We use cookies to improve your experience on our site and to show you relevant advertising. By browsing this website, you agree to our use of cookies. Learn more   

AtoZmath.com
Support us
(New) All problem can be solved using search box
I want to sell my website www.AtoZmath.com with complete code
Home What's new College Algebra Games Feedback About us
Algebra Matrix & Vector Numerical Methods Statistical Methods Operation Research Word Problems Calculus Geometry Pre-Algebra


Home > Matrix & Vector calculators > Solving systems of linear equations using LU decomposition using Crout's method example
10. LU decomposition using Crout's method example ( Enter your problem )
( Enter your problem )
  1. Example `2x+5y=21,x+2y=8`
  2. Example `2x+5y=16,3x+y=11`
  3. Example `2x+3y-z=5,3x+2y+z=10,x-5y+3z=0`
  4. Example `x+y+z=3,2x-y-z=3,x-y+z=9`
 		Other related methods
  1. Inverse Matrix method
  2. Cramer's Rule method
  3. Gauss-Jordan Elimination method
  4. Gauss Elimination Back Substitution method
  5. Gauss Seidel method
  6. Gauss Jacobi method
  7. Elimination method
  8. LU decomposition using Gauss Elimination method
  9. LU decomposition using Doolittle's method
  10. LU decomposition using Crout's method
  11. Cholesky decomposition method
  12. SOR (Successive over-relaxation) method
  13. Relaxation method
1. Example `2x+5y=21,x+2y=8`
(Previous example)		3. Example `2x+3y-z=5,3x+2y+z=10,x-5y+3z=0`
(Next example)

2. Example `2x+5y=16,3x+y=11`


Solve Equations 2x+5y=16,3x+y=11 using Crout's method

Solution:
Total Equations are `2`

`2x+5y=16 -> (1)`

`3x+y=11 -> (2)`

Now converting given equations into matrix form
`[[2,5],[3,1]] [[x],[y]]=[[16],[11]]`

Now, A = `[[2,5],[3,1]]`, X = `[[x],[y]]` and B = `[[16],[11]]`

Crout's method for LU decomposition
Let `A=LU`

`2``5`
`3``1`
 = 
`l_(11)``0`
`l_(21)``l_(22)`
 `xx` 
`1``u_(12)`
`0``1`


`2``5`
`3``1`
 = 
`l_(11)``l_(11)u_(12)`
`l_(21)``l_(21)u_(12) + l_(22)`


This implies
`l_(11)=2`

`l_(11)u_(12)=5=>2xxu_(12)=5=>u_(12)=5/2`



`l_(21)=3`

`l_(21)u_(12) + l_(22)=1=>3xx5/2 + l_(22)=1=>l_(22)=-13/2`



`:.A=L xx U=LU`

`2``5`
`3``1`
 = 
`2``0`
`3``-13/2`
 `xx` 
`1``5/2`
`0``1`
 = 
`2``5`
`3``1`



Now, `Ax=B`, and `A=LU => LUx=B`

let `Ux=y`, then `Ly=B =>`

`2``0`
`3``-13/2`
 `xx` 
`y_1`
`y_2`
 = 
`16`
`11`


```2y_1``=``16```
```3y_1``-``13/2y_2``=``11```


Now use forward substitution method
From (1)
`2y_1=16`

`=>2y_1=16`

`=>y_1=(16)/(2)=8`

From (2)
`3y_1-13/2y_2=11`

`=>3(8)-(13y_2)/(2)=11`

`=>24-(13y_2)/(2)=11`

`=>-(13y_2)/(2)=11-24`

`=>-(13y_2)/(2)=-13`

`=>y_2=-13xx-2/13=2`

Now, `Ux=y`

`1``5/2`
`0``1`
 `xx` 
`x`
`y`
 = 
`8`
`2`


```x``+``5/2y``=``8```
```y``=``2```


Now use back substitution method
From (2)
`y=2`

From (1)
`x+5/2y=8`

`=>x+(5(2))/(2)=8`

`=>x+5=8`

`=>x=8-5`

`=>x=3`

Solution by Crout's method is
`x=3 and y=2`

This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then Submit Here


1. Example `2x+5y=21,x+2y=8`
(Previous example)		3. Example `2x+3y-z=5,3x+2y+z=10,x-5y+3z=0`
(Next example)


Share this solution or page with your friends.


Home What's new College Algebra Games Feedback About us
 
Copyright © 2024. All rights reserved. Terms, Privacy
 
 

.