FindRoot[f, {x, x0}]
searches for a numerical root of f, starting from the point .

FindRoot[lhs==rhs, {x, x0}]
searches for a numerical solution to the equation .

FindRoot[{f1, f2, ...}, {{x, x0}, {y, y0}, ...}]
searches for a simultaneous numerical root of all the .

FindRoot[{eqn1, eqn2, ...}, {{x, x0}, {y, y0}, ...}]
searches for a numerical solution to the simultaneous equations .

Details and OptionsDetails and Options

  • If the starting point for a variable is given as a list, the values of the variable are taken to be lists with the same dimensions.
  • FindRoot returns a list of replacements for x, y, ... , in the same form as obtained from Solve.
  • FindRoot first localizes the values of all variables, then evaluates f with the variables being symbolic, and then repeatedly evaluates the result numerically.
  • FindRoot has attribute HoldAll, and effectively uses Block to localize variables.
  • FindRoot[lhs==rhs, {x, x0, x1}] searches for a solution using and as the first two values of x, avoiding the use of derivatives.
  • FindRoot[lhs==rhs, {x, xstart, xmin, xmax}] searches for a solution, stopping the search if x ever gets outside the range to .
  • If you specify only one starting value of x, FindRoot searches for a solution using Newton methods. If you specify two starting values, FindRoot uses a variant of the secant method.
  • If all equations and starting values are real, then FindRoot will search only for real roots. If any are complex, it will also search for complex roots.
  • You can always tell FindRoot to search for complex roots by adding 0.I to the starting value.
  • The following options can be given:
  • AccuracyGoalAutomaticthe accuracy sought
    EvaluationMonitorNoneexpression to evaluate whenever equations are evaluated
    JacobianAutomaticthe Jacobian of the system
    MaxIterations100maximum number of iterations to use
    PrecisionGoalAutomaticthe precision sought
    StepMonitorNoneexpression to evaluate whenever a step is taken
    WorkingPrecisionMachinePrecisionthe precision to use in internal computations
  • The default settings for AccuracyGoal and PrecisionGoal are WorkingPrecision/2.
  • The setting for AccuracyGoal specifies the number of digits of accuracy to seek in both the value of the position of the root, and the value of the function at the root.
  • The setting for PrecisionGoal specifies the number of digits of precision to seek in the value of the position of the root.
  • FindRoot continues until either of the goals specified by AccuracyGoal or PrecisionGoal is achieved.
  • If FindRoot does not succeed in finding a solution to the accuracy you specify within MaxIterations steps, it returns the most recent approximation to a solution that it found. You can then apply FindRoot again, with this approximation as a starting point.
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