MATLAB Help - Multi-Dimensional Newton Rapshon

How to use Newton Raphson for more than just one dimension.
Things to look out for in N-Dimensional Newton-Raphson
1.) Stationary points - gradient is 1D or det(jacobian) = 0
2.) Poor initial conditions - Try a few different initial conditions to prevent local minima.
3.) Overshoot - Change that alfa parameter to make sure you don't blow up
4.) There are many other problems but those are the most important.

[@88ahh]

Thank you very much man. I watched it 3 times ;)

[@denizdurer6295]

I have got two function fval(1,1)=r2cos(q)+r3cos(x)-r4cos(y)-r1; fval(2,1)=r2sin(q)+r3sin(x)-r4sin(y); r values are given and q is at [0,2pi] and delta q is 2.5 degrees. I have to plot q versus x and q versus y graph. How can I do that??

[@ManhaNAsad]

Dear CM, It has been a pleasure watching your tutorials. With your kind help, I have started learning much more, I expected to do by my own. Could you please find some time to make a tutorials on Gauss Seidel/Newton Raphson for a system of form AX=B? Thanks

[@CarlosMontalvo251]

+zhk I've never done this type of system using Newton-Raphson mainly because the system is linear. Typically AX=B is solved the easiest using row reduction such as using rref(). kzbin.info/www/bejne/oJbCcmqgpMyrl7M NR is typically reserved for non-linear systems.

[@ManhaNAsad]

+Carlos Montalvo Thanks dear for your prompt reply. In the back of my head, I was thinking to learn to use NR while solving a PDE. For this purpose, I thought that AX=B, will be a good starting point. How stupid of me, as you have mentioned correctly, the application of NR is to solve nonlinear systems. I have watched the rref() video many times and for the first time, I learned how to code rref manually, Thanks to you. By the way, you can use this example (x^2+x*y=10, y+3*x*y^2=57 with the solution x=2 and y=3), if you are plannning to do a tutorials on NR for a nonlinear system?

[@CarlosMontalvo251]

+zhk Ok that's a standard non-linear system with two equations. The video here is just 1 equation with two unknowns. I'm actually working on a video right now using NR and Broyden's method that I should be finished with sometime this week. If you can wait a few days I can solve that problem you just posted. You subscribed to my channel so you should get an email about it once I post it.

[@ManhaNAsad]

Sure. I can wait. Thanks

[@CarlosMontalvo251]

I realize I took 5 months to get to this but here is a video on Broyden's method kzbin.info/www/bejne/l6Clomxrf7aep68

[@saddamhussainsoomro2810]

why you use xx or yy and zz, why not single x y or z??? why you show some extra skills here to confuse the new learners, please make it simple and straight forward, and how to solve n equations with n variables? Thanks and please dont mind sir

[@CarlosMontalvo251]

Variable names can be whatever you want. Single x,y, and z is totally valid but I use it to denote single dimensional vectors. xx,yy, and zz are matrices. If you read the documentation on meshgrid www.mathworks.com/help/matlab/ref/meshgrid.html they use capital letters to denote matrices but I don't like having to deal with case sensitive variables. So it's more of a personal preference. This youtube video is straightforward and has helped many people. It is by no means a "new learner" video. You need to be familiar with : plotting in 3D - kzbin.info/www/bejne/foSroaJtft1mncU meshing - kzbin.info/www/bejne/hmeZhKOsn5l7f68 and of course single dimensional newton raphson - kzbin.info/www/bejne/iZzLmJSNndOYZq8 Good luck in your studies. As for solving n equations and n unknowns, it depends on if the equations are nonlinear or not. If they are linear you just need to solve a standard linear set of equations Ax = b where the solution is just A\b If the equations are nonlinear then you have to use an optimization technique. NR is a good start.

[@saddamhussainsoomro2810]

@@CarlosMontalvo251 Thanks sir

[@harsharajendran41088]

Hello, I have a quadratic function to solve. with 4 variables and there are 4 equations. How do I iterate in matlab?

[@CarlosMontalvo251]

This is a tough problem. The solution I presented above is with 1 equation and 2 unknowns. That is z = f(x,y) and you're trying to solve where z =0. Your situations is 4 independent function fn(x,y,z,y) = 0 where fn has 4 function f1,f2,f3,f4. Unfortunately the only way I have come close to solutions like that is where I minimize a scalar cost function. In your case I would set z = f1^2 + f2^2 + f3^2 + f4^2 and then find x,y,z,t that minimizes z or in this case finds where z = 0.

