Newton Raphson implementation, not converging, maybe error in implementation

I have implemented Newton-Raphson method for my non-linear function with one variable "aa" and others are constant.

I read that Newton-Raphson method does not depend much on the initial guess. My implementation is not converging and it is not getting me a solution. I know the solution for this problem based on other method as 0.0142. I would like to know the mistakes in implementation, please take a look at it and let me know your suggestions.

% Predefined parameters
dd = 0.1069;
vv = 0.7889;
L = 1;

% Define the function and its derivative
func = @(aa) ( (-(sqrt(L^2-vv^2)/2)) + (aa*sinh((dd/2)/aa)) );
func_derivative = @(aa) (sinh(dd/(2*aa)) - ((-dd/(2*aa))*(cosh( (dd/(2*aa))/aa )))  );

% Initial guess
aa_guess = 0.01; % You can choose any initial guess

% Tolerance and maximum iterations
tol = 1e-6;
max_iter = 1000;

% Newton-Raphson method
iter = 0;
while true
    iter = iter + 1;
    
    % Evaluate the function and its derivative at the current guess
    f = func(aa_guess);
    %f_prime = func_derivative(aa_guess);
    f_prime = (func(aa_guess+1e-6)-f)*1e6
    
    % Newton-Raphson update rule
    aa_new = aa_guess - f / f_prime;
    
    % Display iteration results
    fprintf('Iteration %d: aa = %f, f(aa) = %f, f''(aa) = %f\n', iter, aa_guess, f, f_prime);
    
    % Check for convergence
    if abs(aa_new - aa_guess) < tol || iter >= max_iter
        break;
    end
    
    % Update the guess
    aa_guess = aa_new;
end

f_prime = -455.1300

Iteration 1: aa = 0.010000, f(aa) = 0.740505, f'(aa) = -455.130017

f_prime = -178.3815

Iteration 2: aa = 0.011627, f(aa) = 0.269331, f'(aa) = -178.381458

f_prime = -89.7550

Iteration 3: aa = 0.013137, f(aa) = 0.076755, f'(aa) = -89.754985

f_prime = -64.3456

Iteration 4: aa = 0.013992, f(aa) = 0.011643, f'(aa) = -64.345556

f_prime = -60.2285

Iteration 5: aa = 0.014173, f(aa) = 0.000376, f'(aa) = -60.228463

f_prime = -60.0928

Iteration 6: aa = 0.014179, f(aa) = 0.000000, f'(aa) = -60.092768

if iter >= max_iter
    disp('Newton-Raphson method did not converge within maximum iterations.');
else
    fprintf('Root found at aa = %f after %d iterations.\n', aa_guess, iter);
end

Root found at aa = 0.014179 after 6 iterations.

[Torsten]

Seems your analytical derivative of the function is incorrect (see above for the result after substitution by a finite-difference quotient).

[VIGNESH BALAJI]

@Torsten I looked into your code and saw that you have changed the analytical derivative to finite difference. I am unable to understand why you multiplied it with 10^-6 after taking the difference ?

f_prime = (func(aa_guess+1e-6)-f)*1e6;

I also saw that you changed the initial guess from 0.1 to 0.01 . I made the intial guess at 0.1 and the code did not work. It gives me an conclusion that Newton-Raphson method highly depends on inital guess ? Am I correct ? Is there any other method which is not highly affected by the initial guess ?

[Dyuman Joshi]

"I am unable to understand why you multiplied it with 10^-6 after taking the difference ?"

That is not multiplication with 10^-6, but division by 10^-6, which results in multiplication by 10^6. Note the sign of the exponent.

Basically that is f' = (f(x+del) - f(x))/del

[John D'Errico]

@VIGNESH BALAJI

Are there methods that depend less on the initial guess? YES, and sometimes, no. And this is big reason why you don't want to write your own solvers, but also why you need to learn about the solvers, and how to use them properly.

For example, fzero is a better choice of solver almost always than writing a Newton method of your own. It will converge rapidly when the problem is well behaved. But it also has ways of avoiding problems.

dd = 0.1069;
vv = 0.7889;
L = 1;
% Define the function and its derivative
func = @(aa) ( (-(sqrt(L^2-vv^2)/2)) + (aa*sinh((dd/2)/aa)) );

[xval,fval,exitflag] = fzero(func,0.1)

xval = 0.0142

fval = -5.5511e-17

exitflag = 1

Even better is if you will provide a bracket around the root, as that will insure it finds a proper solution.

