Question

solve the system 2x + y +z=2;
x+3y + 2z= 2; 3x+ y +2z=2 by LU decomposition method.​

Answer 1

This is going to be incredibly long. We write the system as a matrix equation[1][2]. We will have

⎧⎩⎨2x+3y+z=9x+2y+3z=83x+y+2z=8

Ok, so this becomes

⎛⎝⎜213321132⎞⎠⎟⎛⎝⎜xyz⎞⎠⎟=⎛⎝⎜988⎞⎠⎟

Now we find the LU decomposition for the matrix. I’ll let

A=⎛⎝⎜213321132⎞⎠⎟

and we want to find  A=LU . Now there are multiple ways to do this but this will be longer because … I can. The LU matrices look like

⎛⎝⎜1L21L3101L32001⎞⎠⎟⎛⎝⎜U1100U12U220U13U23U33⎞⎠⎟=⎛⎝⎜U11L21U11L31U11U12L21U12+U22L31U12+L32U22U13L21U13+U23L31U13+L32U23+U33⎞⎠⎟

Now set that equal to the  A  matrix

⎛⎝⎜U11L21U11L31U11U12L21U12+U22L31U12+L32U22U13L21U13+U23L31U13+L32U23+U33⎞⎠⎟=⎛⎝⎜213321132⎞⎠⎟

This means that  U11=2  and  U12=3  and  U13=1 . Now work with the next row.

L21U11=1⟹L21×2=1⟹L21=12

Now the next one

L21U12+U22=2⟹12×3+U22=2⟹U22=12tag∗

Now the final part of the second row

L21U13+U23=3⟹12×1+U23=3⟹U23=52

Then the final row is

L31U11=3⟹L31×2=3⟹L31=32

L31U12+L32U22=32×3+L32×12=1⟹L32=−7

L31U13+L32U23+U33=32×1+−7×52+U33=2⟹U33=18

Then we get

⎛⎝⎜⎜1123201−7001⎞⎠⎟⎟⎛⎝⎜200312015218⎞⎠⎟=⎛⎝⎜213321132⎞⎠⎟

Now to solve this we need to do forward substitution and backward substitution. First solve  Ly=b  for  y . So it becomes

⎛⎝⎜⎜1123201−7001⎞⎠⎟⎟⎛⎝⎜y1y2y3⎞⎠⎟=⎛⎝⎜988⎞⎠⎟

Therefore we get that  y1=9  then  12y1+y2=8⟹y2=72  and  32y1−7y2+y3=8⟹y3=19

Now we want to solve

⎛⎝⎜200312015218⎞⎠⎟⎛⎝⎜x1x2x3⎞⎠⎟=⎛⎝⎜97219⎞⎠⎟

If you solve that you should get

x=⎛⎝⎜⎜251831181918⎞⎠⎟⎟

Now if you multiply  Ax  is it equal to  b ?

⎛⎝⎜213321132⎞⎠⎟⎛⎝⎜⎜251831181918⎞⎠⎟⎟=????