# Factored

Objects kept in factored form

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

## Syntax

```Factored(`list`, <`type`>, <`ring`>)
Factored(`f`, <`type`>, <`ring`>)
```

## Description

`Factored` is the domain of objects kept in factored form, such as prime factorization of integers, square-free factorization of polynomials, or the factorization of polynomials in irreducible factors.

The argument `list` must be a list of odd length and of the form `[u, f1, e1, f2, e2, ..., fr, er]`, where the entries u and fi are elements of the domain `ring`, or can be converted into such elements. The ei must be integers. Here, i ranges from 1 to r.

See section "Operands" below for the meaning of the entries of that list.

An error message is reported, if one of the list entries is of wrong type.

An arithmetical expression `f` given as the first argument is the same as giving the list ```[ring::one, f, 1]```.

See section "Operands" below for the meaning of the entries of that list.

`f` must be an element of the domain `ring`, or must be convertible into such an element, otherwise an error message would be given.

The argument `type` indicates what is known about the factorization. Currently, the following types are known:

• `"unknown"` – nothing is known about the factorization.

• `"irreducible"` – the fi are irreducible over the domain `ring`.

• `"squarefree"` – the fi are square-free over the domain `ring`.

If this argument is missing, then the type of the created factored object is set to `"unknown"`.

The type of factorization is known to any element of `Factored`. Use the methods `"getType"` and `"setType"` (see below) to read and set the type of factorization of a given factored object.

The argument `ring` is the ring of factorization. It must be an integral domain, i.e., a domain of category `Cat::IntegralDomain`.

If this argument is missing, then the domain `Dom::ExpressionField()` is used.

The ring of factorization is known to any element of `Factored`. Use the methods `"getRing"` and `"setRing"` (see below) to read and set the ring of factorization of a given factored object.

You can use the index operator `[ ]` to extract the factors of an element `f` of the domain `Factored`. E.g., for f = uf1e1f2e2 …, you have f [1] = u, f [2] = f1e1, f [3] = f2e2 etc.

You can also use the methods `"factors"` and `"exponents"` (see below) to access the operands, i.e., the call `Factored::factors(f)` returns a list of the factors fi, and `Factored::exponents(g)` returns a list of the exponents ei (1 ≤ ir).

The system functions `ifactor`, `factor` and `polylib::sqrfree` are the main application of this domain, they return their result in form of such factored objects (see their help pages for information about the type and ring of factorization).

There may be no need to explicitly create factored objects, but to work with the results of the mentioned system functions.

Note that an element of `Factored` is printed like an expression and behaves like that. As an example, the result of `f := factor(x^2 + 2*x + 1)` is an element of `Factored` and printed as `(x + 1)^2`. The call `type(f)` returns `"_power"` as the expression type of `f`.

For an element `f` of `Factored`, the call `Factored::convert(f, DOM_LIST)` gives a list of all operands of `f`.

## Examples

### Example 1

The following computes the prime factorization of the integer 20:

`f := ifactor(20)`

The result is an element of the domain `Factored`:

`domtype(f)`

which consists of the following five operands:

`op(f)`

They represent the integer 20 in the following form: 20 = 1 22 5. The factors are prime numbers and can be extracted via `Factor::factors`:

`Factored::factors(f)`

`ifactor` kept the information that the factorization ring is the ring of integers (represented by the domain `Dom::Integer`), and that the factors of `f` are prime (and therefore irreducible, because is an integral domain):

`Factored::getRing(f), Factored::getType(f)`

We can convert such an object into different forms, such as into a list of its operands:

`Factored::convert_to(f, DOM_LIST)`

or into an unevaluated expression, keeping the factored form:

`Factored::convert_to(f, DOM_EXPR)`

or back into an integer:

`Factored::convert_to(f, Dom::Integer)`

You may also use the system function `coerce` here, which has the same effect.

### Example 2

We compute the factorization of the integers 108 and 512:

`n1 := ifactor(108); n2 := ifactor(512)`

The multiplication of these two integers gives the prime factorization of 55296 = 108 512:

`n1*n2`

Note that the most operations on such objects lead to an un-factored form, such as adding these two integers:

`n1 + n2`

You may apply the function `ifactor` to the result, if you are interested in its prime factorization:

`ifactor(%)`

You an apply (almost) each function to factored objects, functions that mainly expect arithmetical expressions as their input. Note that, before the operation is applied, the factored object is converted into an arithmetical expression in un-factored form:

`Re(n1)`

### Example 3

The second system function which deals with elements of `Factored`, is `factor`, which computes all irreducible factors of a polynomial.

For example, if we define the following polynomial of 101:

`p := poly(x^12 + x + 1, [x], Dom::IntegerMod(101)):`

and compute its factorization into irreducible factors, we get:

`f := factor(p)`

If we multiply the factored object with an element that can be converted into an element of the ring of factorization, then we get a new factored object, which then is of the factorization type `"unknown"`:

`x*f`

`Factored::getType(%)`

You may use the function `expand` which returns the factored object in expanded form as an element of the factorization ring:

`expand(f)`

### Example 4

The third system function which return elements of `Factored` is `polylib::sqrfree`, which computes the square-free factorization of polynomials. For example:

`f := polylib::sqrfree(x^2 + 2*x + 1)`

The factorization type, of course, is `"squarefree"`:

`Factored::getType(f)`

## Parameters

 `list` A list of odd length `f` An arithmetical expression `type` A string (default: `"unknown"`) `ring` A domain of category `Cat::IntegralDomain` (default: `Dom::ExpressionField()`)

## Function Calls

Calling a factored object as a function yields the object itself, regardless of the arguments. The arguments are not evaluated.

## Operations

You can apply (almost) every function to factored objects, functions that mainly expect arithmetical expressions as their input.

For example, one may add or multiply those objects, or apply functions such as `expand` and `diff` to them. But the result of such an operation then is usually not any longer of the domain `Factored`, as the factored form could be lost due to the operation (see examples below).

Call `expr(f)` to convert the factored object `f` into an arithmetical expression (as an element of a kernel domain).

The call `coerce(f, DOM_LIST)` returns a list of operands of the factored object `f` (see method `"convert_to"` below).

## Operands

An element f of `Factored` consists of the r + 1 operands u, f1, e1, f2, e2, …, fr, er, such that f = uf1e1f2e2 … frer.

The first operand u and the factors fi are elements of the domain `ring`. The exponents ei are integers.

expand all

## Mathematical Methods

### `_mult` — Multiply factored objects

`_mult(f, g, …)`

Suppose that `g` is an element of the domain `ring` (or can be converted into such an element).

If `g` is a unit of `ring` or a factor of `f`, then the result is a factored object of the same factorization type as `f`. Otherwise, the result is an element of `Factored` with the factorization type `"unknown"`.

If both `f` and `g` are factored objects with factorization type `"irreducible"`, then the result is again a factored object of this type, i.e., the result is still in factored form.

Otherwise, the factored form of `f` is lost, and the result of this method is an element of `ring`.

This method overloads the function `_mult` for factored objects, i.e., one may use it in the form `f*g*...`, or in functional notation: `_mult(f, g, ...)`.

### `_power` — Raise a factored object to a certain power

`_power(f, n)`

If `n` is a positive integer and `f` a factored object with factorization type `"irreducible"` or `"squarefree"`, then the result is still a factored object of this type.

Otherwise, the factored form of `f` is lost, and the result of this method is an element of `ring`.

This method overloads the function `_power` for factored objects, i.e., one may use it in the form `f^n`, or in functional notation: ```_power(f, n)```.

### `expand` — Expand a factored object

`expand(f)`

### `exponents` — Get the exponents of a factored object

`exponents(f)`

### `factor` — Factor a factored object

`factor(f)`

If `f` already is of the factorization type `"irreducible"`, then this method just return `f`.

Otherwise, this method converts `f` into an element of the domain `ring` and calls the method `"factor"` of `ring`.

This method returns a factored object of the domain `Factored` with factorization type `"irreducible"`, if the factorization of `f` can be computed (otherwise, `FAIL` is returned).

This method overloads the function `factor` for factored objects, i.e., one may use it in the form `factor(f)`.

### `factors` — Get the factors of a factored object

`factors(f)`

### `irreducible` — Test if a factored object is irreducible

`irreducible(f)`

The test on irreducible is trivial, if `f` has the factorization type `"irreducible"`.

Otherwise, this method converts `f` into an element of `ring` and calls the method `"irreducible"` of `ring`. The value `FAIL` is returned, if the domain `ring` cannot test if `f` is irreducible.

### `iszero` — Test on zero for factored objects

`iszero(f)`

This method overloads the function `iszero` for factored objects, i.e., one may use it in the form `iszero(f)`.

### `sqrfree` — Compute a square-free factorization of a factored object

`sqrfree(f)`

If `f` already is of the factorization type `"squarefree"`, then this method just return `f`.

Otherwise, this method converts `f` into an element of the domain `ring` and calls the method `"squarefree"` of `ring`.

This method returns a factored object of the domain `Factored` with factorization type `"squarefree"`, if the square-free factorization of `f` can be computed (otherwise, `FAIL` is returned).

This method overloads the function `polylib::sqrfree` for factored objects, i.e., one may use it in the form `polylib::sqrfree(f)`.

## Access Methods

### `_index` — Extract a term of a factored object

`_index(f, i)`

Responds with an error message, if `i` is greater than the number of terms of `f`.

This method overloads the index operator `[ ]` for factored objects, i.e., one may use it in the form `f[i]`.

