New Algorithms for Computing the Matrix Sine and Cosine Separately or Simultaneously
New Algorithms for Computing the Matrix Sine and Cosine Separately or Simultaneously
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DOI：
10.1137/140973979
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发表时间：
2015-02
期刊：
影响因子：
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通讯作者：
Awad H. Al-Mohy;N. Higham;S. Relton
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文献类型：
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作者：
Awad H. Al-Mohy;N. Higham;S. Relton
用于计算矩阵余弦的几种现有算法采用与缩放和使用双角公式相结合的多项式或有理近似。他们的推导是基于前向误差界。我们推导出新的算法，用于计算矩阵余弦，矩阵正弦，同时，这是向后稳定的精确算术和行为在浮点运算的前向稳定的方式。我们的新算法采用了$\sin x$的Pad\'e逼近和$\cos x$和$\sin x$的新的有理逼近，它们是从$e^x$的Pad\' e逼近得到的。缩放量和近似的程度被选择为最小化精确算法中向后稳定的计算成本。数值实验表明，新算法的前向误差和后向误差都可与现有算法相媲美或超过现有算法，并且对三角矩阵的计算尤为有利。
Several existing algorithms for computing the matrix cosine employ polynomial or rational approximations combined with scaling and use of a double angle formula. Their derivations are based on forward error bounds. We derive new algorithms for computing the matrix cosine, the matrix sine, and both simultaneously, that are backward stable in exact arithmetic and behave in a forward stable manner in floating point arithmetic. Our new algorithms employ both Pad\'e approximants of $\sin x$ and new rational approximants to $\cos x$ and $\sin x$ obtained from Pad\'e approximants to $e^x$. The amount of scaling and the degree of the approximants are chosen to minimize the computational cost subject to backward stability in exact arithmetic. Numerical experiments show that the new algorithms have backward and forward errors that rival or surpass those of existing algorithms and are particularly favorable for triangular matrices.