Interpolace hladkých funkcí pomocí kvadratických a kubických splinů

Transcription

1 Univerzita Karlova v Praze Matematicko-fyzikální fakulta BAKALÁŘSKÁ PRÁCE Jiří Eckstein Interpolace hladkých funkcí pomocí kvadratických a kubických splinů Katedra numerické matematiky Vedoucí bakalářské práce: RNDr. Václav Kučera, Ph.D. Studijní program: Matematika Studijní obor: Obecná matematika Praha 01

2 I would like to thank my family and my friends for their support. I would also like to thank my supervisor RNDr. Václav Kučera, Ph.D. for his time dedicated to consultations and reading this bachelor thesis and for many valuable suggestions and comments.

3 Prohlašuji, že jsem tuto bakalářskou práci vypracoval(a) samostatně a výhradně s použitím citovaných pramenů, literatury a dalších odborných zdrojů. Beru na vědomí, že se na moji práci vztahují práva a povinnosti vyplývající ze zákona č. 11/000 Sb., autorského zákona v platném znění, zejména skutečnost, že Univerzita Karlova v Praze má právo na uzavření licenční smlouvy o užití této práce jako školního díla podle 60 odst. 1 autorského zákona. V Praze dne.8.01 Jiří Eckstein

4 Název práce: Interpolace hladkých funkcí pomocí kvadratických a kubických splinů Autor: Jiří Eckstein Katedra: Katedra numerické matematiky Vedoucí bakalářské práce: RNDr. Václav Kučera, Ph.D., Katedra numerické matematiky Abstrakt: V této bakalářské práci se zabýváme základními vlastnostmi interpolace pomocí kvadratických a kubických splinů. Nejprve definujeme pojem interpolace a splinu. Ty poté spojíme a zabýváme se postupně kubickou a kvadratickou splinovou interpolací. Vždy nejprve uvedeme nejznámější typy, pak ukážeme postup sestrojení vybraného interpolačního splinu a shrneme základní vlastnosti. Následně prezentujeme program vytvořený na základě uvedených poznatků a algoritmů. Program využijeme pro interpolaci některých ukázkových funkcí. Spočítáme chyby takto vzniklých interpolací a porovnáváme je s teoretickými výsledky. Klíčová slova: interpolace, kubický spline, kvadratický spline Title: Interpolation of smooth functions using quadratic and cubic splines Author: Jiří Eckstein Department: Department of Numerical Mathematics Supervisor: RNDr. Václav Kučera, Ph.D., Department of Numerical Mathematics Abstract: In this thesis, we study properties of cubic and quadratic spline interpolation. First, we define the notions of spline and interpolation. We then merge them to study cubic and quadratic spline interpolations. We go through the individual spline interpolation types, show an algorithm for constructing selected types and sum up their basic properties. We then present a computer program based on the provided algorithms. We use it to construct spline interpolations of some sample functions and we calculate errors of these interpolation and compare them with theoretical estimates. Keywords: interpolation, cubic spline, quadratic spline

5 Contents Introduction 1 Elementary Notions and Notation Splines Interpolation Theory 5.1 Cubic Spline Interpolation Definitions and Overview Cubic Spline Construction Properties Quadratic Spline Interpolation Definitions and Overview Quadratic Spline Construction Properties Applications Implementation Results Polynomials C k functions Convex functions L norm Conclusion 4 Bibliography 5 1

6 Introduction Interpolation is one of the great branches of numerical mathematics. It is a tool to bridge the gap between discrete points of data and estimate what lies between them. Because of that, it has found use in many scientific disciplines, in many different applications. We shall examine a type of interpolation employing piecewise polynomial functions. In the last fifty years, these functions, called spline functions or splines for short, became quite a popular means of interpolation. They are relatively easy for a computer to both construct and work with and also carry many useful properties. Thanks to these characteristics their usage spans from computer graphics and architecture to numerical integration and solving differential equations. We will focus on quadratic and cubic splines piecewise defined functions consisting of polynomials of degree two and three respectively. In the first chapter, we go through some basic notions, define the concept of a polynomial spline function and introduce the space of polynomial spline functions. We also address the problem of interpolation. In the second chapter, we present various kinds of cubic and quadratic splines. We uncover why quadratic splines require a slightly different approach when constructing an interpolation and we show how to interpolate given data by a spline function, in other words how to construct selected types of splines. We then present some theoretical results concerning error estimates found in literature and, as an example of qualitative results, derive conditions for a cubic spline to preserve convexity of an interpolated function. In the last chapter, we implement algorithms for spline construction, theoretically derived in chapter two, into a computer program. We then use it to produce several examples of spline interpolations and compare their properties with our theoretical findings.

7 1. Elementary Notions and Notation Let us first go through basic notation used throughout this text. The set of all real numbers shall be denoted by R and similarly, the set of all natural numbers by N. By definition, 0 / N. We shall denote open intervals with endpoints a,b R,a < b by (a,b) and closed intervals by [a,b]. In this work, we shall assume finite intervals only. To distinguish vectors and matrices from scalars, we denote vectors by v, matrices by A and scalars simply by x. The first derivative of a function f at a point x in its domain shall be denoted by f (x). Similarly second derivative by f (x) and general n-th by f (n) (x). As for function spaces, the space of functions continuous on an interval I is denoted by C(I), the space of functions whose k-th derivative exists and is continuous is denoted by C k (I), the space of Lebesgue p-integrable functions on an interval I is denoted by L p (I) and finally the Sobolev space over an interval I is denoted by W k,p (I) for definition, see [6]. 1.1 Splines Definition 1.1. Let [a,b] be a closed real interval. Then any set π of the form {x 0,x 1,...,x N+1 R;a = x 0 < x 1... < x N+1 = b} is called a partition of the interval [a,b] and the elements x i,i = 0,...,N + 1 are called nodes of partition π. Moreover, let us denote ν(π) := max{(x i x i 1 );i = 1,...,N + 1} and call it the norm of such a partition. Definition 1.. Let π = {x i ;i = 0,...,N + 1} be a partition of a closed interval [a,b]. Furthermore, let there be h R such that (x i x i 1 ) = h holds for every i = 1,...,N + 1. We say that such a partition is h-uniform. It is clear that h = b a N+1 = ν(π). Definition 1.3. Let [a,b] be a closed finite interval, π be partition of [a,b] with nodes x i,i = 0,...,N+1, let z be a vector in N N with components z i,i = 1,...,N and let m N. We say that a real function s(x) on a domain [a,b] is a polynomial spline function of degree m 1 if the following conditions hold: (i) There are polynomial functions s j,j = 0,...,N of degree m 1, such that s [xj,x j+1 ] = s j. (ii) For every i {1,...,N}, components z i satisfy z i m. (iii) If z i < m for any i {1,...,N}, then for every j {0,...,m 1 z} s (j) i 1 (x i) = s (j) i (x i ). 3

