Quadrature Formulae

INTERNATIONALE SCHRIFTENREIHE ZUR NUMERISCHEN MATHEMATIK SERlE INTERNATIONALE D'ANALYSE NUMERIQUE
Editors:
Ok. Blanc, Lausanne; A. Gkizzetti, Roma; A. Ostrowski, Montagnola; J. Todd, Pasadena; H. Unger, Bonn; A. von Wijngaarden, Amsterdam
VOL. 13
QUADRATURE FORMULAE
1970 SPRINGER BASEL AG
ISBN 978-3-0348-5837-3 ISBN 978-3-0348-5836-6 (eBook)
All Rights reserved No Part of this Book may be reproduced in any form, by Photostat, Microfilm.
or any other means, without written permission from the Publishers Library of Congress Catalog Card Number: 76 124441
© Springer Basel AG 1970 Originally published by Birkhiiuser Verlag Basel, in 1970
Softcover reprint of the hardcover 1st edition 1970
DOI 10.1007/978-3-0348-5836-6
PREFACE
Quadrature formulae are generally obtained substituting the integral to b b
evaluate J f(x) dx with the integral J <p(x) dx where <p(x) is a function a a
approximating f(x), whose indefinite integral is expressible in elementary functions. The choice of <p(x) is performed using interpolation methods. Thus the problem of approximating the integral of f(x) is reduced to the approximation of f(x) itself, that is to a problem not equivalent to the first one. Therefore, it is reasonable to think that it possible to obtain quadrature formulae, without using interpolation methods. In a paper of 1913, G. PEANO (see G. PEANO (1]) first made an attempt in this way and succeeded in obtaining Cavalieri-Simpson's formula, with an integral ex pression of the remainder, only by means of integration by parts. This method was systematically employed by R. VON MISES, who (in a paper of 1935; see R. VON MISES [1]) showed how it is possible to get every qua drature formulae with the only tool of integration by parts.
In 1935, also J. RADON (1] showed how the integral expression of the remainder can be obtained by means of the Green-Lagrange identity, relative to a linear differential operator and its adjoint. However, these fundamental works remained nearly unknown and still to-day the ele mentary method suggested by them is not used in every book concerning to quadrature formulae. From 1954, one of us (see A. GHIZZETTI [1], [2], [3], [4]) had the: opportunity to reconsider and to probe the study of Radon's procedure, pointing out that it allows to obtain not only the expression of the remainder, but also of the part approximating the integral. This result can be attained in more general conditions than the usual ones, that is making use of Lebesgue integrals for finite or infinite intervals. In this way a method, which allows to obtain all the known quadrature for mulae and many others, is carried out by a systematic application of some simple rules. It is worthwhile to note that Radon's method includes that of von Mises, since the integrations by parts used by von Mises, are already performed in the Green-Lagrange identity, used by Radon.
In 1965, we have reconsidered the study of these methods and their applications (see A. GHIZZETTI [5]; A. OSSICINI [1], [2], [3]; A. GHIZZETTI and A. OSSICINI [1]), availing us of the valued help of some collaborators of ours. In this way we succeeded in collecting all necessary elements to draw up this book, in which we expound the theory of quadrature formulae, only by means of the above mentioned methods.
6 Preface
This book is divided into six chapters. Chapter 6 must be considered independently; it contains exclusively the solutions to all the problems, proposed to the reader in every preceding chapter. Chapter 1 is devoted to the exposition of some preliminary notions about differential linear operators which possess adjoint operator. In Chapter 2, as application uf the preceding chapter, the general rule for constructing all the quadrature formulae, having a structure analogous to the classical ones, is given and discussed. We call these formulae elementary to distinguish them from the generalized ones, considered in Chapter 5. These formulae are obtained dividing the integration interval in partial intervals and applying an ele mentary formula to everyone of them. These generalized formulae are just those, which we use for the numerical computation of integrals; on this subject some questions arise about convergence, which are considered in Chapter 5 itself.
Chapters 1, 2, 5 are the most important and this book could be limited to them, since Chapters 3, 4 are exclusively devoted to some applications of the general rule expounded in Chapter 2. They could be considered just an collection of problems. However also Chapters 3, 4 are important, since not only they explain, with concrete examples, the power of the general theory, but in addition they present some very difficult problems, whose solutions requires the knowledge of notions and special techniques, useful in some other cases. For this reason, in Chapter 3 we have collected, for reader's convenience, some elements about certain special functions, which OCCUr in the extensive collection of examples of Chapter 4. These examples have been treated by the same method; what varies from one example to another is the technique of remainder estimate, by means of the integral represent ation provided by the method.
Since in this book we make use of a unique method, which few Authors have considered, the closely related Bibliography would be not large; therefore we have added the list of the most recent books on quadrature formulae; to these books we refer readers for many practical questions (for example, numerical table about nodes and coefficients of the classical for mulae). In some paragraphs, we have put a bibliographic note to point out some papers on which we have founded the draft of the paragraph itself. We are grateful to "Birkhiiuser Verlag" which has received this books in one of its valuable collections.
Rome, October, 1969 A. GHIZZETTI . A. OSSICINI
REMARK
Every chapter is divided into paragraphs, which are denoted by a symbol as § 3.4 (to indicate the fourth paragraph of Chapter 3). In every paragraph, formulas are indicated by three numbers into brackets; for instance (3.4.7) means the 7th formula of § 3.4. For theorems, an analogous notation is used, without brackets, and having the 3th number Roman; for instance, theor. 3.4.II means the second theorem of § 3.4. In the end of every chapter 1,2,3,4,5 there is a paragraph concerning problems proposed to readers; their solutions are collected in Chapter 6. These problems are denoted as § 1.5., Problem 2, but of course the corresponding solution of Chapter 6 must be considered.
TABLE OF CONTENTS
Chapter 1. Additional results on linear differential equations . . 11
1.1 Linear differential operators and their adjoints . 11 1.2 The Green-Lagrange identity. . . . . . . . . 12 1.3 Relations between the solutions of a linear differential
equation and those of the adjoint equation .... 13 1.4 Linear differential equations considered in infinite
intervals 18 1.5 Pro blems . . . . 25
Chapter 2. Elementary quadrature formulae and a general procedure for constructing them. . . . . . . . . . . . . . . . . . . 27
2.1 The elementary quadrature formulae on a finite interval 27 2.2 The elementary quadrature formulae on the interval
[a, +(0). . . . . . . . . . . . . . . . . . . . . 32 2.3 Elementary quadrature formulae on the interval (- 00,
+00) . . . . . . . . . . . . . . . . . . 37 2.4 Some remarks on the rule for obtaining quadrature
formulae . . . . 38 2.5 Gauss problem. . 41 2.6 Tchebychef problem 43 2.7 Problems . . . . . 45
Chapter 3. Special functions 47
3.1 Bernouilli polynomials and numbers. 47 3.2 Euler functions. . . . . . . . . 50 3.3 Orthogonal polynomials connected to a given weight
function. . . . . . . . . . . . 52 3.4 Jacobi polynomials. . . . . . . 58 3.5 Particular cases of J aco bi polynomials. 64 3.6 Laguerre polynomials. . . . . . . 68 3.7 Hermite polynomials . . . . . . . . 70 3.8 Some notions on divided differences. . 71 3.9 The s-orthogonal polynomials connected to a given
weight . 74 3.10 Problems 78
10 Table of contents
Chapter 4. Various examples of elementary quadrature formulae 80
4.1 General remarks on the contents of this chapter 80 4.2 Deduction of three classical formulae . . . . . 80 4.3 The Euler-MacLaurin formula . . . . . . . . 84 4.4 Other examples of application of the generale rule. 86 4.5 The Gauss-Jacobi formulae . . . . . . . . 92 4.6 Particular cases of the Gauss-Jacobi formulae 97 4.7 The Bouzitat formulae of the first kind . 101 4.8 The Bouzitat formulae of the second kind 106 4.9 Gauss-Laguerre formulae 110 4.10 Gauss-Hermite formulae. . . . . 114 4.11 Newton-Cotes formulae . . . . . 116 4.12 Examples of Tchebychef problems 124 4.13 Quadrature formulae connected to s-orthogonal poly-
nomials . . . . . . . . . . . . . . . . . . . . . 131 4.14 Some examples of quadrature formulae exact for trigo
nometric polynomials . . . . . . . . . . . . . . . 139 4.15 Problems . . . . . . . . . . . . . . . . . . . . 145
Chapter 5. Generalized quadrature formulae and questions of convergence . . . . . . . . . . . . . . .. .... 148
5.1 Generalized quadrature formulae in the case of the constant weight . . . . . . . . . . . . . .. 148
5.2 Generalized quadrature formulae with any weight 153 5.3 An example of generalized quadrature formula . 159 5.4 Problems . . . 163
Chapter 6. Solutions to problems 164
6.1 Solutions to problems of § 1.5 164 6.2 Solutions to problems of § 2.7 167 6.3 Solutions to problems of § 3.10 . 169 6.4 Solutions to problems of § 4.15 174 6.5 Solutions to problems of § 5.4 183
Bibliography. 191
CHAPTER 1
Additional results on linear differential equations
1.1. Linear differential operators and their adjoints. In this chapter we shall consider exclusively real-valued functions. If [a, b] is a finite interval, we shall call L[a, b] the class of functions f(x) Lebesgue-integrable (summable) in [a, b] and A Ck[a, b] the class offunctions f(x) whose k-th derivative f(k)(x) is absolu tely continuous in [a, b] (k = 0, 1,2, .. .).1) It is known that if f(x) E A Ck[a, b] the derivative t(k+l)(X) exists almost everywhere in [a, b] and that f(k+1)(x) E L[a, b].
Let us consider in [a, b] a linear differential operator of order n (n = 1, 2,3, ... ):
n dn - k
with the following conditions on the coefficients ak(x)
(1.1.1)
ao(x) = 1; ak(x) E A Cn- k- 1 [a, b], (k = 1,2, ... , n - 1); n,,(x) E L[a, b). (1.1.2)
The operator E can be applied to the functions u(x) E A cn-l[a, b], obtaining the function (defined almost everywhere):
n E[u(x)] = E(u) = 2: ak(x) u(n-k)(x) E L[a, b] .
k=O (1.1.3)
In the following ~e shall have occasion to associate with the operator E the reduced operators
, dr - k E, = L ak(x) dxr - k ' (r = 0,1, ... , n - 1) .2) (1.1.4)
k=O
By virtue of the hypotheses (1.1.2) we can consider the so-called adjoint operator E* of the operator E, defined by
n dn - k
n dn - k E* = '\' a*(x)--
.:::... k dxn - k k=O
(1.1.5)
(1.1.6)
1) For k = 0 we have the class A Cora, b] of absolutely continuous functions; this class is also denoted by A C[a, b].
2) For T = n we obtain En = E.
12 1. Additional results on linear differential equations
and an easy computation shows that
at(x) = ~ (- l)n-i (; == ~) a~k-i)(x) , (k = 0, 1, ... , n). (1.1.7)
From (1.1.2) and (1.1.7) easily follow
a~(x) = (- l)n; at(x) E A on-k-1 [a, b], (k = 1,2, ... , n - 1); a!(x) E L[a, b],
(1.1.8)
analogous to (1.1.2), so that the operator E* can be applied to the functions v(x) E A on-1[a, b] obtaining a function (defined almost everywhere):
n E*[v(x)] = E*(v) = 2: (- l)n-k [ak(x) v(x)]<n-k) E L[a, b]. (1.1.9)
k~O
From (1.1.8) it follows that we can consider the operator E**, adjoint to E*, but we do not obtain a new operator, since the reader will easily prove that E** = E.
We shall also have the occasion to consider the adjoint operator E: of the reduced operators E F , for which we obviously have
r
E:(v) = 2: (- 1),-k [ak(x) v(X)](,-k) , (r = 0, 1, ... , n - 1). (1.1.10) k~O
Note also that there exists the recurrence formula
* d * E, (v) = - dx E,_dv) + a,(x) v(x) , (r = 1, 2, ... , n) .1) (1.1.11)
1.2. The Green-Lagrange identity. We shall begin by proving that, if u(x), v(x) E A Ok-1[a, b], (k = 1,2, 3, ... ) the relation
d k-1 v(x) U(k)(x) - (- l)k u(x) v(k)(x) = - .r: (- l)k-i-1 u(i)(x) v(k-i-1)(X) (1.2.1)
dx i~O
holds almost everywhere in [a, b]. We prove it by induction, observing first of all that for k = 1, (1.2.1)
is evident since it becomes v u' + u v' = ! (u v) •
To pass from k to k + 1, it is thus sufficient to write
v u(k+1L (- 1)k+1 u V(k+1) = ! [(- l)k U V(k)] + [v (:: t) -(- l)k :: V(k)]
=~[(-I)kUV(k) + 1;1 (_I)k-i_l(dU)(i)v(k_i_1)] dx i~O dx
= ~ [( - l)k u V(k) + f (- l)k-i U(i) V(k-i)] = ~ f (- l)k-i u(i) V(k-i). dx i~1 dx i~O
1) For r = n, remember that E: = E*.
1.3 Relations between the solutions 13
Now we can prove the Green-Lagrange identity, relative to a linear dif ferential operator E of order n of the type (1.1.1) under the hypotheses (1.1.2) and to its adjoint E* [see (1.1.5)]. The Green-Lagrange identity states that for every pair of functions u(x), v(x) E A on-I [a, b], the following equation bolds almost everywhere in [a, b]:
d n-I
v(x) E[u(x)] - u(x) E*[v(x)] = dx k~ u(k)(X) E:_k_I[V(X)] , (1.2.2)
where the operators E:-k - I are defined by (1.1.10). We have n n
vE(u) - uE*(v) = v J.: aj u(n-i) - u J.: (- l)n-; (at v)(n-i) ;=0 ;=0
n-I = J.: [aj v . u(n-i) - (- l)n-i u . (at v)(n-i)] ,
i=O
and applying (1.2.1) to the functions u E A on-I [a, b], at v E A on-i-I [a, b]
n-I d n-i-I
V E(u) - u E*(v) = I: - E (- l)n-i-k-I U(k) (at v)(n-i-k-I) i=O dx k=O
d n-I n-k-I = - I: u(k) I: (- l)n-k-I-i (at v)(n-k-I-i) ;
dx k=O i=O
from which, using (1.1.10), (1.2.2) follows.
1.3. Relations between the solutions of a linear differential equation and those of the adjoint equation. Let us use the symbols E, E* with the meaning already given in (1.1.1), (1.1.2) and (1.1.5). Let us consider in [a, b] the non-homogeneous linear differential equation
E(u) =!(x) (1.3.1)
with !(x) E L[a, b]. Let [uI(x), u2(x), ... , un(x)], where Ut E A On-Ira, b], be a fundamental system of solutions of the homogeneous equation E(u) = 0, and let U(x) =f. 0 their Wronskian. It is known that the general solution of (1.3.1) is expressed by
n '" u(x) = 2: lXi Ui(X) + J K(x, ;) !(;) d; ,
i=I a (1.3.2)
where lXI' lX2' ••• , lX,. are arbitrary constants and K(x,;) is the so-called Oauchy resolvent kernel, which is (as function of x) the particular solution of the homogeneous equation E(u) = 0 which satisfies, at the point;, the following initial conditions:
[:~ K(x, ;) 1=; = dh, n-I , (h = 0, 1, ... , n - 1) .1) (1.3.3)
1) ~rs is the Kronecker delta = 0 (if r =1= 8)
= 1 (if r = 8).
It may be written n K(x, ~) = L: u,(x) vd~) , (1.3.4)
i=1
where the functions v1(x), v2(x), ... , vn(x) are given by the system of n linear equations n
L: U~h)(X) v,(x) = fih, n-l , (h = 0, I, ... , n - I) , (1.3.5) i=1
whose determinant is U(X).l) Therefore (1.3.2) may also be written
u(x) =ig u,(x) [(Xi + ! vM)f(~) d~] . (1.3.6)
As to the derivatives of u(x), it is well known that from (1.3.2) and (1.3.6) follows
(1.3.7)
We observe another property of the functions Vi(X) introduced by (1.3.5); we have almost everywhere
that is n L: u~n)(x) Vi(X) = - a1(x) .
;=1 (1.3.8)
The relations (1.3.5), (1.3.8) may be generalized in the way shown by the following theorem:
Theorem 1.3.1. The function8 Vj(x) defined by the 8y8tem (1.3.5) belong to the clas8 A on-1[a, b] and 8ati8fy the following relations
n L: U~h)(X) E:[vi(x)] = fih, n-k-l , (h, k = 0, I, ... , n - I), (1.3.9)
;=1
n L: u~n)(x) Et[Vi(X)] = - Uk+l(X) , (k = 0, I, ... , n - I) ,2) (1.3.10)
;=1
where E: are the operator8 defined by (1.1.10).
Proof. Since Et(Vi) = Vi' for k = ° the equation (1.3.9) reduce to (1.3.5) (which assures Vi E A G) and (1.3.10) reduces to (1.3.8). We proceed by
1) From (1.3.5) and from the fact that Ui(X) E A Cn-l[a, b] it follows immediately that v;(x) E A C[a, b]; but we shall prove shortly that Vi(X) E A cn-l[a, b] (see theorem 1.3.1).
2) (1.3.10) holds almost everywhere.
induction to assume that for a certain k (with 0 < k < n - I) the equation (1.3.9) hold, that is Et(Vi) E A 0 or Vi E A Ok. We prove that as a conse quence (1.3.10) holds, with the same k and (1.3.9) with k + I instead of k; thus Vi E A Ok+!. We may prove (1.3.10) immediately by observing that
n = - 2: a" I5n- h, n-k-l = - ak+l .
h=l
To prove the other assertion, we start from (1.3.9) and differentiate both members; there results almost everywhere
n n d ~ uV'+ 1) E*(v ) + Y' U<.h) - E*(v ) = 0 L,-, ki ...J'dxkt ,
i=l ;=1 (h = 0, I, ... , n - I) . (1.3.11)
The value of the first term is 151t+1, n-k-1 if h = 0, 1, ... , n - 2 [from (1.3.9)] and - ak +! if h = n - I [from (1.3.10)]. For the second term we have from (1.1.11)
vVith this, (1.3.11) yields
= - L u~h) Et+!(v;) + aU1 15k , n-1 . i=1
n
l51t+l, n-k-1 - 2: t4lt ) Et+!(v;) + 0 = 0 , i=1 '
(h = 0, I, ... , n - 2) , I n
- ak+l - 2: u~n-l) Et+l(Vi) + aU1 = 0 i=1
which may be gathered in
n
f (1.3.12)
2: u~h) Et+1(V;j = bh, n-k-2 , (h = 0, 1, ... , n - 1) , (1.3.13) i=1
which is precisely (1.3.9) with k + 1 instead of k, q.e.d.
\Ve now prove another theorem:
Theorem 1.3.II. The functions [v1(.c), v2(x), ... ,vn(x)] defined by the system (1.3.5) form a fundamental system of solutions of the homogeneous adjoint equation
E*(v) = 0
1 V(:t;) = U(x) .
The general solution of the non-homogeneous adjoint equation
E *(v) = g(x) , (g(x) E L[a, b]) , (1.3.16) is given by
n x v(x) = ~ fJi Vi(X) + J K*(x,~) g(~) d~ , (1.3.17)
i=l a
where fJI> fJ2' .. , fJn are arbitrary constants, while for the resolvent kernel K *(x, ~) we have n
K*(x,~) = - K(~, x) = - ~ Vi(X) Ui(~) . (1.3.18)
Therefore (1.3.17) may also be written
v(x) = i~ Vi(X) [fJi - J Ui(~) g(~) d~] . (1.3.19)
Proof. We begin by proving that E*(Vi) = 0 (i = 1,2, ... , n). Starting from (1.3.9) written with k = n - 1, we may proceed in the same fashion as in (1.3.11), (1.3.12), (1.3.13) (with k = n - 1, E! = E*) thus obtaining
n ~ U~h) E*(Vi) = 0 , (h = 0, 1, ... , n - 1) ,
i=l
which, together with U(x) i= 0, give E*(Vi) = O. To prove (1.3.15) we observe that from (1.1.10) it easily follows that the
derivative v(r)(x), (r = 0, 1, ... ,n - 1) of a v( ) E A en- 1[a, b] may be expressed by means of E~(v), E~_l(V), ... ,Et(v) with a formula of the type r
v(r)(x) = (- 1)' E~[v(x)] + L crs(x) E;_8[V(X)] . (1.3.20)
Thus we may write
(
u~ . .. u\n_l») :: u; ... u~n-l) Un u~... U~n-l)
we get, by virtue of (1.3.9)
t5 t5 00 ... 0] t50,n-l t51,n-l ... ~n-1,n-l 00 ... 10
Vex) U(x) = (-I)(~) O,n-2 1,n-2'" n-l,n-2 = (-I)(~) . . . . . = 1 .
01. .. 00 10 ... 00
Finally we prove (1.3.18). Considering that in the first member of the diffe rential equation (1.3.16) the coefficient of v(n) is not equal to 1, but to (_I)n [see (1.1.8)], we may assert that the resolvent kernel K*(x, ~) is given by
n 1: Vi(X) Wi(~)' where the functions w1(x), W2(x) , ... , wn(x) are defined by the
i=1
system of linear equations n
1: v~h) W, = (_I)n t5h,n-1, (n = 0, 1, ... ,n - 1). i=l
By application of equation (1.3.20), this system may also be written
ig [( _I)h Et(Vi) + 8~ Clis Et_8(Vi)] Wi = (_I)n ~h, n-1'
(n=~I, .. . ,n- 1),
or also, by subtracting from every equation an obvious linear combination of the preceding ones,
n 1: Et(Vi) Wi = (_l)n-h bh n-l ,
i=l ' (n = 0, 1, ... , n - 1) .
Finally another transformation leads to n
1: Et(Vi) (- Wi) = ~h,n-l' (n = 0,1, ... ,11- 1). (1.3.21) i=1
But from (1,3.9) we have n
1: Et(Vi) Ui = bo,n-h-l = bh,n-l, (n = 0,1, ... , n - 1), (1.3.22) i=1
and since the determinant of Et(vj) is (-I)(~) Vex) =1= 0, the comparison of (1.3.21) with (1.3.22) leads to Wi = - Uj. Equation (1.3.18) follows imme diately.
2 Ghizzetti/Ossicini: Quadrature
To complete this theorem, we note that from (1.3.17) and (1.3.19) we get n z
E:_h _ 1[V(X)] = }; fJI E:_h _ 1[Vi(X)] + J E:_h _ 1[K*(x, m g(~) d~ 'i=l a
= ig E:_h_1[t'i(X)] [fJi - ! uM) g(~) d~] . (1.3.23)
A fundamental system (Ut, u2 , ••• ,un) of solutions of E(u) = ° and a funda mental system (VI' v2 , ••• ,vn) of solutions of E*(v) = ° will be called associated if the latter can be obtained from the first using (1.3.5).
The following theorem will also be useful:
Theorem 1.3.111. With the meaning previously given to Uf(X) and Vj(x) the formula
holds.
n-l
(i, j = 1,2, ... , n) (1.3.24)
Proof. Let us denote by Ail the first member of (1.3.24). Holding the index i fixed, for every h = 0,1, ... ,n - 1 we may write
n n n-l n-l n
}; U~h) Ail = }; U}h) }; U~k) E:_k _ 1(vl) = }; U~lc) }; u)hl E:-k - 1(t'l) j=l j=l k=O lc=O j=l
and thus by (1.3.9) n n-l
}; U}hl Ail = }; 'U~k) ~h,lc = u~h) , (h = 0, 1, ... , n - 1) . j=l k=O
This system of n linear equations in n unknowns Ail' Ai2' .•• ,Ain in virtue of the fact that U(x) =F 0, has only the solution Ail = ~il which establishes (1.3.24), q.e.d.
1.4. Linear differential equations considered in infinite intervals.1) The statements made in § 1,1, 1.2 and 1.3 pertain to the case in which the inde pendent variable x varies in a finite interval [a, b]. With slight modifications they hold also when x varies in an infinite interval, for instance [a, +00).2) It suffices to substitute for the class L[a, b] the class 40c [a, + 00) of locally 8ummable functions in [a, + 00) and for the class A Glc[ a, b] the class A G~oc [a, + 00) of functions whose k-th derivative is locally absolutely continuous in [a, + 00).3) With only these modifications all the formulae and theorems of § 1.1, 1.2, 1.3 remain valid. We shall now give some other basic ideas essentially connected with the infinite interval.
1) See A. GmzZETTI [2], [3], [4]. 2) Here we only refer to the case of [a, + (0); similar considerations hold for the
interval (- 00, +(0), and we shall summarily state them at the end of the paragraph. 3) A function f(x) is called locally 8ummable (or locally ab80lutely continuoUB) in
[a, +(0) when it is summable (or absolutely continuous) in every finite interval con tained in [a, +(0).
1.4 Linear differential equations considered in infinite intervals 19
Writing f(x) E I [a, + 00), that is, f(x) is integrable in [a, + 00), we shall mean thatf(x) is locally summable in [a, + 00) and moreover that the integral +00
f f(x) dx is convergent'!) In a particular case it may happen that this integral a
be absolutely convergent; we note that this is the same as saying that f(x) is Lebesgue-integrable (summable) on the interval [a, + 00), that is f(x) E L [a, + 00). Let us now consider again the differential equations (1.3.1) and (1.3.16), namely
E(u) = f(x)
under the new hypothesesf(x), g(x) E L\oc[a, + 00).
