On the Error in QR Integration

Abstract

An important change of variables for a linear time varying system $\dot x=A(t)x, t\ge 0$, is that induced by the QR-factorization of the underlying fundamental matrix solution: $X=QR$, with Q orthogonal and R upper triangular (with positive diagonal). To find this change of variable, one needs to solve a nonlinear matrix differential equation for Q. Practically, this means finding a numerical approximation to Q by using some appropriate discretization scheme, whereby one attempts to control the local error during the integration. Our contribution in this work is to obtain global error bounds for the numerically computed Q. These bounds depend on the local error tolerance used to integrate for Q, and on structural properties of the problem itself, but not on the length of the interval over which we integrate. This is particularly important, since—in principle—Q may need to be found on the half-line $t\ge 0$.