Ejemplo de factorización PLU - Egor...

Ejemplo de factorizacion PLU

Egor Maximenkohttp://www.egormaximenko.com

Instituto Politecnico Nacional,Escuela Superior de Fısica y Matematicas

Mexico, D.F.

5 de enero de 2015

PrerrequisitosAlgoritmo de factorizacion LU

(guardando L y U juntas)Matrices de permutacion

Factorizacion PLU

Problema de factorizacion PLU

Hallar una factorizacion PLU de la matriz

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

Se buscan matrices P, L, U tales quePA = LU,P es una matriz de permutacion,L es unitriangular inferior,U es triangular superior.

Una matriz puede tener varias factorizaciones PLU.En este ejemplo una respuesta posible es

1 0 0 00 0 1 00 0 0 10 1 0 0

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

1 0 0 00 0 1 00 0 0 10 1 0 0

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

Una matriz puede tener varias factorizaciones PLU.

En este ejemplo una respuesta posible es

1 0 0 00 0 1 00 0 0 10 1 0 0

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

1 0 0 00 0 1 00 0 0 10 1 0 0

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

Solucion (en cada paso PA = LU)

P1,2,3,4A =

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

R2 +=−3R1R3 +=−6R1R4 += 5R1

=========

5 2 4 33 0 0 −26 3 −1 −3−5 −12 5 12

R2↔R3−−−−→

P1,3,2,4 A =

5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12

R3 += 0R2R4 += 4R2

=========

5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0

R3↔R4−−−−→

P1,3,4,2 A =

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

R4 += 0R3=========

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

P1,2,3,4A =

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

R2 +=−3R1R3 +=−6R1R4 += 5R1

=========

5 2 4 3

3 0 0 −26 3 −1 −3−5 −12 5 12

R2↔R3−−−−→

P1,3,2,4 A =

5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12

R3 += 0R2R4 += 4R2

=========

5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0

R3↔R4−−−−→

P1,3,4,2 A =

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

R4 += 0R3=========

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

P1,2,3,4A =

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

R2 +=−3R1

R3 +=−6R1R4 += 5R1

=========

5 2 4 33

0 0 −26 3 −1 −3−5 −12 5 12

R2↔R3−−−−→

P1,3,2,4 A =

5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12

R3 += 0R2R4 += 4R2

=========

5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0

R3↔R4−−−−→

P1,3,4,2 A =

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

R4 += 0R3=========

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

P1,2,3,4A =

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

R2 +=−3R1R3 +=−6R1

R4 += 5R1

=========

5 2 4 33

0 0 −2

3 −1 −3−5 −12 5 12

R2↔R3−−−−→

P1,3,2,4 A =

5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12

R3 += 0R2R4 += 4R2

=========

5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0

R3↔R4−−−−→

P1,3,4,2 A =

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

R4 += 0R3=========

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

P1,2,3,4A =

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

R2 +=−3R1R3 +=−6R1R4 += 5R1=========

5 2 4 33

0 0 −2

3 −1 −3

−12 5 12

R2↔R3−−−−→

P1,3,2,4 A =

5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12

R3 += 0R2R4 += 4R2

=========

5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0

R3↔R4−−−−→

P1,3,4,2 A =

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

R4 += 0R3=========

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

P1,2,3,4A =

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

R2 +=−3R1R3 +=−6R1R4 += 5R1=========

5 2 4 33 0

0 −2

3 −1 −3

−12 5 12

R2↔R3−−−−→

P1,3,2,4 A =

5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12

R3 += 0R2R4 += 4R2

=========

5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0

R3↔R4−−−−→

P1,3,4,2 A =

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

R4 += 0R3=========

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

P1,2,3,4A =

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

R2 +=−3R1R3 +=−6R1R4 += 5R1=========

5 2 4 33 0 0

3 −1 −3

−12 5 12

R2↔R3−−−−→

P1,3,2,4 A =

5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12

R3 += 0R2R4 += 4R2

=========

5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0

R3↔R4−−−−→

P1,3,4,2 A =

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

R4 += 0R3=========

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

P1,2,3,4A =