(1.4.1)
(1.4.2)
From § 1.3 we know that having introduced two associated systems, (uv ... ,un), (vv ... ,vn) of solutions of the homogeneous equations E(u) = 0, E*(v) = 0, we can derive from (1.4.1) and (1.4.2) the equations [see (1.3.7) and (1.3.23)]:
u(h)(x) = i~ u<f')(x) [tXi + ! f(~) vM) d~], (h = 0, 1, ... , n - 1) ,
E:_h_1[V(X)] =ig E:_h_1[Vi(X)] [Pi - J g(~) uM) d~], (h = 0, 1, ... , n - 1) ,
where tXi, Pi are arbitrary constants.
( 1.4.3)
(1.4.4 )
Now let us assume that the term denoted by g(x) satisfies the additional hypotheses
g(x) Ui(X) E I [a, + 00) , (i = 1,2, ... ,n) . (1.4.5)
We choose the arbitrary constants Pi so that lim [Pi - f g Ui d~] = 0; it is evident that we must put :1:->-+00 a
+00
Pi = f g(~) uM) d~ , (i = 1,2, ... , n) . (1.4.6) a
In this way we specify a certain particular solution of (1.4.2) that we shall denote by V(x) and for which (1.4.4) becomes
n +00 E:_h_1[V(X)] = 1: E::-h- 1 [Vi(X)] f g(~) Ui(~) d~ ,
i~l z
(h = 0, 1, ... , n - 1) , (1.4.7)
z 1) This means that the lim f f(x) dx is finite. In the case oftheinterval (- 00, + 00)
o z both the limits lim f f(x) dx and lim f f(x) dx must be finite.
x-+-oo x X-J-+OO 0
2·
and in particular for h = n - 1
n +00 V(x) = ~ Vj(x) J g(;) '11M) d; (1.4.8)
i=1 x that is, recalling (1.3.18)
+00 V(x) = J K(;, x) gm d; . (1.4.9)
x
This particular solution V(x) will be denoted as that particular solution of (1.4.2) which satisfies the initial conditions null at the point + 00. This sentence is not to be taken literally; in fact generally it is not true that
lim V(h)(X) = 0, (h = 0, 1, ... ,n - 1) . 1) (1.4.10) X-++CX>
However, we shall use it for the following three reasons:
1°) if it is true that in general (1.4.10) do not hold, although n linear in dependent combinations of V(h)(x) do exist (with coefficient functions of x) which tend to zero for x -+ + 00, we have in fact
n-l
~ U~h)(X) E!_h_l[V(X)] = [using (1.4.7)] h=O
n-l n +00 = ~ U~h)(X) ~ E!_h_l[Vt(X)] r g(;) uM) d;
h=O ;=1 i n +00 n-l
= ~ J g(;) uM) d; ~ U~h)(X) E!_h_l[Vt(X)] = [using (1.3.24)] ;=1 x h=O
n +00 +00 = ~ J g(;) uM) d; . bjt = J g(;) uM) d;
;=1 x x
and therefore n-l
lim ~ U~h)(x) E!_h_l[V(X)] = 0, (i = 1,2, ... , n); 2) (1.4.11) x-++oo h=O
1 1 1) For instance, having assumed n = 1, E(u) = u' + - u, E*(v) = - v' + - v,
x x 1 1
g(x) = - in the interval [1, + 00) we have u1(x) = -, v1(x) = x. The hypothesis x x
g u1 E 1 [1, + 00) is satisfied and it turns out that
+00 f 1 1 V(x) = x T . T dt; = 1 .
x
2) In the example of the preceding footnote, (1.4.11) becomes
1 u1(x) V(x) = -·1 ..... 0 (for x ..... + 00).
x
x, 2°) the expression (1.4.9) of V(x) is similar to the expression J K(~, x) g(~) d~
x of the solution of (1.4.2) which satisfies the initial conditions null at the point xo; 1)
3°) having introduced the above mentioned sentence, in Chapter 2, we shall have the advantage to be able to use the same language for the quadrature formulae on finite intervals and for those on infinite intervals.
In connection with this solution V(x) of (1.4.2) and with any solution u(x) of (1.4.1.), we consider the following function
n-1
(1.4.12)
In the next chapter we shall see that, in the theory of quadrature formulae on infinite intervals, it is interesting to discover under what other conditions [besides (1.4.5)] we may assert that
lim w(x) = 0 . (1.4.13)
Thus we now give the following theorem:
Theorem 1.4.1. If the given function g(x) in equation (1.4.2) verifies the following hypotheses [a particular case of (1.4.5)]
g(x) Ui(X) E L [a, + (0), (i = 1,2, ... , n) (1.4.14)
and if the given function f(x) in equation (1.4.1) satisfies these other hypotheses +00
f(x) Vi(X) J Ig(~) uM)1 d~ E L [a, + (0) , (i = 1,2, ... , n) , (1.4.15)
then, for every solution u(x) of (1.4.1) and for that particular solution V(x) of (1.4.2) which satisfies the initial conditions null at the point + 00, we have
g(x) u(x) E L [a, + (0) ,
f(x) V(x) E L [a, + (0) , n-l
lim w(x) = lim 2 U(h)(X) E!_h_l[V(X)] = 0 .
Proof. Equation (1.4.15) says that +00 +00 J If(x) Vi(X) I dx J Ig(~) uM)1 d~ < + 00 ,
a x
(1.4.16)
(1.4.17)
(1.4.18)
whence, by a well-known theorem of Tonelli, the function f(x) Vi(X) g(~) uM)
1) If we write (1.4.10) and (1.4.11) with lim , they are equivalent; on the contrary X~Xo
they are not equivalent in the case of the point + 00.
2) This function is the same which occurs in the right member of the Green-Lagrange identity (1.2.2).
of the two variables x and ~ is summable in the unbounded domain (~ > x ~ a) of the (x, ~) plane. Then.applying the Fubini theorem we may write
+00 +00 +00 I;
J f(x) Vi(X) dx J g(~) uM) d~ = J g(~) Ui(~) d~ J f(x) Vi(X) dx a x a a
and also, interchanging the variables x and ~ in the second member:
+00 +00 +00 x
J f(x) Vi(X) dx J g(~) Ui(~) d~ = J g(x) Ut(X) dx J f(~) vM) d~. (1.4.19) a x a a
Thus we may say that from a hypothesis given above we draw not only the obvious conclusion
+00 f(x) Vt(X) J g(~) Ui(~) d~ E L [a, + 00 ) , (i = 1,2, ... ,n) (1.4.20)
but also x
g(a..) Ui(X) J f(~) vM) d~ E L [a, + (0), (i = 1,2, ... , n) (1.4.21) a
showing the validity of (1.4.19). Once that is done, we observe that using (1.4.3), with h = 0, we obtain
g(x) u(x) = i~ [Xi g(x) Ui(X) + g(x) Ui(X) !f(~) vM) d~] , which, together with (1.4.14) and (1.4.21) make (1.4.16) hold and establishes the validity of the formula
From (1.4.8) we get n +00
f(x) V(x) = 2: f(x) vlv) J g(~) uM) d~ i=1 x
which, together with (1.4.20), proves (1.4.17) and the existence of
+00 n +00 +00 J f(x) V(:t) dx = 2: J f(x} Vi(X) dx J g(~) uM) d~ . a i=l a x
Now we observe that the Green-Lagrange identity (1.2.2) gives
d V E(u) - U E*(V) = dx w(x)
that is d
f(x) V(x) - g(x) u(x) = dx w(x) .
There follows x x J f(x) Vex) dx - J g(x) u(x) dx = w(x) - w(a}
a a
and therefore, with a passage to the limit for x --+ + 00, and using (1.4.22) and (1.4.23):
n [+00 +00 ",.!.i:oo w(x) = w(a) +i~ ! f(x) v,(x) dx 1 g(~) uM) d~
+00 +00 '" ] - ex, J g(x) Uj(X) dx - J g(x) u,(x) dx J f(~) vM) d~ .
a a a
In virtue of (1.4.19) this formula reduces to' n +00
lim w(x) = w(a) - J; ex, J g(x) Uj(X) dx "'-++ 00 i=l a
and to obtain (1.4.18) we need to prove that the value of this last expression is O.
