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TWO Directions

This document explores a novel approach to developing a 3-term conjugate gradient (CG) method tailored for nonsymmetric real matrices, specifically focusing on SPD (symmetric positive definite) normal matrices. It discusses the properties of normal matrices and their implications for convergence in iterative methods. The inquiry extends to the conditions under which iterations can be executed through a 3-term recursion. The findings utilize insights from established works, including Faber and M, and address practical considerations for hermitian matrices as well.

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TWO Directions

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  1. TWO Directions Gene Golub: for the construction of a 3-term conjugate gradient like (CG) method for nonsymmetric real matrices SPD

  2. Normal(1) Matrices • Remark: • A symmetric?? • A is normal matrix ???

  3. Normal(1) Matrices

  4. A normal + A normal + If eigs not line wathen CG(3)=matrices for which iterations can be carried out using 3-term recursion Any idea??? Faber&M def F&M slides

  5. A is the translation and rotation of a -self adjoint matrix A normal if A is B-self adjoint + Use A normal + Use If eigs not line Faber & M wathen for HSPD wathen wathen Faber&M def F&M slides F&M slides BA Hermition CG(3)=matrices for which iterations can be carried out using 3-term recursion

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