Error formulas for block rational Krylov approximations of matrix functions
This work provides incremental improvements in error analysis for numerical linear algebra, specifically benefiting researchers and practitioners in computational mathematics.
The paper tackles the problem of approximating matrix functions using block rational Krylov methods by proposing two explicit error formulas derived from block FOM residual characterizations, and it derives a posteriori error bounds that avoid quadrature for practical evaluation.
This paper investigates explicit expressions for the error associated with the block rational Krylov approximation of matrix functions. Two formulas are proposed, both derived from characterizations of the block FOM residual. The first formula employs a block generalization of the residual polynomial, while the second leverages the block collinearity of the residuals. A posteriori error bounds based on the knowledge of spectral information of the argument are derived and tested on a set of examples. Notably, both error formulas and their corresponding upper bounds do not require the use of quadratures for their practical evaluation.