8 In the special case of z i = m, no relation between values or derivatives of the two polynomial functions neighboring with x i is required. We shall denote the set of all polynomial spline functions of degree m 1 with nodes x i and vector z by S(m,π,z). The vector z from Definition 1.3 is sometimes called the multiplicity vector of the spline function and affects the smoothness of the spline. The set S(m,π,z) is clearly also a vector space. By counting the degrees of freedom, it can be seen that its dimension is N dims(m,π,z) = m + z i. For a detailed proof, see [1]. Since z used in practice is often constant, i.e. z 1 =... = z N = z N, let us denote S(m,π,z) = S(m,π, (z,...,z)) by S(m,π,z). 1. Interpolation Let us consider the following problem. Let there be given a finite set of data consisting of ordered pairs {(x i ;f i )} N+1 i=0 R, such that there are no i,j {0,...,N + 1},i j, such that x i = x j while f i f j. We seek a real function f X, such that for every i = 0,...,N +1, its domain contains x i and f satisfies f(x i ) = f i, where X is a given function space. Definition 1.4. Let there be given a set D = {(x i ;f i )} N+1 i=0, a function space X and a real function f, such that D, X and f satisfy all the conditions stated above. We shall call such an f an interpolating function of the given set in the space X. Without loss of generality, we shall assume from now on that the data given consists of N + unique pairs, which are moreover ordered with respect to x i, so that x 0 < x 1 <... < x N+1. This is for the sake of practicality and also ensures the existence of N + different conditions on values of the sought function. Interpolation problem can be approached in different ways. Note that in our definition, we only require values of an interpolating function to fit given data. Such an interpolation is then called a Lagrange interpolation. On the other hand, we could pose conditions on its derivatives, demanding their existence and fitting of given data. That is fairly common and such an approach is then called Hermite interpolation. It is often needed for the resulting function f to have some additional useful properties. This can be ensured by a proper choice of the space X. For example we could choose the space of polynomials of degree at most N +1. Note that the degree is chosen so that we have just enough data to uniquely determine such a polynomial. This is a very straightforward approach to obtain a very well- -behaved interpolating function for small N. However, if N is large, undesired oscillations in the interpolating polynomial are to be expected in general. To avoid this kind of problems, we may naturally choose to limit the degree of polynomials used to interpolate. That would generally mean over-defining the polynomial, so instead we can choose to use more polynomials at once a piecewise-polynomial interpolating function, such as a spline function. 4 i=1

9 . Theory.1 Cubic Spline Interpolation.1.1 Definitions and Overview Definition.1. Let there be given a finite real interval [a,b] and f W 4, ([a,b]). Let π = {x i } N+1 0 be a partition of the interval [a,b] and let z satisfy all the conditions of Definition 1.3 with m = 4. The function s : [a,b] R is called a cubic spline interpolation of the function f if the following conditions are met: (i) The function s is an element of S(4,π,z). (ii) We have s (j) (x i ) = f (j) (x i ) for every i {1,...,N} and every j {0,...,z i 1}. The special case of s S(4,π, ) is called a Hermite cubic spline. With our definition of interpolation in mind, we will only study splines s S(4,π, 1). Note that the definition above poses only N i=1 z i conditions on the function s, but dims(4,π,z) = 4 + N i=1 z i. That means we need to prescribe four more conditions. There are many ways of choosing these; for example, the following seven are mentioned in []: (i) Conditions for type I spline interpolation: s(x 0 ) = f(x 0 ), s (x 0 ) = f (x 0 ), s(x N+1 ) = f(x N+1 ), s (x N+1 ) = f (x N+1 ). (ii) Conditions for type II spline interpolation: s(x 0 ) = f(x 0 ), s (x 0 ) = f (x 0 ), s(x N+1 ) = f(x N+1 ), s (x N+1 ) = f (x N+1 ). (iii) Conditions for type III spline interpolation: s(x 0 ) = f(x 0 ), s(x N+1 ) = f(x N+1 ), s (x 0 ) = s (x N+1 ), s (x 0 ) =s (x N+1 ). (iv) Conditions for periodic spline interpolation: s(x 0 ) = s(x N+1 ), s (x 0 ) = s (x N+1 ), s (x 0 ) =s (x N+1 ), s (x 0 ) = s (x N+1 ). (v) Conditions for natural spline interpolation: (vi) Conditions (vii) Conditions s(x 0 ) = s(x 1 ) = s(x N ) = s(x N+1 ) = 0. s(x 0 ) = f(x 0 ), s(x N+1 ) = f(x N+1 ), s (x 0 ) = s (x N+1 ) = 0. s(x 0 ) = f(x 0 ), s(x N+1 ) = f(x N+1 ), x1 xn+1 x 0 s(x)dx = x N s(x)dx = x1 x 0 xn+1 x N f(x)dx, f(x)dx. 5