[xval,fval,exitflag] = fzero(func,[1e-2,0.1])

xval = 0.0142

fval = -5.5511e-17

exitflag = 1

USE EXISTING TOOLS WHENEVER POSSIBLE. Unless of course, you can do a better job than the professional who wrote that tool.

Finally, you should note that had you used fzero here, it would have saved you perhaps hours of time writing code, and trying then to figure out where you went wrong. And it took me one line of code to solve the problem. I never even needed to compute the derivative, which you apparently got wrong.

Answer 1

[John D'Errico]

There are some significant misunderstandings on your part that I want to clear up, as well as perhaps help you solve your problem.

First, that you can always choose any start point. WRONG.

It is not true that a Newton scheme can start anywhere. Some start points may cause your method to diverge to +/- inf. Some start points may cause your scheme to converge to a non-solution. Some start points may cause numerical problems in the evaluation. So it is not at all true that any point is acceptable!

For example, consider the simple cubic polynomial...

syms x
y = expand((x+1)*(x-1)*(x-2) + 1)

y = 

fplot(y,[-2,3])
yline(0)

Clearly, it has only one real root, near -1.5. If you start the solver in the region of that root, it will converge to the root we want.

But imagine you decide to start the solver out around x==2? Now it will probably get stuck in that valley. A Newton scheme may just oscillate in that valley around 1.5, and never escape.

And suppose we decide to start the solver at one of two flat spots on the curve, where the slope is exactly zero? It will diverge to plus or minus infinity at the first step, due to the resulting divide by zero.

The concept of a basin of attraction is an important one. It is the set of starting points that will converge to a valid solution. As long as you start the solver in a spot that lies in the basin of attraction, AND that does not create a numerical problem, like an underflow or overflow in double precision arithmetic, then yes, you can succeed. But you stated several times that it does not matter where you start. And that is simply wrong.

Now we can look at your specific problem.

dd = 0.1069;
vv = 0.7889;
L = 1;
% Define the function and its derivative
% Use the dotted operators to allow the code to be vectorized
func = @(aa) ( (-(sqrt(L^2-vv^2)/2)) + (aa.*sinh((dd/2)./aa)) );
% PLOT THE FUNCTION!!!!!!!!!!!! ALWAYS PLOT EVERYTHING!!!!!!!!!
fplot(func)

It looks like your function has a spike in it around zero. Zoom in.

fplot(func,[-.1,.1])
ylim([-.2,.2])
yline(0)
grid on

So 0.01 might not be a terrible guess as a start point. A little low, but not bad.

Next, check if your derivative is correct. I can drop the first term, since it is not a function of aa.

syms DD AA
diff(AA*sinh((DD/2)/AA),AA)

ans = 

And you wrote this:

(sinh(dd/(2*aa)) - ((-dd/(2*aa))*(cosh( (dd/(2*aa))/aa ))) );

which appears to be wrong. So fix your derivative. It looks like you divided by aa too many times.

And use fewer parens. Using more parens is not always a good thing, if it makes your code unreadable, to the extent that you cannot read it yourself.

[VIGNESH BALAJI]

@John D'Errico Thanks a lot for the detailed answer and clearing my wrong understanding. I understand the need and reason for starting in the right region. I guess the valley you are referring to is like a local minima and a function can have many local minima / valleys and if i start there then it will get struck there as the derivative will not be able to yield a direction from that point. I will also make sure that I use only required number of brackets and follow BODMAS rule more efficiently.

I mainly want to use the available tools and I have a problem statement of changing parameters for which I cannot choose a constant search region in my simulation hence, I have been trying 3 ways - fzero, fsolve and newton-raphson.

I will explain you my problem in detail here.

The 2 constant parameters of d (horizontal distance) and v (vertical distance) can vary to any number. Their maximum variation is constrained by L (physical length) following pythogoras theoreom and minimum can be very less close to 0. Due to this variation, I am unable to keep a constant search region, I believe as the parameters vary the search region also varies because I have this function

fun = @(a)  ( (-(sqrt(L^2-v^2)/2)) + (a*sinh((d/2)/a)) ) ;

In this function, the second term a*sinh will not have a zero crossing (I need to plot and check it), whereas the first part of the equation makes the function to have a zero crossing. The parameter v is affecting it the zero crossing and the parameter d is affecting the shape of the curve.