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

R2 +=−3R1R3 +=−6R1R4 += 5R1=========

5 2 4 33 0 0 −26

3 −1 −3

−12 5 12

R2↔R3−−−−→

P1,3,2,4 A =

5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12

R3 += 0R2R4 += 4R2

=========

5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0

R3↔R4−−−−→

P1,3,4,2 A =

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

R4 += 0R3=========

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

P1,2,3,4A =

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

R2 +=−3R1R3 +=−6R1R4 += 5R1=========

5 2 4 33 0 0 −26 3

−1 −3

−12 5 12

R2↔R3−−−−→

P1,3,2,4 A =

5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12

R3 += 0R2R4 += 4R2

=========

5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0

R3↔R4−−−−→

P1,3,4,2 A =

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

R4 += 0R3=========

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

P1,2,3,4A =

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

R2 +=−3R1R3 +=−6R1R4 += 5R1=========

5 2 4 33 0 0 −26 3 −1

−12 5 12

R2↔R3−−−−→

P1,3,2,4 A =

5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12

R3 += 0R2R4 += 4R2

=========

5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0

R3↔R4−−−−→

P1,3,4,2 A =

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

R4 += 0R3=========

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

P1,2,3,4A =

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

R2 +=−3R1R3 +=−6R1R4 += 5R1=========

5 2 4 33 0 0 −26 3 −1 −3−5

−12 5 12

R2↔R3−−−−→

P1,3,2,4 A =

5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12

R3 += 0R2R4 += 4R2

=========

5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0

R3↔R4−−−−→

P1,3,4,2 A =

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

R4 += 0R3=========

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

P1,2,3,4A =

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

R2 +=−3R1R3 +=−6R1R4 += 5R1=========

5 2 4 33 0 0 −26 3 −1 −3−5 −12

R2↔R3−−−−→

P1,3,2,4 A =

5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12

R3 += 0R2R4 += 4R2

=========

5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0

R3↔R4−−−−→

P1,3,4,2 A =

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

R4 += 0R3=========

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

P1,2,3,4A =

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

R2 +=−3R1R3 +=−6R1R4 += 5R1=========

5 2 4 33 0 0 −26 3 −1 −3−5 −12 5

R2↔R3−−−−→

P1,3,2,4 A =

5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12

R3 += 0R2R4 += 4R2

=========

5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0

R3↔R4−−−−→

P1,3,4,2 A =

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

R4 += 0R3=========

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

P1,2,3,4A =

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

R2 +=−3R1R3 +=−6R1R4 += 5R1=========

5 2 4 33 0 0 −26 3 −1 −3−5 −12 5 12

R2↔R3−−−−→

P1,3,2,4 A =

5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12

R3 += 0R2R4 += 4R2

=========

5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0

R3↔R4−−−−→

P1,3,4,2 A =

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

R4 += 0R3=========

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

P1,2,3,4A =

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

R2 +=−3R1R3 +=−6R1R4 += 5R1=========

5 2 4 33 0 0 −26 3 −1 −3−5 −12 5 12

R2↔R3−−−−→

P1,3,2,4 A =

5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12

R3 += 0R2R4 += 4R2

=========

5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0

R3↔R4−−−−→

P1,3,4,2 A =

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

R4 += 0R3=========

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

P1,2,3,4A =

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

R2 +=−3R1R3 +=−6R1R4 += 5R1=========

5 2 4 33 0 0 −26 3 −1 −3−5 −12 5 12

R2↔R3−−−−→

P1,3,2,4 A =

5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12

R3 += 0R2R4 += 4R2

=========

5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0

R3↔R4−−−−→

P1,3,4,2 A =

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

R4 += 0R3=========

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

P1,2,3,4A =

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

R2 +=−3R1R3 +=−6R1R4 += 5R1=========

5 2 4 33 0 0 −26 3 −1 −3−5 −12 5 12

R2↔R3−−−−→

P1,3,2,4 A =

5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12

R3 += 0R2R4 += 4R2

=========

5 2 4 36 3 −1 −33

0 0 −2

−4 1 0