10 .1. Cubic Spline Construction In this section, we will construct a cubic spline interpolation from given data, specifically type II interpolation. We will study construction of a type I cubic interpolation later on, though not in great detail. The types I and II will be our main focus as we present some of their properties in section.1.3. Let there be a real interval [a,b] and let π = {x i } N+1 i=0 be its partition. There is no need for an actual function to be given since we only need to know the values in knots of π. Having that in mind, let there be given values f i R,i = 0,...,N +1 representing an interpolated function on the interval [a,b] and also f 0 and f N+1, standing for f (x 0 ) and f (x N+1 ). We have now all the data needed for a type II interpolation. Let us now introduce the following practical notation. For s S(4,π, 1), let us denote: s i : = s(x i ), s i : = s (x i ), s i : = s (x i ), for i = 0,...,N + 1. Moreover let us denote the left and right third derivatives of s in node x i respectively, by s i : = lim x x i s (x) and s i+ : = lim x x i + s (x). Lemma.. Let [a,b] be a closed real interval, π = {x 0,x 1,...,x N+1 } its partition and s S(4,π, (1,...,1)). Let us set h i := x i+1 x i,i = 0,...,N.Then for every i = 1,...,N we have 6 s i+1 s i h i 6 s i s i 1 h i 1 = h i 1 s i 1 + (h i 1 + h i )s i + h i s i+1. Proof. Let i {1,...N}. Using a Taylor series expansion of the function s(x) about x i, we have By simple manipulation, we obtain s i+1 = s i + s ih i + h i s i + h3 i 6 s i+, s i 1 = s i s ih i 1 + h i 1 s i h3 i 1 6 s i. 6 s i+1 s i h i = 6s i + 3h i s i + h is i+, 6 s i s i 1 h i 1 = 6s i + 3h i 1 s i + h i 1s i. And by adding the two equations, we have 6 s i+1 s i h i 6 s i s i 1 h i 1 = 3s i (h i + h i 1 ) + h is i+ h i 1s i. (.1) Expanding s (x) about x i then gives us s i+1 = s i + h i s i+, s i 1 = s i h i 1 s i, 6

11 from which we get h is i+ = h i s i + h i s i+1, h i 1s i = h i 1 s i h i 1 s i 1. Finally, substituting this into equation (.1), we have 6 s i+1 s i h i 6 s i s i 1 h i 1 = 3s i (h i + h i 1 ) h i s i + h i s i+1 h i 1 s i + h i 1 s i 1. We will mostly focus on h-uniform partitions, in which case we can rewrite the statement of Lemma. as s i 1 + 4s i + s i+1 = 6 s i+1 s i + s i 1 h. (.) We will continue to use the notation of h i in the case of non-uniform partition from now on. We only need to find values of s i,i {1,...,N} to be able to construct the spline, since we can then uniquely define all the polynomials, of which the spline consist, as is shown in the following lemma. Lemma.3. Let p(x) be a cubic polynomial and let us write p(x) = ax 3 + bx + cx + d, where a,b,c,d R; let x 1,x,p 1,p,p 1,p R and x1 < x. Then p(x) satisfies p(x 1 ) = p 1, p(x ) = p, p (x 1 ) = p 1, and p (x ) = p if and only if its coefficients satisfy a = p 1 p 6(x 1 x ), b = p 1 6ax 1, c = p 1 p a(x 3 1 x 3 ) b(x 1 x ) x 1 x, d = p 1 ax 3 1 bx 1 cx 1. Proof. By differentiating p twice, we have By subtracting (.4) from (.3), we then have p (x 1 ) = p 1 = 6ax 1 + b, (.3) p (x ) = p = 6ax + b. (.4) And then 6a(x 1 x ) = p 1 p, a = p 1 p 6(x 1 x ). b = p 1 6ax 1. 7

12 Now, by evaluating p(x) at x 1 and x, we obtain p(x 1 ) = p 1 = ax bx 1 + cx 1 + d, (.5) p(x ) = p = ax 3 + bx + cx + d. (.6) In a similar fashion, subtracting of (.6) from (.5) gives us c = p 1 p a(x 3 1 x 3 ) b(x 1 x ) x 1 x and finally d = p 1 ax 3 1 bx 1 cx 1. We now need to find the values of s i,i {1,...,N}. Using Lemma., we see we can do this by solving a system of linear equations of the form Ay = b, where y T = (s 1,s,...,s N), (h 0 + h 1 ) h h 1 (h 1 + h ) h... A = h N (h N + h N 1 ) h N h N 1 (h N 1 + h N ) is an N N b i = (.7) matrix and b T = (b i ) N i=1 R N is a vector, whose components are 6 f i+1 f i h i 6 f f 1 h 1 6 f N+1 f N h N 6 f i f i 1 h i 1, i {,...,N 1}, 6 f 1 f i 0 h 0 h 0 f 0, i = 1, 6 f N f N 1 h N 1 h N f N+1, i = N. In the case of an h-uniform partition, A and b can be rewritten in a simpler form: A = (.8) b i = 6 f i+1 f i h 6 f f 1 h 6 f N+1 f N h 6 f i f i 1, i {,...,N 1}, h 6 f 1 f i 0 f h 0, i = 1, 6 f N f N 1 f h N+1, i = N. 8

