Introduction to Numerical Analysis I. Conjugate Gradient Methods. MATH/CMPSC 455. A-Orthogonal Basis. form a basis of , where is the i-th row of the identity matrix. They are orthogonal in the following sense:.
Conjugate Gradient Methods
form a basis of , where
is the i-th row of the identity matrix. They are orthogonal in the following sense:
Introduce a set of nonzero vectors ,
They satisfy the following condition:
We say they are A-orthogonal, or conjugate w.r.t A.
They are linearly independent, and form a basis.
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?
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.
Theorem: Let A be a symmetric positive-definite matrix. In the Conjugate Gradient Method, we have