WOLFRAM

Wolfram Language & System Documentation Center
Wolfram Language Home Page »

NProduct
✖
NProduct

✖
NProduct[f,{i,imin,imax}]

gives a numerical approximation to the product .

✖
NProduct[f,{i,imin,imax,di}]

uses a step di in the product.

Details and Options

  • NProduct can be used for products with both finite and infinite limits.
  • NProduct[f,{i,…},{j,…},…] can be used to evaluate multidimensional products.
  • The following options can be given:
  • AccuracyGoalInfinitynumber of digits of final accuracy sought
    EvaluationMonitor Noneexpression to evaluate whenever f is evaluated
    Method Automaticmethod to use
    PrecisionGoalAutomaticnumber of digits of final precision sought
    VerifyConvergence Truewhether to explicitly test for convergence
    WorkingPrecision MachinePrecisionthe precision used in internal computations
  • Possible settings for the Method option include:
  • "EulerMaclaurin"Euler–Maclaurin summation method
    "WynnEpsilon"Wynn epsilon extrapolation method
  • With the Euler–Maclaurin method, the options AccuracyGoal and PrecisionGoal can be used to specify the accuracy and precision to try and get in the final answer. NProduct stops when the error estimates it gets imply that either the accuracy or precision sought has been reached.
  • You should realize that in sufficiently pathological cases, the algorithms used by NProduct can give wrong answers. In most cases, you can test the answer by looking at its sensitivity to changes in the setting of options for NProduct.
  • VerifyConvergence is only used for products with infinite limits.
  • N[Product[…]] calls NProduct.
  • NProduct first localizes the values of all variables, then evaluates f with the variables being symbolic, and then repeatedly evaluates the result numerically.
  • NProduct has attribute HoldAll, and effectively uses Block to localize variables.

Examples

open allclose all

Basic Examples  (1)Summary of the most common use cases

Approximate an infinite product numerically:

In[1]:=1
✖
https://wolfram.com/xid/0y8osy-hgm0j
Out[1]=1

The error versus the exact value of :

In[2]:=2
✖
https://wolfram.com/xid/0y8osy-xlxpz
Out[2]=2

Scope  (5)Survey of the scope of standard use cases

Approximate the value of a finite product:

In[1]:=1
✖
https://wolfram.com/xid/0y8osy-bhymr8
Out[1]=1

This is effectively the ratio of two infinite products:

In[2]:=2
✖
https://wolfram.com/xid/0y8osy-zcyib
Out[2]=2

Approximate a multidimensional product:

In[1]:=1
✖
https://wolfram.com/xid/0y8osy-je2e8x
Out[1]=1

Approximate a multidimensional product with the second index depending on the first:

In[1]:=1
✖
https://wolfram.com/xid/0y8osy-pw7nwp
Out[1]=1

Complex infinite product approximation:

In[1]:=1
✖
https://wolfram.com/xid/0y8osy-ohg49o
Out[1]=1

Multiply the even factors of an infinite product:

In[1]:=1
✖
https://wolfram.com/xid/0y8osy-fewhjy
Out[1]=1

An equivalent way of specifying the same multiplication:

In[2]:=2
✖
https://wolfram.com/xid/0y8osy-kwz6o5
Out[2]=2

Options  (8)Common values & functionality for each option

AccuracyGoal and PrecisionGoal  (1)

Approximate a product with the default tolerances:

In[1]:=1
✖
https://wolfram.com/xid/0y8osy-iu4wep
Out[1]=1

Find the error versus the exact value:

In[2]:=2
✖
https://wolfram.com/xid/0y8osy-bv1rtv
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0y8osy-cflxjl
Out[3]=3

The error with smaller absolute and relative tolerances:

In[4]:=4
✖
https://wolfram.com/xid/0y8osy-b0xwq6
Out[4]=4

The error with larger absolute and relative tolerances:

In[5]:=5
✖
https://wolfram.com/xid/0y8osy-gs1rof
Out[5]=5

EvaluationMonitor  (3)

Get the number of evaluation points used in a product approximation:

In[1]:=1
✖
https://wolfram.com/xid/0y8osy-l16plx
Out[1]=1

The evaluation points used in a product approximated by an integration method:

In[1]:=1
✖
https://wolfram.com/xid/0y8osy-k1c2o6
Out[1]=1

The evaluation points used in a product approximated by a sequence extrapolation method:

In[1]:=1
✖
https://wolfram.com/xid/0y8osy-e9546p
Out[1]=1

Method  (1)

Use the Wynn epsilon method to approximate an infinite product:

In[1]:=1
✖
https://wolfram.com/xid/0y8osy-evwiuu
Out[1]=1

Compare to the exact value:

In[2]:=2
✖
https://wolfram.com/xid/0y8osy-v25dv
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0y8osy-k3ak
Out[3]=3

The error is smaller with the default method:

In[4]:=4
✖
https://wolfram.com/xid/0y8osy-nktbd
Out[4]=4

NProductFactors  (1)

By default NProduct uses 15 factors at the beginning before approximating the tail:

In[1]:=1
✖
https://wolfram.com/xid/0y8osy-y9wilg
Out[1]=1

The error for this example is large because the factors peak at 20:

In[2]:=2
✖
https://wolfram.com/xid/0y8osy-1gb3n
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0y8osy-n0n8nx
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0y8osy-lnvw3c
Out[4]=4

Increasing NProductFactors to include this feature improves the approximation:

In[5]:=5
✖
https://wolfram.com/xid/0y8osy-i0jcsm
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0y8osy-gekq3d
Out[6]=6

VerifyConvergence  (1)

By default the factor's convergence is verified:

In[1]:=1
✖
https://wolfram.com/xid/0y8osy-1qw9sd
Out[1]=1

Generally, if the convergence is not verified the computation is faster:

In[2]:=2
✖
https://wolfram.com/xid/0y8osy-186ogj
Out[2]=2

WorkingPrecision  (1)

Use higher precision to get a better approximation:

In[1]:=1
✖
https://wolfram.com/xid/0y8osy-8zptc4
Out[1]=1

Find the error versus the exact value:

In[2]:=2
✖
https://wolfram.com/xid/0y8osy-i8v4z5
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0y8osy-oujqf3
Out[3]=3

Applications  (2)Sample problems that can be solved with this function

Estimate the infinite product of a BesselJ sequence:

In[1]:=1
✖
https://wolfram.com/xid/0y8osy-rfovo
Out[1]=1

Use a product representation to approximate the Sin function:

In[1]:=1
✖
https://wolfram.com/xid/0y8osy-dyhyux
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0y8osy-dv5h05
Out[2]=2

Properties & Relations  (2)Properties of the function, and connections to other functions

Use Product to compute exact formulas:

In[1]:=1
✖
https://wolfram.com/xid/0y8osy-x3v3j7
Out[1]=1

The results of Product and NProduct agree closely:

In[2]:=2
✖
https://wolfram.com/xid/0y8osy-d5dpzh
Out[2]=2

A product is equivalent to the exponential of a sum of logarithms of factors:

In[1]:=1
✖
https://wolfram.com/xid/0y8osy-dm579i
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0y8osy-glkpxs
Out[2]=2

Possible Issues  (2)Common pitfalls and unexpected behavior

NProduct may not detect divergence for some infinite products:

In[1]:=1
✖
https://wolfram.com/xid/0y8osy-3sflac
Out[1]=1

Convergence verification is based on a ratio test that is inconclusive when equal to 1:

In[2]:=2
✖
https://wolfram.com/xid/0y8osy-4lnp0x
Out[2]=2

You should realize that in sufficiently pathological cases, the algorithms used by NProduct can give wrong answers:

In[1]:=1
✖
https://wolfram.com/xid/0y8osy-lz8x26
Out[1]=1

Compare to the result from Product:

In[2]:=2
✖
https://wolfram.com/xid/0y8osy-8u6n6h
Out[2]=2

One option is increasing the WorkingPrecision and NProductFactors:

In[3]:=3
✖
https://wolfram.com/xid/0y8osy-634amg
Out[3]=3
Wolfram Research (1988), NProduct, Wolfram Language function, https://reference.wolfram.com/language/ref/NProduct.html (updated 2003).
✖
Wolfram Research (1988), NProduct, Wolfram Language function, https://reference.wolfram.com/language/ref/NProduct.html (updated 2003).

Text

Wolfram Research (1988), NProduct, Wolfram Language function, https://reference.wolfram.com/language/ref/NProduct.html (updated 2003).

✖
Wolfram Research (1988), NProduct, Wolfram Language function, https://reference.wolfram.com/language/ref/NProduct.html (updated 2003).

CMS

Wolfram Language. 1988. "NProduct." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2003. https://reference.wolfram.com/language/ref/NProduct.html.

✖
Wolfram Language. 1988. "NProduct." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2003. https://reference.wolfram.com/language/ref/NProduct.html.

APA

Wolfram Language. (1988). NProduct. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/NProduct.html

✖
Wolfram Language. (1988). NProduct. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/NProduct.html

BibTeX

@misc{reference.wolfram_2025_nproduct, author="Wolfram Research", title="{NProduct}", year="2003", howpublished="\url{https://reference.wolfram.com/language/ref/NProduct.html}", note=[Accessed: 21-May-2025 ]}

✖
@misc{reference.wolfram_2025_nproduct, author="Wolfram Research", title="{NProduct}", year="2003", howpublished="\url{https://reference.wolfram.com/language/ref/NProduct.html}", note=[Accessed: 21-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_nproduct, organization={Wolfram Research}, title={NProduct}, year={2003}, url={https://reference.wolfram.com/language/ref/NProduct.html}, note=[Accessed: 21-May-2025 ]}

✖
@online{reference.wolfram_2025_nproduct, organization={Wolfram Research}, title={NProduct}, year={2003}, url={https://reference.wolfram.com/language/ref/NProduct.html}, note=[Accessed: 21-May-2025 ]}