While, I was running the simulation for a trajectory, for few configurations example the data point I provided above made the simulation to break as fzero was not able to find the right roots due to incorrect search region. This lead to the search for other methods to solve it.

If I am correct fsolve (True region method), will also suffer from this incorrect search region problem just like fzero and newton-raphson. Am I correct ?

I have explained my problem in detail. I would really like your suggestion to solve this problem for all possible range of configurations for my nonlinear function. I am looking forward to hearing from you.

[Torsten]

If I am correct fsolve (True region method), will also suffer from this incorrect search region problem just like fzero and newton-raphson. Am I correct ?

Yes.

I have explained my problem in detail. I would really like your suggestion to solve this problem for all possible range of configurations for my nonlinear function.

Find two points xl and xr for a with f(xl)*f(xr) < 0 and call fzero with the interval [xl xr] as initial guess. To find these points, you could successively scan the range of values for "a" that are physically senseful.

[VIGNESH BALAJI]

@Torsten Thanks for your answer.

I have few questions in your suggestion to understand it more.

1. Why do you prefer fzero over folve ?
2. I understand the reason behind f(xl)*f(xr) < 0 as it should have change of sign. Is the way to find the xl (lower limit) and xr (upper limit) based on random guess by keep searching ? I have an idea if write a script to vary all range of values for these 2 parameters (v and d) and record the zero crossings in an array. Probably this could give me a guess for selecting xl and xr.

Let me know your views on this.

[Torsten]

dd = 0.1069;
vv = 0.7889;
L = 1;
func = @(a) ( (-(sqrt(L^2-vv^2)/2)) + (a.*sinh((dd/2)./a)) );

amin = 1e-5;                 % minimum lower limit for a 
amax = 1;                    % maximum upper limit for a
deltaa = 1e-3;               % Scan size for the interval [amin amax]
a = amin:deltaa:amax;
fa = func(a);
index = find(diff(fa>0),1);  % find first sign change position
a = fzero(func,[a(index) a(index+1)])

a = 0.0142

[John D'Errico]

fzero is a more robust solver (as long as there is only one variable) than fsolve. With more than 1 variable, you have no choice as fzero would not apply. If you have a bracket around the solution where there is a change of sign, then fzero will always generate a solution.

If you have no bracket around the root, then fzero uses a fairly simple scheme to try to find a bracket (but it is not terribly intelligent in this respect.) Once it has one, then it can just drop into the default algorithm.

Is there some magic way you can be assured of finding a bracket? NO. I can always generate a function where a bracket can never be found, even though technically, one exists. There is no mathematical trick that will insure a bracket will be found. And one thing I observe with your function, if you go too far out, then you will see numerical problems.

Can you just sample the function at some set of points, and look to see if there is a pair that form a brcket? Yes. Of course. This is a simple scheme. As long as your function is reasonably smooth, you do not even need the points to be terribly close together. And if it is fairly smooth, very often the simple scheme fzero uses internally is sufficient to find the bracket on its own.

You said that sometimes, you had issues with fzero, on some problems. Is there any way to ALWAYS insure a solution will be found for ANY problem? OF COURSE NOT! For some problems, numerical issues may interfere. You may be experiencing underflows or overflows. You may be able to scale things to avoid that, but we have not been shown which which cases had a problem. Possibly, for some choices of parameters, no solution exists at all.

[VIGNESH BALAJI]

@Torsten Thanks a lot for the awesome code, detailing the way to do it :) :)

[VIGNESH BALAJI]

@John D'Errico Okay, I get it you mean fzero (Bisection method) is better than fsolve (True Region method).

The main way to solve my problem is to find a right Bracket. Can you elaborate your statement if I go too far in my function, then I will see numerical problems.

I just got an idea, my variation in the 2 constant parameters of the function are based on real world scenarios (have a length constraint, thereby they have a minimum and a maximum range). In that case, I can plot this function with the minimum (dd = 0.01, vv=0.98) and maximum range(dd=0.98, vv = 0) and also I can get some points in the middle. This can help me to plot and see how the function varies, a visual representation is always good to understand it. I would really like to see the existence of valleys (local minimas) and flat points (I didn't understand the concept of flat points earlier, can you please explain it again).

