This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
 FUNCTION APPROXIMATIONS PACKAGE SYMBOL

# 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:
 AccuracyGoal Automatic the accuracy sought MaxIterations 15 maximum number of iterations to use ShowProgress False whether progress is to be monitored WorkingPrecision 40 the 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:
 accuracy estimate of the accuracy of the current approximation to the solution x current approximation to the solution precision current working precision extraprecision number of extra digits of precision being used delta predicted change in the approximation during the next iteration
Check:
Needs["FunctionApproximations`"]
 Out[2]=
Check:
 Out[3]=
 Scope   (2)
can find complex roots:
Check:
An expression expr is interpreted as expr==0:
 Applications   (1)
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: