Introduction to Numerical Analysis I
MATH/CMPSC 455
Conjugate Gradient Methods
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.
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?
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.
CG (ORIGINAL VERSION)
While
End While
Theorem: Let A be a symmetric positive-
definite matrix. In the Conjugate Gradient
Method, we have
CG (PRACTICAL VERSION)
While
End While
Example:
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