I made a small code for this visualisation

close all
clear all
clc

x1 = 0;
y1 = 1.0;

x2 = 0.01;
y2 = 1.98;
 %% plot sinh
dd = abs(x1-x2);
vv = abs(y1-y2);
L = 1.0;

func = @(a) ( (-(sqrt(L^2-vv^2)/2)) + (a.*sinh((dd/2)./a)) );

figure();
[X,Y] = fplot(func);

idx = find( diff( sign( Y(1,:) ) ) ) + 1;

plot(X,Y);

yline(0, '--','Color', 'r');
ylim([-0.5 1.5])                                        % ADDED
grid   

When I looked at the plots for some of the range of values for dd (horizontal distance) and vv (vertical distance). I was able to understand the funcction better and I was unable to see the valley or flat point in these plots. Please do help me spot them.

1. Normal case dd = 0.5, vv = 0

2. Maximum range dd = 0.98, vv = 0

3. This is the plot where the problem started with incorrect search region dd = 0.1069 , vv = 0.788932 .

In this plot, I earlier had a start search region of 0.2, I guess this is in the flat line and when it was made to 0.01 fzero was able to find the solution as it can see the direction of the curve in that region. I guess the problem with flat line is it does not give the direction of the curve, Am I correct ?

4. This dataset d = 0.01, v = 0.98 has a problem in zero crossing as it does not cross zero. I am not sure whether any method can find this corner case

In this case, fzero did not find a solution whereas fsolve found a solution. I guess the reason could be fzero looks for exact zero crossing and fsolve is looking for a region close to zero. Based on this I am not sure why you still call fzero as robust when compared to fsolve ? Please let me know your suggestion on this case ?

This is the code i used to arrive at a solution for this corner case

dd = 0.01; %abs(x1-x2);
vv = 0.98; %abs(y1-y2);
L = 1.0;
x0 = 0.1;

func = @(a) ( (-(sqrt(L^2-vv^2)/2)) + (a.*sinh((dd/2)./a)) );
a = abs(fsolve(func, x0));

Answer a = 9.322285907071188e-04

Thanks a lot for your answers, I believe I am very close to having a complete understanding. Please do answer me the questions I have asked above about flat points, fsolve and my approach to visualise and select a good enough bracket (as it is not always possible to get a right bracket).

I am looking forward for your answers :) :)

[Matt J]

First, that you can always choose any start point. WRONG.

In general, that is true, but I believe that Newton-Raphson is globally convergent for strictly convex functions, which in this case it is (assuming, aa>0, dd>0). You would have to threshold the iteration updates to some lower bound lb where fun(lb)>0,

aa_new=max(lb, aa_guess - f / f_prime;)

but this should converge from any starting aa>0.

[Torsten]

It might be a problem to get a suitable lb or to formulate more complicated updates for aa:

% Predefined parameters
dd = 0.1069;
vv = 0.7889;
L = 1;
% Define the function and its derivative
func = @(aa) ( (-(sqrt(L^2-vv^2)/2)) + (aa.*sinh((dd/2)./aa)) );
func_derivative = @(aa) sinh(dd./(2*aa)) - dd*cosh( dd./(2*aa))./(2*aa);
% Initial guess
aa_guess = 1; % You can choose any initial guess
lb = 0.1;
%lb = 0.01;
% Tolerance and maximum iterations
tol = 1e-6;
max_iter = 1000;
% Newton-Raphson method
iter = 0;
while true
    iter = iter + 1;
    
    % Evaluate the function and its derivative at the current guess
    f = func(aa_guess);
    f_prime = func_derivative(aa_guess);
    
    % Newton-Raphson update rule
    aa_new = max(lb,aa_guess - f / f_prime);
    
    % Display iteration results
    fprintf('Iteration %d: aa = %f, f(aa) = %f, f''(aa) = %f\n', iter, aa_guess, f, f_prime);
    
    % Check for convergence
    if (abs(aa_new - aa_guess) < tol && abs(func(aa_new)) < tol) || iter >= max_iter
        break;
    end
    
