Example: Introduction to Krylov Subspace Methods DEF: Krylov sequence

Example: Introduction to Krylov Subspace Methods DEF: Krylov subspace Krylov subspace DEF: Krylov matrix

Example: Introduction to Krylov Subspace Methods Remark: DEF: Krylov matrix

Conjugate Gradient Method We want to solve the following linear system

0K=1K=2K=3K=4 x1 x2 x3 X4 Conjugate Gradient Method Example: Solve: Conjugate Gradient Method

constants vectors Conjugate Gradient Method

We want to solve the following linear system Define: quadratic function Example:

Conjugate Gradient Method Example: Remark: Why not max?

Conjugate Gradient Method Remark: IDEA: Search for the minimum Problem (1)

Conjugate Gradient Method Example: minimum

Conjugate Gradient Method Method: “search direction” “step length” Method:

Conjugate Gradient Method Method:

Conjugate Gradient Method Method: Conjugate Gradient Method

INNER PRODUCT

Inner Product DEF: We say that Is an inner product if Example:

Inner Product DEF: We say that Is an inner product if Example: where H is SPD We define the norm

Inner Product DEF: We say that Is symmetric bilinear form if Example: where H is Symmetric

Inner Product DEF: Example: DEF: where H is SPD

Conjugate Gradient

Conjugate Gradient Method Method: Conjugate Gradient Method

Method:

Conjugate Gradient Method Method: Conjugate Gradient Method

Lemma :[Elman,Silvester,Wathen Book]

Conjugate Gradient Method