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# 67. mnewton

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## 67.1 Introduction to mnewton

`mnewton` is an implementation of Newton's method for solving nonlinear equations in one or more variables.

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@ref{Category: Numerical methods} · @ref{Category: Share packages} · @ref{Category: Package mnewton}

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## 67.2 Functions and Variables for mnewton

Option variable: newtonepsilon

  Default value: `10.0^(-fpprec/2)`

  Precision to determine when the `mnewton` function has converged towards the solution. If `newtonepsilon` is a bigfloat, then `mnewton` computations are done with bigfloats. See also `mnewton`.

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  @ref{Category: Package mnewton}

Option variable: newtonmaxiter

  Default value: `50`

  Maximum number of iterations to stop the `mnewton` function if it does not converge or if it converges too slowly.

  See also `mnewton`.

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  @ref{Category: Package mnewton}

Function: mnewton (FuncList,VarList,GuessList)

  Multiple nonlinear functions solution using the Newton method. FuncList is the list of functions to solve, VarList is the list of variable names, and GuessList is the list of initial approximations.

  The solution is returned in the same format that `solve()` returns. If the solution is not found, `[]` is returned.

  This function is controlled by global variables `newtonepsilon` and `newtonmaxiter`.

      (%i1) load("mnewton")$

      (%i2) mnewton([x1+3*log(x1)-x2^2, 2*x1^2-x1*x2-5*x1+1],
                    [x1, x2], [5, 5]);
      (%o2) [[x1 = 3.756834008012769, x2 = 2.779849592817897]]
      (%i3) mnewton([2*a^a-5],[a],[1]);
      (%o3)             [[a = 1.70927556786144]]
      (%i4) mnewton([2*3^u-v/u-5, u+2^v-4], [u, v], [2, 2]);
      (%o4) [[u = 1.066618389595407, v = 1.552564766841786]]

  The variable `newtonepsilon` controls the precision of the approximations. It also controls if computations are performed with floats or bigfloats.

      (%i1) load(mnewton)$

      (%i2) (fpprec : 25, newtonepsilon : bfloat(10^(-fpprec+5)))$

      (%i3) mnewton([2*3^u-v/u-5, u+2^v-4], [u, v], [2, 2]);
      (%o3) [[u = 1.066618389595406772591173b0,
                                     v = 1.552564766841786450100418b0]]

  To use this function write first `load("mnewton")`. See also `newtonepsilon` and `newtonmaxiter`.

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  @ref{Category: Package mnewton}

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