    % Update the guess
    aa_guess = aa_new;
end

Iteration 1: aa = 1.000000, f(aa) = -0.253785, f'(aa) = -0.000051
Iteration 2: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 3: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 4: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 5: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 6: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 7: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 8: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 9: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 10: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 11: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 12: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 13: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 14: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 15: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 16: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 17: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 18: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 19: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 20: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 21: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 22: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 23: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 24: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 25: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 26: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 27: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 28: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 29: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 30: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 31: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 32: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 33: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 34: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 35: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 36: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 37: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 38: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 39: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 40: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 41: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 42: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 43: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 44: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 45: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 46: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 47: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 48: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 49: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 50: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 51: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 52: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 53: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 54: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 55: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 56: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 57: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 58: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 59: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 60: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 61: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 62: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 63: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 64: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 65: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 66: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 67: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 68: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 69: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 70: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 71: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 72: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 73: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 74: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 75: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 76: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 77: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 78: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 79: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 80: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 81: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 82: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
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Iteration 547: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 548: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 549: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 550: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 551: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 552: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 553: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 554: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 555: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 556: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 557: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 558: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 559: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 560: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 561: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 562: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 563: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 564: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 565: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 566: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 567: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 568: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 569: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 570: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 571: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 572: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 573: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 574: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 575: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 576: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 577: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 578: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 579: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 580: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 581: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 582: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 583: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 584: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 585: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 586: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 587: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 588: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 589: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 590: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 591: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 592: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 593: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 594: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 595: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 596: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 597: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 598: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 599: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 600: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 601: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 602: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 603: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 604: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 605: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 606: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 607: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 608: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 609: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 610: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 611: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 612: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 613: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 614: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 615: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 616: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 617: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 618: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 619: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 620: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 621: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 622: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 623: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 624: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 625: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 626: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 627: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 628: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 629: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 630: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 631: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 632: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 633: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 634: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 635: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 636: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 637: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 638: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 639: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 640: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 641: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 642: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 643: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 644: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 645: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 646: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 647: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 648: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 649: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 650: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 651: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 652: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 653: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 654: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 655: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 656: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 657: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 658: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 659: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 660: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 661: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 662: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 663: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 664: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 665: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 666: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 667: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 668: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 669: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 670: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 671: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 672: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 673: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 674: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 675: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 676: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 677: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 678: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 679: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 680: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 681: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 682: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 683: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 684: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 685: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 686: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 687: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 688: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 689: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 690: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 691: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 692: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 693: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 694: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 695: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 696: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 697: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 698: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 699: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 700: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 701: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 702: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 703: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 704: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 705: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 706: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 707: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 708: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 709: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 710: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 711: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 712: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 713: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 714: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 715: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 716: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 717: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 718: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 719: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 720: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 721: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 722: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 723: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 724: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 725: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 726: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 727: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 728: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 729: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 730: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 731: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 732: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 733: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 734: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 735: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 736: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 737: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 738: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 739: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 740: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 741: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 742: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 743: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 744: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 745: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 746: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 747: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 748: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 749: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 750: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 751: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 752: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 753: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 754: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 755: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 756: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 757: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 758: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 759: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 760: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 761: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 762: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 763: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 764: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 765: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 766: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 767: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 768: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 769: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 770: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 771: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 772: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 773: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 774: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 775: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 776: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 777: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 778: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 779: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 780: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 781: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 782: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 783: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 784: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 785: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 786: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 787: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 788: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 789: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 790: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 791: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 792: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 793: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 794: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 795: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 796: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 797: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 798: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 799: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 800: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 801: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 802: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 803: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 804: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 805: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 806: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 807: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 808: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 809: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 810: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 811: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 812: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 813: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 814: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 815: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 816: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 817: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 818: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 819: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 820: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 821: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 822: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 823: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 824: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 825: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 826: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 827: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 828: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 829: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 830: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 831: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 832: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 833: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 834: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 835: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 836: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 837: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 838: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 839: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 840: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 841: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 842: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 843: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 844: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 845: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 846: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 847: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 848: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 849: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 850: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 851: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 852: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 853: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 854: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 855: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 856: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 857: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 858: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 859: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 860: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 861: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 862: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 863: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 864: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 865: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 866: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 867: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 868: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 869: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 870: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 871: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 872: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 873: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 874: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 875: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 876: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 877: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 878: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 879: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 880: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 881: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 882: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 883: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052370
Iteration 884: aa = 0.100000, f(aa) = -0.251229, f'(aa) = -0.052...

if iter >= max_iter
    disp('Newton-Raphson method did not converge within maximum iterations.');
else
    fprintf('Root found at aa = %f after %d iterations.\n', aa_guess, iter);
end

Newton-Raphson method did not converge within maximum iterations.

