Introduction to Numerical Analysis I MATH/CMPSC 455 Conjugate Gradient Methods.
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Transcript of Introduction to Numerical Analysis I MATH/CMPSC 455 Conjugate Gradient Methods.
![Page 1: Introduction to Numerical Analysis I MATH/CMPSC 455 Conjugate Gradient Methods.](https://reader036.fdocuments.us/reader036/viewer/2022071710/56649dc45503460f94ab7ba1/html5/thumbnails/1.jpg)
Introduction to Numerical Analysis I
MATH/CMPSC 455
Conjugate Gradient Methods
![Page 2: Introduction to Numerical Analysis I MATH/CMPSC 455 Conjugate Gradient Methods.](https://reader036.fdocuments.us/reader036/viewer/2022071710/56649dc45503460f94ab7ba1/html5/thumbnails/2.jpg)
A-ORTHOGONAL BASIS
form a basis of ,
where
is the i-th row of the identity matrix. They
are orthogonal in the following sense:
They are linearly independent, and form a basis.
Introduce a set of nonzero vectors ,
They satisfy the following condition:
We say they are A-orthogonal, or conjugate w.r.t A.
![Page 3: Introduction to Numerical Analysis I MATH/CMPSC 455 Conjugate Gradient Methods.](https://reader036.fdocuments.us/reader036/viewer/2022071710/56649dc45503460f94ab7ba1/html5/thumbnails/3.jpg)
CONJUGATE DIRECTION METHOD
Theorem: For any initial guess, the sequence
generated by the above iterative method,
converges to the solution of the linear system
in at most n iterations. Question: How to find the A-orthogonal bases?
![Page 4: Introduction to Numerical Analysis I MATH/CMPSC 455 Conjugate Gradient Methods.](https://reader036.fdocuments.us/reader036/viewer/2022071710/56649dc45503460f94ab7ba1/html5/thumbnails/4.jpg)
CONJUGATE GRADIENT METHOD
Answer:
Each conjugate direction is chosen to be a linear
combination of the residual and the previous
direction
Conjugate Gradient Method: Conjugate direction method on this particular basis.
![Page 5: Introduction to Numerical Analysis I MATH/CMPSC 455 Conjugate Gradient Methods.](https://reader036.fdocuments.us/reader036/viewer/2022071710/56649dc45503460f94ab7ba1/html5/thumbnails/5.jpg)
CG (ORIGINAL VERSION)
While
End While
![Page 6: Introduction to Numerical Analysis I MATH/CMPSC 455 Conjugate Gradient Methods.](https://reader036.fdocuments.us/reader036/viewer/2022071710/56649dc45503460f94ab7ba1/html5/thumbnails/6.jpg)
Theorem: Let A be a symmetric positive-
definite matrix. In the Conjugate Gradient
Method, we have
![Page 7: Introduction to Numerical Analysis I MATH/CMPSC 455 Conjugate Gradient Methods.](https://reader036.fdocuments.us/reader036/viewer/2022071710/56649dc45503460f94ab7ba1/html5/thumbnails/7.jpg)
CG (PRACTICAL VERSION)
While
End While
![Page 8: Introduction to Numerical Analysis I MATH/CMPSC 455 Conjugate Gradient Methods.](https://reader036.fdocuments.us/reader036/viewer/2022071710/56649dc45503460f94ab7ba1/html5/thumbnails/8.jpg)
Example: