Checking Correctness of LU Decomposition

[twoski]

Homework Statement

compute the LU decomposition of the 3x3 matrix:

A= 2, 1, 1/2
1/2, 2, 1
1, 1/2, 2

Let f be the vector (5,3,6)

Solve Lg=f and then Ux=g. You can check your answer with A_{3}x=f

The Attempt at a Solution

I finished the calculations:

U = 1, 1/2, 1/4
0, 1, 1/2
0, 0, 1

L = 1, 0, 0
-1/4, 1, 0
-1/2, 0, 1

g = (5, 17/4, 17/2)
f = (23/8, 0, 17/2)

I want to check my answer. What does the subscript 3 signify?

 

[vela]

Probably a typo.

 

[twoski]

Hm, okay, so if that's a typo then checking is as easy as doing Ax=f?

When i check, i end up with (10, 159/16, 159/8) which is certainly wrong.

I am trying to follow some instructions on LU decomposition because our professor's notes are really lacking, and it says:

To get L, start with the idenity matrix and use the following rules. Any row operations that involves adding a multiple of one row to another, for example, Ri + kRj, put the value –k in the ith-row, jth-column of the identity matrix.


Any row operations that involves getting a leading one on the main diagonal, for example, kRi, put the value 1/k in the position of the identity matrix where the leading one occurs.

So, if i did 1/2R1 when working on my U matrix, would i multiply the leading 1 in R1 by 2 when getting my L matrix? I keep trying to recalculate the g vector and every time i end up with very specific fractions like 19/14 which seems wrong to me. The vector f has no fractions in it so ideally my x vector shouldn't contain any weird fractions.

 

[SteamKing]

First and foremost, A = LU. I don't think you have the correct decomposition, certainly not by trying to calculate a11.

 

[twoski]

I followed these directions exactly and I don't even end up with proper L and U matrices. How frustrating.

If A =
2, 1, 1/2
1/2, 2, 1
1, 1/2, 2

Then all I need to do is:

R2-1/4R1
R3-1/2R1
1/2R1
4/7R2
4/7R3

Now according to the notes, i have to use these as a reference for computing L.

A2,1 = 1/4
A3,1 = 1/2
A1,1 = 2
A2,2 = 7/4
A3,3 = 7/4

Am i doing anything wrong?

 

[vela]

The instructions you're following are flawed or incomplete.

Each row operation can be represented by a matrix. For example, R2 = R2-1/4 R1 can be represented by the lower-triangular matrix
$$O_1 = \begin{pmatrix} 1 & 0 & 0 \\ -\frac{1}{4} & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}.$$ If you multiply A on the left by O₁, you'll see you get the result of performing the row operation. So you did a bunch of row operations to get the upper triangular matrix. In terms of the matrices, you have
$$U = O_n O_{n-1} \dots O_2 O_1 A.$$ Each of the row operations is invertible, so its corresponding matrix is invertible. That means you can multiply by the inverses to get
$$O_1^{-1} O_2^{-1}\dots O_{n-1}^{-1}O_n^{-1} U = A.$$ The product of the inverses gives you L.

The instructions you were given tell you how to write down the inverses for each row operation. For instance, if you invert O₁, you'd get
$$O_1^{-1} = \begin{pmatrix} 1 & 0 & 0 \\ \frac{1}{4} & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}.$$ So rather than try to write down L in one fell swoop as you've tried to do, you want to write down the five inverse matrices separately, and then multiply them together in the correct order to get L.