[@raghundon]

Thank you for the presentation. How to solve a set of transcendental equations using newton raphson? Can you please make a video for that?

[@CarlosMontalvo251]

Can you give me an example of a set of transcendental equations?

[@raghundon]

Something like this - cos x1 + cos x2 + cosx 3 = 3M cos5 x1 + cos5 x2 + cos5x 3 = 0 cos7 x1 + cos7 x2 + cos7x 3 = 0 M ranges from 0 to 1.

[@CarlosMontalvo251]

Are those three separate equations? Or do you want the minimum of all three simultaneous equations? Your second and third equations don't make sense. What is cos5 x2? Is that cos(5*x2) where x2 is a variable? Or is it cos(5)*x2? Also, functions like sine and cosine are cyclic and thus have multiple solutions. Are you sure this is what you want?

[@raghundon]

Thank you for the reply, sorry for the poor typing. This should make it clear - cos a + cos b + cos c = 3M cos 5a + cos 5b + cos 5c = 0 cos 7a + cos 7b + cos 7c = 0 M varies b/w 0 to 1. a, b and c are variables. It should be solved simultaneously. Yes, u r right they result in multiple solutions. I want exactly that. The resultant contours aren't continuous either. This makes the programming challenging Thank you!

[@CarlosMontalvo251]

Ok yea so basically you have an equation of the form f(x) = 0 where f is a vector function and x is a vector as well. It's almost the same as the video, the only different is that f is now a vector. Let me think about it for a bit and get back to you.

[@mehmetkaraaslan1649]

Do you have any video for modified Newton Raphson method.?

[@CarlosMontalvo251]

Do you mean like secant method. kzbin.info/www/bejne/nGici3WMmJiHY68

[@abdullahomair2811]

Hey man, you are the one. Could we have the code . Thanks.

[@CarlosMontalvo251]

+Abdullah Omair function newton2D() clc close all x = -5:0.1:5; y = -5:0.1:5; [xx,yy] = meshgrid(x,y); zz = f(xx,yy); mesh(xx,yy,zz) %%% f = 0 hold on r0 = [-1;1]; %%initial guess on x is 10 and my initial guess on y is 10 alfa = 0.1; while abs(f(r0(1),r0(2))) > 1e-5 r0 = r0 - alfa*f(r0(1),r0(2))./fprime(r0(1),r0(2)); plot3(r0(1),r0(2),f(r0(1),r0(2)),'rs','MarkerSize',20) end r0 f(r0(1),r0(2)) theta = linspace(0,2*pi,100); r = 5; xc = r*cos(theta); yc = r*sin(theta); %plot3(xc,yc,0*xc,'r-','LineWidth',5) hold on %%%Find the minimum r0 = [2;2]; %%initial guess on x is 10 and my initial guess on y is 10 alfa = 0.1; for index = 1:1000 plot3(r0(1),r0(2),f(r0(1),r0(2)),'r*','MarkerSize',20) r0 = r0 - alfa*inv(fdblprime(r0(1),r0(2)))*fprime(r0(1),r0(2)); end r0 f(r0(1),r0(2)) function z = f(x,y) %z = (x+2).*(x-2).*(x-4).*(x+4) + y.^2; %z = (x.^2 - 4).*(x.^2-16) + y.^2; z = x.^4 - 20.*x.^2 + 64 + y.^2; function zprime = fprime(x,y) %%Gradient which is a vector dfdx = 4*x.^3 - 40*x; dfdy = 2*y; zprime = [dfdx;dfdy]; function zdblprime = fdblprime(x,y) %Jacobian which is a matrix df2dx2 = 12*x.^2 - 40; df2dxdy = 0; df2dy2 = 2; df2dydx = 0; zdblprime = [df2dx2 df2dxdy ; df2dydx df2dy2];

[@abdullahomair2811]

Thanks dude

[@alibaskan9200]

Hello Dr Carlos, thanks for the video and it is really helpful for me. Im new at Matlab so I want to ask you about function command. I know that it should have ended with "end" but you didnt use "end" in their last line. I have tried them with "end" and it has already smulated it. But in some another projects it doesnt work without "end" Why? can you explain this for me ?

[@CarlosMontalvo251]

You just have to be consistent. If you use end in one function you have to use end in another. I prefer not to use end just because it's less typing

[@alibaskan9200]

@@CarlosMontalvo251 Thanks for your reply.

[@alibaskan9200]

​@@CarlosMontalvo251 I have another question for you :) what is alfa = 0.2 there is no anyhing like this in general formula.

