In this paper,we study asymptotic power series of the composition f(x)=h(g(x)),where g(x)=∑_(n=0)^(∞)b_(n)x^(-n),b_(n)∈R,and h is a given elementary function.The asymptotic expansions have been obtained for the com...In this paper,we study asymptotic power series of the composition f(x)=h(g(x)),where g(x)=∑_(n=0)^(∞)b_(n)x^(-n),b_(n)∈R,and h is a given elementary function.The asymptotic expansions have been obtained for the composition with an exponential or logarithmic function.Using the re-cursive method,we present the asymptotic expansions for the composition with seven trigonometric functions,respectively.As an application,the asymptotic expansions of roots of some equations are given.Computational results show that our recursive formula is more efficient than the method of Lagrange's inverse theorem.展开更多
用5个定理给出最小一乘线性回归的相关性质,为其工程应用奠定了基础。文中首先证明了“由“最小一乘”准则确定的直线y=b1x1+ b2x2经过其两个样本点”以及“由最小一乘准则确定的直线y=b1x1+ b2x2+a经过其三个样本点”。然后应用数学归...用5个定理给出最小一乘线性回归的相关性质,为其工程应用奠定了基础。文中首先证明了“由“最小一乘”准则确定的直线y=b1x1+ b2x2经过其两个样本点”以及“由最小一乘准则确定的直线y=b1x1+ b2x2+a经过其三个样本点”。然后应用数学归纳法得到如下定理:设有n(n>P)个样本点(x1i, x2i, ? xP i, yi,),则由最小一乘准则确定的线性非奇次模型y=b1x1+b2x2+?bPxP+a经过其P+1个样本点,而相应的奇次模型必经过其P个样本点。通过大量工程实例证实了最小一乘具有较强的稳健性,同时也证实了定理的正确性。展开更多
基金Supported by The Innovation Fund of Postgraduate,Sichuan University of Science&Engineering(Y2024336)NSF of Sichuan Province(2023NSFSC0065).
文摘In this paper,we study asymptotic power series of the composition f(x)=h(g(x)),where g(x)=∑_(n=0)^(∞)b_(n)x^(-n),b_(n)∈R,and h is a given elementary function.The asymptotic expansions have been obtained for the composition with an exponential or logarithmic function.Using the re-cursive method,we present the asymptotic expansions for the composition with seven trigonometric functions,respectively.As an application,the asymptotic expansions of roots of some equations are given.Computational results show that our recursive formula is more efficient than the method of Lagrange's inverse theorem.
文摘用5个定理给出最小一乘线性回归的相关性质,为其工程应用奠定了基础。文中首先证明了“由“最小一乘”准则确定的直线y=b1x1+ b2x2经过其两个样本点”以及“由最小一乘准则确定的直线y=b1x1+ b2x2+a经过其三个样本点”。然后应用数学归纳法得到如下定理:设有n(n>P)个样本点(x1i, x2i, ? xP i, yi,),则由最小一乘准则确定的线性非奇次模型y=b1x1+b2x2+?bPxP+a经过其P+1个样本点,而相应的奇次模型必经过其P个样本点。通过大量工程实例证实了最小一乘具有较强的稳健性,同时也证实了定理的正确性。
