Extended Krylov subspace methods generalize classical Krylov, since also products with the inverse of the matrix under consideration are allowed. Recent advances have shown how to efficiently construct an orthogonal basis for the extended subspace, as well as how to build the highly structured projected matrix, named an extended Hessenberg matrix. It was shown that this alternative can lead to faster convergence for particular applications.In this text biorthogonal extended Krylov subspace methods are considered for general nonsymmetric matrices. We will show that the data resulting from the oblique projection can be expressed as a nonsymmetric tridiagonal matrix-pair. This is a direct generalization of the classical biorthogonal Krylov subspace method where the projection becomes a single nonsymmetric tridiagonal matrix. To obtain this result we first need to revisit the classical extended Krylov subspace algorithm and prove that the projected matrix can be written efficiently as a structured matrix-pair, where the structure can take several forms, such as, e.g., Hessenberg or inverse Hessenberg. Based on the compact storage of a tridiagonal matrix-pair in the biorthogonal setting, we can develop short recurrences.The numerical experiments confirm the validity of the approach.

The solution of linear non-autonomous ordinary differential equation systems (also known as the time-ordered exponential) is a computationally challenging problem arising in a variety of applications. In this work, we present and study a new framework for the computation of bilinear forms involving the time-ordered exponential. Such a framework is based on an extension of the non-Hermitian Lanczos algorithm to 4-mode tensors. Detailed results concerning its theoretical properties are presented. Moreover, computational results performed on real-world problems confirm the effectiveness of our approach.

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The problem of computing recurrence coefficients of sequences of rational functions orthogonal with respect to a discrete inner product is formulated as an inverse eigenvalue problem for a pencil of Hessenberg matrices. Two procedures are proposed to solve this inverse eigenvalue problem, via the rational Arnoldi iteration and via an updating procedure using unitary similarity transformations. The latter is shown to be numerically stable. This problem and both procedures are generalized by considering biorthogonal rational functions with respect to a bilinear form. This leads to an inverse eigenvalue problem for a pencil of tridiagonal matrices. A tridiagonal pencil implies short recurrence relations for the biorthogonal rational functions, which is more efficient than the orthogonal case. However the procedures solving this problem must rely on nonunitary operations and might not be numerically stable.

We introduce a new method with spectral accuracy to solve linear non-autonomous ordinary differential equations (ODEs) of the kind d dt ũ(t) = f (t)ũ(t), ũ(−1) = 1, with f (t) an analytic function. The method is based on a new expression for the solution ũ(t) given in terms of a convolution-like operation, the ⋆-product. This expression is represented in a finite Legendre polynomial basis translating the initial problem into a matrix problem. An efficient procedure is proposed to approximate the Legendre coefficients of ũ(t) and its truncation error is analyzed. We show the effectiveness of the proposed procedure through some numerical experiments. The method can be easily generalized to solve systems of linear ODEs.