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The interplay between classical analysis and (numerical) linear algebra  a tribute to Gene H. Golub
 ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS
, 2002
"... Much of the work of Golub and his collaborators uses techniques of linear algebra to deal with problems in analysis, or employs tools from analysis to solve problems arising in linear algebra. Instances are described of such interdisciplinary work, taken from quadrature theory, orthogonal polynomia ..."
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Much of the work of Golub and his collaborators uses techniques of linear algebra to deal with problems in analysis, or employs tools from analysis to solve problems arising in linear algebra. Instances are described of such interdisciplinary work, taken from quadrature theory, orthogonal polynomials, and least squares problems on the one hand, and error analysis for linear algebraic systems, elementwise bounds for the inverse of matrices, and eigenvalue estimates on the other hand.
c ○ 2005 Society for Industrial and Applied Mathematics QUADRATURE RULES BASED ON THE ARNOLDI PROCESS ∗
"... Abstract. Applying a few steps of the Arnoldi process to a large nonsymmetric matrix A with initial vector v is shown to induce several quadrature rules. Properties of these rules are discussed, and their application to the computation of inexpensive estimates of the quadratic form 〈f, g 〉: = v ∗ (f ..."
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Abstract. Applying a few steps of the Arnoldi process to a large nonsymmetric matrix A with initial vector v is shown to induce several quadrature rules. Properties of these rules are discussed, and their application to the computation of inexpensive estimates of the quadratic form 〈f, g 〉: = v ∗ (f(A)) ∗g(A)v and related quadratic and bilinear forms is considered. Under suitable conditions on the functions f and g, the matrix A, and the vector v, the computed estimates provide upper and lower bounds of the quadratic and bilinear forms.