Solving 3 simultaneous cubic equations i 1 3 L 1 + i 1 K + V 1 + ( i 2 + i 3 + C ) Z n = 0 i 2 3 L 2 + i 2 K + V 2 + ( i 1 + i 3 + C ) Z n = 0 i 3 3 L 3 + i 3 K + V 3 + ( i 1 + i 2 + C ) Z n = 0where L 1 , L 2 , L 3 , K , V 1 , V 2 , V 3 , C and Z n are all known constants.What methods can I use to obtain the values of i 1 , i 2 and i 3 ?

Question

Solving 3 simultaneous cubic equations      i    1    3        L    1    +      i    1    K  +      V    1    +  (      i    2    +      i    3    +  C  )      Z    n    =  0      i    2    3        L    2    +      i    2    K  +      V    2    +  (      i    1    +      i    3    +  C  )      Z    n    =  0      i    3    3        L    3    +      i    3    K  +      V    3    +  (      i    1    +      i    2    +  C  )      Z    n    =  0where       L    1    ,      L    2    ,      L    3    ,  K  ,      V    1    ,      V    2    ,      V    3    ,  C and       Z    n   are all known constants.What methods can I use to obtain the values of       i    1    ,      i    2   and      i    3  ?

Brendon Fernandez · Accepted Answer

A numerical way to solve this would be to use the Newton-Raphson method. This method can be extended to 3 dimensions as follows:            i              n        +        1              →    =            i      n        →    −      J          −      1        (      i    n    )      f    →    (      i    n    )Where   J is the Jacobian matrix of the system:  J  =      [                            3                      i            1            2                                L            1                    +          K                                      Z            n                                                Z            n                                                            Z            n                                    3                      i            2            2                                L            2                    +          K                                      Z            n                                                            Z            n                                                Z            n                                    3                      i            3            2                                L            3                    +          K                      ]  Choose an initial &quot;guess&quot;             i      0        →  , and repeat this process. Since it&#039;s an iterative process, the more times you evaluated it, the closer you get to the solution.

Misael Matthews · Accepted Answer

Any interactive method for solve this equations is difficult to calculate each interate manually. I recommend an already established method that has many theorems that guarantee convergence. Newton&#039;s method with guaranteed convergence theorems for Kantorovich is easy to implement in many computational software for numerical or algebraic computing.

Solving 3 simultaneous cubic equations <msubsup> i 1 3 </msubsup> L

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