[@CarlosMontalvo251]

@@alibaskan9200 Alfa is a user parameter that is dependent on your system. Think of it like a control gain or a knob you can tune. If the number is big the system could go unstable. If it's too small the system may take too long to converge. There is a happy medium to ensure convergence and not create an instability.

[@alibaskan9200]

@@CarlosMontalvo251 Thank you, you have really helped me.

[@zeepan4443]

Hi, How to solve if any function have four variables , etc f(w,x,y,z) ?

[@CarlosMontalvo251]

You just add more dimensions to the gradient and the jacobian.

[@zeepan4443]

I tried but failed. Sir, would you please help me? its my project part.

[@zeepan4443]

Sir, for example in following example, if i added one more variable named 'm' as given in following example, function newton2D() clc close all x=-10:0.1:10; y=-10:0.1:10; m=-10:0.1:10; [xx,yy,mm]=meshgrid(x,y,m); zz=f(xx,yy,mm); mesh(xx,yy,mm,zz) %find that where f-0? r0=[10;10;10]; %Initial guess on X, and Y are 10 alfa=0.2; while abs(f(r0(1),r0(2),r0(3)))>1e-6 r0=r0-alfa*f(r0(1),r0(2),r0(3))./fprime(r0(1),r0(2),r0(3)); end hold on plot3(r0(1),r0(2),r0(3),0,'rs','MarkerSize',20) function z=f(x,y,m) z=x.^2+y.^2-25+m.^2; function zprime= fprime(x,y,m) z=x.^2+y.^2+m.^2-25; dfdx=2*x; dfdy=2*y; dfdm=2*m; zprime=[dfdx;dfdy;dfdm]; then i received following error while simulating it.

[@zeepan4443]

Error using matlab.graphics.chart.primitive.Surface/set While setting the 'XData' property of 'Surface': Value must be a vector or 2D array of numeric type Error in matlab.graphics.chart.internal.ctorHelper (line 6) set(obj, pvpairs{:}); Error in matlab.graphics.chart.primitive.Surface Error in mesh (line 139) hh = matlab.graphics.chart.primitive.Surface('XData',x,'YData',y,'ZData',z,'CData',c,... Error in newton2D (line 10) mesh(xx,yy,mm,zz)

[@CarlosMontalvo251]

You can't just arbitrarily add an extra variable everywhere. Meshgrid is soley used for plotting and you can't plot 3 dimensional functions. You're getting an error because of that. What I'm saying is that the mesh function only takes 3 arguments. If you are missing that then you do not understand what the underlying mechanics are of the problem and I can't help you from my keyboard or from youtube. I suggest you talk to one of your professors or colleagues in person to help you.

[@andressuarez7503]

hi good night how can I contact you

[@metroidandroid]

"If you can do it in 2 dimensions you can do it in 5 dimensions", that's the equivalent of 2setviolin "if you can play it slow, you can play it fast" lol

[@CarlosMontalvo251]

Haha. Programming is a bit different. Once you've vectorized you're code there is no limit to how big you can make the vector unless you max out on computer memory

[@keithkelly6197]

Make the font size bigger

[@CarlosMontalvo251]

gitlab.com/Montalvo/MATLAB_Games.git just download the code

[@sebastianossandon2532]

I have to program this method in a way you can input N numbers of variables and equations and the program calculates its derivatives (not using matlab functions) and solution... I don't even know how to start lol.

[@CarlosMontalvo251]

What you probably want to do is start with computing derivatives numerically kzbin.info/www/bejne/qnWTin6cltWmg5Y

[@sebastianossandon2532]

I do have a method to do that, my main issue is to expand things to work for n variables, specially the jacobian.

[@CarlosMontalvo251]

Oh man. Yea that is tough. I feel like I've done this before though. I wonder if I dig through my notes I can find something

[@sebastianossandon2532]

Oh that would be so helpful... any hint is welcome lol.

[@whentutv7415]

Hi Sebastian, Did you succeed ? I am also having the same problem. Could you help me please

[@mathematical_fun_with_Sumbel]

Increase font size plz

[@CarlosMontalvo251]

Too late now. I'll do it next time

[@mathematical_fun_with_Sumbel]

Please do something sir, my research work base on this code. Sir can u please send m the video if possible

[@CarlosMontalvo251]

@@mathematical_fun_with_Sumbel It would probably be easier for you to just download the code. It should be on this repo gitlab.com/Montalvo/MATLAB_Games.git

[@danielcm81]

click, click, click, blah, blah, blah, click, click, click, blah.