x1,y1]=rungekutta(v1,v2,v3,v4,v5);
%checking if condition is being met
for i=1:length(x1)
if (abs(11.8872-x1(i))<0.01) && (abs(18.288-x1(end))<0.01)
break
end
end
for p=1:length(x0)
if (abs(11.8872-x0(p))<0.01) && (abs(18.288-x0(end))<0.01)
break
end
end
% takes the "i" & "p" from the condition and picks a good(hopefully constant for x and y
y1=y1(i)-1.001; y0=y0(p)-1.001; x1=x1(i)-11.8872; x0=x0(p)-11.8872;
%approximate derivative
dxdtheta= (x1-x0)/(theta1-theta0);
dydtheta = (y1-y0)/(theta1-theta0);
dxdV = (x1-x0)/(V1-V0);
dydV = (y1-y0)/(V1-V0);
f=[x1;y1];
J = [dxdtheta,dxdV;dydtheta,dydV];
t=J\f;
disp(' theta V f t')
disp([theta1 norm(V1) norm(f) norm(t)])
theta0=theta1; V0=V1; theta1=theta1-t; V1=V1-t(2); it=it+1;
end
if it<maxit
theta=theta1; V=V1;
else
disp('Ingen konvergens!')
theta=[];
end
  9 Comments
Helen Hailegesesse
Helen Hailegesesse on 7 Dec 2023
We arent sure of where you mean we should change our Runga-Kutta to 4 updates? Its either in our function for newton or our function for Runga-Kutta and either way we arent sure of how to implement it. Sorry for our confusion, we're very grateful for this help!
Helen Hailegesesse
Helen Hailegesesse on 7 Dec 2023
The time of flight should be determined by the initial angle and velocity in (x0,y0). Besides ((x1,y1) = (11.8872,1.001) we also have a third point the curve needs to pass through (x2,y2) = (18.288, 0), the landing location. Its possible that there are several different combinations of angle and velocity that go through those three points, but we suspect that theres one unique curve that does. We were tasked to find the smallest angle that would achieve this.

