Lecture 19

Eugene E. Tyrtyshnikov
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Eugene E. Tyrtyshnikov: Russian Academy of Sciences, Institute of Numerical Mathematics

A chapter in A Brief Introduction to Numerical Analysis, 1997, pp 165-176 from Springer

Abstract: Abstract If $$ f(x) = \tfrac{1} {2}(Ax,x) - \operatorname{Re} (b,x), A = A^* \in \mathbb{C}^{n \times n} $$ , then the boundedness of f from below is equivalent to the nonnegative definiteness of A (prove this). Let us assume that A > 0. In this case, a linear system Ax = b has a unique solution z, and, for any x, $$ f(x) - f(z) = \frac{1} {2}(A(x - z),x - z) \equiv E(x). \Rightarrow $$ z is the single minimum point for f (x). ⇒ A minimization method for f can equally serve as a method of solving a linear system with the Hermitian positively definite coefficient matrix.

Keywords: Conjugate Gradient Method; Krylov Subspace; Arnoldi Method; Minimal Residual; Hessenberg Matrix (search for similar items in EconPapers)
Date: 1997
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DOI: 10.1007/978-0-8176-8136-4_19

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