2.5.11 Defining Numerical ValuesIf you make a definition such as f[x_] := value, Mathematica will use the value you give for any f function it encounters. In some cases, however, you may want to define a value that is to be used specifically when you ask for numerical values.
| expr = value | define a value to be used whenever possible | | N[expr] = value | define a value to be used for numerical approximation |
Defining ordinary and numerical values. | This defines a numerical value for the function f. | |
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N[f[x_]] := Sum[x^-i/i^2, {i, 20}]
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| Defining the numerical value does not tell Mathematica anything about the ordinary value of f. | |
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| If you ask for a numerical approximation, however, Mathematica uses the numerical values you have defined. | |
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You can define numerical values for both functions and symbols. The numerical values are used by all numerical Mathematica functions, including NIntegrate, FindRoot and so on.
| N[expr] = value | define a numerical value to be used when default numerical precision is requested | | N[expr, {n, Infinity}] = value | define a numerical value to be used when n-digit precision and any accuracy is requested |
Defining numerical values that depend on numerical precision. | This defines a numerical value for the symbol const, using 4n + 5 terms in the product for n-digit precision. | |
In[4]:=
N[const, {n_, Infinity}] := Product[1 - 2^-i, {i, 2, 4n + 5}]
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| Here is the value of const, computed to 30-digit precision using the value you specified. | |
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Mathematica treats numerical values essentially like upvalues. When you define a numerical value for f, Mathematica effectively enters your definition as an upvalue for f with respect to the numerical evaluation operation N.
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