Controlling the Precision of Results

In doing numerical operations like NDSolve and NMinimize, the Wolfram Language by default uses machine numbers. But by setting the option WorkingPrecision->n you can tell it to use arbitraryprecision numbers with ndigit precision.

This does a machineprecision computation of a numerical integral:
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This does the computation with 30digit arbitraryprecision numbers:
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When you give a setting for WorkingPrecision, this typically defines an upper limit on the precision of the results from a computation. But within this constraint you can tell the Wolfram Language how much precision and accuracy you want it to try to get. You should realize that for many kinds of numerical operations, increasing precision and accuracy goals by only a few digits can greatly increase the computation time required. Nevertheless, there are many cases where it is important to ensure that high precision and accuracy are obtained.

WorkingPrecisionthe number of digits to use for computations
PrecisionGoalthe number of digits of precision to try to get
AccuracyGoalthe number of digits of accuracy to try to get

Options for controlling precision and accuracy.

This gives a result to 25digit precision:
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50digit precision cannot be achieved with 30digit working precision:
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Given a particular setting for WorkingPrecision, each of the functions for numerical operations in the Wolfram Language uses certain default settings for PrecisionGoal and AccuracyGoal. Typical is the case of NDSolve, in which these default settings are equal to half the settings given for WorkingPrecision.

The precision and accuracy goals normally apply both to the final results returned, and to various norms or error estimates for them. Functions for numerical operations in the Wolfram Language typically try to refine their results until either the specified precision goal or accuracy goal is reached. If the setting for either of these goals is Infinity, then only the other goal is considered.

In doing ordinary numerical evaluation with N[expr,n], the Wolfram Language automatically adjusts its internal computations to achieve ndigit precision in the result. But in doing numerical operations on functions, it is in practice usually necessary to specify WorkingPrecision and PrecisionGoal more explicitly.