LU 分解

ピボット選択つきの LU 分解が利用できる.

octave:1> a=[1 2; 3 4] 
a =

  1  2
  3  4

octave:2> [l u p] = lu(a)
l =

  1.00000  0.00000
  0.33333  1.00000
octave:1> a=[1 2; 3 4] 
a =

  1  2
  3  4

octave:2> [l u p] = lu(a)
l =

  1.00000  0.00000
  0.33333  1.00000

u =

  3.00000  4.00000
  0.00000  0.66667

p =

  0  1
  1  0

octave:3> p*a
ans =

  3  4
  1  2

octave:4> l*u
ans =

  3  4
  1  2

PA = LU より PAx = LUx = Pf. よって, x = U^-1L^-1Pf.

octave:9> a=[1, -1; 2, 1]
a =

   1  -1
   2   1

octave:10> b=[0; 3]
b =

  0
  3

octave:13> u\(l\(p * b))
ans =

  1
  1