[VIGNESH BALAJI]

@Matt J and @Torsten Based on the examples above and trials I belive although my function is a convex function, without proper search region (lb) the newton-raphson does not converge.

[VIGNESH BALAJI]

@John D'Errico and @Torsten Both of your suggestions and code snippets helped me to solve my actual problem of finding the roots of the equation when the parameters vary. I final settled with by understanding the need for finding a good search region and toggling between fzero (if I have a zero crossing, sign change) and if not I started using fsolve with a decent initial guess. I will accept this answer and close the thread. Thanks a lot :)

[Matt J]

It might be a problem to get a suitable lb or to formulate more complicated updates for aa:

Well, bisection faces the same problem. The algorithm has to find an interval where a sign change occurs at the end points. In this case, we know the function goes to Inf at aa=0, so it should be sufficient to try lb od descending magnitude, lb=0.1, 0.01,0.001,... Until a qualified lower bound is found. As shown below, it didn't take long to find one.

% Predefined parameters
dd = 0.1069;
vv = 0.7889;
L = 1;
% Define the function and its derivative
func = @(aa) ( (-(sqrt(L^2-vv^2)/2)) + (aa.*sinh((dd/2)./aa)) );
func_derivative = @(aa) sinh(dd./(2*aa)) - dd*cosh( dd./(2*aa))./(2*aa);
% Initial guess
aa_guess = 1; % You can choose any initial guess
lb = 0.01;
%lb = 0.01;
% Tolerance and maximum iterations
tol = 1e-6;
max_iter = 1000;
% Newton-Raphson method
iter = 0;
while true
    iter = iter + 1;
    
    % Evaluate the function and its derivative at the current guess
    f = func(aa_guess);
    f_prime = func_derivative(aa_guess);
    
    % Newton-Raphson update rule
    aa_new = max(lb,aa_guess - f / f_prime);
    
    % Display iteration results
    fprintf('Iteration %d: aa = %f, f(aa) = %f, f''(aa) = %f\n', iter, aa_guess, f, f_prime);
    
    % Check for convergence
    if (abs(aa_new - aa_guess) < tol && abs(func(aa_new)) < tol) || iter >= max_iter
        break;
    end
    
    % Update the guess
    aa_guess = aa_new;
end

Iteration 1: aa = 1.000000, f(aa) = -0.253785, f'(aa) = -0.000051
Iteration 2: aa = 0.010000, f(aa) = 0.740505, f'(aa) = -455.279643
Iteration 3: aa = 0.011626, f(aa) = 0.269426, f'(aa) = -178.474720
Iteration 4: aa = 0.013136, f(aa) = 0.076826, f'(aa) = -89.802472
Iteration 5: aa = 0.013992, f(aa) = 0.011672, f'(aa) = -64.368193
Iteration 6: aa = 0.014173, f(aa) = 0.000380, f'(aa) = -60.240739
Iteration 7: aa = 0.014179, f(aa) = 0.000000, f'(aa) = -60.103653

if iter >= max_iter
    disp('Newton-Raphson method did not converge within maximum iterations.');
else
    fprintf('Root found at aa = %f after %d iterations.\n', aa_guess, iter);
end

Root found at aa = 0.014179 after 7 iterations.

Answer 2

[Matt J]

A simple transformation of variables z=dd/2/aa makes this problem, and its solution by NR, much easier. In particular, you can then just chose 0 as the lower bound, lb, and any non-negative initial guess z>=0 should work..

% Predefined parameters
dd = 0.1069;
vv = 0.7889;
L = 1;
const=  -sqrt(L^2-vv^2)/2;

% Define the function and its derivative
func = @(z) const*z + dd.*sinh(z)/2;
func_derivative = @(z) 1+dd*cosh(z)/2;
% Initial guess
z_guess = 1; % You can choose any initial guess
lb = 0;
%lb = 0.01;
% Tolerance and maximum iterations
tol = 1e-6;
max_iter = 1000;
% Newton-Raphson method
iter = 0;
while true
    iter = iter + 1;
    
    % Evaluate the function and its derivative at the current guess
    f = func(z_guess);
    f_prime = func_derivative(z_guess);
    
