We present a practical algorithm to approximate the exponential of skew-Hermitian matrices up to round-off error based on an efficient computation of Chebyshev polynomials of matrices and the corresponding error analysis. It is based on Chebyshev polynomials of degrees 2, 4, 8, 12 and 18 which are computed with only 1, 2, 3, 4 and 5 matrix-matrix products, respectively. For problems of the form $\exp(-iA)$, with $A$ a real and symmetric matrix, an improved version is presented that computes the sine and cosine of $A$ with a reduced computational cost. The theoretical analysis, supported by numerical experiments, indicates that the new methods are more efficient than schemes based on rational Pad\'e approximants and Taylor polynomials for all tolerances and time interval lengths. The new procedure is particularly recommended to be used in conjunction with exponential integrators for the numerical time integration of the Schr\"odinger equation.
翻译:我们提出了一个实际算法,根据对Chebyshev 矩阵多式矩阵的有效计算和相应的误差分析,将skew-Hermitian矩阵的指数接近到圆差,该算法以基比舍夫 2、4、8、12和18度的多式矩阵为基础,分别以1、2、3、4和5个矩阵矩阵矩阵-矩阵产品计算。对于美元表(iA)$的问题,以美元为真实和对称矩阵的问题,我们提出了一个改进的版本,用较低的计算成本计算美元正正正弦和正辛酸。理论分析得到数字实验的支持,表明新方法比以理性帕德约式和泰勒多式模型为基础的所有可容度和时间间隔长度方案更有效。特别建议与指数式集成器一起使用新程序,用于Schr\“加比方程”的数字时间整合。