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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:.
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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: 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.
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
