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Computing \(f(A)b\) via least squares polynomial approximations. (English) Zbl 1234.65027

In order to compute \(f(A)b\) efficiently for a diagonalizable matrix \(A\) with only real eigenvalues and a vector \(b\), this paper starts with approximating \(f(t)\) by a spline \(s(t)\) on an interval that contains the extreme eigenvalues of \(A\). Special attention is given to use a function inner product that eases the spline computation and allows the least squares polynomial to be found without numerical integration. The method can be adapted to a large variety of functions \(f\). The paper investigates the method’s behavior for \(f(t) = \sqrt{t}\) and sparse positive definite \(A\). It derives error bounds and shows numerical results and comparisons with other methods.

MSC:

65F50 Computational methods for sparse matrices
65F35 Numerical computation of matrix norms, conditioning, scaling
15A16 Matrix exponential and similar functions of matrices
65F60 Numerical computation of matrix exponential and similar matrix functions

Software:

SparseMatrix
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