用差商代替导数的非线性最小二乘估计

李桂苓; 万剑华; 陶华学

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用差商代替导数的非线性最小二乘估计

李桂苓, 万剑华, 陶华学

文章导航 > 武汉大学学报 ● 信息科学版 > 2001 > 26(2): 118-121

李桂苓, 万剑华, 陶华学. 用差商代替导数的非线性最小二乘估计[J]. 武汉大学学报 ● 信息科学版, 2001, 26(2): 118-121.

引用本文:

李桂苓, 万剑华, 陶华学. 用差商代替导数的非线性最小二乘估计[J]. 武汉大学学报 ● 信息科学版, 2001, 26(2): 118-121.

LI Guiling, WAN Jianhua, TAO Huaxue. Nonlinear Least Squares Adjustment of Non-derivative[J]. Geomatics and Information Science of Wuhan University, 2001, 26(2): 118-121.

Citation:

LI Guiling, WAN Jianhua, TAO Huaxue. Nonlinear Least Squares Adjustment of Non-derivative[J]. Geomatics and Information Science of Wuhan University, 2001, 26(2): 118-121.

用差商代替导数的非线性最小二乘估计

1 石油大学(华东)测绘工程系, 山东东营, 257061;
2 山东科技大学地球信息科学与工程学院, 山东泰安, 271019

基金项目: 国家自然科学基金资助项目(49874001)。

详细信息

作者简介:
李桂苓,硕士,讲师。主要研究方向为非线性测量数据处理。代表成果:不同类型观测值非线性平差模型的降维解算;用气温梯度进行折光改正的三角高程测量及应用;多类观测值非线性平差参数的方差-协方差传播。E-mail:kyk@hdpu.edu.cn

中图分类号: P207

Nonlinear Least Squares Adjustment of Non-derivative

1 University of Petroleum, Department of Surveying Engineering, Dongying, Shandong, China, 257061;
2 School of Geoinformation Science and Technology, Shandong University of Science and Technology, Tai'an, Shandong, China, 271019

摘要: 针对不同类型观测值的非线性最小二乘平差,介绍一种不使用导数的解析方法。在这种解算中,由于只使用函数值,避免了二阶和二阶偏导数的计算,使原本复杂的计算得以简化。实例验证了本方法的有效性和可靠性。
- 差商代替导数 /
- 非线性 /
- 最小二乘估计
Abstract: In classical adjustment,most of observation values are still expanded to carry out linear adjustment at their approximations in terms of Taylor's series of which the first order term as a linear function is taken out for the treatment of nonlinear functions of observation values existing broadly in the field of surveying and mapping science and technology.This treatment is based on undetermined quantities which are very close to their true values.In effect,a number of nonlinear models can not be tackled as usual.It is more and more highly requiring for the quality of observation accomplishments,treatment of observation datum and accuracy appraise,especially in the rocket-sky development of today's high technology.Apart from this,traditional and single geodetic surveying or non-geodetic surveying datum has been expanded into the combination of observation quantities on geodetic surveying or lots of types,different accuracy observation quantities on non-geodetic surveying with the unending advent of modern observation apparatus,the continuous enhancement and improvement of surveying means. This paper gives a computational method of nonlinear least squares adjustment without derivative for different types of observation value,in which a function value instead of its derivative is only used.As far as we know,finding the partial value of a function is a troublesome task,especially when the components f_i(x)(i=1,2,…,m+n) of a nonlinear function f(x) are very complicated.The essence of the method is to simplify its computations for a Jacobi's matrix,which determines the coefficient matrix with relation to a linear equation set by utilizing algebraic interpolation method with function values.In a geometrical sense,what is different from that in Guass-Newton's method is that a series of super-tangent planes can be converted into super-cut ones or that it can be seen as a discrete deformation of Guass-Newton's method.Especially while every difference step size is taken as a constant,the calculation is even more convenient.Examples show that the method is an effective algorithm and is of super-linear convergence velocity.
- non-derivative /
- nonlinear /
- least squares adjustment

