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Introduction to Numerical Analysis IPowerPoint Presentation

Introduction to Numerical Analysis I

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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.

Theorem: Let A be a symmetric positive-definite matrix. In the Conjugate Gradient Method, we have

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