R3↔R4−−−−→

P1,3,4,2 A =

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

R4 += 0R3=========

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

P1,2,3,4A =

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

R2 +=−3R1R3 +=−6R1R4 += 5R1=========

5 2 4 33 0 0 −26 3 −1 −3−5 −12 5 12

R2↔R3−−−−→

P1,3,2,4 A =

5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12

R3 += 0R2

R4 += 4R2

=========

5 2 4 36 3 −1 −33 0

0 −2

−4 1 0

R3↔R4−−−−→

P1,3,4,2 A =

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

R4 += 0R3=========

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

P1,2,3,4A =

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

R2 +=−3R1R3 +=−6R1R4 += 5R1=========

5 2 4 33 0 0 −26 3 −1 −3−5 −12 5 12

R2↔R3−−−−→

P1,3,2,4 A =

5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12

R3 += 0R2R4 += 4R2=========

5 2 4 36 3 −1 −33 0

0 −2

−5 −4

R3↔R4−−−−→

P1,3,4,2 A =

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

R4 += 0R3=========

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

P1,2,3,4A =

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

R2 +=−3R1R3 +=−6R1R4 += 5R1=========

5 2 4 33 0 0 −26 3 −1 −3−5 −12 5 12

R2↔R3−−−−→

P1,3,2,4 A =

5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12

R3 += 0R2R4 += 4R2=========

5 2 4 36 3 −1 −33 0 0

−5 −4

R3↔R4−−−−→

P1,3,4,2 A =

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

R4 += 0R3=========

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

P1,2,3,4A =

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

R2 +=−3R1R3 +=−6R1R4 += 5R1=========

5 2 4 33 0 0 −26 3 −1 −3−5 −12 5 12

R2↔R3−−−−→

P1,3,2,4 A =

5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12

R3 += 0R2R4 += 4R2=========

5 2 4 36 3 −1 −33 0 0 −2−5 −4

R3↔R4−−−−→

P1,3,4,2 A =

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

R4 += 0R3=========

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

P1,2,3,4A =

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

R2 +=−3R1R3 +=−6R1R4 += 5R1=========

5 2 4 33 0 0 −26 3 −1 −3−5 −12 5 12

R2↔R3−−−−→

P1,3,2,4 A =

5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12

R3 += 0R2R4 += 4R2=========

5 2 4 36 3 −1 −33 0 0 −2−5 −4 1

R3↔R4−−−−→

P1,3,4,2 A =

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

R4 += 0R3=========

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

P1,2,3,4A =

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

R2 +=−3R1R3 +=−6R1R4 += 5R1=========

5 2 4 33 0 0 −26 3 −1 −3−5 −12 5 12

R2↔R3−−−−→

P1,3,2,4 A =

5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12

R3 += 0R2R4 += 4R2=========

5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0

R3↔R4−−−−→

P1,3,4,2 A =

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

R4 += 0R3=========

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

P1,2,3,4A =

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

R2 +=−3R1R3 +=−6R1R4 += 5R1=========

5 2 4 33 0 0 −26 3 −1 −3−5 −12 5 12

R2↔R3−−−−→

P1,3,2,4 A =

5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12

R3 += 0R2R4 += 4R2=========

5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0

R3↔R4−−−−→

P1,3,4,2 A =

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

R4 += 0R3=========

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

P1,2,3,4A =

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

R2 +=−3R1R3 +=−6R1R4 += 5R1=========

5 2 4 33 0 0 −26 3 −1 −3−5 −12 5 12

R2↔R3−−−−→

P1,3,2,4 A =

5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12

R3 += 0R2R4 += 4R2=========

5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0

R3↔R4−−−−→

P1,3,4,2 A =

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

R4 += 0R3=========

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

P1,2,3,4A =

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

R2 +=−3R1R3 +=−6R1R4 += 5R1=========

5 2 4 33 0 0 −26 3 −1 −3−5 −12 5 12

R2↔R3−−−−→

P1,3,2,4 A =

5 2 4 36 3 −1 −33 0 0 −2−5 −12 5 12

R3 += 0R2R4 += 4R2=========

5 2 4 36 3 −1 −33 0 0 −2−5 −4 1 0

R3↔R4−−−−→

P1,3,4,2 A =

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

R4 += 0R3=========

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

El ultimo estado de la solucion es

P1,3,4,2 A =

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

Respuesta:1 0 0 00 0 1 00 0 0 10 1 0 0

︸︷︷︸

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

︸︷︷︸

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

︸︷︷︸

5 2 4 30 3 −1 30 0 1 00 0 0 −2

︸︷︷︸

El ultimo estado de la solucion es

P1,3,4,2 A =

5 2 4 36 3 −1 −3−5 −4 1 0

3 0 0 −2

Respuesta:1 0 0 00 0 1 00 0 0 10 1 0 0

︸︷︷︸

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

︸︷︷︸

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

︸︷︷︸

5 2 4 30 3 −1 30 0 1 00 0 0 −2

︸︷︷︸

Comprobacion

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

5 2 4 330 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

P1,3,4,2 A = P1,3,4,2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

5 2 4 330 15 23 15−25 −22 −15 −3

15 6 12 7

Comprobacion

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

5 2 4 330 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

P1,3,4,2 A = P1,3,4,2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

5 2 4 330 15 23 15−25 −22 −15 −3

15 6 12 7

Comprobacion

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

2 4 330 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

P1,3,4,2 A = P1,3,4,2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

5 2 4 330 15 23 15−25 −22 −15 −3

15 6 12 7

Comprobacion

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

4 330 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

P1,3,4,2 A = P1,3,4,2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

5 2 4 330 15 23 15−25 −22 −15 −3

15 6 12 7

Comprobacion

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

330 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

P1,3,4,2 A = P1,3,4,2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

5 2 4 330 15 23 15−25 −22 −15 −3

15 6 12 7

Comprobacion

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

5 2 4 3

30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

P1,3,4,2 A = P1,3,4,2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

5 2 4 330 15 23 15−25 −22 −15 −3

15 6 12 7

Comprobacion

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

5 2 4 3

12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

P1,3,4,2 A = P1,3,4,2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

5 2 4 330 15 23 15−25 −22 −15 −3

15 6 12 7

Comprobacion

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

5 2 4 3

30 12 + 3

24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

P1,3,4,2 A = P1,3,4,2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

5 2 4 330 15 23 15−25 −22 −15 −3

15 6 12 7

Comprobacion

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

5 2 4 3

30 12 + 3 24− 1

18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

P1,3,4,2 A = P1,3,4,2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

5 2 4 330 15 23 15−25 −22 −15 −3

15 6 12 7

Comprobacion

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

5 2 4 3

30 12 + 3 24− 1 18− 3

−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

P1,3,4,2 A = P1,3,4,2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

5 2 4 330 15 23 15−25 −22 −15 −3

15 6 12 7

Comprobacion

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

5 2 4 3

30 12 + 3 24− 1 18− 3−25

− 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

P1,3,4,2 A = P1,3,4,2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

5 2 4 330 15 23 15−25 −22 −15 −3

15 6 12 7

Comprobacion

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

5 2 4 3

30 12 + 3 24− 1 18− 3−25 − 10− 12

− 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

P1,3,4,2 A = P1,3,4,2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

5 2 4 330 15 23 15−25 −22 −15 −3

15 6 12 7

Comprobacion

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

5 2 4 3

30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1

− 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

P1,3,4,2 A = P1,3,4,2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

5 2 4 330 15 23 15−25 −22 −15 −3

15 6 12 7

Comprobacion

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

5 2 4 3

30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 0

15 6 + 0 12 + 0 + 0 9 + 0 + 0− 2

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

P1,3,4,2 A = P1,3,4,2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

5 2 4 330 15 23 15−25 −22 −15 −3

15 6 12 7

Comprobacion

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

5 2 4 3

30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015

6 + 0 12 + 0 + 0 9 + 0 + 0− 2

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

P1,3,4,2 A = P1,3,4,2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

5 2 4 330 15 23 15−25 −22 −15 −3