To this purpose note that (1.4.12) in connection with (1.4.3) and (1.4.7) gives
w(x) = :#: i~ U~h)(X) [exi + /f(~) vM) d~ tR E!_h_l[vf(x)] JOOg(~) uM) d~
= iR ig [ex i + J f(~) vM) d~] JOOg(~) uM) d~ :#olU}h)(X) E!_h_l[v1(x)]
and then using (1.3.24)
w(x) = i~ it [exi + I f(~) vM) d~] T")g(~) uM) d~ i5"
= ig [ex i + J f(~) vM) d~] JOOg(~) uM) d~ . n +00
For x = a we obtain in particular w(a) = X exi J g(~) u .• m d~, q.e.d. i=l a
In the case of the interval (- 00, + 00) the hypotheses analogous to (1.4.5) is
g(x) u,(x) E 1(- 00, + 00), (i = 1,2, ... , n) . (1.4.24)
If it is satisfied, we may give in (1.4.4)1) to the arbitrary constants {Ji the values
o +00
(Ji = - J g(~) uM) d~ or (J, = J g(~) uM) d~ -00 0
and therefore, with regard to the differential equation (1.4.2) consider the particular solution
n '" '" V_(x) = - J; v,(x) J g(~) u,(~) d~ = - J K(~, x) g(~) d~ (1.4.25)
i=l -00 -00
1) In (1.4.3) and (1.4.4) the symbol a denotes now an arbitrary fixed point; we shall assume for instance a = O.
which satisfies the initial conditions null at the point - 00, or the particular solution
n +00 +00 V +(x) = l: Vi(X) f g(;) uM) d; = f K(;, x) g(;) d;
i=l x x (1.4.26)
mhich satisfies the initial conditions null at the point + 00.1)
Then, the theorem analogous to theorem 1.4.1, can be stated as follows:
Theorem 1.4.II. If the given function g(x) of the equation (1.4.2) satisfies the following hypotheses [particular case of (1.4.24)]:
g(x) Ui(X) E L (- 00, + (0) , (i = 1, 2, ... , n) (1.4.27)
and if the givenfunctionf(x) of the equation (1.4.1) satisfies these other hypotheses
'" f(x) Vi(X) f Ig(;) uM) I d; E L (- 00, 0] ,
(i d,2, ... , n) , } . -00 +00
'"
(1.4.28)
then for every solution u(x) of (1.4.1) and for the particular solutions V _(x), V +(x) of (1.4.2) above introduced, we have
g(x) u(x) E L (- 00, + (0),
f(x) V _(x) E L (- 00, 0] , f(x) V +(x) E L [0, + (0) ,
n-l
lim w_(x) = lim l: u(h)(x) E:-h-1[V _(x)] = ° , X-'I>- -00 x-+ -00 It=o
n-l
lim w+(x) = lim l: U(h)(X) E:_h_1[V +(x)] = ° . "'-++00 "'-++00 h~O I
(1.4.29)
(1.4.30)
(1.4.31)
The proof of this theorem is to be carried out reasoning separately on the intervals (- 00, 0], [0, + (0) analogously to the proof of theorem 1.4.I. It is only to be noted that the statement (1.4.29) is a consequence of g(x) u(x) E L (- 00, 0], and g(x) u(x) E L [0, + (0) which are deduced separately. Note that in the last two theorems, the n hypotheses (1.4.14) or (1.4.27) generally are not mutually independent; some may be consequence of the others. The same holds for (1.4.15) or (1.4.28).
1) The justification of these two sentences (which are not to be taken literally) is analogous to the one previously given. Instead of (1.4.11) we have now
n-l lim l.,' u~h)(x) E:_h_1[V-(x)] = 0,
"'-+-00 h~O n-l
lim }; u~h)(x) E!_h_l[V +(x)] = 0, "'-++00 h~O
(i = 1,2, ... , n)·
n E(u) = L: aTe u(n-k) = 0 ,
k=O
n aTe u(n-k) E(u) = E ( _ k)' -k- = 0, k=O n . x
(aTe constant) ,
(aTe constant) ,
25
(1)
(2)
(3)
and write their adjont equations E*(v) = O. In each one of the three cases, devise pairs of associated systems [~(x), ... , un(x)], [vl(x), ... ,vn(x)] of solutions of the two equations E(u) = 0, E*(v) = 0 (see § 1.3), In the cases (2) and (3) assume that the roots .xl' £X2' ••• , £Xn of the respective characteristic equations
be all simple.
i' aTe ( £x k) = 0 k=O n-
2. Prove that, if U(x) = [~(x), ... ,un(x)), V(x) = [vl(x), ... ,vn(x)] are two associated systems of solutions of the equations E(u) = 0, E*(v) = 0 (see § 1.3), then, fixing arbitrarily a matrix a = (CtTe), (i, k = 1, 2, ... , n), with constant elements and det a =F 0, there result associated also the
- - systems U(x) = [ul(x), ... ,un(x)], V(x) = [vl(x), ... , vn(x)] defined by
U(x) = V(x) aT , V(x) = V(X) 0-1 .
3. A linear differential operator E of order n:
n dn-k E = E aTe(x) dxn-k
Te=O
E* = (_I)R E.
(1)
(2)
In order to construct such an operator we may fix arbitrarily the coefficients
a2.(x) E A cn-2 8-1,(8 = 0, 1, ... , [; ])and then obtain the others coefficients
a2 r+1(x) by means of
_ ')' (2,-28+1) , 22 ,-28+2 (n-28) a2r+1(X) - :=OT _ 8 + I B2r- 2s+2 2r'-28 + 1 a28 (x), (3)
(r = 0, 1, ... , [n ; I ]),
where B2 ,-2 8 +2 are the Bernouilli's numbers (see § 3.1).
4. Given a function f(x) E OR[a, b] and a linear differential operator E of order n [see (1.1.1) and (1.1.2)] we want construct a solution u(x) of the
differential equation E(u) = 0 which interpolate f(x) in the sense that we must get
u(k)(Xi) = j<k)(X() , (i = 1,2, ... ,rn; k = 0, 1, ... ,Pi) , (1) with
a < Xl < Xa < ... < Xm < b , m :::: 1, Pi:::: 0 ,
m + PI + P2 + ... + Pm = n .
Give the condition for the existence and the unicity of u(x) [with any f(x)], the expression of u(x) and, using the Green-Lagrange identity (1.2.2), an integral expression of the remainder
R(x) = f(x) - u(x) . (2)
Elementary quadrature formulae and a general procedure for constructing them
2.1. The elementary quadrature formulae on a finite interval.1) In this chapter, as in the last, we shall consider exclusively real-valued functions. Let [a, b] be a finite interval on the axis x and n a positive integer; we shall consider an integral of the type
b
J g(x) u(x) dx (2.1.1) a
where the functions g(x) (weight functions which are assumed non zero on a set of positive measure) and u(x) (argument function) satisfy the following hypotheses:
g(x) E L[a, b] , u(x) E A en-Ira, b] . (2.1.2)
Let us fix in [a, b] a certain number m 2: 1 of points Xv x2 , ••• , Xm (nodes) in such a way that
(2.1.3)
and let us fix also a linear differential operator E of order n [the same n as in (2.1.2)] expressed by a formula of the type (1.1.1) with coefficients which satisfy the hypotheses (1.1.2). We shall call elementary quadrature formula of the integral (2.1.1) relative to the nodes Xl' x2 , ••• , Xm and to the linear differential operator E, any formula of the type
b n-1 m J g(x) u(x) dx = L; L; Ahi u(h)(Xj) + R[u(x)] , (2.1.4)
a h=Oi=l
where the constants coefficients A/Ii are independent of u(x) and the linear functional R[u(x)] (remainder) is null when u(x) is a solution of the homogeneous linear differential equation E[u(x)] = O.
In other words (2.1.4) is intended to be accompanied by the condition
E(u) = 0 => R(u) = 0 . (2.1.5)
In practice, every formula of the type (2.1.4) is used considering the linear functional
n-1 m A(u) = L; L; A"i u(h)(xd
h=Oi=l
(2.1.6)
2) For sake of uniformity we shall sometimes write Xo instead of a and Xm+l in stead of b.
28 2. Elementary quadrature formulae
as an approximate value of the integral (2.1.1) and the other functional R(u) as the corresponding error of approximation. The error is null, that is the quadrature formula is exact, when u(x) satisfies the differential equation E(u) = 0. 1)
In the literature we find innumerable examples of formulae of the type (2.1.4). Most of them use in A(u) only the values u(x/) of the argument function at the nodes Xi and not (when n> 1) values of the derivatives U'(Xi), ... , u(n-l)(Xi); but this means only that in such formulae the coefficients Ali' ... ' An-I, i turn out to be zero. Moreover almost all classical formulae are exact when u(x) is a polynomial or a trigonometric polynomial; this corresponds to the fact that such formulae are relative to the following particular differential operators:
(exact formula for polynomials of degree n - 1)
d • (d2 ) E=-[] -2+k2 dx k=l dx
(n = 2'/1 + 1; exact formula for trigonometric poly nomials of order '/I).