    % Newton-Raphson update rule
    z_new = max(lb,z_guess - f / f_prime);
    
    % Display iteration results
    fprintf('Iteration %d: aa = %f, f(aa) = %f, f''(aa) = %f\n', iter, z_guess, f, f_prime);
    
    % Check for convergence
    if (abs(z_new - z_guess) < tol && abs(func(z_new)) < tol) || iter >= max_iter
        break;
    end
    
    % Update the guess
    z_guess = z_new;
end

Iteration 1: aa = 1.000000, f(aa) = -0.244446, f'(aa) = 1.082478
Iteration 2: aa = 1.225821, f(aa) = -0.293440, f'(aa) = 1.098895
Iteration 3: aa = 1.492853, f(aa) = -0.345781, f'(aa) = 1.124926
Iteration 4: aa = 1.800234, f(aa) = -0.395843, f'(aa) = 1.166131
Iteration 5: aa = 2.139684, f(aa) = -0.433511, f'(aa) = 1.230221
Iteration 6: aa = 2.492068, f(aa) = -0.444921, f'(aa) = 1.325216
Iteration 7: aa = 2.827803, f(aa) = -0.418580, f'(aa) = 1.453454
Iteration 8: aa = 3.115793, f(aa) = -0.355863, f'(aa) = 1.603869
Iteration 9: aa = 3.337670, f(aa) = -0.274083, f'(aa) = 1.753351
Iteration 10: aa = 3.493990, f(aa) = -0.194671, f'(aa) = 1.880519
Iteration 11: aa = 3.597510, f(aa) = -0.130451, f'(aa) = 1.976387
Iteration 12: aa = 3.663514, f(aa) = -0.084114, f'(aa) = 2.042911
Iteration 13: aa = 3.704688, f(aa) = -0.052930, f'(aa) = 2.086691
Iteration 14: aa = 3.730053, f(aa) = -0.032807, f'(aa) = 2.114575
Iteration 15: aa = 3.745568, f(aa) = -0.020147, f'(aa) = 2.131982
Iteration 16: aa = 3.755018, f(aa) = -0.012303, f'(aa) = 2.142718
Iteration 17: aa = 3.760759, f(aa) = -0.007487, f'(aa) = 2.149290
Iteration 18: aa = 3.764243, f(aa) = -0.004547, f'(aa) = 2.153297
Iteration 19: aa = 3.766354, f(aa) = -0.002758, f'(aa) = 2.155732
Iteration 20: aa = 3.767634, f(aa) = -0.001671, f'(aa) = 2.157210
Iteration 21: aa = 3.768409, f(aa) = -0.001013, f'(aa) = 2.158106
Iteration 22: aa = 3.768878, f(aa) = -0.000613, f'(aa) = 2.158649
Iteration 23: aa = 3.769162, f(aa) = -0.000371, f'(aa) = 2.158978
Iteration 24: aa = 3.769334, f(aa) = -0.000225, f'(aa) = 2.159177
Iteration 25: aa = 3.769438, f(aa) = -0.000136, f'(aa) = 2.159297
Iteration 26: aa = 3.769501, f(aa) = -0.000082, f'(aa) = 2.159370
Iteration 27: aa = 3.769539, f(aa) = -0.000050, f'(aa) = 2.159414
Iteration 28: aa = 3.769562, f(aa) = -0.000030, f'(aa) = 2.159441
Iteration 29: aa = 3.769576, f(aa) = -0.000018, f'(aa) = 2.159457
Iteration 30: aa = 3.769585, f(aa) = -0.000011, f'(aa) = 2.159467
Iteration 31: aa = 3.769590, f(aa) = -0.000007, f'(aa) = 2.159473
Iteration 32: aa = 3.769593, f(aa) = -0.000004, f'(aa) = 2.159477
Iteration 33: aa = 3.769595, f(aa) = -0.000002, f'(aa) = 2.159479
Iteration 34: aa = 3.769596, f(aa) = -0.000001, f'(aa) = 2.159480

aa= dd/2/z_guess; %undo change of variables

if iter >= max_iter
    disp('Newton-Raphson method did not converge within maximum iterations.');
else
    fprintf('Root found at aa = %f after %d iterations.\n', aa, iter);
end

Root found at aa = 0.014179 after 34 iterations.