LU Factorization - Solve with Carter Barker

In summary, the conversation discusses the steps involved in LU-Factorization, including switching rows, multiplying by constants, and adding rows together. The final question is whether a mistake was made in the process. However, the starting matrix is not mentioned, making it difficult to determine the accuracy of the steps.
  • #1
cbarker1
Gold Member
MHB
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Dear Everyone,
I have some trouble with LU-Factorization. The problem work is below:
Original Matrix

First row- 3,0,1
2nd row- 6,1,1
3rd row- -3,1,0

Elementary Matrix 1
1st- 1,0,0
2nd- 0,1,0
3rd- 0,0,1
Switch R1 to R3

1st- 0,0,1
2nd- 0,1,0
3rd- 1,0,0​

The row operation:
Interchange R1 and R3

1st- -3,1,0
2nd- 6,1,1
3rd- 3,0,1

Elementary 2
1st- 1,0,0
2nd- 0,1,0
3rd- 0,0,1

Multiply -1/3R1

1st- -1/3,0,0
2nd- 0,1,0
3rd- 0,0,1​

The Row Operation
Multiplying -1/3 R1

1st- 1, -1/3, 0
2nd- 6, 1, 1
3rd- 3,0,1

Elementary 3

1st- 1,0,0
2nd- 0,1,0
3rd- 0,0,1

-6R1+R2

1st- 1,0,0
2nd- -6,1,0
3rd- 0,0,1​

The Row Operation
-6R1+R2

1st- 1,-1/3,0
2nd- 0,-1,1
3rd- 3,0,1

Elementary 4
1st-1,0,0
2nd- 0,1,0
3rd- 0,0,1

-3R1+R3
1st-1,0,0
2nd-0,1,0
3rd- -3,0,1

​

Row Operation
-3R1+R3

1st- 1,-1/3,0
2nd-0, -1, 1
3rd- 0, 1, 1

Elementary 5
1st- 1,0,0
2nd- 0,1,0
3rd- 0, 0,1

R2+R3

1st- 1,0,0
2nd- 0,1,0
3rd- 0,1,1

​
Row Operation:
R2+R3

1st- 1,-1/3,0
2nd- 0,-1,1
3rd- 0,0,2

Did I make a mistake somewhere? Thank you,

Carter Barker
 
Last edited:
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  • #2
Cbarker1 said:
Dear Everyone,

I have some trouble with LU-Factorization. The problem work is below
Elementary Matrix

In first row- 3,0,1
1st- 1,0,0 1st- 0,0,1
2nd row- 6,1,1 2nd- 0,1,0 2nd- 0,1,0 Switch R1 to R3
3rd row- -3,1,0 3rd- 1,0,0
Elementary Matrix
1st- 1,0,0
2nd- 0,1,0
3rd- 0,0,1

1st- 0,0,1
2nd- 0,1,0 Switch R1 to R3

The row operation:
Interchange R1 and R3

1st- -3,1,0 1st- 1,0,0 1st- -1/3,0,0 Multiplying by -1/3 R1
2nd- 6,1,1 2nd- 0,1,0 2nd- 0,1,0
3rd- 3,0,1 3rd- 0,0,1 3rd- 0,0,1

The Row Operation
Multiplying -1/3 R1

1st- 1, -1/3, 0 1st- 1,0,0 1st- 1,0,0 -6R1+R2
2nd- 6, 1, 1 2nd- 0,1,0 2nd- -6,1,0
3rd- 3,0,1 3rd- 0,0,1 3rd- 0,0,1

The Row Operation
-6R1+R2

1st- 1,-1/3,0 1st 1,0,0 1st- 1,0,0 -3R1+R3
2nd- 0,-1,1 2nd 0,1,0 2nd- 0,1,0
3rd- 3,0,1 3rd 0,0,1 3rd- -3,0,1

Row Operation
-3R1+R3

1st- 1,-1/3,0
2nd-0, -1, 1
3rd- 0, -1, 1 Did I made a mistake somewhere? Thank you,

Carter Barker

From what you have written, I can't even tell what your starting matrix that you are trying to find the LU composition of is...
 

1. What is LU factorization and what is its purpose?

LU factorization is a numerical method used to solve systems of linear equations. Its purpose is to decompose a matrix into two triangular matrices, making it easier to solve the system of equations.

2. How does LU factorization work?

LU factorization works by decomposing a matrix into an upper triangular matrix and a lower triangular matrix. This is done by manipulating the original matrix using elementary row operations, such as row additions and row swaps.

3. Why is LU factorization important in scientific computing?

LU factorization is important in scientific computing because it allows for the efficient solution of large systems of linear equations. It also provides a more stable and accurate solution compared to other methods, such as Gaussian elimination.

4. What is the difference between LU factorization and Gaussian elimination?

LU factorization and Gaussian elimination are both methods used to solve systems of linear equations. The main difference is that LU factorization decomposes the original matrix, while Gaussian elimination directly manipulates the original matrix. This makes LU factorization more efficient and accurate compared to Gaussian elimination.

5. Can LU factorization be used to solve systems of non-linear equations?

No, LU factorization can only be used to solve systems of linear equations. For systems of non-linear equations, other numerical methods, such as Newton's method, must be used.

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