### `getRing` — Get the ring of factorization

`getRing(f)`

### `getType` — Get the type of factorization

`getType(f)`

### `has` — Existence of an object in a factored object

`has(f, x, …)`

This method overloads the function `has` for factored objects, i.e., one may use it in the form `has(f, x, ...)`.

### `map` — Map a function to the operands of factored objects

`map(f, func, …)`

See the system function `map` for details.

This method overloads the function `map` for factored objects, i.e., one may use it in the form `map(f, func, ...)`.

### `nops` — Number of operands of a factored object

`nops(f)`

This method overloads the function `nops` for factored objects, i.e., one may use it in the form `nops(f)`.

### `op` — Extract an operand of a factored object

`op(f, i)`

Returns `FAIL`, if `i` is greater than the number of operands of `f`.

This method overloads the function `op` for factored objects, i.e., one may use it in the form `op(f, i)`.

### `select` — Select operands of a factored object

`select(f, func, …)`

This method overloads the function `select` for factored objects, i.e., one may use it in the form `select(f, func, ...)`.

### `set_index` — Set/change a term of a factored object

`set_index(f, i, x)`

Responds with an error message, if `i` is greater than the number of terms of `f`.

 Note:   Make sure that `x` either is an element of the domain `ring`, or an integer.

This method overloads the index operator `[ ]` for factored objects, i.e., one may use it in the form `f[i] := x`.

### `setRing` — Set the ring of factorization

`setRing(f, ring)`

 Note:   Use this method with caution! Make sure that the factorization of `f` is still valid over the new ring, and that the operands of `f` have the correct domain type.`ring` must be a domain of category `Cat::IntegralDomain`, which is not checked by this method.

### `setType` — Set the type of factorization

`setType(f, type)`

 Note:   Use this method with caution! Make sure that the factorization type corresponds with the factorization of `f`.

### `subs` — Substitute subexpressions in the operands of a factored object

`subs(f, x = a, …)`

This method overloads the function `subs` for factored objects, i.e., one may use it in the form `subs(f, x = a, ...)`.

### `subsop` — Substitute operands of a factored object

`subsop(f, i = a, …)`

This method overloads the function `subsop` for factored objects, i.e., one may use it in the form `subsop(f, i = a, ...)`.

### `type` — Expression type of factored objects

`type(f)`

## Conversion Methods

### `convert` — Convert an object into a factored object

`convert(x)`

If the conversion fails, then `FAIL` is returned.

`x` may either be a list of the form ```[u, f1, e1, ..., fr, er]``` of odd length (where ```u, f1, ..., fr``` are of the domain type `ring`, or can be converted into such elements, and `e1, ..., er` are integers), or an element that can be converted into the domain `ring`. The latter case corresponds to the list `[ring::one,x,1]`.

### `convert_to` — Convert factored objects into other domains

`convert_to(f, T)`

If the conversion fails, then `FAIL` is returned.

If `T` is the domain `DOM_LIST`, then the list of operands of `f` is returned.

If `T` is the domain `DOM_EXPR`, then the unevaluated expression `u*f1^e1*f2^e2*...*fr^er` is returned, where `u, f1, e1, ...` are the operands of `f`.

Otherwise, the method `"convert"` of the domain `T` is called to convert `f` into an element of the domain `T` (which could return `FAIL`).

Use the function `expr` to convert `f` into an object of a kernel domain (see below).

### `create` — Create simple and fast a factored objects

`create(list)`

`create(x)`

This method creates a new factored object with the operands ```ring::one, x, 1```.

### `expr` — Convert a factored object into a kernel domain

`expr(f)`

 Note:   Note that the factored form of `f` may be lost due to this conversion.

### `expr2text` — Convert a factored object into a string

`expr2text(f)`

### `testtype` — Type testing for factored objects

`testtype(f, T)`

This method is called from the system function `testtype`.

### `TeX` — LaTeX formatting of a factored object

`TeX(f)`

The method `"TeX"` of the domain `ring` is used to get the LaTeX-representation of the corresponding operands of `f`.

This method is called from the system function `generate::TeX`.

## Technical Methods

### `_concat` — Concatenate operands of factored objects

`_concat(f, g)`

`f` and `g` must have the same factorization type and factorization ring, otherwise an error message is given.

### `maprec` — Allow recursive mapping for factored objects

`maprec(f, x, …)`

First `f` is converted into the unevaluated expression `u*f1^e1*f2^e2*...*fr^er`, where ```u, f1, e1, ...``` are the operands of `f`. Then the function `misc::maprec` is called with this expression as its first parameter.

Note that the result of this method is not longer an object of `Factored`!

### `print` — Pretty-print routine for factored objects

`print(f)`

### `unapply` — Create a procedure from a factored object

`unapply(f, <x>)`

This method overloads the function `fp::unapply` for factored objects, i.e., one may use it in the form `fp::unapply(f)`. See `fp::unapply` for details.