How a non linear system of equations be solved if non linearity is due to sine cosine function.
Please write down the system of equations you are trying to solve.
There are variety of ways to solve this nonlinear system of equations. However, independent of the method you choose, first try to eliminate unknowns by combining them to produce a single function, if applicable. The methods are:
1) Multi-dimensional version of the Newton-Raphson method
2) Iterative approach
3) Gauss-Seidel elimination
4) Gauss-Jordan method (variant of 3rd))
You can find the theory and numerical implementation of these methods in any standard book on Numerical analysis.
Also if required you can get rid of "sine" behavior by two methods depending on what is applicable in your problem.
1) If the argument is small, then there is a Taylor series approximation for sine.
2) Even if the argument is not negligible, you can always break a sine cycle into 4 quadrants where they are linear! So Sin(x) could be replaced by mx where m is the calculated slope.
Hope it helps!
For non linear equations with single variable use bisection, Newton Raphson,iteration technique, secent etc. methods to get approximate solution. Better use computer programs to solve them appromately with the algorithm of the methods stated here.
Right.I am thankful to all of you for sharing your views. I will try those methods mentioned above and Sir G.L.Wilson I have written down my system of equations.What about this now.We have unknowns twice.
AC-µF=0 is my non linear system of equation to be solved, where C=[C1,C2,…,C6]T are the unknowns, F=[f1, f2, … ,f6]T, A is the matrix(6×6) of co-efficients given below.µ=0.5 is a constant and F is also given below.
-21.56402 -2.71596 5.69487 2.12701 0.27564 0.01254
-3.10123 -9.68194 -0.51508 3.89857 1.60293 0.20072
A= 3.20584 -5.17804 -6.60787 2.01335 3.37763 1.01613
3.36818 1.03184 -6.583 -2.01273 3.18902 3.21148
1.59034 3.6999 -0.46155 -1.84654 -3.18008 7.84053
0.27041 2.09451 5.71391 13.66226 -21.75329 16.25812
f1=µsinh[µ(0.39664C1)],
f2=µsinh[µ(0.31843C1+0.31887C2)]
f3=µsinh[µ(0.15624C1+0.29346C2+0.29397C3)]
f4= µsinh[µ(0.04929C1+0.16469C2+0.29346C3+0.29415C4)]
f5= µsinh[µ(0.00807C1+0.05059C2+0.16923C3+0.31842C4+0.31954C5)]
f6= µsinh[µ(0.0003C1+0.00449C2+0.03617C3+0.16391C4+0.39616C5+0.39895C6)]
@ Shakeel
You definitely need to use matrix algebra and employ Gauss Seidel or Gauss Jordan method. Your equation is quite complicated though. Be careful while implementing it.
A solution to your system is simply C=0. This trivial solution may or may not be the only solution however.
You can attempt to find other solutions by experimenting with one of the numerical methods mentioned above. My inclination would be to automate the generation of many thousands of initial guesses into Newton's method in an attempt to find non-trivial solutions. Initial guesses could be generated randomly using a pseudo random number generator. You might look for solutions to AC-µF=0 where C is restricted to a 6 dimensional hyper-cube -a
Thanks a lot Sir G L Wilson, u have replied in detail, will follow these instructions.
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