The bulge-chasing kernel in the multi-shift QR algorithm is invoked repeatedly on the critical path. The bulge-chasing kernel operates on a computational window embedded in a bigger matrix. It requires multiple calls to the bulge-chasing kernel to chase all bulges; a highly optimised routine therefore reduces the length of the critical path and potentially the overall execution time. Off-diagonal blocks are updated with DGEMM operations. We provide an optimised sequential implementation that runs at 50% of the peak performance and is 5–15 times faster than DLAQR5, LAPACK’s standard bulge-chasing routine.