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Gauss quadrature can be naturally generalized to approximate quasi-definite linear functionals where 3 the interconnections with (formal) orthogonal polynomials, (complex) Jacobi matrices and Lanczos algorithm are 4 analogous to those in the positive definite case. In this survey we review these relationships with giving references 5 to literature that presents them in several related contexts. In particular, the existence of the n-weight (complex) 6 Gauss quadrature corresponds to successfully performing the first n steps of the Lanczos algorithm for generating 7 the biorthogonal bases of the two associated Krylov subspaces. The Jordan decomposition of the (complex) Jacobi 8 matrix can be explicitly expressed in terms of the Gauss quadrature nodes and weights and the associated orthogonal 9 polynomials. Since the output of the Lanczos algorithm can be made real whenever the input is real, the value of the 10 Gauss quadrature is a real number whenever all relevant moments of the quasi-definite linear functional are real. 11

The Lanczos algorithm and complex Gauss quadrature