How can I take the value from an equation and plug it back into the same equation? (Newton-Raphson)

I'm trying to implement a Newton-Raphson function into my script. The code I've created so far is;

function [ALI] = down(A, B, C, D, E, n)
A = 0.012;
B = 0.04;
C =0.06;
D = sqrt(A)/2;
n =20
for i = 1:n
    E = D - F(D)/F'(D)
    ALI = D-E;
end
end

The equation for E has been edited into a simpler form of the newton-raphson equation; the equation is extremely long and includes the variables A, B, & C too. I want to perform equation E in a loop, in which the value from each iteration is placed back into itself, until D minus E is within the required tolerance. However, the way I've written the function means that it just repeats the same equation over and over and I'm not sure how to change this, please could someone help?

Also, variables A, B, C and D depend on other variables, which are calculated earlier in the script.. is there a way in which I can define the variables in the function so that they take the values already calculated in the script? I'm currently having to use a caluclator for the values and then plugging them into the function. This will be a major problem later when im trying to performs loops etc. Thank you.

Answer 1

[DGM]

Something like this:

function D = down(A, B, C, D, maxiter, tol)

% things that are in the argument list don't get rewritten

% function parameters
% A, B, C

% initial condition
% D

% process parameters
% maxiter, tol

stepsize=1;
iter=1;
while stepsize > tol
    E = D - F(D)/Fprime(D); % this estimate
    stepsize = abs(D-E);
    D = E; % is the next point to evaluate at
	
    iter = iter+1;
    % just in case we don't converge
    if iter > maxiter
        break;
    end
end
end

If the parameters and intial conditions are input arguments that depend on external variables, why not just precalculate them before passing them to the function?

[Bob Wake]

Thanks DGM, but unfortunately I get the error that there aren't enough input parameters when I use this code and i'm not sure why. I think I was poor in my description. I've retyped the code i want to perform in FORTRAN in the comments above, and also my attempt at translation. If you get the chance please could you see if this makes any sense or guide me as to where i'm going wrong?

For the second part of my question, when I try to precalculate the values before passing them to the function, I also get the same error message. That is until I include said variables inside the function. Thanks again I really appreciate this.

[DGM]

In the first example:

 SUBROUTINE DOWN(A,B,C,D) 
C INPUT A, B, C
C OUTPUT D

This is essentially the same thing as this:

function D = down(A, B, C)

Both of these lines define A,B,C as input arguments and D as the output argument. Reassigning values to them within the function scope overwrites the inputs.

There are some errors in my post. That's the hazard of posting stuff off the top of my head. I'll fix them in a moment, but let's start with a working example instead.

function D = testdown(A, B, C, D, maxiter, tol)

% define function and derivative
% i'm going to do these as anonymous functions
% that's not strictly necessary
% since the function you gave isn't a function of D
% then I don't know what you're actually trying to solve
% so i'll just make up an example function
F = @(x) A-B*x^C; % the function
Fprime = @(x) B*C*-x^(C-1); % its derivative

stepsize=1;
iter=1;
while stepsize > tol
    E = D - F(D)/Fprime(D); % this estimate
    stepsize = abs(D-E);
    D = E; % is the next point to evaluate at
	iter = iter+1;
	
    % just in case we don't converge
    if iter > maxiter
        break;
    end
end

end

So we have a function, and we call it like so:

clf
% parameters
A = 2;
B = 1;
C = 2;
% initial estimate
D = 1;
% process parameters
stepsize = 1E-6;
n = 20;

% specify all arguments
rootest = testdown(A,B,C,D,n,stepsize);

x=linspace(0,5,100);
y=A-B*x.^C;
plot(x,y); grid on; hold on;
plot(rootest,A-B*rootest^C,'ko')

This will plot the test curve, and it will calculate the root and display it.

When input arguments are given explicitly in a function definition, they don't all have to be supplied when calling the function, but if the function execution ever reaches a line where one of the missing arguments is required for calculation, it will throw an error about not having enough arguments. The way I wrote testdown() requires that all arguments be specified. There are ways around this, but for the moment, just specify all the arguments.

