Numerical Linear Approximation in C - TiERA

950 Pages·2008·4.01 MB·English

Numerical
Linear
Approximation
in C
© 2008 by Taylor & Francis Group, LLC
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© 2008 by Taylor & Francis Group, LLC
Numerical
Linear
Approximation
in C
Nabih N. Abdelmalek, Ph.D.
William A. Malek, M.Eng.
© 2008 by Taylor & Francis Group, LLC
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Library of Congress Cataloging-in-Publication Data
Abdelmalek, Nabih N.
Numerical linear approximation in C / Nabih Abdelmalek and William A.
Malek.
p. cm. -- (CRC numerical analysis and scientific computing)
Includes bibliographical references and index.
ISBN 978-1-58488-978-6 (alk. paper)
1. Chebyshev approximation. 2. Numerical analysis. 3. Approximation theory.
I. Malek, William A. II. Title. III. Series.
QA297.A23 2008
518--dc22 2008002447
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© 2008 by Taylor & Francis Group, LLC
v
Contents
List of figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
Acknowledgements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii
Warranties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii
About the authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv
PART 1
Preliminaries and Tutorials
Chapter 1 Applications of Linear Approximation
1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Applications to social sciences and economics. . . . . . . . . . . . . . . . 5
1.2.1 Systolic blood pressure and age . . . . . . . . . . . . . . . . . . . 6
1.2.2 Annual teacher salaries. . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.3 Factors affecting survival of island species . . . . . . . . . . 7
1.2.4 Factors affecting fuel consumption. . . . . . . . . . . . . . . . . 8
1.2.5 Examining factors affecting the mortality rate. . . . . . . . 8
1.2.6 Effects of forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.7 Factors affecting gross national products. . . . . . . . . . . . 9
1.3 Applications to industry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.1 Windmill generating electricity . . . . . . . . . . . . . . . . . . 10
1.3.2 A chemical process. . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Applications to digital images. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4.1 Smoothing of random noise in digital images . . . . . . . 12
1.4.2 Filtering of impulse noise in digital images . . . . . . . . . 14
1.4.3 Applications to pattern classification . . . . . . . . . . . . . . 15
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vi Numerical Linear Approximation in C
1.4.4 Restoring images with missing high-frequency
components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4.5 De-blurring digital images using the Ridge equation. . 18
1.4.6 De-blurring images using truncated eigensystem. . . . . 19
1.4.7 De-blurring images using quadratic programming. . . . 20
Chapter 2 Preliminaries
2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Discrete linear approximation and solution of overdetermined
linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Comparison between the L , the L and the L norms by a
1 2 ∞
practical example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.1 Some characteristics of the L and the Chebyshev
1
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4 Error tolerances in the calculation. . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5 Representation of vectors and matrices in C. . . . . . . . . . . . . . . . . 33
2.6 Outliers and dealing with them . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.6.1 Data editing and residual analysis. . . . . . . . . . . . . . . . . 36
Chapter 3 Linear Programming and the Simplex Algorithm
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1.1 Exceptional linear programming problems. . . . . . . . . . 41
3.2 Notations and definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3 The simplex algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.1 Initial basic feasible solution . . . . . . . . . . . . . . . . . . . . 47
3.3.2 Improving the initial basic feasible solution. . . . . . . . . 48
3.4 The simplex tableau. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.5 The two-phase method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.5.1 Phase 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.5.2 Phase 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.5.3 Detection of the exceptional cases . . . . . . . . . . . . . . . . 56
3.6 Duality theory in linear programming. . . . . . . . . . . . . . . . . . . . . . 57
3.6.1 Fundamental properties of the dual problems. . . . . . . . 59
3.6.2 Dual problems with mixed constraints. . . . . . . . . . . . . 60
3.6.3 The dual simplex algorithm . . . . . . . . . . . . . . . . . . . . . 61
3.7 Degeneracy in linear programming and its resolution . . . . . . . . . 62
3.7.1 Degeneracy in the simplex method. . . . . . . . . . . . . . . . 62
3.7.2 Avoiding initial degeneracy in the simplex algorithm . 63
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Contents vii
3.7.3 Resolving degeneracy resulting from equal θ . . . . . 63
min
3.7.4 Resolving degeneracy in the dual simplex method. . . . 64
3.8 Linear programming and linear approximation. . . . . . . . . . . . . . . 64
3.8.1 Linear programming and the L approximation. . . . . . 64
1
3.8.2 Linear programming and Chebyshev approximation. . 66
3.9 Stability of the solution in linear programming . . . . . . . . . . . . . . 67
Chapter 4 Efficient Solutions of Linear Equations
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 Vector and matrix norms and relevant theorems. . . . . . . . . . . . . . 72
4.2.1 Vector norms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2.2 Matrix norms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2.3 Hermitian matrices and vectors . . . . . . . . . . . . . . . . . . 73
4.2.4 Other matrix norms. . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2.5 Euclidean and the spectral matrix norms . . . . . . . . . . . 77
4.2.6 Euclidean norm and the singular values. . . . . . . . . . . . 78
4.2.7 Eigenvalues and the singular values of the sum and
the product of two matrices. . . . . . . . . . . . . . . . . . . . . . 79
