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AcknowledgmentsThe authors and editors gratefully acknowledge the valuable contributions of the following individuals, organizations and programs. Organizations and ProgramsThese organizations and programs provided grants, graduate fellowships, other funding, and resources.
The information presented here does not necessarily reflect the position or the policy of the funding agencies and no official endorsement should be inferred. IndividualsThe authors and editors wish to thank the following individuals.
The authors and editors are grateful to the SIAM staff for their great job of production, and particularly wish to thank:
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Software RepositoryList by Chapter / Problem ClassClick on a chapter link to see a short description of that chapter's problem type and get a list of software for solving that problem.
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Software RepositoryChapter 10: Common IssuesThis chapter describes issues common to all algorithms in the book, such as sparse matrix representation and computation, both sequential and parallel. The efficiency of most iterative methods in the book is primarily determined by the performance of the matrix-vector product code and therefore on the storage format of the matrix. Each chapter section addresses a different issue.
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Software RepositoryChapter 11: Preconditioning TechniquesPreconditioning replaces the system Ax = lambda x with the system M^-1 Ax = M^-1 lambda x. A good choice of M makes the condition number of M^-1 A much smaller than the condition number of A, thus accelerating convergence. M is an approximation to A such that
Many iterative methods depend in part on preconditioning to improve performance and ensure fast convergence. Preconditioning eigenproblems can be difficult. The matrices being preconditioned are often nearly singular, interior eigenvalues are difficult to find, and the preconditioned eigenvalue method may need restarting. Unlike Krylov subspace methods, preconditioning methods generally find just one eigenvalue at a time. However, this can sometimes be alleviated by using a block approach. In spite of the difficulties, preconditioned is more competitive if not too many eigenvalues are desired and if computing the eigenvectors is required. Software
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Software RepositoryChapter 4: Hermitian Eigenvalue ProblemsIn this chapter, we record the available software for the standard Hermitian, or most often real symmetric, eigenvalue (HEP), A x = lambda x, where A = A*. It is the most commonly occuring algebraic eigenvalue problem, for which we have the most reliable theory and the most powerful algorithms and software available. Software
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Software RepositoryChapter 5: Generalized Hermitian Eigenvalue ProblemsA generalized Hermitian eigenvalue problem (GHEP) is given by, Ax= lambda B x, where A and B are Hermitian, A* = A and B^* = B. We call the pair {A,B} of matrices matrix pencil. In this chapter we make the additional assumption which A or B or alpha*A + beta*B for some scalars alpha and beta, is positive definite, in that case we talk about a Hermitian definite pencil. This assumption is true for a wide class of practically important cases, and the theory is very closely related to the standard Hermitian eigenproblem, as expounded in Chapter 4. If no positive definite combination exists, we could as well regard {A,B} as a general pencil and use the theory and algorithms described in Chapter 8. Software
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Software RepositoryChapter 6: Singular Value DecompositionIn this chapter we consider the singular value decomposition (SVD) of the m-by-n matrix A. We assume without loss of generality that m > = n; if m < n consider A^*. As described in section 2.4, this decomposition may be written A = U Sigma V^*, where U = [u_1,...,u_m] is an m-by-m unitary matrix, V = [v_1,...,v_n] is an n-by-n unitary matrix, and Sigma is an m-by-n diagonal matrix with entries Sigma_{ii} = sigma_i for i=1,..,n. The u_i are the left singular vectors, the v_i are the right singular vectors, and the sigma_i are the singular values. The singular values are nonnegative and sorted so that sigma_1 >= sigma_2 >= ... >= sigma_n >= 0. The number r of nonzero singular values is the rank of A. If A is real, U and V are real and orthogonal. Software
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Software RepositoryChapter 7: Non-Hermitian Eigenvalue ProblemsIn this chapter we record the software for the non-Hermitian eigenvalue problem (NHEP), A x = lambda x, where the square matrix A \= A^*. x \= 0 is called a right eigenvector. A vector y \= 0 satisfying y^* A = lambda y^* is called a left eigenvector of A. Software
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Software RepositoryChapter 8: Generalized Non-Hermitian Eigenvalue ProblemsThis chapter records the software for the numerical solution of the right generalized non-Hermitian eigenvalue problem (GNHEP), A x = lambda B x, where A and B are general n-by-n matrices. Occasionally, one may also seek the solution of the left generalized non-Hermitian eigenvalue problem, y^* A = lambda y^* B. Software
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Software RepositoryChapter 9: Nonlinear Eigenvalue ProblemsIn this chapter, we discuss the important class of quadratic eigenproblems, with a small sidestep to higher order polynomial eigenproblems. The scope of the algebraic eigenvalue problem is also widened to geometrical properties of invariant subspaces. A template is presented that can be used to solve variational problems that are defined over sets of subspaces. Software
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