13 Definition.4. Let M = (m i,j ) n i,j=1 be a matrix. If a ii j i a ij, for all i {1,...,n} we say that A is diagonally dominant. Moreover, if all the inequalities are strict, we call matrix A strictly diagonally dominant. Both of the matrices (.7) and (.8) are tridiagonal, so it is easy to see that both of them are also strictly diagonally dominant. Let us now recall a classical result from linear algebra. Lemma.5. Let A be a strictly diagonally dominant matrix. Then A is regular. Knowing this, we see that our system has a unique solution. Once we have it, Lemma.3 enables us to construct all the polynomials needed for the spline function we sought. The system can be solved in a very standard way using Gaussian elimination. Moreover, since the matrices are strictly diagonally dominant, no pivoting is necessary, as is shown in [3]. Other types can be constructed in a similar way, or through only small modifications of the system of equations. Let us now show a way of constructing a type I cubic spline interpolation. For simplicity, we will restrict ourselves to h-uniform partitions only and will not go into details, since it is similar to the construction of the previous type. Lemma.6. Let [a,b] be a closed real interval, π = {x 0,x 1,...,x N+1 } its h- -uniform partition and s S(4,π, (1,...,1)). Then for every i = 1,...,N we have 3 s i+1 s i 1 = s i 1 + 4s i + s h i+1. Proof. Using the Taylor expansion of s(x) about x i gives us s i+1 = s i + s ih + h s i + h3 6 s i+, s i 1 = s i s ih + h s i h3 6 s i, s i+1 s i 1 = hs i + h3 6 (s i+ + s i ). Next, using the Taylor expansion of s (x) about x i gives us And therefore s i+1 = s i + s i h + h s i+, s i 1 = s i s i h + h s i, h (s i+ + s i ) = s i+1 + s i 1 hs i. s i+1 s i 1 = hs i + h 3 (s i+1 + s i 1 hs i). The lemma s statement follows immediately. 9

14 Let there be given data f i i = 0,...,N + 1, f 0 and f N+1 This then allows us to construct a type I cubic spline interpolation using the following system: s 1 s. s N 1 s N = 3 f f 0 h 3 f 3 f 1 h. 3 f N f N h 3 f N+1 f N 1 h (.9) The matrix is tridiagonal and strictly diagonally dominant and due to Lemma.5 also regular, so a unique solution exists. Similar to Lemma.3, coefficients of the spline s polynomials can be then calculated using the values s i and s i, i = 0...,N Properties The three following theorems are adopted from []. They provide estimates for error of cubic spline interpolation. Theorem.7. Let there be given a function f C 4 ([a,b]). Let π = {x i } N+1 i=0 be a h-uniform partition of [a,b] and h = b a. Let s S(4,π, 1) be a type II cubic N+1 spline interpolation of the function f, i.e. s(x i ) = f(x i ),i {0,...,N + 1},s (x 0 ) = f (x 0 ),s (x N+1 ) = f (x N+1 ). Let us set M := max x [a,b] f(4) (x). Then the following estimates hold: (i) max x [a,b] f(x) s(x) 7 3 Mh4, (ii) max x [a,b] f (x) s (x) 7 16 Mh3, (iii) max f (x) s (x) 7 x [a,b] 8 Mh, (iv) sup{ f (x) s (x) : x [a,b] \ {x i } N+1 i=0 } Mh. Theorem.8. Let j = or j = 3 and let there be given a function f C j ([a,b]). Let π = {x i } N+1 i=0 be a h-uniform partition of [a,b] and h = b a. Let s S(4,π, 1) N+1 be a type II cubic spline interpolation of the function f. Let us set ω(f (j),h) := sup f (j) (x) f (j) (y), C := 4 and C 3 := 5. Then the following estimates hold: 4 x y h x,y [a,b] (i) max f(x) s(x) 1C x [a,b] 4 jh j ω(f (j),h), (ii) max x [a,b] f (x) s (x) 1 C jh j 1 ω(f (j),h), (iii) max f (x) s (x) C j h j ω(f (j),h). x [a,b] 10

15 Theorem.9. Let k {, 3, 4}, let π = {x i } N+1 i=0 be a partition of an interval [a,b] and let there be given a function f W k, ([a,b]). Let s S(4,π, 1) be a type I interpolation of the function f and let us set h := max{(x i x i 1 ); i = 1,...,N + 1}. Then there exists C R, C > 0, so that the following estimates hold: (f s) (j) L ([a,b]) Chk j f (k) L, j {0, 1, }, ([a,b]) (f s) (j) L ([a,b]) Chk j 1 f (k) L, j {0, 1}. ([a,b]) Let us present a theorem found in [5], but first, let us define the following function space of functions with absolutely continuous n-th derivative: { AC n+1 ([a,b]) := f C n ([a,b]) : f (n+1) L ([a,b]) f (n) (β) f (n) (α) = β α f(n+1) (ξ)dξ, α,β [a,b] } Theorem.10. Let there be given a function f AC 4 ([a,b]) and an arbitrary partition π of the interval [a,b]. Let s S(4,π, 1) be a type I or a type II cubic spline interpolation of the function f. Let us set h := max (x h i+1 x i ), β :=, i=0,...,n min (x i+1 x i ) i=0,...,n C 0 := 8, C := 1, C 4 := 3, C 8 3 := β+β 1. Let k {0, 1,, 3}. Then we have the following estimate: f (k) s (k) L ([a,b]) C kh 4 k f (4) L ([a,b]). Moreover C 0 and C 1 are optimal. It is also possible to study qualitative properties. We shall attempt to formulate conditions sufficient to ensure that a type II spline interpolation preserves convexity of an interpolated function. Let us denote the fact that a matrix A (or a vector u) have all components non-negative by A 0 (or u 0). Definition.11. Let A = (a ij ) n i,j=1 be a strictly diagonally dominant matrix satisfying { > 0, i = j, a ij = 0, i j, for all i,j = 1,...,n. Then A is called an M-matrix. Lemma.1. Let A be a M-matrix. Then A 1 exists and all its elements are non-negative. Proof. The existence of A 1 follows from its strict diagonal dominance and Lemma.5. By way of contradiction, let us suppose that a given matrix A satisfies all premises, but it is not true that A 1 0. Let us have B = (b kl ) n k,l=1 = A 1, then there must exist at least one element b ij < 0. Moreover, let us select the least one, so then b ij < b kl for all k,l = 1,...,n. Let us write e k = (0,...,0, 1, 0,...,0) 11

