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InterpolateRoot


searches for a numerical solution to the equation using and as the first two values of x.
  • gives the solution as a rule of the form .
  • will search for a root of the equation expr==0.
  • searches for a solution using inverse cubic interpolation of the last four data points. It does not use derivative information.
  • works very slowly when the solution is a multiple root.
  • is not as robust as FindRoot. However, it is useful when the location of the root is approximately known, each evaluation of the function is expensive, and high precision is desired.
  • If the equation and starting values are real, then will search only for real roots, otherwise it will search for complex roots.
  • The following options can be given:
AccuracyGoalAutomaticthe accuracy sought
MaxIterations15maximum number of iterations to use
ShowProgressFalsewhether progress is to be monitored
WorkingPrecision40the precision to use in internal computations
  • The setting for AccuracyGoal refers to the accuracy of the root rather than the magnitude of the residual at the root.
  • The precision used in internal computations typically varies from a little more than machine precision at the beginning to the setting for WorkingPrecision at the end.
  • If does not succeed in finding a solution to the desired accuracy within MaxIterations steps, it returns the most recent approximation found.
  • With True, will print followed by , where:
accuracyestimate of the accuracy of the current approximation to the solution
xcurrent approximation to the solution
precisioncurrent working precision
extraprecisionnumber of extra digits of precision being used
deltapredicted change in the approximation during the next iteration
Check:
Needs["FunctionApproximations`"]
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Out[2]=
Check:
In[3]:=
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can find complex roots:
Check:
An expression expr is interpreted as expr==0:
The Riemann hypothesis states that all zeros of the function Zeta[z] lie on the line of the complex plane. The first few zeros:
Define a function that evaluates the Zeta function and increments a counter:
Check the real part of the root near to 400 digits:
The number of function evaluations (and steps) needed:
Using FindRoot with the secant method:
Using FindRoot with Newton's method:
Multiple roots converge slowly: