## 76 Package mnewton ¶

### 76.1 Introduction to mnewton ¶

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

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### 76.2 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.

  When `newtonepsilon` is a bigfloat, `mnewton` computations are done with bigfloats; otherwise, ordinary floats are used.

  See also `mnewton`.

  Categories: 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`.

  Categories: Package mnewton ·

Option variable: newtondebug ¶

  Default value: `false`

  When `newtondebug` is `true`, `mnewton` prints out debugging information while solving a problem.

  Categories: Package mnewton ·

Function: mnewton
`mnewton` (FuncList, VarList, GuessList)
`mnewton` (FuncList, VarList, GuessList, DF) ¶

  Approximate solution of multiple nonlinear equations by Newton’s method.

  FuncList is a list of functions to solve, VarList is a list of variable names, and GuessList is a list of initial approximations. The optional argument DF is the Jacobian matrix of the list of functions; if not supplied, it is calculated automatically from FuncList.

  FuncList may be specified as a list of equations, in which case the function to be solved is the left-hand side of each equation minus the right-hand side.

  If there is only a single function, variable, and initial point, they may be specified as a single expression, variable, and initial value; they need not be lists of one element.

  A variable may be a simple symbol or a subscripted symbol.

  The solution, if any, is returned as a list of one element, which is a list of equations, one for each variable, specifying an approximate solution; this is the same format as returned by `solve`. If the solution is not found, `[]` is returned.

  Functions and initial points may contain complex numbers, and solutions likewise may contain complex numbers.

  `mnewton` is governed by global variables `newtonepsilon` and `newtonmaxiter`, and the global flag `newtondebug`.

  `load("mnewton")` loads this function.

  See also `realroots`, `allroots`, `find_root` and `newton`.

  Examples:

      (%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]]

  Categories: Package mnewton ·