A High-Precision DQDS Algorithm

With the rise of high-performance computing, the precision requirements for numerical calculations are getting higher and higher. Rounding errors cannot be avoided in floating point operations, and therefore the calculation result can be inaccurate and even incorrect when rounding errors are not taken into account. We study and analyze in detail the Differential Quotient Difference with Shifts (DQDS) algorithm, used in algorithms to calculate the eigenvalues of tridiagonal matrices, and propose a novel High Precision DQDS algorithm (HDQDS) based on Error Free Transformations (EFT). We use EFT to reduce the accumulation of errors and obtain a much more precise algorithm. The relative error of the HDQDS algorithm remains constant at the machine precision error level, while the relative error of the DQDS algorithm has fluctuations in the same experimental scenario. Numerical experiments show that the proposed high-precision HDQDS algorithm allows for higher precision and more stability than the original alagrithm.