With this as introduction, we are in a position to arrive quickly at a simple procedure to get a formula of type (2.1.4) under condition (2.1.5).
Let us consider the non-homogeneous linear differential equation
E*[<p(x)] = g(x) , (2.1.7)
where E* denotes the adjoint operator of E [see (1.1.5)]. Now we fix arbitrarily m - 1 solutions
<PI(X), <P2(X), ... , <Pm-leX) E A en-l[a, b] (2.1.8)
of (2.1.7) and consider also the two solutions <Po(x), <Pm(x) ofthe same equation (2.1.7) which are respectively determined by the initial conditions null at the points a, b; namely <p~k)(a) = 0, <p<::')(b) = 0, (h = 0, 1, ... ,n - 1) and therefore, recalling (1.3.17) and (1.3.18):
'" b
<Po (x) = - J K(~, x) g(~) d~ , <Pm(x) = J K(~, x) g(~) d~ • (2.1.9) a '"
We now make use of the Green-Lagrange identity (1.2.2), substituting vex) = <Pi(X), (i = 0, 1, ... , m).
Considering that E*(<pt) = g, we get
(i = 0, 1, ... , m)
and therefore, integrating on the interval [Xi> Xi+1]:
J g u dx = - 1: U(h) E:-k- 1(<Pt) + J <Pi E(u) dx , "'HI (n-l ]"'Hl "'HI • k=O ••
(i=O,l, ... ,m).
1) The solutions of E(u) = 0 are not the only functions u(x) for which (2.1.4) is exact; we shall make precise later the whole of such functions (see Remark 4).
2.1 The elementary quadrature formulae on a finite interval 29
Summing with respect to the index i, we arrive at
b m [n-l ]XHI m XHI ! g u dx = - i~ h~O U(h) E;_h_l(CP;) Xi + i~ xI CPi E(u) dx . (2.1.10)
Bearing in mind that from cp~h)(a) = 0, cpC::,>(b) = ° follows
[E;-h-l(CPO)]X~xo = 0, [E;-h-l(CPm)]X~Xm+1 = 0, (h = 0, 1, ... , n - 1) ,
we may then write
m Xi+l
This formula is precisely of type (2.1.4) with
A"i = {E;-h-l [CPi(X) - CPi-l(X)]}X~Xi' (h = 0, 1, ... , n - 1; i = 1,2, ... , m), (2.1.12)
m Xi+l
R[u(x)] = ~ J CPi(X) E[u(x)] dx , (2.1.13) i=O Xi
and from these relations it is evident that the two desired conditions are satisfied [coefficients A"i independent of u(x) and validity of (2.1.5)].
Thus we have the following rule for constructing an elementary quadrature formula of type (2.1.4) relative to the nodes Xl' X2' ••• , Xm and to the linear differential operator E:
1) we consider the linear differential equation E*(cp) = g and its solutions CPo(x) and CPm(x) defined by (2.1.9);
2) we fix as we like m - 1 other solutions CPI(X), ... , CPm-l(X) of the same equation E*(cp) = g;
3) we compute the coefficients Ahi by means of (2.1.12); 4) we write the expression for the remainder R(u) using (2.1.13).
This rule is not only reasonably simple; it is also the only one possible, as is proved by the following theorem:
Theorem 2.1.1. If, having fixed m nodes Xl' x2' ••• , Xm and the mn constants Atli , the linear functional
b n-l m R(u) = J g u dx - ~ ~ Ani U(h)(X;) (2.1.14)
a h~Oi~l
is null when u is a solution of the homogeneous linear differential equation E(u) = 0, then there are uniquely determined m - Isolution8 rpl(X), ... , rpm-I(X) of the differential equation E*(rp) = g which, together with the two rpo(x) and rpm (X) given by (2.1.9), make valid (2.1.12) and (2.1.13). Proof - Equations (2.1.12) may also be written
[E:-h-l(tpi)]X~Xi = Ahi + [E:-h-l(rpi-l)]X~Xi ' (h = 0, 1, ... , n - 1; i = 1,2, ... , m) ,
and therefore, to have them satisfied, itis necessary that rpl(X) be the solution of E*(rp) = g which is characterized by the following initial conditions at the point Xl: I)
[E:-h-l(rpl)]X~Xl = Ahl + [E:-h-I(q;O)]X~Xl ' (h = 0, 1, ... , n - 1) ,
and successively that rp2(X) be the one characterized by the following initial conditions at the point x2 :
[E:-h-l(rp2)]X~X' = Ah2 + [E:-h-l(rpl)]X~X. ' (h = 0, 1, ... , n - 1) ,
and so on for rp3(X), ... ,rpm-l(X), Then we see that rpm (X) must coincide with that solution of E *(tp) = g which satisfies the following initial conditions at the point Xm:
[E:-h-l(rpm)]X~Xrn = A hm + [E:-h-l(rpm-I)]X~Xrn' (h = 0,1, ... , m - 1) .
It remains only to see that the solution tpm(x) coincides with that considered in the statement of the theorem, namely that defined by the initial conditions
[E:-h-l(tpm)]x~b = 0, (h = 0, 1, ... ,n - 1) (2.1.15)
and we have also to prove (2.1.13). To this purpose we observe that, making use of the solutions rpl(X), ... ,
... , rpm-l(X), tpm(x) just introduced, we may do again the same computation which led to (2.1.10). However, we cannot pass from (2.1.10) to (2.1.11) since we do not yet know whether tpm(x) satisfies (2.1.15); therefore instead of (2.1.11) we must write
b n-l m J g u dx = L: L: [E:_h_ 1 (rpi - tpi-l)]X~XI U(h)(Xi) a h~Oi~l
n-l m XHI
- L: [E:-h-l(tpm)]x~b u(h)(b) + L: J rpi E(u) dx . (2.1.16) h~O i~O Xi
Taking into account equation (2.1.14) and the fact that the solutions rpo, tpl' ... . . . ,rpm have been chosen so as to satisfy (2.1.12), (2.1.16) may also be written as
n-l m XHI
R(u) = - L: [E:-h-l(rpm)]x~b u(h)(b) + L: J rpi E(u) dx. (2.1.17) h~O i~O XI
1) Please note that, here and in the following, to assign at a point of [a, b] the values of E!-h-I(/fJ), (h = 0, 1, ... , n - 1) is the same thing as to give the values of /fJ(h) and that if the former are all null, the latter too are null and viceversa [see formula (1.3.20)].
2.1 The elementary quadrature formulae on a finite interval 31
We now recall that by hypothesis R(u) must be null when u is a solution of the differential equation E(u) = o. Therefore, denoting by (ul , Us, ... , un) a fundamental system of solutions of that equation, it follows from (2.1.17), putting u = Uj, (j = 1, 2, ... , n), that
n-l 1: u~h)(b) [E:-h-l(CPm)]x~b = 0 , h~O
(j = 1, 2, ... , n) .
Since the Wronskian det (u~h)(b)) is different from zero, we conclude that (2.1.15) holds and finally that (2.1.17) reduces to (2.1.13), q.e.d.
On the basis of this theorem, the above stated rule allows us to find as particular cases of (2.1.4) all the known quadrature formulae1) and to get several others. We intend to come back to this in § 2.4. Now we add some remarks on the above mentioned rule.
Remark 1. If we wish to be effectively present in the quadrature formula (2.1.4) all the prefixed nodes Xi (i = 1,2, ... , m) we require that, for every value of the index i at least one of the coefficients Ahi be non-zero. Keeping into account (2.1.12) it is obvious that this is equivalent to requiring that the function tpi(X) - CPi-l(X) must not be identically zero. We achieve our objective by choosing the solution CPl(X) different from CPo(x); CPI(X) different from CPl(X); ... ; CPm-2(X) different from CPm-3(X) and finally tpm-l(X) different from CPm-2(X) and from CPm(x).
Remark 2. Having fixed the nodes Xl' XI' ... , Xm and the linear differential operator E, we may write the quadrature formula in oo(m-l)n different ways, since (m - 1) n is the number of arbitrary constants on which the m - 1 solutions CPI(X), .•. , CPm-l(X) of the differential equation E*(cp) = g of order n depend.
Remark 3. If Xl = a, (2.1.12) and [E:-h-l(CPo)]x~a = 0, (h = 0, 1, ... , n - 1), imply
Xl
besides in the expression (2.1.13) for R(u) the first term J CPo E(u) dx vanishes. a
Therefore, if Xl = a, it is useless to consider the solution CPo(x). Analogously, if Xm = b we have
A hm = [E:-h- 1 (cpm - CPm-l)]x~b = - [E:-h-l(CPm-l)]x~b , b
while in R(u) the last term J cpm E(u) dx vanishes; therefore if Xm = b it 0I:m
is useless to consider the solution CPm(x).
