Factorization may also refer to more general decompositions of a mathematical object into the product of smaller or simpler objects.
Faktorisasi juga dapat merujuk ke dekomposisi objek matematika lain ke hasil perkalian objek-objek yang lebih kecil atau sederhana.
For example, every function may be factored into the composition of a surjective function with an injective function.
Sebagai contoh, setiap fungsi dapat difaktorkan menjadi komposisi fungsi surjektif dan fungsi injektif.
Matrices possess many kinds of matrix factorizations.
Matriks memiliki banyak jenis faktorisasi.
For example, every matrix has a unique LUP factorization as a product of a lower triangular matrix L with all diagonal entries equal to one, an upper triangular matrix U, and a permutation matrix P; this is a matrix formulation of Gaussian elimination.
Sebagai contoh, setiap matriks memiliki hasil faktorisasi LUP yang unik, dengan matriks segitiga bawah L dengan entri pada diagonal utama bernilai 1, matriks segitiga atas U , dan matriks permutasi P ; ketiga matriks ini didapatkan dari formulasi eliminasi Gaussian.
When an expression can be factored, the factors are often much simpler, and may thus offer some insight on the problem.
Ketika sebuah ekspresi dapat difaktorkan, faktor-faktor tersebut umumnya lebih sederhana, dan dapat memberikan wawasan dari masalah tersebut.
For example, x 3 − a x 2 − b x 2 − c x 2 + a b x + a c x + b c x − a b c -ax^{2}-bx^{2}-cx^{2}+abx+acx+bcx-abc} having 16 multiplications, 4 subtractions and 3 additions, may be factored into the much simpler expression ( x − a ) ( x − b ) ( x − c ) , with only two multiplications and three subtractions.
Sebagai contoh, x 3 − a x 2 − b x 2 − c x 2 + a b x + a c x + b c x − a b c -ax^{2}-bx^{2}-cx^{2}+abx+acx+bcx-abc} yang secara total memiliki 16 perkalian, 4 pengurangan, dan 3 penjumlahan, dapat difaktorkan menjadi ekspresi yang lebih sederhana ( x − a ) ( x − b ) ( x − c )
For example, the LU decomposition gives a matrix as the product of a lower triangular matrix by an upper triangular matrix.
Sebagai contoh, faktorisasi LU memfaktorkan matriks menjadi hasil perkalian matriks segitiga bawah dan matriks segitiga atas.
[1912], The Theory of Equations with an introduction to the theory of binary algebraic forms (Volume one), Dover Dickson, Leonard Eugene (1922), "First Course in the Theory of Equations", Nature, 109 (2746), New York: John Wiley & Sons: 773, Bibcode:1922Natur.109R.773., doi:10.1038/109773c0 Fite, William Benjamin (1921), College Algebra (Revised), Boston: D. C. Heath & Co. Klein, Felix (1925), Elementary Mathematics from an Advanced Standpoint; Arithmetic, Algebra, Analysis, Dover Selby, Samuel M. (1970), CRC Standard Mathematical Tables (18th ed.), The Chemical Rubber Co. Wolfram Alpha can factorize too.
[1912], The Theory of Equations with an introduction to the theory of binary algebraic forms (Volume one), Dover Dickson, Leonard Eugene (1922), First Course in the Theory of Equations, New York: John Wiley & Sons Fite, William Benjamin (1921), College Algebra (Revised), Boston: D. C. Heath & Co. Klein, Felix (1925), Elementary Mathematics from an Advanced Standpoint; Arithmetic, Algebra, Analysis, Dover Selby, Samuel M., CRC Standard Mathematical Tables (Edisi 18th), The Chemical Rubber Co. Hazewinkel, Michiel, ed. (2001)