Numerical methods for the interpolation and approximation of data by spline functions

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by  Neil Stewart

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Citation: Cox, M G (1975). Numerical methods for the interpolation and approximation of
data by spline functions. (Unpublished Post-Doctoral thesis, City, University of London)
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IKE CUT UNIVERSITY'
Department of Mathematics
NUMERICAL METHODS FOR THE jTTTERPOLATRON
AND AJPHIOXLMA.TION OF DATA BY SPLINTS FUNCTIONS
by
M a COY. BSc, AFINA
Thesis submitted for the Decree of Doctor of Philosophy
to the City University, St John Street, London
July 1975
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To ray wife Rosalie who suffered
my moods, and had only a part-time
husband during the preparation
of this work.
I l l
ABSTRACT
NUMERICAL METHODS FOR THE INTERPOLATION AND
APPROXIMATION OF DATA BY SPLINE FUNCTIONS
It is often important in practice to obtain approximate representations
of physical data by relatively simple mathematical functions. The
approximating functions are usually required to meet certain criteria
relating to accuracy and smoothness. In the past, polynomials have
frequently been used for this task, but it has long been recognised that
there are many types of data set for which polynomial approximations are
unsatisfactory in that a very high degree may be required to achieve the
required accuracy. Moreover, even if such a polynomial can be computed,
it frequently tends to exhibit spurious oscillations not present in the
data itself.
In an attempt to overcome these difficulties attention has turned in
recent years to the use of piecewise polynomials or spline functions. A
spline function, or simply a spline, is composed of a set of polynomial
arcs, usually of low degree, joined end to end in such a way as to form
a smooth function. Splines tend to have greater flexibility than
polynomials in the approximation of physical data and much attention has
been devoted in the last decade to the theory of splines. The development
of robust numerical methods for computing with splines lias, however,
lagged somewhat behind the theory. The main objective of this work is
the construction and analysis of such methods. In order to obtain
efficient and stable metnods a representation of splines that is well-
conditioned and that results in fast computational schemes is required.
Representations in terms of B-splines prove to be eminently suitable and
accordingly we study B-splines in some detain and give various algorithms
for calculations in which they are involved.
when B-splines arc used as a basis for interpolation oi’ least-squares
data fitting the resulting linear algebraic systems to be solved for the
spline coefficients have a special structure. Stable numerical methods
that exploit this structure to the full are presented.
Cur algorithms are used to obtain spline approximations to a variety of
data sets drawn from practical applications. Their performance on these
problems illustrates the power of splines over more conventional
approximating functions.
ACIOiOWLEDGEMENTS
This thesis is bassd on work carried out between i 969 and 1975 while
the author was employed at the National Physical Laboratory and
registered at the City University as a part-time student for the
Degree of Doctor of Philosophy.
I am indebted to my internal supervisor Professor Y E Price and my
external supervisor Mr J & Hayes whose guidance and encouragement
enabled me to complete this work.
I also wish to acknowledge many fruitful discussions with
Mr E L ATbasiny, Mr & T Anthony, Professor C XI Clenshaw, Professor
W M Gentleman, Dr J H Wilkinson and my supervisors on various aspects
of linear algebra, ei'ror analysis, spline functions, data approximation
and numerical methods in general.
Finally I thank Mrs Joan Mann for her accurate typing of the manuscript.
vi
CONTENTS
Title page 5.
Abstract iii
Acknowledgements v
Contents vi
Introduction xi
Chapter 1 Pleating-point arithmetic and error analysis 1
1.1 Floating-point arithmetic
1.2 Floating-point error analysis 4
1.3 Algoritluns and numerical stability 14
Chapter 2 The numerical solution of linear algebraic equations 19
2.1 The solution, of triangular systems 21
2.2 The linear least-squares problem 24
2.3 Cholesky decomposition of the normal equations 27
2.4 Gaussian elimination 23
2.5 The use of orthogonal transformations 34
2.6 The modified Gram-Schmidt method 3&
2.7 The method of Householder transformations ■ 40
2.8 Classical plane rotations 43
2.9 Modern plane rotations 40
2.10 A comparison of the plane-rotation methods with
other methods based upon orthogonal transformations 61
2.11 Stepped-banded matrices 69
2.12 Triangular!2ation of stepped-banded matrices using
Gaussian elimination 70
2.13 Triangu.larizatioii of stepped-banded matrices using
stab ill zed elementary transformations 74
v ii
2.14 Orthogonal t r*i angular i z at :i. on of stepped-bandcd
matrices using plane rotations 79
2.15 The singular value decomposition 82
2.16 Perturbation bounds for the solution of linear
systems 87
Chapter 3 B--splines and their numerical evaluation $2
3.1 Definition of a spline function 93
3.2 The definition of a B-spline 93
3.3 The conventional method of evaluating B-splines 101
3.4 A recurrence relation for B-splines 103
3.5 The values of B-splinos at the ends of the range 111
3.6 The sum of normalised B-splines and bounds for
their values 112
3*7 A posteriori error bounds for the values of
'B-splines computed from divided differences 115
3.8 A posteriori error bounds for the values of
B-splines computed by the method of convex
c ombinat ions 117
3.9 A priori error bounds for the values of B-splines
computed by the method of convex combinations ’ 119
3.10 The effects of perturbations in the data 122
3.11 Numerical examples 123
3.12 The evaluation, for a prescribed argument, of all
non-zero B-splines of order n 129
3.13 Other methods for evaluating B-splines 134
Chapter 4 Differentiation and integration of B-splines 137
4.1 Recurrence relations for the derivatives of
B-splines ", 37
The derivatives of the E-splines at the ends of
the range 140
The derivatives of B-splines at the knots 145
Algorithms for the evaluation of B-spline
derivatives 149
The definite and indefinite integrals of
B-splines 150
The B-spline representation of splines and
polynomials 157
The B-spline representation of splines 156
The numerical evaluation of a spline from its
B-spline representation 162
Error analyses of algorithms for evaluating a
spline from its B-spline representation 164
The effect of errors in the B-spline coefficients
on the computed value of the spline 170
The B-spline representation of powers 172
Algorithms for computing the B-spline coefficients 175
The B-spllne representation of polynomials 181
Error analyses of the algorithms for computing
B-spline coefficients 18?
The derivatives of a spline represented dn
E-spline form 195
The indefinite integral of a spline represented in
B-spline fora 203
Representation in piecewise - Chebyshev-series
form
205
Spline interpolation 208
The spline interpolation pi'obiem 209