cos(x1)+cos(x2)+cos(x3)+cos(x4)+cos(x5)=0.9;;
cos(5(x1))+cos(5(x2))+cos(5(x3))+cos(5(x4))+cos(5(x5))=0;;
cos(7(x1)))+cos(7(x2))+cos(7(x3))+cos(7(x4))+cos(7(x5))=0;;
cos(11(x1))+cos(11(x2))+cos(11(x3))+cos(11(x4))+cos(11(x5))=0;;
cos(13(x1))+cos(13(x2))+cos(13(x3))+cos(13(x4))+cos(13(x5))=0;;
0
Hello. You could use a least squares based technique. You define an objective function that is the sum of squares of your equations and seek to minimise this function with respect to x1, ..., x5. There are numerous optimisation techniques that can be applied. A common one is iterated least-squares (since you have a nonlinear objective function). The simplest technique would probably be gradient descent, where you only need the matrix of first derivatives of the objective function. Newton-based methods would normally use second derivative (Hessian) information also.
For the particular problem you have, I might be inclined to write everything in terms of cos(x_i) i = 1...5, since you can use multiple angle formulas to expand the terms. You then only have to work with polynomials. The 90 degree constraint will mean that all the cos(x_i) terms are positive. You could add this as a constraint and use a constrained optimisation technique.
One of the good way to solve these equations is optimization technique. You can use the least-square method as discussed Graham W Pulford.
To solve these equations by Newton Rapshon method, use the conventional procedure with few modifications. At the step of error checking, if any variable angle (say Xy) is more than 90 then set X5 to 90 and Xy to 0. If any variable angle is less than 0 set that angle 0 for next iteration.
Use optimization toolbox of Matlab and apply fmincon and select SQP algorithm.
You need to formulate the objective function as
function F = myfun(x)
F = [(cos(x(1))+cos(x(2))+cos(x(3))+cos(x(4))+cos(x(5))-0.9).^2+...
(cos(5*(x(1)))+cos(5*(x(2)))+cos(5*(x(3)))+cos(5*(x(4)))+cos(5*(x(5)))).^2+...
(cos(7*(x(1)))+cos(7*(x(2)))+cos(7*(x(3)))+cos(7*(x(4)))+cos(7*(x(5)))).^2+...
(cos(11*(x(1)))+cos(11*(x(2)))+cos(11*(x(3)))+cos(11*(x(4)))+cos(11*(x(5)))).^2+...
(cos(13*(x(1)))+cos(13*(x(2)))+cos(13*(x(3)))+cos(13*(x(4)))+cos(13*(x(5)))).^2
];
The final answer for your problem will be (x1,x2,...,x5);
0.09394722636872234
0.6284710204878169
0.7517307864723759
1.4261827951912638
1.5707963267948966
All these values are in radian.
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