We consider Lipschitz-continuous nonlinear maps in finite-dimensional Banach and Hilbert spaces. Boundedness and monotonicity of the operator are characterized quantitatively in terms of certain functionals. These fun...
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We consider Lipschitz-continuous nonlinear maps in finite-dimensional Banach and Hilbert spaces. Boundedness and monotonicity of the operator are characterized quantitatively in terms of certain functionals. These functionals are used to assess qualitative properties such as invertibility, and also enable a generalization of some well-known matrix results directly to nonlinear operators. Closely related to the numerical range of a matrix, the Gerschgorin domain is introduced for nonlinear operators. This point set in the complex plane is always convex and contains the spectrum of the operator's Jacobian matrices. Finally, we focus on nonlinear operators in Hilbert space and hint at some generalizations of the von Neumann spectral theory.
We consider the general class of power series where the terms may be expressed as the Laplace transforms of known functions. The sum of the series can then be evaluated efficiently and accurately by means of quadratur...
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We consider the general class of power series where the terms may be expressed as the Laplace transforms of known functions. The sum of the series can then be evaluated efficiently and accurately by means of quadrature schemes, recently published by Frank Stenger. The method works also far outside the region of convergence as will be illustrated by numerical examples.
Estimates concerning the spectrum of a graded matrix and other information useful for a reliable and efficient handling of certain complications in the numerical treatment of some stiff ODE's, can be inexpensively...
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Estimates concerning the spectrum of a graded matrix and other information useful for a reliable and efficient handling of certain complications in the numerical treatment of some stiff ODE's, can be inexpensively obtained from the factorized Jacobian. The validity of the estimates is studied by considering them as the first step in a block LR algorithm, which may be of interest in its own right. Its convergence properties are examined.
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