﻿ product – Algosim documentation
Algosim documentation: product

# product

Computes the product of all elements in a container or in a sequence. The elements may be numbers or square matrices.

## Syntax

• `product(X)`

• `X` is a vector, matrix, list, or set

• `product(expr, var, a, b)`

• `expr` is an expression in one variable `var`

• `var` is the variable in `expr`

• `a` is the lower bound

• `b` is the upper bound

## Description

### Computing the product of all elements in a container

If `X` is a container, then `product(X)` is the product of all elements in `X`.

The following container types are supported:

• vectors

• the result is the product of all components

• matrices

• the result is the product of all entries

• lists containing numbers

• the result is the product of all numbers

• lists containing square matrices

• the result is the matrix product of all matrices

• sets containing numbers

• the result is the product of all numbers

Typewise, the factors may be either real or complex, and the result is complex iff at least one of the factors is complex.

### Computing the product of all elements in a sequence

`product(expr, var, a, b)` computes the product of the expression `expr` in one variable `var` as `var` takes all integer values from `a` to `b` (inclusively). This precisely implements the pi notation for n-ary products.

`expr` must return a number or a matrix.

## Notes

These note apply to both modes of operation: computing the product of all elements in a container and computing the product of all elements in a sequence.

The empty product is `1`. Notice that the product of an empty collection of matrices is also equal to the scalar `1`, because the system cannot tell that the collection indeed is an empty collection of matrices. And even if it could, it wouldn’t know the size of the matrices.

When computing the product of a collection of objects, the objects must be of the same kind (numbers or matrices). If they are matrices, they must all be of the same size.

You cannot compute the product of all matrices in a set because matrix multiplication is not commutative and a set is not an ordered container.

## Examples

### Computing the product of all elements in a container

`product(❨2, 1, 3, −4, 5, 2, 1, 3, −2❩)`
`1440`
`product(❨❨4, 2, 1❩, ❨3, 2, 5❩❩)`
`240`
`product('(❨❨1, 3❩, ❨2, 1❩❩, ❨❨−2, 1❩, ❨0, 3❩❩, ❨❨1, 0❩, ❨−1, i❩❩))`
```⎛−12  10⋅i⎞
⎝ −9  5⋅i ⎠
```
`product(unique(PrimeFactors(19871226)))`
`6623742`

### Computing the product of all elements in a sequence

`abs(product(1 − 1/prime(n)^2, n, 1, 100000) − 6/π^2)`
`3.11339510472⋅10^−8`