Iterative Solution of Large Sparse Systems of Equations by Wolfgang Hackbusch

By Wolfgang Hackbusch

Re-creation offers emphasis at the algebraic constitution of linear generation, now not frequently incorporated in such a lot literature
Completely renewed references
Content grew out of a sequence of lectures given via author
Extensive and priceless appendices included

In the second one version of this vintage monograph, entire with 4 new chapters and up-to-date references, readers will now have entry to content material describing and analysing classical and smooth tools with emphasis at the algebraic constitution of linear new release, that is often overlooked in different literature.

The useful quantity of labor raises dramatically with the dimensions of structures, so one has to go looking for algorithms that the majority successfully and appropriately clear up structures of, e.g., numerous million equations. the alternative of algorithms is determined by the designated homes the matrices in perform have. a massive type of huge structures arises from the discretization of partial differential equations. subsequently, the matrices are sparse (i.e., they comprise typically zeroes) and well-suited to iterative algorithms.

The first variation of this publication grew out of a chain of lectures given by means of the writer on the Christian-Albrecht college of Kiel to scholars of arithmetic. the second one version contains really novel approaches.

Topics
Numerical Analysis
Linear and Multilinear Algebras, Matrix Theory
Partial Differential Equations

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Additional resources for Iterative Solution of Large Sparse Systems of Equations

Sample text

2. More interesting than a single value ρm+1,m is the geometric mean ρm+k,m := [ρm+k,m+k−1 · ρm+k−1,m+k−2 · . . 23a) can more easily be represented by ρm+k,m := em+k / em 1/k . 23b) The properties of ρm+k,m are summarised below. 22. (a) Denote the dependence of the magnitude ρm+k,m on the starting value x0 by ρm+k,m (x0 ). Then lim max{ρm+k,m (x0 ) : x0 ∈ KI } = ρ(M ) k→∞ for all m. 23c) holds, provided that x0 does not lie in the subspace U ⊂ KI of dimension < #I spanned by all eigenvectors and possibly existing principal vectors of the matrix M corresponding to eigenvalues λ with |λ| < ρ(M ).

As a characteristic quantity we choose the effective amount of work Eff(Φ) := It(Φ)CΦ = −CΦ / log(ρ(M )). 31a) Eff(Φ) measures the amount of work for an error reduction by 1/e in the unit ‘CA n arithmetic operations’. Correspondingly, the effective amount of work for the error reduction by the factor of 1/e is given by Eff(Φ, ε) := −It(Φ)CΦ log(ε) = CΦ log(ε)/ log(ρ(M )). 28. In the case of the model problem, the cost factor of the Gauss–Seidel iteration is CΦ = 1 (because of CA = 5, cf. 14). 99039 for the grid size h = 1/32.

3) exclusively for the matrix A. 1. An iterative method is a (in general nonlinear) mapping Φ : KI × KI × KI×I → KI . 3) with a starting value x0 = y ∈ KI : x0 (y, b, A) := y , xm+1 (y, b, A) := Φ(xm (y, b, A), b, A) for m ≥ 0. 4) If A is fixed, we write xm (y, b) instead of xm (y, b, A). If all parameters y, b, A are fixed, we write xm . If Φ is called an iteration method, we expect that the method is applicable to a whole class of matrices A. Here ‘applicable’ means that Φ is well defined (including the case that the sequence xm diverges).

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