16 where the 1 is on the k-th position. Then A 1 e j is the j-th column of A 1 = B, in other words, a vector with b ij on its i-th position. We then have e j = AA 1 e j = A(A 1 e j ). First, let us assume that i j the case of i = j will be considered later. Let us now compare the i-th row of both sides of the equation. Using the non-negativity of off diagonal a ik and the diagonal dominance, we have: 0 = n a ik b kj k=1 = a ii b ij + a ik b kj + k i, k i, b kj 0 b kj <0 = a ii b ij + a ik b kj + k i, k i, b kj 0 b kj <0 a ik b kj a ik b kj a ii b ij + a ik b kj + b ij a ik k i, k i, b kj 0 }{{ } b kj <0 0 < a ii b ij b ij a ii = a ii b ij b ij a ii = 0, so we obtain that 0 < 0 which is a contradiction. Now, we shall study the case of i = j. We again compare the i-th row of both sides of the equation: 1 = n a ik b ki k=1 Using essentially identical calculations as seen above, we would finally obtain, that 1 < 0. Therefore A 1 must be non-negative. If A 1 0, the following implication clearly holds for all u R n : Au 0 u 0. Let us now recall the matrix A from (.8) and the corresponding vector b from (.1.). We will rewrite the system in a slightly different form: s f 0 0 s f 0 f 1 +f s h 6 f 1 f +f 3 h = s N 1 6 f N f N 1 +f N s h N 6 f N 1 f N +f N s h N+1 f N+1 1

17 Let us denote this extended matrix by C and the right-hand side vector by d = (d 0,...,d N+1 ) T. And now we will transform it into an M-matrix, so that we can use Lemma.1. Lines of the transformed matrix are obtained in the following way: The first and the last line are unchanged. The second line is obtained as a result of linear combination of the first three lines of the matrix C with coefficients 1, 1 and 1 in that order. 4 The second to last line is obtained as a result of linear combination of the last three lines of the matrix C with coefficients 1, 1 and 1 in that order. 4 For i = 3,...,N, the i-th line is obtained as a result of linear combination of the (i 1)-th, i-th and (i + 1)-th line of the matrix C with coefficients 1, 1 and 4 1 in that order. 4 The resulting matrix is then (4 1) (4 1) =: M (4 1 4 ) (4 1) Lines of the right-hand side vectors are transformed in the same way, so that the resulting system is equivalent to the original one. Let us denote this new vector by p = (p 0,...,p N+1 ) T. We can see that Lemma.1 is now applicable and therefore the matrix M is an M-matrix. We can now consider sufficient conditions enabling a type II interpolating spline to preserve convexity of an interpolated function. First let us note that since second derivative of a cubic polynomial is a linear function, thus we have s i > 0 and s i+1 > 0 s (x) > 0,x [x i,x i+1 ] for all i = 0,...,N. That means that s i > 0 i 0,...,N + 1 is a sufficient condition for a cubic spline to be convex. Let the interpolated function be convex on [x 0,x N+1 ]. Since the matrix M is an M-matrix, the type II interpolation will preserve the convexity if p 0. That gives us the following conditions: p 0 = f 0 0, p N+1 = f N+1 0, p 1 = 6 (4f 4h 0 9f 1 + 6f f 3 ) f 0 0, p i = 6 ( f 4h i + 6f i 1 10f i + 6f i+1 f i+ ) 0, i =,...,(N 1), p N = 6 (4f 4h N+1 9f N + 6f N 1 fn ) f N

18 Let us now assume f C 3 ([x 0,x N+1 ]). Employing Taylor expansion, we can reformulate the last three conditions as h f (x i ) + O(h 3 ) 0, i = 1,...,N. This is satisfied e.g. for f (x i ) > 0 and a sufficiently small h. We can sum up our findings in the following theorem. Theorem.13. Let f C 3 ([a,b]) and let f (x) > 0, x [a,b]. Then there exists h 0 > 0, so that for all 0 < h h 0 such that π = {a + ih} N+1 i=0 is an h- uniform partition of [a, b], a type II cubic spline interpolation s(x) of the function f(x) satisfies s (x) > 0, x [a,b].. Quadratic Spline Interpolation..1 Definitions and Overview In the case of a quadratic spline interpolation on a given partition π = {x i } N+1 i=1, we consider the space S(3,π,z). As in section.1.1, we shall only study the case of z = (1, 1,...,1), or in other words the space S(3,π, 1). Not only is it a good choice for interpolation but also it is the only option if we want our spline function to be an element of at least C 1. Since dims(3,π, 1) = N + 3 and we would like to prescribe N values for the interpolating function, we are left with only three remaining conditions to set. Moreover, since we usually wish to control the values in the endpoints x 0 and x N+1 as well, only a single condition remains. Since the commonly used condition are often placed at the endpoints, we are then forced to pick only one of them. To enable a more symmetrical set of data to be used, we will prescribe values for the spline function on a new set of N + 1 knots. These new knots are called spline knots to distinguish them from the partition knots the elements of the partition π. Spline knots {t i } N+1 i=1 on an interval [a,b] with a partition π = {x 0,...,x N+1 } need to satisfy: a = x 0 < t 1 < x 1 <... < x N < t N+1 < x N+1 = b Values for the quadratic spline interpolating a function f C 1 ([a,b]) are then prescribed in these N + 1 points as s(t i ) = f(t i ), i = 1,...,N + 1. We have now two conditions to pose on the spline. The most common choices listed in [] are: (i) Conditions for type I quadratic interpolation: s(x 0 ) = s(x N+1 ), s (x 0 ) = s (x N+1 ). (ii) Conditions for type II quadratic interpolation: s(x 0 ) = f(x 0 ), s(x N+1 ) = f(x N+1 ). (iii) Conditions for type III quadratic interpolation: s (x 0 ) = f (x 0 ), s (x N+1 ) = f (x N+1 ). 14