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Accepted Answer

Torsten
Torsten on 7 Dec 2023
Edited: Torsten on 8 Dec 2023
This is quite a tricky problem - even with the MATLAB solvers available.
For given V and theta at t = 0, integrate until x becomes 11.8872 and use the difference between y from the integration and y1 as first nonlinear equation to be satisfied. Then continue integrating until x becomes 18.288 and use the difference between y from the integration and y2 as second nonlinear equation to be satisfied.
This gives a system of two nonlinear equations in the unknowns V and theta.
I think it will become quite messy to set it up with your own ode integrator and nonlinear solver.
x0 = 0;
y0 = 2.3 ;
x1 = 11.8872;
y1 = 1.001;
x2 = 18.288;
y2 = 0;
g = 9.82;
kx = 0.01;
ky = 0.02;
m = 0.058;
fun_ode = @(t,u)[u(3);u(4);-kx*(sqrt(u(3)^2+u(4)^2))^1.5 / m;(-m*g - ky*(sqrt(u(3)^2+u(4)^2))^1.5) / m];
tspan = [0 100];
V0 = 300/3.6;
theta0 = -4*pi/180;
sol = fsolve(@(z)fun_nle(z,x0,y0,x1,y1,x2,y2,fun_ode,tspan),[V0; theta0]);
Equation solved. fsolve completed because the vector of function values is near zero as measured by the value of the function tolerance, and the problem appears regular as measured by the gradient.
V = sol(1)
V = 5.3614e+03
theta = sol(2)
theta = -0.0793
event = @(t,u)deal([u(2),1,0]);
options = odeset('Events',event);
u0 = [x0;y0;V*cos(theta);V*sin(theta)];
[~,U] = ode45(fun_ode,tspan,u0,options);
plot(U(:,1),U(:,2))
grid on
function res = fun_nle(z,x0,y0,x1,y1,x2,y2,fun_ode,tspan)
V = z(1);
theta = z(2);
xp = V*cos(theta);
yp = V*sin(theta);
u0 = [x0;y0;xp;yp];
event1 = @(t,u)deal([u(1)-x1,1,0]);
options = odeset('Events',event1);
[~,~,TE,UE,IE] = ode45(fun_ode,tspan,u0,options);
res(1,1) = UE(2) - y1;
u0 = UE;
event2 = @(t,u)deal([u(1)-x2,1,0]);
options = odeset('Events',event2);
[~,~,TE,UE,IE] = ode45(fun_ode,tspan,u0,options);
res(2,1) = UE(2) - y2;
end
  2 Comments
Helen Hailegesesse
Helen Hailegesesse on 7 Dec 2023
The assignment was to solve it without Matlabs integrators and solvers, that’s why we made our own Runga Kutta and Newton functions. But thank you for this extensive answer, you’ve helped a great deal!
Torsten
Torsten on 7 Dec 2023
Edited: Torsten on 8 Dec 2023
If you replace the calls to "fsolve" and "ode45" by calls to your own nonlinear solver and ode integrator, the code from above will also work for self-written code.
This structure - a clear separation of nonlinear solver and ode integrator - will force you not to mix parts of the ode integrator with the nonlinear solver (as you did in your code from above).
The below code e.g. replaced "fsolve" by the self-written code "nls" - simply by changing a single name:
sol = nls(@(z)fun_nle(z,x0,y0,x1,y1,x2,y2,fun_ode,tspan),[V0; theta0]);
instead of
sol = fsolve(@(z)fun_nle(z,x0,y0,x1,y1,x2,y2,fun_ode,tspan),[V0; theta0]);
Do the same for "rungekutta" instead of "ode45", and you are done.
x0 = 0;
y0 = 2.3 ;
x1 = 11.8872;
y1 = 1.001;
x2 = 18.288;
y2 = 0;
g = 9.82;
kx = 0.01;
ky = 0.02;
m = 0.058;
fun_ode = @(t,u)[u(3);u(4);-kx*(sqrt(u(3)^2+u(4)^2))^1.5 / m;(-m*g - ky*(sqrt(u(3)^2+u(4)^2))^1.5) / m];
tspan = [0 100];
V0 = 300/3.6;
theta0 = -4*pi/180;
sol = nls(@(z)fun_nle(z,x0,y0,x1,y1,x2,y2,fun_ode,tspan),[V0; theta0]);
V = sol(1)
V = 5.3614e+03
theta = sol(2)
theta = -0.0793
event = @(t,u)deal([u(2),1,0]);
options = odeset('Events',event);
u0 = [x0;y0;V*cos(theta);V*sin(theta)];
[~,U] = ode45(fun_ode,tspan,u0,options);
plot(U(:,1),U(:,2))
grid on
function res = fun_nle(z,x0,y0,x1,y1,x2,y2,fun_ode,tspan)
V = z(1);
theta = z(2);
xp = V*cos(theta);
yp = V*sin(theta);
u0 = [x0;y0;xp;yp];
event1 = @(t,u)deal([u(1)-x1,1,0]);
options = odeset('Events',event1);
[~,~,TE,UE,IE] = ode45(fun_ode,tspan,u0,options);
res(1,1) = UE(2) - y1;
u0 = UE;
event2 = @(t,u)deal([u(1)-x2,1,0]);
options = odeset('Events',event2);
[~,~,TE,UE,IE] = ode45(fun_ode,tspan,u0,options);
res(2,1) = UE(2) - y2;
end
function u = nls(f,uold)
u = zeros(numel(uold),1);
itermax = 30;
eps = 1e-6;
error = 1.0e5;
uiter = uold;
iter = 0;
while error > eps && iter < itermax
u = uiter - Jac(f,uiter)\f(uiter);
error = norm(u-uiter);
iter = iter + 1;
uiter = u;
end
end
function J = Jac(f,u)
y = f(u);
h = 1e-6;
for i = 1:numel(u)
u(i) = u(i) + h;
yh = f(u);
J(:,i) = (yh-y)/h;
u(i) = u(i) - h;
end
end

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More Answers (1)

Matt J
Matt J on 7 Dec 2023
Edited: Matt J on 7 Dec 2023
Landing on a point with ill-conditioned J is a hazard of plain-vanilla Newton's method. That's why people usually don't use it. They use fsolve instead.
  2 Comments
Helen Hailegesesse
Helen Hailegesesse on 7 Dec 2023
Hey, thank you for answering!
We arent allowed to use fsolve, so our problem is rather to make J well conditioned using Newton's method. At the moment it stops after two iterations, and theta and V become NaN.
Matt J
Matt J on 7 Dec 2023
Edited: Matt J on 7 Dec 2023
Well, there are several things you can try,
  1. Use actual derivatives instead of finite differences
  2. Reduce the finite difference stepsize
  3. Choose a different initial point. Since it's only a 2 variable problem,you should be able to plot the objective function to see approximately where the solution lies.You could also plot cond(J) as a surface to see where the singularities are, and avoid them.

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