1) At least the most common formulae, which approximate the integral (2.1.1) with a linear functional of the argument function u(x) of type (2.1.6). There exist a few examples of quadrature formulae of other types which do not fall within the theory here stated.
Remark 4. From (2.1.13) it is clear that R(u) = 0 for all and only the func tions u for which E(u) E L[a, b] is orthogonal on [a, b] to the so-called influence function W(x) defined as
W(x) = <Pi(X) in [Xi, Xi +1] , (i = 0,1, ... , m), (2.1.18)
that is, for which b
f fP(~) F(~) d~
E[ u(x)] = F(x) - a b W(x) ,1) .<2.1.19) f (/)2(~) d~
a
where F(x) is an arbitrary function summable on [a, b]. Hence, the quadra ture formula (2.1.4) is exact if and only if the argument function u(x) is the solution of the linear differential equation (2.1.19) where F(x) E L[a, b] is arbitrary and W(x) is given by (2.1.18). Note that the condition (2.1.5) made at the be ginning corresponds to the particular choices F(x) = 0 or F(x) = W(x).
2.2. The elementary quadrature formulae on the interval [a, + 00).2) We shall use here the same terminology and notations as in § 1.4. Let us consider an integral of the type
+00 J g(x) u(x) dx (2.2.1) a
where the functions g(x) (weight function) and u(x) (argument function) satisfy the following hypotheses:
g(x) E L1oc[a, +00) , u(x) E A Cfo-;;1[a, +00) , (2.2.2)
g(x) u(x) E I[a, +00) . (2.2.3)
Let us fix some points xl> x2' ••• , Xm (nodes) such that
(2.2.4)
and a linear differential operator E of order n expressed by a formula of the type (1.1.1) where the coefficients ak(x) satisfy the hypotheses
ao(x) = 1; ak(x) E A Cfo-;;k-l[a, +00), (k = 1,2, ... , n - 1);
an(x) E L1oc[a, +00) . (2.2.5)
We shall call elementary quadrature formula for the convergent integral (2.2.1) relative to nodes xl> x2, ••• , Xm and to the linear differential opemtor E, any formula of the type
+00 n-l m J g(x) u(x) dx = 2: 2: Ahi U(h)(Xi) + R[u(x)] (2.2.6) a h=O i=l
b
1) We have f fP2(~) d~ > 0 since the case in which every solution tpi(X) of the a
equation E*(tp) = g may be identically zero in [Xi' Xi+1] is excluded since the weight function g(x) was supposed different from zero on a set of positive measure.
2) See A. GHIZZETTI [2], [3], [4].
2.2 The elementary quadrature formulae on the interval [a, + (0) 33
whose coefficients Alii are independent of u(x) and such that
E(u) = ° ~ R(u) = 0 . (2.2.7)
To construct such a formula it is sufficient to give a rule analogous to the one of § 2.1, however with some clarifications and remarks.
If we introduce a fundamental system (uv u2 , ••• , un) of the solutions of E(u) = 0, it is obvious that in order that the condition (2.2.7) make sense, the hypotheses
g(X) Ui(X) E I[a, +00) , (i = 1,2, ... , n) , (2.2.8)
is indispensable. With this condition established, and fixing one point b with b > XII"
we may perform, with respect to the interval [a, b], a computation ana logous to that of § 2.1.
Namely, we denote by CPo(x) that solution of the differential equation E *[cp(x)] = g(x) which satisfies the initial conditions null at the point X = a and by CPl(X), ... , CPm(x) m other solutions arbitrarily fixed.
We apply the Green-Lagrange identity (1.2.2), putting vex) = CPi(X) (i = 0, 1, ... , m). This leads to
d n-l CPi E(u) - g u = -d E U(h) E!-h-1(CPi) , (i = 0, 1, ... , m - 1, m) ,
xh=o
from which, integrating them on the intervals [Xi, Xi+d (i = 0, 1, ... , m - 1), [xm' b] respectively and summing the resulting equations, we obtain
! g u dx = _ ~i:l [~1U(h) E!_h_1(CPi)]>:HI _ [~1U(h) E!_h_1(cpm)]b a .=0 h=O >:1 h=O >:m
m-l "'HI b
+ I J CPi E(u) dx + J cpm E(u) dx , i=O Xi lim
from which, with a simple transformation and use of the fact that cp~h)(a) = 0, (h = 0, 1, ... , n - 1),
b n-1 m J g u dx = I I [E!-h-1 (CPi - CPi-1)]",=",/ U(h)(Xi) a h=Oi=1
- [h~IU(h) E!-h-1(cpm)1=b+ ~~11+lcpi E(u) dx +",[ f{Jm E(u) dx. (2.2.9)
This relation, by means of a passage to the limit with b -'>- + 00, will evidently lead to a formula of the type (2.2.6), if it is possible to choose f{Jm(x) in such a way that
lim [~IU(h) E!-h-1(CPm)] = 0 , b-++oo h=O x=b
cpm E(u) E I[a, + 00). (2.2.10)
It is possible to construct examples (see A. GHIZZETTI [2]) which show that (2.2.3) and (2.2.8) are generally not sufficient to assure that the differential
equation E*(qJ) = g possesses a solution satisfying (2.2.10). On the other hand it does not seem easy, confining ourselves to the field of integrals which are simply convergent and avoiding any particularization of the operator E, to arrive at a formulation of hypotheses which can assure the validity of the procedure but not be too restrictive. Therefore we think it appropriate to confine ourselves to giving a general theorem for the case of absolutely convergent integrals.
In the case of simply convergent integrals, we think it better to study (2.2.10) case by case, that is for any fixed choice of the operator E.
The theorem for absolutely convergent integrals is immediately derived from theorem 1.4.1 and can be stated as follows:
Theorem 2.2.1. Under hypotheses (2.2.2) and letting (uv ... , un), (VI' ... , vn) be two associated systems of solutions of the differential equations E(u) = 0, E*(v) = 0, we assume moreover that the weight function g(x) satisfy the following hypotheses
g(x) Ui(X) E L[a, +00) , (i = 1,2, ... , n) , (2.2.11 )
and that the argument function u(x) satisfy also +00
E[u(x)] Vi(X) J ig(~) uM)i d~ E L[a, +00), (i = 1,2, ... , n). (2.2.12)
Then g(x) u(x) E L[a, +00) (2.2.13)
and, if as solution qJm(x) of E*(qJ) = g we take that V(x) which satisfies the initial conditions null at the point +00 [see (1.4.9)], (2.2.10) is satisfied amd (2.2.6) follows from (2.2.9) together with
Ahi = [E:- h - 1 (qJi - qJi-l)]X~xl' (h = 0, 1, ... , n - 1; i = 1, 2, ... , m) , (2.2.14)
m Xi+l
R(u) = J: J qJ! E(u) dx, (with Xo = a, xm +1 = + 00). (2.2.15) i=O Xi
We have then the following rule to construct an elementary quadrature formula of type (2.2.6) relative to the nodes Xl' x2 , ••• , Xm and to the linear differential operator E [when for instance the hypotheses (2.2.2), (2.2.11) and (2.2.12) are satisfied]:
1) we consider the linear differential equation E*(qJ) = g and its solutions x +00
qJo(x) = - J K(~, x) g(~) d~ , qJm(x) = J K(~, x) g(~) d~; (2.2.16) a
2) we fix arbitrarily m - 1 other solutions qJl(X), ... , qJm-l(X) of the same equation E*(qJ) = g;
3) we compute the coefficients Ahi by means of (2.2.14) ;
4) we write the expression of the remainder R(u) using (2.2.15).
2.2 The elementary quadrature formulae on the interval [a, + 00) 35
As in the preceding § 2.1 this rule is essentially the only possible one, as shown by the following theorem:
Theorem 2.2.11. Having fixed the linear differential operator E, we assume that the functions g(x), u(x) satisfy the hypotheses (2.2.2), (2.2.11) and (2.2.12). Having then fixed the m node8 Xl' x2, ... , Xm and the mn constants A hi, if the linear functional
+00 n-l m R(u) = J g u dx - L: L: Ahi u(h)(xd
a h~O i~l
(2.2.17)
is null when u is a solution of the differential equation E(u) = 0, then m - 1 solutions 971(X), ... , 97m-l(X) of the differential equation E*(97) = g are uniquely determined, which, together with the two solutions 97o(x) and 97m(x) defined by (2.2.16), imply (2.2.14) and (2.2.15).