19 .. Quadratic Spline Construction We will show the algorithm for constructing a type III quadratic spline interpolation, since we will present a standard error estimate for this type, found in []. Let there be given a finite real interval [a,b], its partition π = {x 0,...,x N+1 } and a suitable set of spline knots {t 1,...,t N+1 }. Moreover let there be given a set of data {f 1,...,f N +1} representing interpolated function values. Finally, let there be given f a,f b R. We seek a function s S(3,π, 1) satisfying s (x 0 ) = f a, s(t i ) = f i, i 1,...,N + 1, s (x N+1 ) = f b. We shall introduce the following practical notation. Let s S(3, π, 1) and i {0,...,N + 1}. We then write s i := s(x i ), s i := s (x i ), s i+ := lim x x i + s (x), s i := lim x x i s (x). Lemma.14. Let there be given a finite real interval [a,b], its partition π = {x 0,...,x N+1 } and spline knots {t 1,...,t N+1 } satisfying a = x 0 < t 1 < x 1 <... < x N < t N+1 < x N+1 = b. Let s S(3,π, 1) and i {1,...,N}. Let us set g i := s(t i ) We have ) g i+1 g i = s (t i+1 x i ) i+1 + (x i+1 x i ) s i (t i+1 t i (t i+1 x i ) (t i x i ) (x i+1 x i ) (x i x i 1 + s ) i 1 Proof. First, by expanding s(x) about x i into a Taylor series, we have By subtracting (.10) - (.11), we get (t i x i ). (x i x i 1 ) g i+1 = s i + s i(t i+1 x i ) + s (t i+1 x i ) i+, (.10) g i = s i s i(x i t i ) + s (x i t i ) i. (.11) g i+1 g i = s i(t i+1 t i ) + s (t i+1 x i ) i+ s (x i t i ) i. (.1) Next, we expand s (x) about x i and get And therefore s i+1 = s i + s i+(x i+1 x i ), s i 1 = s i + s i (x i x i 1 ). s i+ = s i+1 s i x i+1 x i, s i = s i s i 1 x i x i 1. Finally, by substituting these into (.1) we obtain g i+1 g i = s i(t i+1 t i ) + (s i+1 s i )(t i+1 x i ) (x i+1 x i ) (s i s i 1 )(x i t i ) (x i x i 1 ). The lemma s proposition follows directly. 15

20 Lemma.15. Let p(x) be a quadratic polynomial and let us write p(x) = ax + bx + c, where a,b,c R; let x 1,x,t,f,p 1,p R and x 1 < t < x. Then p(x) satisfies p (x 1 ) = p 1, p (x ) = p and p(t) = f if and only if its coefficients satisfy Proof. By differentiating p, we have a = p p 1 (x x 1 ), b = p 1 ax 1, c = f at bt. p 1 = ax 1 + b, p = ax + b. Subtraction of these two equations gives us And then we obtain Finally, by evaluating p at t, we obtain a(x x 1 ) = p p 1, a = p p 1 (x x 1 ). b = p 1 ax 1 f = at + bt + c, c = f at bt. To construct the spline, we now only need to find the values of s i,i {1,...,N}. Let us set k i = (t i x i ) (x i x i 1 ), l i = (t i+1 t i (t i+1 x i ) (x i+1 x i ) (t i x i ) (x i x i 1 ) m i = (t i+1 x i ) (x i+1 x i ). ), Using Lemma.14, we see we can once again do this by solving a system of linear equations of the form Ay = b, where y T = (s 1,s,...,s N), 16

21 l 1 m k l m.... A = k N 1 l N 1 m N k N l N and b R N with components f i+1 f i, i {,...,N 1}, b i = f f 1 f ak 1, i = 1, f N+1 f N f b m N, i = N. (.13) Let us now consider a h-uniform partition of the given interval. Additionally, let us consider the following special set of spline knots: t i := x i 1+x i,i = 1,...,N. In this case, A and b can be rewritten in the following simple form: A = (f h i+1 f i ), i {,...,N 1}, 8 b i = h (f f 1 f ak 1 ), i = 1, 8 (f h N+1 f N f b m N), i = N. We will now study diagonal dominance of these matrices. We have l i = t i+1 t i (t i+1 x i ) (x i+1 x i ) (t i x i ) (x i x i 1 ) = (t i+1 x i ) + (x i t i ) (t i+1 x i ) (x i+1 x i ) (t i x i ) (x i x i 1 ) (.14) = (t i+1 x i ) (x i+1 x i ) (x i+1 x i ) + (x i t i ) (x i x i 1 ) (x i x i 1 ) (t i+1 x i ) (x i+1 x i ) (t i x i ) (x i x i 1 ) = (t i+1 x i ) (x i+1 x i ) (x i+1 x i t i+1 + x i ) + (x i t i ) (x i x i 1 ) (x i x i 1 t i + x i ) = (t i+1 x i ) (x i+1 x i ) (x i+1 x i t i+1 + x i ) + (x i t i ) }{{} (x i x i 1 ) (x i x i 1 t i + x i ), }{{} > m i + k i >(t i+1 x i )>0 >(x i t i )>0 so we see that the matrix (.13), as well as the matrix (.14), is strictly diagonally dominant. Then using Lemma.5, we know that a unique solution of the system exists and can be found. The solution can be obtained using Gaussian elimination. Thanks to the strict diagonal dominance no pivoting is necessary, again referring to [3]. Finally, we can use Lemma.15 to construct the spline function. Other types of quadratic splines can be constructed via modifications of this algorithm or using similar procedure. 17

