Solve numerical system of nonlinear equations?I need to solve a nonlinear system of equations that looks like this 1 n 4 ( c 4 b 4 R − ln ⁡ ( 1 − x 1 − x 2 − x 3 ) ) = 1 n 3 ( c 3 b 3 R − ln ⁡ ( x 3 ) ) = 1 n 2 ( c 2 b 2 R − ln ⁡ ( x 2 ) ) = 1 n 1 ( c 1 b 1 R − ln ⁡ ( x 1 ) )The numerical equivalent of what i'm trying to solve right now is 4100 4100 / 297 − 1.987 ln ⁡ ( x 1 ) = 6200 6200 / 303 − 1.987 ln ⁡ ( x 2 ) = 7000 7000 / 327.5 − 1.987 ln ⁡ ( x 3 ) = 4100 4100 / 404 − 1.987 ln ⁡ ( 1 − x 1 − x 2 − x 3 ) What algorithm should i use to solve this? If anyone can solve this, could you post how you did it? I tried using a multivariable newton's algorithm but my jacobian exploded

Question

Solve numerical system of nonlinear equations?I need to solve a nonlinear system of equations that looks like this      1          n      4        (            c      4                      b        4            R        −  ln  ⁡  (  1  −      x    1    −      x    2    −      x    3    )  )  =      1          n      3        (            c      3                      b        3            R        −  ln  ⁡  (      x    3    )  )  =      1          n      2        (            c      2                      b        2            R        −  ln  ⁡  (      x    2    )  )  =      1          n      1        (            c      1                      b        1            R        −  ln  ⁡  (      x    1    )  )The numerical equivalent of what i&#039;m trying to solve right now is      4100          4100              /            297      −      1.987      ln      ⁡      (              x        1            )        =      6200          6200              /            303      −      1.987      ln      ⁡      (              x        2            )        =      7000          7000              /            327.5      −      1.987      ln      ⁡      (              x        3            )        =      4100          4100              /            404      −      1.987      ln      ⁡      (      1      −              x        1            −              x        2            −              x        3            )      What algorithm should i use to solve this? If anyone can solve this, could you post how you did it? I tried using a multivariable newton&#039;s algorithm but my jacobian exploded

penzionekta · Accepted Answer

HintI think that you could reduce your problem to a single equation of a single variable.Suppose that you define  A  =      1          n      4        (            c      4                      b        4            R        −  l  n  (  1  −      x    1    −      x    2    −      x    3    )  )So, you have  A  =      1          n      3        (            c      3                      b        3            R        −  l  n  (      x    3    )  )  =      1          n      2        (            c      2                      b        2            R        −  l  n  (      x    2    )  )  =      1          n      1        (            c      1                      b        1            R        −  l  n  (      x    1    )  )So       x    1    ,      x    2    ,      x    3   can be expressed as a function of   A      x    i    (  A  )  =  exp  ⁡  (                    c        i                              b          i                         R              −    A               n      i        )and you end with a single nonlinear equation   A  =  f  (  A  ) where only   A appears.  A  =      1          n      4        (            c      4                      b        4            R        −  l  n  (  1  −      x    1    (  A  )  −      x    2    (  A  )  −      x    3    (  A  )  )  )You could notice that your equation in   A also write  F  (  A  )  =      ∑          i      =      1              i      =      4        exp  ⁡  (                    c        i                              b          i                         R              −    A               n      i        )  =  1which, more than likely, would much better conditionned if written  G  (  A  )  =  log  ⁡            (            ∑          i      =      1              i      =      4        exp  ⁡  (                    c        i                              b          i                         R              −    A               n      i        )            )        =  0Assuming all       n    i    &amp;gt;  0, this shows that an upper bound of   A corresponds to the maximum,over   i, of the   (            c      i                      n        i                           b        i                   R        ) terms. This gives a starting value for unknown   ASolve it for   A (Newton method seems to be very appropriate) and go backwards to get       x    1    ,      x    2    ,      x    3  In such a manner, you could notice that you could have as many equations as you could wish and apply the same procedure.Using the data from your problem, we can easily find that the solution is   A  =  271.069 from which we get       x    1    =  0.514479  ,      x    2    =  0.297290  ,      x    3    =  0.106527 which are your answers.

Grace Jefferson · Accepted Answer

Ok i actually found the answer and i&#039;m just gonna share it since no one has posted yet      x    1    =  .514  ,      x    2    =  .297  ,      x    3    =  .107Rearrange the equation for easier Jacobian            4100              /            (      1.987      ∗      297      )      −      l      n      (              x        1            )              4100              /            1.987        =            6200              /            (      1.987      ∗      303      )      −      l      n      (              x        2            )              6200              /            1.987        =            7000              /            (      1.987      ∗      327.5      )      −      l      n      (              x        3            )              7000              /            1.987        =            4100              /            (      1.987      ∗      404      )      −      l      n      (      1      −              x        1            −              x        2            −              x        3            )              4100              /            1.987      Construct a system of equations in the form of       f    i    (      x    1    ,      x    2    ,      x    3    )  =  0 for   i  =  1  ,  2  ,  3      f    1    (      x    1    ,      x    2    ,      x    3    )  =            4100              /            (      1.987      ∗      297      )      −      l      n      (              x        1            )              4100              /            1.987        −            4100              /            (      1.987      ∗      404      )      −      l      n      (      1      −              x        1            −              x        2            −              x        3            )              4100              /            1.987            f    2    (      x    1    ,      x    2    ,      x    3    )  =            6200              /            (      1.987      ∗      303      )      −      l      n      (              x        2            )              6200              /            1.987        −            4100              /            (      1.987      ∗      404      )      −      l      n      (      1      −              x        1            −              x        2            −              x        3            )              4100              /            1.987            f    3    (      x    1    ,      x    2    ,      x    3    )  =            7000              /            (      1.987      ∗      327.5      )      −      l      n      (              x        3            )              7000              /            1.987        −            4100              /            (      1.987      ∗      404      )      −      l      n      (      1      −              x        1            −              x        2            −              x        3            )              4100              /            1.987        F  (  X  )  =  [      f    1    (      x    1    ,      x    2    ,      x    3    )  ,      f    2    (      x    1    ,      x    2    ,      x    3    )  ,      f    3    (      x    1    ,      x    2    ,      x    3    )      ]    T    X  =  [      x    1    ,      x    2    ,      x    3        ]    T  Construct Jacobian for the system      F    ′    (  X  )  =  [            ∂              f        1                    ∂              x        1              ,            ∂              f        1                    ∂              x        2              ,            ∂              f        1                    ∂              x        3              ;            ∂              f        2                    ∂              x        1              ,            ∂              f        2                    ∂              x        2              ,            ∂              f        2                    ∂              x        3              ;            ∂              f        3                    ∂              x        1              ,            ∂              f        3                    ∂              x        2              ,            ∂              f        3                    ∂              x        3              ]Solve   F  (  X  )  =  −      F    ′    (  X  )  H for   H using Gaussian eliminationUpdate   X by using       X          (      k      +      1      )        =      X          (      k      )        +      H          (      k      )      , in this case using   X  =  [  .01  ,  .01  ,  .01      ]    T  Repeat until reached acceptable error boundMy jacobian exploded because i didn&#039;t give a good initial guess, but then i guessed slightly lower and everything worked.

I need to solve a nonlinear system of equations that looks like this 1/(n_4)(c_4/(b_4R)−ln(1−x_1−x_2−x_3))=1/n_3(c_3/(b_3R)−ln(x_3))=1/n_2(c_2/(b_2R)−ln(x_2))=1/n_1(c_1/(b_1R)−ln(x_1))

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