Extremal signatures for bivariate Chebyshev approximation problems

1993 ◽  
Vol 5 (4) ◽  
pp. 215-225
Author(s):  
S. Thiry
Keyword(s):  
Chebyshev Approximation ◽  
Approximation Problems

scholarly journals Using Parameter Elimination to Solve Discrete Linear Chebyshev Approximation Problems

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2210
Author(s):  
Nikolai Krivulin
Keyword(s):  
Linear Regression ◽  
Algebraic Approach ◽  
Chebyshev Approximation ◽  
Direct Solution ◽  
Unknown Parameters ◽  
Absolute Deviation ◽  
Maximum Absolute Deviation ◽  
Regression Problems ◽  
Approximation Problems ◽  
Parameter Elimination

We consider discrete linear Chebyshev approximation problems in which the unknown parameters of linear function are fitted by minimizing the least maximum absolute deviation of errors. Such problems find application in the solution of overdetermined systems of linear equations that appear in many practical contexts. The least maximum absolute deviation estimator is used in regression analysis in statistics when the distribution of errors has bounded support. To derive a direct solution of the problem, we propose an algebraic approach based on a parameter elimination technique. As a key component of the approach, an elimination lemma is proved to handle the problem by reducing it to a problem with one parameter eliminated, together with a box constraint imposed on this parameter. We demonstrate the application of the lemma to the direct solution of linear regression problems with one and two parameters. We develop a procedure to solve multidimensional approximation (multiple linear regression) problems in a finite number of steps. The procedure follows a method that comprises two phases: backward elimination and forward substitution of parameters. We describe the main components of the procedure and estimate its computational complexity. We implement symbolic computations in MATLAB to obtain exact solutions for two numerical examples.

Chebyshev approximation to data by geometric elements

1993 ◽  
Vol 5 (10) ◽  
pp. 509-522 ◽  
Author(s):  
Rudolf Drieschner
Keyword(s):  
Chebyshev Approximation ◽  
Geometric Elements

Algorithm for Spatial Straightness Evaluation Using Theories of Linear Complex Chebyshev Approximation and Semi-infinite Linear Programming

2005 ◽  
Vol 128 (1) ◽  
pp. 167-174 ◽  
Author(s):  
LiMin Zhu ◽  
Ye Ding ◽  
Han Ding
Keyword(s):  
Linear Programming ◽  
Chebyshev Approximation ◽  
Approximation Problem ◽  
Linear Complex ◽  
Infinite Linear Programming ◽  
Minimum Zone ◽  
Effectiveness And Efficiency ◽  
Primal And Dual ◽  
Straightness Error ◽  
Infinite Linear

This paper presents a novel methodology for evaluating spatial straightness error based on the minimum zone criterion. Spatial straightness evaluation is formulated as a linear complex Chebyshev approximation problem, and then reformulated as a semi-infinite linear programming problem. Both models for the primal and dual programs are developed. An efficient simplex-based algorithm is employed to solve the dual linear program to yield the straightness value. Also a general algebraic criterion for checking the optimality of the solution is proposed. Numerical experiments are given to verify the effectiveness and efficiency of the presented algorithm.

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