4.2.8 Accuracy of the solution of linear equations . . . . . . . . 80
4.3 Elementary matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.4 Gauss LU factorization with complete pivoting. . . . . . . . . . . . . . 84
4.4.1 Importance of pivoting . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.4.2 Using complete pivoting. . . . . . . . . . . . . . . . . . . . . . . . 86
4.4.3 Pivoting and the rank of matrix A. . . . . . . . . . . . . . . . . 88
4.5 Orthogonal factorization methods. . . . . . . . . . . . . . . . . . . . . . . . . 90
4.5.1 The elementary orthogonal matrix H . . . . . . . . . . . . . . 90
4.5.2 Householder(cid:146)s QR factorization with pivoting. . . . . . . 90
4.5.3 Pivoting in Householder(cid:146)s method . . . . . . . . . . . . . . . . 92
4.5.4 Calculation of the matrix inverse A(cid:150)1. . . . . . . . . . . . . . 93
4.6 Gauss-Jordan method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.7 Rounding errors in arithmetic operations . . . . . . . . . . . . . . . . . . . 95
4.7.1 Normalized floating-point representation. . . . . . . . . . . 95
4.7.2 Overflow and underflow in arithmetic operations . . . . 96
4.7.3 Arithmetic operations in a d.p. accumulator. . . . . . . . . 97
4.7.4 Computation of the square root of a s.p. number . . . . 100
4.7.5 Arithmetic operations in a s.p. accumulator. . . . . . . . 100
4.7.6 Arithmetic operations with two d.p. numbers. . . . . . . 101
4.7.7 Extended simple s.p. operations in a d.p. accumulator 101
4.7.8 Alternative expressions for summations and inner-
product operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
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viii Numerical Linear Approximation in C
4.7.9 More conservative error bounds. . . . . . . . . . . . . . . . . 104
4.7.10 D.p. summations and inner-product operations . . . . . 105
4.7.11 Rounding error in matrix computation. . . . . . . . . . . . 106
4.7.12 Forward and backward round-off error analysis. . . . . 107
4.7.13 Statistical error bounds and concluding remarks . . . . 109
PART 2
The L Approximation
1
Chapter 5 Linear L Approximation
1
5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.1.1 Characterization of the L solution. . . . . . . . . . . . . . . 116
1
5.2 Linear programming formulation of the problem. . . . . . . . . . . . 116
5.3 Description of the algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.4 The dual simplex method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.5 Modification to the algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.6 Occurrence of degeneracy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.7 A significant property of the L approximation . . . . . . . . . . . . . 129
1
5.8 Triangular decomposition of the basis matrix. . . . . . . . . . . . . . . 130
5.9 Arithmetic operations count . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.10 Numerical results and comments. . . . . . . . . . . . . . . . . . . . . . . . . 134
5.11 DR_L1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.12 LA_L1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.13 DR_Lone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.14 LA_Lone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Chapter 6 One-Sided L Approximation
1
6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
6.1.1 Applications of the algorithm. . . . . . . . . . . . . . . . . . . 186
6.1.2 Characterization and uniqueness. . . . . . . . . . . . . . . . . 186
6.2 A special problem of a general constrained one . . . . . . . . . . . . . 186
6.3 Linear programming formulation of the problem. . . . . . . . . . . . 187
6.4 Description of the algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
6.4.1 Obtaining an initial basic feasible solution. . . . . . . . . 188
6.4.2 One-sided L solution from above . . . . . . . . . . . . . . . 191
1
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Contents ix
6.4.3 The interpolation property . . . . . . . . . . . . . . . . . . . . . 191
6.5 Numerical results and comments. . . . . . . . . . . . . . . . . . . . . . . . . 191
6.6 DR_Loneside. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
6.7 LA_Loneside. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
Chapter 7 L Approximation with Bounded Variables
1
7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
7.1.1 Linear L approximation with non-negative
1
parameters (NNL1). . . . . . . . . . . . . . . . . . . . . . . . . . . 216
7.2 A special problem of a general constrained one . . . . . . . . . . . . . 216
7.3 Linear programming formulation of the problem. . . . . . . . . . . . 217
7.3.1 Properties of the matrix of constraints . . . . . . . . . . . . 220
7.4 Description of the algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
7.5 Numerical results and comments. . . . . . . . . . . . . . . . . . . . . . . . . 222
7.6 DR_Lonebv. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
7.7 LA_Lonebv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
Chapter 8 L Polygonal Approximation of Plane Curves
1
8.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
8.1.1 Two basic issues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
8.1.2 Approaches for polygonal approximation . . . . . . . . . 246
8.1.3 Other unique approaches. . . . . . . . . . . . . . . . . . . . . . . 248
8.1.4 Criteria by which error norm is chosen. . . . . . . . . . . . 249
8.1.5 Direction of error measure . . . . . . . . . . . . . . . . . . . . . 249
8.1.6 Comparison and stability of polygonal
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
8.1.7 Applications of the algorithm. . . . . . . . . . . . . . . . . . . 251
8.2 The L approximation problem. . . . . . . . . . . . . . . . . . . . . . . . . . 251
1
8.3 Description of the algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
8.4 Linear programming technique. . . . . . . . . . . . . . . . . . . . . . . . . . 253
8.4.1 The algorithm using linear programming. . . . . . . . . . 255
8.5 Numerical results and comments. . . . . . . . . . . . . . . . . . . . . . . . . 255
8.6 DR_L1pol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
8.7 LA_L1pol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
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