15 6 12 7

Comprobacion

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

5 2 4 3

30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0

12 + 0 + 0 9 + 0 + 0− 2

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

P1,3,4,2 A = P1,3,4,2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

5 2 4 330 15 23 15−25 −22 −15 −3

15 6 12 7

Comprobacion

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

5 2 4 3

30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0

9 + 0 + 0− 2

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

P1,3,4,2 A = P1,3,4,2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

5 2 4 330 15 23 15−25 −22 −15 −3

15 6 12 7

Comprobacion

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

5 2 4 3

30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

P1,3,4,2 A = P1,3,4,2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

5 2 4 330 15 23 15−25 −22 −15 −3

15 6 12 7

Comprobacion

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

5 2 4 3

30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

P1,3,4,2 A = P1,3,4,2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

5 2 4 330 15 23 15−25 −22 −15 −3

15 6 12 7

Comprobacion

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

5 2 4 3

30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

P1,3,4,2 A = P1,3,4,2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

5 2 4 330 15 23 15−25 −22 −15 −3

15 6 12 7

Comprobacion

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

5 2 4 3

30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

P1,3,4,2 A = P1,3,4,2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

5 2 4 330 15 23 15−25 −22 −15 −3

15 6 12 7

Comprobacion

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

5 2 4 3

30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

P1,3,4,2 A = P1,3,4,2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

5 2 4 330 15 23 15−25 −22 −15 −3

15 6 12 7

Comprobacion

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

5 2 4 3

30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

P1,3,4,2 A = P1,3,4,2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

Comprobacion

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

5 2 4 3

30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

P1,3,4,2 A = P1,3,4,2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

Comprobacion

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

5 2 4 3

30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

P1,3,4,2 A = P1,3,4,2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

5 2 4 3

30 15 23 15

−25 −22 −15 −315 6 12 7

Comprobacion

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

5 2 4 3

30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

P1,3,4,2 A = P1,3,4,2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

5 2 4 3

30 15 23 15

−25 −22 −15 −315 6 12 7

Comprobacion

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

5 2 4 3

30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

P1,3,4,2 A = P1,3,4,2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

Comprobacion

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

5 2 4 3

30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

P1,3,4,2 A = P1,3,4,2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

Comprobacion

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

5 2 4 3

30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

P1,3,4,2 A = P1,3,4,2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

Comprobacion

1 0 0 06 1 0 0−5 −4 1 0

3 0 0 1

5 2 4 30 3 −1 30 0 1 00 0 0 −2

5 2 4 3

30 12 + 3 24− 1 18− 3−25 − 10− 12 − 20 + 4 + 1 − 15 + 20 + 015 6 + 0 12 + 0 + 0 9 + 0 + 0− 2

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

P1,3,4,2 A = P1,3,4,2

5 2 4 3

15 6 12 730 15 23 15−25 −22 −15 −3

5 2 4 3

30 15 23 15−25 −22 −15 −3

15 6 12 7

EjercicioConstruir una factorizacion PLU de la siguiente matriz y hacer la comprobacion.

0 −2 1 −23 1 5 66 0 11 66 8 1 20

¡Ejercer para aprender!

Una respuesta posible:

0 1 0 01 0 0 00 0 0 10 0 1 0

1 0 0 00 1 0 02 −3 1 02 1 0 1

3 1 5 60 −2 1 −20 0 −6 20 0 0 −4

0 −2 1 −23 1 5 66 0 11 66 8 1 20

0 1 0 01 0 0 00 0 0 10 0 1 0

1 0 0 00 1 0 02 −3 1 02 1 0 1

3 1 5 60 −2 1 −20 0 −6 20 0 0 −4

0 −2 1 −23 1 5 66 0 11 66 8 1 20

0 1 0 01 0 0 00 0 0 10 0 1 0

1 0 0 00 1 0 02 −3 1 02 1 0 1

3 1 5 60 −2 1 −20 0 −6 20 0 0 −4

Ejemplo de factorización PLU - Egor...

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