trigonometric non linear equation using newton's iteration process

Find a,b,c,d,e,f

cos(5*a)+cos(5*b)+cos(5*c)+cos(5*d)+cos(5*e)+cos(5*f)=0

cos(7*a)+cos(7*b)+cos(7*c)+cos(7*d)+cos(7*e)+cos(7*f)=0

cos(11*a)+cos(11*b)+cos(11*c)+cos(11*d)+cos(11*e)+cos(11*f)=0

cos(13*a)+cos(13*b)+cos(13*c)+cos(13*d)+cos(13*e)+cos(13*f)=0

cos(17*a)+cos(17*b)+cos(17*c)+cos(17*d)+cos(17*e)+cos(17*f)=0

cos(19*a)+cos(19*b)+cos(19*c)+cos(19*d)+cos(19*e)+cos(19*f)=0

0 <a < b < c < d < e < f < 90

X0 = [0.001;0.001;0.001;
0.001;0.001;0.001];
maxIter =100;
tolX = .01;
% Computation
X = X0;
Xold = X0;
 R = inv(j),
for i =1:maxIter
    [f,j]=new(X);   
    X = (X-R*f);
    err(:,i) = (X-Xold)
    Xold = X;
  X =[0 pi/2; 0 pi/2; 0 pi/2; 0 pi/2; 0 pi/2; 0 pi/2;];
    if  (err(:,i)<tolX)
        break;
    end
end

[Walter Roberson]

I suspect that those angles are degrees and so need cosd

[Avismit Dutta]

But sir how can I able to add (0 <a < b < c < d < e < f < 90) logic on my program, its seem to me very difficalt to add this logic in my program. Can you help me to add this logic?

[Avismit Dutta]

Functon

function [fval,jac]=new(X)

a = X(1);
b = X(2);
c = X(3);
d = X(4);
e = X(5);
f = X(6);
 
fval(1,1)=cosd(5*a)+cosd(5*b)+cosd(5*c)+cosd(5*d)+cosd(5*e)+cosd(5*f);
fval(2,1)=cosd(7*a)+cosd(7*b)+cosd(7*c)+cosd(7*d)+cosd(7*e)+cosd(7*f);
fval(3,1)=cosd(11*a)+cosd(11*b)+cosd(11*c)+cosd(11*d)+cosd(11*e)+cosd(11*f);
fval(4,1)=cosd(13*a)+cosd(13*b)+cosd(13*c)+cosd(13*d)+cosd(13*e)+cosd(13*f);
fval(5,1)=cosd(17*a)+cosd(17*b)+cosd(17*c)+cosd(17*d)+cosd(17*e)+cosd(17*f);
fval(6,1)=cosd(19*a)+cosd(19*b)+cosd(19*c)+cosd(19*d)+cosd(19*e)+cosd(19*f);

jac=  [-5*sind(5*X(1)),-5*sind(5*X(2)),-5*sind(5*X(3)),-5*sind(5*X(4)),-5*sind(5*X(5)),-5*sind(5*X(6));
     -7*sind(7*X(1)),-7*sind(7*X(2)),-7*sind(7*X(3)),-7*sind(7*X(4)),-7*sind(7*X(5)),-7*sind(7*X(6));
     -11*sind(11*X(1)),-11*sind(11*X(2)),-11*sind(11*X(3)),-11*sind(11*X(4)),-11*sind(11*X(5)),-11*sind(11*X(6));
     -13*sind(13*X(1)),-13*sind(13*X(2)),-13*sind(13*X(3)),-13*sind(13*X(4)),-13*sind(13*X(5)),-13*sind(13*X(6));
     -17*sind(17*X(1)),-17*sind(17*X(2)),-17*sind(17*X(3)),-17*sind(17*X(4)),-17*sind(17*X(5)),-17*sind(17*X(6));
     -19*sind(19*X(1)),-19*sind(19*X(2)),-19*sind(19*X(3)),-19*sind(19*X(4)),-19*sind(19*X(5)),-19*sind(19*X(6))];

[Walter Roberson]

sort() thepparameters at each point that you update them. The equations are symmetric so reordering the parameters does not matter.

Or simply sort the parameters once after you have finished producing them.

[Alex Sha]

if all cos() is actual cosd() as Walter said, there are multi-solutions in the range of [0, 90] , three of solutions look like below:

1:

a: 18.1410684336521
b: 33.6765676044253
c: 49.5606463686944
d: 58.3268836794621
e: 67.2621448309475
f: 89.0225760155082

Feval:
1.19348975147204E-14
-6.06459327201492E-15
-1.3239409568655E-14
-3.63598040564739E-15
-9.65894031423886E-15
6.21724893790088E-15

2:

a: 4.41898196140848
b: 19.2978768806839
c: 28.882938678063
d: 42.5453097696474
e: 52.2742306633304
f: 70.4614879829698

Feval:
-1.77635683940025E-15
-2.22044604925031E-16
-6.77236045021345E-15
5.44009282066327E-15
-1.55986334959834E-14
7.21644966006352E-16

3:

a: 11.8028358955696
b: 34.1797808927155
c: 39.8848312026887
d: 54.8298062016401
e: 63.4548474846056
f: 82.5007888103137

Feval:
-2.55351295663786E-15
7.54951656745106E-15
-1.50990331349021E-14
-1.90958360235527E-14
1.19904086659517E-14
-1.27675647831893E-14