scholarly journals Amesos2 and Belos: Direct and Iterative Solvers for Large Sparse Linear Systems

2012 ◽  
Vol 20 (3) ◽  
pp. 241-255 ◽  
Author(s):  
Eric Bavier ◽  
Mark Hoemmen ◽  
Sivasankaran Rajamanickam ◽  
Heidi Thornquist
Keyword(s):  
Linear Systems ◽  
Iterative Methods ◽  
Sparse Matrix ◽  
Sparse Matrices ◽  
Direct Methods ◽  
Iterative Solvers ◽  
Software Project ◽  
Sparse Linear Systems ◽  
Scalar Type ◽  
Mixed Precision

Solvers for large sparse linear systems come in two categories: direct and iterative. Amesos2, a package in the Trilinos software project, provides direct methods, and Belos, another Trilinos package, provides iterative methods. Amesos2 offers a common interface to many different sparse matrix factorization codes, and can handle any implementation of sparse matrices and vectors, via an easy-to-extend C++ traits interface. It can also factor matrices whose entries have arbitrary “Scalar” type, enabling extended-precision and mixed-precision algorithms. Belos includes many different iterative methods for solving large sparse linear systems and least-squares problems. Unlike competing iterative solver libraries, Belos completely decouples the algorithms from the implementations of the underlying linear algebra objects. This lets Belos exploit the latest hardware without changes to the code. Belos favors algorithms that solve higher-level problems, such as multiple simultaneous linear systems and sequences of related linear systems, faster than standard algorithms. The package also supports extended-precision and mixed-precision algorithms. Together, Amesos2 and Belos form a complete suite of sparse linear solvers.

scholarly journals A Class of Schur Multipliers on Some Quasi-Banach Spaces of Infinite Matrices

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Nicolae Popa
Keyword(s):  
Banach Spaces ◽  
Infinite Matrices ◽  
Schur Multipliers ◽  
Scalar Type

We characterize the Schur multipliers of scalar type acting on scattered classes of infinite matrices.

scholarly journals Lipschitz continuity of spectral measures

1999 ◽  
Vol 59 (3) ◽  
pp. 369-373
Author(s):  
Werner J. Ricker
Keyword(s):  
Banach Space ◽  
Lipschitz Condition ◽  
Lipschitz Continuity ◽  
Analogous Result ◽  
Spectral Measures ◽  
Scalar Type ◽  
Algebra Of Sets

A characterisation is given of all (finitely additive) spectral measures in a Banach space (and defined on an algebra of sets) which satisfy a Lipschitz condition. This also corrects (slightly) an analogous result in the more specialised setting of resolutions of the identity of scalar-type spectral operators (due to C.A. McCarthy).

scholarly journals On power-bounded prespectral operators

1976 ◽  
Vol 20 (2) ◽  
pp. 173-175
Author(s):  
H. R. Dowson
Keyword(s):  
Growth Condition ◽  
Bounded Operator ◽  
Spectral Operator ◽  
Scalar Type ◽  
Image Position ◽  
Power Bounded Operator ◽  
Power Bounded

Foguel (8) and Fixman (7) independently proved that an invertible spectral operator, which is power-bounded, is of scalar type. Their proofs rely heavily on a result of Dunford on spectral operators whose resolvents satisfy a growth condition. (See Lemma 3.16 of (6, p. 609).) Observe that the resolvent of an invertible power-bounded operator T satisfies an inequality of the form

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