QR Decomposition w/ Householder and Givens Transformations

In summary, QR decomposition is a method of factoring a matrix into an orthogonal matrix Q and an upper triangular matrix R. This can be achieved through the use of Householder and Givens transformations, which are mathematical operations that transform a matrix into a simpler form while preserving its essential properties. QR decomposition is useful in various applications, such as solving linear systems of equations, finding eigenvalues and eigenvectors, and performing least squares regression. It is a powerful tool in linear algebra and numerical analysis, allowing for efficient computation and analysis of complex matrices.
  • #1
Th3HoopMan
8
0
Could anybody link me to some good examples on how to go about doing them? I honestly have no idea how to go about doing these two types of problems.
 
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  • #2
Th3HoopMan said:
Could anybody link me to some good examples on how to go about doing them? I honestly have no idea how to go about doing these two types of problems.
It's not clear what you are looking for here.

Do you want to know how to develop QR decomposition using HH & Givens Transforms?
Or
Are you looking for examples of problems which can be solved using QR decomposition?
 
  • #3
SteamKing said:
It's not clear what you are looking for here.

Do you want to know how to develop QR decomposition using HH & Givens Transforms?
Or
Are you looking for examples of problems which can be solved using QR decomposition?
Examples of problems which can be solving using QR
 
  • #4
Th3HoopMan said:
Examples of problems which can be solving using QR
Just about any regression problem where the number of data points exceeds the degree of the curve being fitted.

You use QR to find the minimum of the residuals in place of forming the normal equations.

Here is an example using linear least squares:

http://www.uta.edu/faculty/rcli/Teaching/math5392/NotesByHyvonen/lecture3.pdf

Note: actual problem starts on p. 11, but there is a good intro. in pp. 1-10. :smile:
 
  • #5
SteamKing said:
Just about any regression problem where the number of data points exceeds the degree of the curve being fitted.

You use QR to find the minimum of the residuals in place of forming the normal equations.

Here is an example using linear least squares:

http://www.uta.edu/faculty/rcli/Teaching/math5392/NotesByHyvonen/lecture3.pdf

Note: actual problem starts on p. 11, but there is a good intro. in pp. 1-10. :smile:
Thank you!
 

Related to QR Decomposition w/ Householder and Givens Transformations

1. What is QR Decomposition?

QR Decomposition is a mathematical technique used to decompose a matrix into an orthogonal matrix (Q) and an upper triangular matrix (R). It is commonly used in linear algebra and is particularly useful for solving systems of linear equations and finding eigenvalues and eigenvectors.

2. What are Householder Transformations?

Householder Transformations are a type of matrix transformation used in the QR Decomposition process. They involve reflecting a vector or matrix across a hyperplane, resulting in a new vector or matrix that is orthogonal to the original one.

3. What are Givens Transformations?

Givens Transformations are another type of matrix transformation used in QR Decomposition. They involve rotating a matrix or vector in a plane to eliminate certain elements, resulting in a new matrix or vector that is orthogonal to the original one.

4. Why are Householder and Givens Transformations used in QR Decomposition?

Householder and Givens Transformations are used in QR Decomposition because they are efficient and stable methods for decomposing a matrix into orthogonal and upper triangular matrices. These transformations also preserve the original matrix's structure, making it easier to solve linear equations and perform other operations.

5. What are the applications of QR Decomposition w/ Householder and Givens Transformations?

QR Decomposition w/ Householder and Givens Transformations has many practical applications in fields such as engineering, physics, and finance. It is commonly used for data analysis, signal processing, image recognition, and optimization problems. It is also used in computer graphics and simulations to solve systems of linear equations efficiently.

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