What Are the Key Components of Gram Schmidt Algorithms for QTQ-I in MATLAB?

[Jamin2112]

Homework Statement

So my assignment is to compare a variety of Graham Schmidt algorithms by looking at ||Q^TQ-I||₁

Homework Equations

So my teacher has the algorithms for the Gram Schmidt and "Modified Gram Schmidt" written as

The Attempt at a Solution

So I'm confused about the q_i which appears in the Classical one. Where did it come from? What does it mean?

Also, what is the r_ij? Just a constant to use in the next line?

Same with the stuff in the Modified one.

I just went ahead and typed into MATLAB even though it ain't makin' sense.

 

[I like Serena]

Hi Jamin2112!

So I'm confused about the q_i which appears in the Classical one. Where did it come from? What does it mean?

q_i is introduced in this algorithm.
The point of the algorithm is to find a set of q_i that form an orthonormal basis.
That is, q_i is the "output" of the algorithm.

Also, what is the r_ij? Just a constant to use in the next line?

Yes, it is just a constant to use in the next line.

Note that it holds the result for the inner product of q_i and a_j, which is ultimately needed to calculate q_j.

 

[Jamin2112]

Hi Jamin2112!



q_i is introduced in this algorithm.
The point of the algorithm is to find a set of q_i that form an orthonormal basis.
That is, q_i is the "output" of the algorithm.

But if you look in the Classical Gram Schmidt algorithm, q is used on the right side of an equality before it has been defined.

 

[I like Serena]

But if you look in the Classical Gram Schmidt algorithm, q is used on the right side of an equality before it has been defined.

I see why you might think that, but no it isn't.
Check the range of the inner loop.

Any q_j is only used after it has been calculated.

In the first iteration of the outer loop (j=1), the inner loop is not executed (since i runs from 1 to j-1).
In every next iteration of the outer loop, only the q_j that have already been calculated are used.

 

[Jamin2112]

I see why you might think that, but no it isn't.
Check the range of the inner loop.

Any q_j is only used after it has been calculated.

In the first iteration of the outer loop (j=1), the inner loop is not executed (since i runs from 1 to j-1).
In every next iteration of the outer loop, only the q_j that have already been calculated are used.

hmmmmm ...

Well, the program seems to running without any errors, so I'll take your word for it!

 

[I like Serena]

hmmmmm ...

Well, the program seems to running without any errors, so I'll take your word for it!

All right. I'll leave it at that then.

Let me know if you have more questions.

 

[Jamin2112]

All right. I'll leave it at that then.

Let me know if you have more questions.

Turned it in and got 10/10 [PLAIN]http://www.fastfatburning.org/wp-content/plugins/RSSPoster_PRO/cache/c2584_success_baby.jpg

But I do have another question. Refer yourself to my QR Decomposition thread.