22 ..3 Properties The following theorem concerning the interpolation error is adopted from []. Theorem.16. Let there be given a function f C 3 ([a,b]). Let π = {x i } N+1 i=0 be a h-uniform partition of [a,b] and h = b a. Let s S(3,π, 1) be a type III N+1 quadratic spline interpolation of the function f with spline knots {t i } N+1 i=1, where t i = x i 1+x i, i.e. s(t i ) = f(t i ),i {1,...,N + 1},s (x 0 ) = f (x 0 ),s (x N+1 ) = f (x N+1 ). Let us set M := max f (3) (x). Then the following estimates hold: x [a,b] (i) max x [a,b] f(x) s(x) 7 4 Mh3, (ii) max x [a,b] f (x) s (x) 7 1 Mh, (iii) sup{ f (x) s (x) : x [a,b] \ {x i } N+1 i=0 } 11 1 Mh. Let us present a quadratic equivalent of Theorem.10, found in [5]. Theorem.17. Let there be given a function f AC 3 ([a,b]) and an arbitrary partition π of the interval [a,b]. Let s S(3,π, 1) be a type I, type II or a type III quadratic spline interpolation of the function f. Let us set h := max (x i+1 x i ), β := i=0,...,n C 0 := 1 4, C 1 := 7 4, C := h, min (x i+1 x i ) i=0,...,n { 5 6 if 1 β 3, β+β 1 if β > 3. Let k {0, 1, }. Then we have the following estimate: f (k) s (k) L ([a,b]) C kh 3 k f (3) L ([a,b]). Moreover C 0 is optimal. 18

23 3. Applications 3.1 Implementation In the previous chapter, algorithms for spline construction were introduced. We have programmed a simple Java-based application which allows user to construct certain types of splines. The runnable program is located on an enclosed CD and is considered an integral part of this thesis. Since its source code, complete with commentary, is too long to be included in this text, it is also present on the CD. After running the application, the user can choose one of the four offered spline interpolation types, namely cubic spline interpolations types I and II and quadratic spline interpolation types II and III. Afterwards, the user should supply appropriate data. One plain-text file and two numerical entries are required for cubic splines and two plain-text files and two numerical entries for quadratic splines. The files are selected via a dialog window or can be specified by their path. The numerical data are supplied via text input fields marked as left and right endpoint data. The meaning of the two numerical entries is dependent on the spline type chosen. In the cases of quadratic type III and cubic type I interpolation, they represent a desired value of the first derivative of the spline function at the interval endpoints. In the case of quadratic type II interpolation, they set the spline values at the endpoints and finally if the selected spline interpolation is cubic type II, they represent the values of the second derivative of the spline function at the endpoints. The files specified by the user need to be of the following format. Cubic splines require only one file containing the partition knots and interpolated values at those knots. Each line of the file represents one pair of a knot and its value, separated by a semicolon. Lines must be in ascending order with respect to the knots. In the case of quadratic splines, one file contains partition knots, while the other contains spline knots and their respective values. The former one is simply a list of ordered knots, one entry per line. The latter is then very similar to the file required for a cubic spline and on each line contains 1 pair consisting of an spline knot and desired value at said knot, once again separated by a semicolon. Sample input files are included on the enclosed CD. The program reads the given data and constructs a matrix appropriate for the selected task, accompanied by a vector of right-hand sides to complete the set of linear equations, as it was described in the previous chapters. We take advantage of the fact, that all considered matrices are tridiagonal, so only four vectors are kept in the memory the three diagonals and the mentioned vector instead of the whole matrix. The program then solves this system via Gaussian elimination without pivoting. We yet again remark, that no pivoting is needed, as is shown in [3]. Once the solution is known, polynomial coefficients are calculated through formulas described in Lemma.3 and Lemma.15. Since the spline function consists of polynomials, it is essentially constructed at this point. There are two forms of output; the constructed spline function is graphed on the computer s monitor and the list of polynomial coefficients is saved to a file output.txt 19

24 located in the same folder as the program. One line is saved for every subinterval of the given partition, in other words for every polynomial part of the spline function. Each line then consists of four coefficients in descending order of degree, separated by a semicolon. In the case of a quadratic spline, the first coefficient is then always zero. If we consider a cubic polynomial of the form ax 3 + bx + cx + d, where a,b,c,d R, then the corresponding line in the output file would be in the form a;b;c;d. 3. Results We shall now study some particular cases of interpolation. We have presented several theorems concerning interpolation error estimates. Let us choose some examples to test the sharpness of these results Polynomials We shall start with polynomial functions. Due to the uniqueness of spline interpolation, cubic splines should exactly interpolate data representing polynomials of degree at most 3. Similarly quadratic splines should be precise for data representing polynomials of degree at most. Figure 3.1: Interpolations of the function x 3 on a 0.5-uniform partition of the interval [ 1, 1]. From left to right: cubic type II and quadratic type III interpolations. Figure 3.1 then shows results of interpolation of a cubic polynomial using a cubic and a quadratic spline on a 0.5-uniform partition of [ 1, 1]. The difference between the graphs is not clear at first sight. Using suitable software, for example Matlab, we can attempt to calculate, or at least approximate, values of max x [ 1,1] x 3 s(x). This approach yielded the following results. The maximal error in the above sense using type II cubic interpolation is approximately The maximal error using type III quadratic interpolation is approximately While the cubic interpolation is precise up to machine precision, the error of quadratic interpolation is substantially greater. This should be attributed to the fact, that the degree of the interpolated polynomial is greater then the degree of the interpolating one. 0

