Factorization Task: Difference between revisions
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A [[Factorization Task]] is a [[transformation task]] that decomposes an [[numerical object]] into a [[product]] of other [[numerical object]]s. | A [[Factorization Task]] is a [[transformation task]] that decomposes an [[numerical object]] into a [[product]] of other [[numerical object]]s. | ||
* <B | * <B>AKA:</B> [[Factorization Task|Factorization]], [[Decomposition]]. | ||
** … | |||
* <B>Example(s):</B> | * <B>Example(s):</B> | ||
** [[Integer Factorization]]. | ** [[Integer Factorization]]. | ||
* <B | * <B>See:</B> [[Matrix Decomposition Task]]. | ||
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===2012=== | == References == | ||
* (Wikipedia, 2012) | |||
** QUOTE: In [[mathematics]], '''factorization | === 2012 === | ||
* (Wikipedia, 2012) ⇒ http://en.wikipedia.org/wiki/Factorization | |||
** QUOTE: In [[mathematics]], '''factorization</B> (''also'' '''factorisation</B> ''in [[American and British English spelling differences#-ise, -ize (-isation, -ization)|British English]]'') or '''factoring</B> is the decomposition of an object (for example, a [[number]], a [[polynomial]], or a [[matrix (mathematics)|matrix]]) into a [[product (mathematics)|product]] of other objects, or ''factors'', which when [[multiplication|multiplied]] together give the original. For example, the number 15 factors into [[prime number|primes]] as 3 × 5, and the polynomial ''x</i><sup>2</sup> − 4 factors as (''x'' − 2)(''x'' + 2). In all cases, a product of simpler objects is obtained. <P> The aim of factoring is usually to reduce something to “basic building blocks”, such as numbers to prime numbers, or polynomials to [[irreducible polynomial]]s. Factoring integers is covered by the [[fundamental theorem of arithmetic]] and [[polynomial factorization|factoring polynomials]] by the [[fundamental theorem of algebra]]. [[Viète's formulas]] relate the coefficients of a polynomial to its roots. <P> The opposite of polynomial factorization is [[polynomial expansion|expansion]], the multiplying together of polynomial [[divisor|factors]] to an “expanded” polynomial, written as just a sum of terms. <P> [[Integer factorization]] for large integers appears to be a difficult problem. There is no known method to carry it out quickly. Its complexity is the basis of the assumed security of some [[public key cryptography]] algorithms, such as [[RSA (algorithm)|RSA]]. <P> A [[matrix (math)|matrix]] can also be factorized into a product of matrices of special types, for an application in which that form is convenient. One major example of this uses an [[orthogonal matrix|orthogonal]] or [[unitary matrix]], and a [[triangular matrix]]. There are different types: [[QR decomposition]], ''LQ'', ''QL'', ''RQ'', ''RZ''. <P> Another example is the factorization of a [[function (mathematics)|function]] as the [[function composition|composition]] of other functions having certain properties; for example, every function can be viewed as the composition of a [[surjective function]] with an [[injective function]]. This situation is generalized by [[factorization system]]s. | |||
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Latest revision as of 18:14, 2 June 2024
A Factorization Task is a transformation task that decomposes an numerical object into a product of other numerical objects.
- AKA: Factorization, Decomposition.
- …
- Example(s):
- See: Matrix Decomposition Task.
References
2012
- (Wikipedia, 2012) ⇒ http://en.wikipedia.org/wiki/Factorization
- QUOTE: In mathematics, 'factorization (also factorisation in British English) or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 × 5, and the polynomial x2 − 4 factors as (x − 2)(x + 2). In all cases, a product of simpler objects is obtained.
The aim of factoring is usually to reduce something to “basic building blocks”, such as numbers to prime numbers, or polynomials to irreducible polynomials. Factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viète's formulas relate the coefficients of a polynomial to its roots.
The opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an “expanded” polynomial, written as just a sum of terms.
Integer factorization for large integers appears to be a difficult problem. There is no known method to carry it out quickly. Its complexity is the basis of the assumed security of some public key cryptography algorithms, such as RSA.
A matrix can also be factorized into a product of matrices of special types, for an application in which that form is convenient. One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types: QR decomposition, LQ, QL, RQ, RZ.
Another example is the factorization of a function as the composition of other functions having certain properties; for example, every function can be viewed as the composition of a surjective function with an injective function. This situation is generalized by factorization systems.
- QUOTE: In mathematics, 'factorization (also factorisation in British English) or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 × 5, and the polynomial x2 − 4 factors as (x − 2)(x + 2). In all cases, a product of simpler objects is obtained.