The function you gave:

D = sqrt(A)/2;

isn't a function of D. It needs to be a function of D so that its value can be calculated at each estimate point. The second expression

A/4./sqrt(B^2+(C+D)*(C+D)))/(1+(C+D)*A/4./(B^2+(C+D)*(C+D))^1.5

Isn't the derivative of the function you gave. I have no idea what you're actually trying to solve, but this isn't going to work.

In the example I gave, D is the sole independent variable. From the perspective of testdown(), it starts out as an input argument (the initial estimate), and it is refined until exit conditions are met, upon which it is the output argument. A,B,and C are constant parameters. The function for which you're trying to find roots needs to be a function of the independent variable. You can call it D or A or whatever, but it has to be consistent.

[Bob Wake]

Hi DGM, thanks again. The function I'm using calculates the inflow ratio across a helicopter main rotor. So D is an assumed value when the helicopter is hovering (sqrt(a)/2), but changes in forward flight. The equation for DA is an equation by Glauert in the form of newton-raphson. Anyway I have managed to converge the calculation by placing the first equation for D before the 'for' loop and it now works, so thank you for your help. I'm sorry for my poor explanation also.

[Bob Wake]

However, I'm still unsure of how to take the variables from the script, into the function..

So like I say, A, B, C are calculated using equations with other variables earlier in the script. But I want to be able to take these precalculated values into the function, without having to rewrite their corresponding equations inside the function..

Answer 2

[Jan]

Replace

E = D - F(D)/F'(D)

by

D = D - F(D)/F'(D)

For the 2nd part: You have defined the function with inputs:

function [ALI] = down(A, B, C, D, E, n)

but A,B,C,D are overwritten directly. E does not seem to be an input at all, but a locally used variable.

If you want to use the inputs, define them outside and provide them as inputs when you call your funcion:

function main
A = 0.012;
B = 0.04;
C = 0.06;
D = sqrt(A) / 2;
n = 20;

ALI = down(A, B, C, D, n)
end

function ALI = down(A, B, C, D, n)

for i = 1:n
    Dold = D;
    D = D - F(D)/F'(D)
    ALI = D - Dold;
end
end

[Bob Wake]

Thanks for your reply Jan, but when I try this the code just repeats the same calculation 20 times. This is my fault for not explaining properly.

I have written the code in FORTRAN which I can now show now.

 SUBROUTINE DOWN(A,B,C,D) 
C INPUT A, B, C
C OUTPUT D
       MAXIT=20 
       D=0.5*SQRT(A) 
       DO I=1,MAXIT 
       DA=(D-A/4./SQRT(B*B+(C+D)*(C+D)))/(1.+(C+D)*A/4./(B*B+(C+D)*(C+D))**1.5) 
       D=D-DA 

I've then tried translating as follows;

       	   function D = down(A, B, C, n)
A = 0.012;
B= 0.04;
C = 0.06;
n=20;
for i = 1:n
D = sqrt(A)/2;
DA = (D-A/4./sqrt(B^2+(C+D)*(C+D)))/(1+(C+D)*A/4./(B^2+(C+D)*(C+D))^1.5)
D = D - DA
end
end

However the equation DA, and D just repeat the same values as though the equation converges instantly, which is n ot true. Do you know why this is?

Also, I've defined variables A to D here as numbers, however this is only for the first leg. I want to use the function 'down' during my second leg of calculations, in which said variables will have different values. Altogether I have 9 legs of calculations and so defining the variables like this would not work because they have to keep changing.

But when I try to remove A, B and C from the function, it states there is 'not enough input arguments'. But if i was to include said variables in there equation form i.e. A = H+F+k^y, this would require me to define basically my whole script in the function, as otherwise there is 'unregocnised variables' or 'not enough arguments'.

Is there a way I can get the variables to take the values that have been precalculated in the script? The function does not start until after variables A to D have already been defined in the script and so there is no problem with undefined variables. Thanks again, I really appreciate your time.

[Jan]

You re-define D with the initial value in each iteration:

for i = 1:n
   D = sqrt(A)/2;
   ...       

Move the line "D = sqrt(A)/2;" before the loop - exactly as in the FORTRAN code.

[Bob Wake]

Thank you!