25 Theorem.7 gives us estimates for functions f C 4 in the case of type II cubic spline interpolation. Since x 3 C, we can apply it in this situation. Similarly, Theorem.16 is applicable to the type III quadratic spline interpolation. Let us set f(x) := x 3 and let us now calculate M 3 := max x [ 1,1] f(3) (x) and M 4 := max x [ 1,1] f (4) (x) the constants used in the mentioned theorems. Since M 4 = 0, the precise interpolation by cubic spline could have been expected. On the other hand, M 3 = 6 and the stated estimate for quadratic spline interpolation therefore is so in this case of such a smooth function, the estimate is about four times too high. 3.. C k functions We shall now study functions that have only a finite number of continuous derivatives. Specifically, let us define f k (x) := x k sgn(x) for k N. Then f k C k 1 but f k / C k. Figure 3. shows the result of interpolating f 4 with a type II cubic spline. Since f 4 / C 4, we can not use Theorem.7 to determine error bounds of such an interpolation. Figure 3.: A type II cubic spline interpolation of f 4 (x) on the interval [ 1, 1] with 0.1-uniform partition. The computed maximal error is max x [ 1,1] f 4 (x) s(x) = We can now compare it with estimates of Theorem.8 with j = 3, which is suitable for these circumstances. Firstly, by direct computation, we have ω(f (3) 4, 0.1) =.4, so the theorem then gives us max x [ 1,1] f 4 (x) s(x) and the estimate in this case is about five times too high Convex functions Let us now study an interpolation of the following function { 10 3 x 6, x [ 1, 0] g(x) = 10 3 x 4, x (0, 1] Such a function is strictly convex, even though g (0) = 0, and also g C 3 ([ 1, 1]) while g / C 4 ([ 1, 1]). In section.1.3, we have formulated Theorem.13 concerning type II cubic interpolations. Note that the function g(x) does not meet 1

26 all of the conditions of the mentioned theorem, namely g (x) > 0 is not true for all x [ 1, 1]. We are interested in seeing, whether the unsatisfied conditions are necessary in this case. Figure 3.3: A type II cubic interpolation of the function g(x) on a 0.-uniform partition of the interval [ 1, 1]. Figure 3.3 shows the output of type II cubic interpolation of the function g(x). It does seem that the interpolation came out convex, but it lacks needed detail. We can use Matlab to graph our spline function in more detail, as is shown in figure 3.4. Figure 3.4: Type II cubic interpolation of the function g(x) on 0.-uniform partition of the interval [ 1, 1] magnified in Matlab to provide more detail. We can now clearly see, that the spline function interpolating the function g(x) is not convex. We can assume that this is either caused by too rough interval partition or can be accounted to the fact that g (0) = L norm Theorem.9 enables us to estimate the error of a type I interpolation in the L -norm, though it is hardly applicable, since no explicit value of the constant

27 C (see the theorem) is provided. We can calculate the value of the error for some particular function though. We can for example choose exp(x). This choice ensure comfortable differentiating, but as a downside, the estimate will probably be quite understated, since this function is infinitely differentiable and thus very far from a worst case scenario function. Figure 3.5: Type I cubic interpolation of the function exp(x) on a 0.-uniform partition of the interval [ 1, 1]. Figure 3.5 shows type I cubic spline interpolation of exp(x) on a 0.-uniform partition of the interval [ 1, 1]. Using Matlab, we will compute errors of derivatives in L -norm. We obtain: exp s L (I) , exp s L (I) , exp s L (I) We can now directly compute how much would C have to be at least for the estimates to hold. We calculate exp L (I) 1.9. We shall set k = 4. Then we obtain by direct calculation from the three computed results, that C 0.014, C 0, 04, C 0.. So we have experimentally determined that C 0. for it to be valid in the estimates of Theorem.9. It shows that the constant C can be expected to be reasonably large. Let us mention that, unlike the previously used polynomials, the exp function is only approximated by standard numerical procedure in the program. This introduces a new possible source of interpolation error. 3

28 Conclusion We have shown basic variants of quadratic and cubic spline interpolations together with the algorithm to construct selected ones of them. While providing a summary of basic approximation and qualitative properties, we have proved the existence and uniqueness of spline interpolation, as well as conditions sufficient for preservation of the convexity in the interpolation. We were also able to test some of the results on sample problems. Specifically, we tested the sharpness of the presented interpolation error estimates for various functions satisfying the theorem sharply, as well as nicer functions than the theorems require. In one case, when explicit values of constant in the estimates are not known, we evaluate these constants for a nontrivial function. As a consequence, it seems, that these constants are not unreasonably large. Furthermore, as an example of qualitative results, we have tested the sharpness of assumptions from Theorem.13, and shown that without satisfying its conditions, the interpolating spline does not indeed have to be convex. The book [4] provides a detailed overview of theoretical findings as well as many different spline types and approaches to spline interpolation. On the other hand, the authors of [1] present many applications of splines and spline interpolation. Moreover, they provide extensive list of available literature on the topic of splines. Both of these books are suitable for a reader wishing to expand on knowledge concerning splines in general and their many applications. 4

29 Bibliography [1] Micula, Gheorghe and Sanda Micula. Handbook of splines. Boston: Kluwer Academic Publishers, ISBN [] Najzar, Karel. Základy teorie splinů. Praha: Karolinum, 006. ISBN [3] Golub, Gene H. and Charles F. Van Loan. Matrix computations. Baltimore: Johns Hopkins University Press, ISBN [4] Kobza, Jiří. Splajny. Olomouc: Univerzita Palackého, ISBN X. [5] Dubuc, Serge and Gilles Deslauriers. Spline functions and the theory of wavelets. Providence: American Mathematical Society, ISBN [6] Evans, Lawrence C. Partial differential equations. Providence: American Mathematical Society, 010. ISBN