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用差商代替导数的非线性最小二乘估计

1 石油大学(华东)测绘工程系, 山东东营, 257061;

2 山东科技大学地球信息科学与工程学院, 山东泰安, 271019

基金项目: 国家自然科学基金资助项目(49874001)。

作者简介:
李桂苓,硕士,讲师。主要研究方向为非线性测量数据处理。代表成果:不同类型观测值非线性平差模型的降维解算;用气温梯度进行折光改正的三角高程测量及应用;多类观测值非线性平差参数的方差-协方差传播。E-mail:kyk@hdpu.edu.cn

收稿日期: 2000-10-24
刊出日期: 2001-02-05

中图分类号: P207

关键词:

差商代替导数 /

非线性 /

最小二乘估计

摘要: 针对不同类型观测值的非线性最小二乘平差,介绍一种不使用导数的解析方法。在这种解算中,由于只使用函数值,避免了二阶和二阶偏导数的计算,使原本复杂的计算得以简化。实例验证了本方法的有效性和可靠性。

English Abstract

Nonlinear Least Squares Adjustment of Non-derivative

1 University of Petroleum, Department of Surveying Engineering, Dongying, Shandong, China, 257061;

2 School of Geoinformation Science and Technology, Shandong University of Science and Technology, Tai'an, Shandong, China, 271019

Received Date: 2000-10-24
Publish Date: 2001-02-05

Keywords:

Abstract: In classical adjustment,most of observation values are still expanded to carry out linear adjustment at their approximations in terms of Taylor's series of which the first order term as a linear function is taken out for the treatment of nonlinear functions of observation values existing broadly in the field of surveying and mapping science and technology.This treatment is based on undetermined quantities which are very close to their true values.In effect,a number of nonlinear models can not be tackled as usual.It is more and more highly requiring for the quality of observation accomplishments,treatment of observation datum and accuracy appraise,especially in the rocket-sky development of today's high technology.Apart from this,traditional and single geodetic surveying or non-geodetic surveying datum has been expanded into the combination of observation quantities on geodetic surveying or lots of types,different accuracy observation quantities on non-geodetic surveying with the unending advent of modern observation apparatus,the continuous enhancement and improvement of surveying means. This paper gives a computational method of nonlinear least squares adjustment without derivative for different types of observation value,in which a function value instead of its derivative is only used.As far as we know,finding the partial value of a function is a troublesome task,especially when the components f_i(x)(i=1,2,…,m+n) of a nonlinear function f(x) are very complicated.The essence of the method is to simplify its computations for a Jacobi's matrix,which determines the coefficient matrix with relation to a linear equation set by utilizing algebraic interpolation method with function values.In a geometrical sense,what is different from that in Guass-Newton's method is that a series of super-tangent planes can be converted into super-cut ones or that it can be seen as a discrete deformation of Guass-Newton's method.Especially while every difference step size is taken as a constant,the calculation is even more convenient.Examples show that the method is an effective algorithm and is of super-linear convergence velocity.

李桂苓, 万剑华, 陶华学. 用差商代替导数的非线性最小二乘估计[J]. 武汉大学学报 ● 信息科学版, 2001, 26(2): 118-121.

引用本文:

李桂苓, 万剑华, 陶华学. 用差商代替导数的非线性最小二乘估计[J]. 武汉大学学报 ● 信息科学版, 2001, 26(2): 118-121.

LI Guiling, WAN Jianhua, TAO Huaxue. Nonlinear Least Squares Adjustment of Non-derivative[J]. Geomatics and Information Science of Wuhan University, 2001, 26(2): 118-121.

Citation:

LI Guiling, WAN Jianhua, TAO Huaxue. Nonlinear Least Squares Adjustment of Non-derivative[J]. Geomatics and Information Science of Wuhan University, 2001, 26(2): 118-121.