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Matrix functions of the form f (A)v, where A is a large symmetric matrix, f is a function, and v = 0 is a vector, are commonly approximated by first applying a few, say n, steps of the symmetric Lanczos process to A with the initial vector v in order to determine an orthogonal section of A. The latter is represented by a (small) n × n tridiagonal matrix to which f is applied. This approach uses the n first Lanczos vectors provided by the Lanczos process. However, n steps of the Lanczos process yield n + 1 Lanczos vectors. This paper discusses how the (n + 1)st Lanczos vector can be used to improve the quality of the computed approximation of f (A)v. Also the approximation of expressions of the form v T f (A)v is considered.

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