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NProduct

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

NProduct[f, {i, imin, imax}]
    gives a numerical approximation to the product of f for i from imin to imax.

NProduct[f, {i, imin, imax, di}] uses step di. imax may be Infinity. NProduct[f, {i, ...}, {j, ...}, ...] evaluates a multidimensional product (an inner bound may depend on an outer index). The index is localised (HoldAll). Evaluated as Exp[NSum[Log[f], ...]], so the NSum engine (Euler-Maclaurin for monotone factors, Wynn's epsilon otherwise) and its convergence test carry over. Machine or arbitrary precision via WorkingPrecision.

Options: Method (Automatic | EulerMaclaurin | WynnEpsilon), WorkingPrecision (default MachinePrecision), NProductFactors (leading factors taken explicitly, default 15), NProductExtraFactors, WynnDegree, VerifyConvergence (default True; a divergent product gives ComplexInfinity), AccuracyGoal, PrecisionGoal.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= NProduct[1 - 1/n^2, {n, 2, Infinity}]
Out[1]= 0.5

In[2]:= NProduct[(n^2)/(n^2 - 1), {n, 2, Infinity}]
Out[2]= 2.0

In[3]:= NProduct[1 + 1/n^2, {n, 1, Infinity}]
Out[3]= 3.67608

Implementation notes

Attributes: HoldAll, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= NProduct[1 - 1/n^2, {n, 2, Infinity}]
Out[1]= 0.5

In[1]:= NProduct[Cos[1/n], {n, 1, Infinity}]
Out[1]= 0.388536

Notes

NProduct[f, {i, imin, imax}] numerically evaluates a product, with imax allowed to be Infinity. The first case is a telescoping product: Product[1 - 1/n^2, {n, 2, Infinity}] = 1/2 exactly. The second converges to about 0.388536. NProduct is evaluated internally as Exp[NSum[Log[f], ...]], so the Euler–Maclaurin / Wynn's-epsilon machinery and convergence test of NSum carry over. With VerifyConvergence -> True (default) a divergent product gives ComplexInfinity. Use WorkingPrecision for arbitrary precision.