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ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones

ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones Professor George H. Born Lecture 28: Givens Transformations. Announcements. Exam 2 – Friday, November 8 E-mail Marco with questions for Wednesday’s lecture Open book, open notes

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ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones

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  1. ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones Professor George H. Born Lecture 28: Givens Transformations

  2. Announcements • Exam 2 – Friday, November 8 • E-mail Marco with questions for Wednesday’s lecture • Open book, open notes • Bring extra paper if you think you will need it • Be sure to bring a calculator!

  3. Least Squares via Givens Transformations

  4. Derivation of LS Solution via Orthogonal Transformation

  5. Derivation of LS Solution via Orthogonal Transformation

  6. Givens Rotation

  7. Givens Rotation

  8. Givens Transformation for n>1 • Consider the desired result • To achieve this, we select the Givens matrix such that • We then use this transformation in top equation

  9. Givens Transformation • After applying the transformation, we get: • Repeat for all remaining non-zero elements in the third column

  10. Application of Given Transformations • Need to find the orthogonal matrix Q to yield a matrix of the form of the RHS • Q is generated using a series of Givens transformations G

  11. Application of Givens Transformations We select G to get a zero for the term in red: To achieve this, we use:

  12. Application of Givens Transformations We select G to get a zero for the term in red: To achieve this, we use:

  13. Application of Givens Transformations We select G to get a zero for the term in red: To achieve this, we use:

  14. Application of Givens Transformations We select G to get a zero for the term in red: To achieve this, we use:

  15. Application of Givens Transformations We select G to get a zero for the term in red: To achieve this, we use:

  16. Application of Givens Transformations We select G to get a zero for the term in red: To achieve this, we use:

  17. Application of Givens Transformations We select G to get a zero for the term in red: To achieve this, we use:

  18. Application of Givens Transformations • We now have the required Q matrix (for this conceptual example):

  19. Givens Transformation Algorithm (Outline)

  20. Batch vs. Givens – An New Example

  21. Problem Statement • Consider the case where:

  22. Givens Example

  23. Givens Example

  24. Givens Example

  25. Givens Result • We then have the matrices needed to solve the system:

  26. Concept Quiz

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