Proof. As in the proof of theorem 2.1.I, first we see that in order that (2.2.14) hold, the above mentioned solutions 971(X), 972(X), ... , 'Pm-I(X) must be neces sarily those determined by the following initial conditions respectively at the points Xl' x2, ••• , Xm-l;
[E:-h-I(97I)]"'~"', = Ahl + [E:-h-I('PO)]Z~"" '
[E:-h-I(97m-I)]"'~"',n_l = A h, m-I + [E:-h-I(97m-2)]"'~"'m_l' (h = 0, 1, ... , n - 1) ,
I (2.2.18)
and moreover that the solution 97m must coincide with the one determined by these initial conditions at the point Xm:
[E:-h-I(97m)]"'~"'m = A hm + [E:-h-I(97m-d]"'~"'m , (2.2.19)
(h = 0, 1, ... , n - 1) .
With the solutions 971>"" 97m of E*(97) = g defined by (2.2.18) and (2.2.19) we may write (2.2,9) and pass to the limit as b --r + 00. In virtue of theorem l.4.I, hypotheses (2.2,11), (2,2,12) assure that g(x) u(x) E L[a, +(0); the passage to the limit in (2.2.9) yields, considering also that we have already proved the validity of (2.2.14):
+00 n-I m m-I "'1+1 J g u dx = L: L: Ahi u(h)(x;) + L: J 97i E(u) dx a h~O i~1 i~O "'I
+ lim {- [~IU(h) E:-h - 1(97 m )] + J 97m E(u) ax } , b-++oo h~O ",~b Xm
that is for (2.2.17)
Since R(u) must be zero when u coincides with the solutions u l , ... , u" of E(u) = 0, we find
n-1 lim 1: U)hl(x) E!_h_1[tpm(X)] = 0, (j = 1,2, ... , n). (2.2.21)
"' ..... +00 h=O
tpm(x) =ig V((X) [Pi - j g(~) uM) d~] , (2.2.22)
the value to the limit given in (2.2.21) is
",~:oo ~:U)hl(X)i~ E!_h_1[V((X)] [Pi - j g(~) uM) d~]
-"'::00 i~ ~i - ! g(~) uM) d~] hn~1U~hl(X) E!_h_I[Vi(X)] .
That is, on the basis of theorem 1.3.III,
"'::00 i~1 [Pi - j g(~) uM) d~] 15(1 = Pi - !~(~) UM) d~ , therefore we must have, for (2.2.21)
+00 Pi = J g(~) Uj(~) d~ , (j = 1,2, ... , n) .
a
Substituting this expression for PI in (2.2.22) we deduce that for the solution tpm(x) we have " +00
tpm(x) = 1: Vi(X) J g(~) uM) d~ . i=1 '"
Now we see from (1.4.8) and (1.4.9) that tpm(x) is exactly that solution which satisfies the initial conditions null at the point +00. With the notations of theorem 1.4.1 we have namely tpm(x) = V(x); using then the two other .conclusions of that theorem, we conclude that
lim ["j;1U(h) E!-h-1(tpm)] = 0 , b ..... +oo h=O ",=b
b +00 lim J tpm E(u) dx = J tpm E(u) dx ,
b-++oo X,n Xm
therefore (2.2.20) reduces to (2.2.15) q.e.d. Note that formulae (2.2.6), (2.2.14) and (2.2.15) in the case of the interval
[a, +(0) are quite similar to (2.1.4), (2.1.12) and (2.1.13) relative to the case of the finite interval [a, b]. Remarks 1, 2, and 3 of the preceding § 2.1 hold here too. Remark 3 is to be limited to the fact that if Xl = a it is useless to consider the solution tpo(x). As to Remark 4, it still stands that (2.2.6) is exact for all and only the functionsu(x) for whichE(u) is in [a, +(0) orthogonal to the influence function 4>(x) determined by (2.1.18), but gen erally (2.1.19) (with b = + (0) has no meaning.
2.3 Elementary quadrature formulae on the interval (- 00, + 00) 37
2.3. Elementary quadrature formulae on the interval (- 00, + 00).1) Let us consider an integral of the type
+00 J g(X) U(X) dx -00
where the functions g(x) and u(x) satisfy the hypotheses
g(x) E L1oc(-00, +00), u(x) E A C~-;;l(_oo, +00),
g(x) u(x) E 1(-00, +00).
Having fixed the nodes Xl' X2, ••• , X", such as to have the ordering
(2.3.1 )
(2.3.2)
(2.3.3)
(2.3.4)
and a linear differential operator E of order n, of type (1.1.1) and whose coefficients a!:(x) satisfy the hypotheses
ao(x) = 1; a!:(x) E A C~;;-k-l(_oo, +00), (k = 1,2, ... , n - 1):
an(x) E L1oc( - 00, + 00) (2.3.5)
we may find a corresponding elementary quadrature formula +00 n-l m J g(x) u(x) dx = 1: 1: Ahi U(h)(Xi) + R[u(x)] (2.3.6)
-00 h~O i~l
under the usual condition
E(u) = 0 =? R(u) = 0 . (2.3.7)
Keeping the notation of § 1.4, we may reason as in § 2.2 requiring [together with (2.3.2)] the hypotheses
g(x) Ui(X) E L(-oo, +00), 1
-00 I (2.3.8)
x
E[ u(x)] Vi(X) J [g($) Ui($) [ d$ E L( - 00, 0] ,
+00
I E[u(x)] Vi(X) J [g($) uM)[ d$ E L[O, +00) , x
(i = 1,2, ... , n) , J which imply (by theorem 1.4.II) g(x) u(x) E L( - 00, + 00). More precisely, having fixed a < Xl and b> Xm and choosing arbitrarily m + 1 solutions gJo(x), gJI(X), ... , gJm(x) of E*(gJ) = g, we may write the following formula, similar to (2.2.9):
b [ n-l ] n-l m J g u dx = 1: U(h) E;-h-l(gJO) + 1:.1: [E;-h_l (gJi-gJi_l)]X~XiU(h)(X;)- a h~O x~a lt~O ,~l
_[~l u(h) E:-lt-l(gJm)l~b + ITO E(u) dx + :~ll;i E(u) dx +x1 gJm E(u) dx
(2.3.9) 1) See A. GHIZZETTI [2], [3], [4].
and take the limits as a ~ - 00, b ~ + 00. Applying theorem 1.4.II we see that with such passages to the limit, (2.3.9) yields a formula of the type (2.3.6) if we take as lPo(X) , IPm(x) the solutions of E*(IP) = g which satisfy the initial conditions null respectively at the points -00, +00.
We thus arrive to the following rule, similar to the one of § 2.2, and valid when the hypotheses (2.3.2) and (2.3.8) are satisfied:
1) we consider the linear differential equation E*(IP) = g and its solution lPo(x) defined by the initial conditions null at the point -00 and thus given by
'" lPo(x) = - J K(;, x) g(;) d; (2.3.10) -00
and the solution IPm(x) which satisfies the initial conditions null at the point +00, expressed by
+00
IPm(x) = J K(;, x) g(;) d;; (2.3.11)
2) we fix as we like m - 1 solutions 1P1(X), ... , IPm-l(X) of the same equation E*(IP) = g;
3) we compute the coefficients Ahi by means of
(h = 0, 1, ... , n - 1; i = 1,2, ... , m) ;
(2.3.12)
4) we write the expression of the remainder R(u) in the following way: m Z'+l
R(u) = ~ J lPi E(u) dx, (with Xo = - 00, xm+1 = + 00) . (2.3.13) i=O "',
This rule is essentially the only possible since there exists a theorem similar to theorem 2.2.II whose statement and proof we leave to the reader.
Finally Remarks 1, 2, 4 of § 2.1 hold, if we remember that (2.1.19) (with a = - 00, b = + 00) has no longer any meaning.
2.4. Some remarks on the rule for obtaining quadrature formulae. If we examine the three rules given in § 2.1,2.2, and 2.3 for obtaining elementary quadrature formulae on a finite interval [a, b], on an interval [a, +00) and on the interval (-00, +00), we see that they are formally ideni/ical, so that we may always refer to (2.1.4), (2.1.9), (2.1.12), (2.1.13) with the understanding that a can also be finite and b = + 00, or a = - 00, b = + 00. The only difference among the three cases is in the formulation of the necessary hypotheses regarding the weight function g(x) and the argument function u(x). In the case of infinite intervals these hypotheses call in also two associated systems [~(x), ... , u,,(x)], [vI(x), ... , v,,(x)] of solutions of the differential equations E(u) = 0, E*(v) = 0. We know that the rules given are the most general possible, wherefrom we may derive systematically all the known quadrature formulae, choosing appropriately case by case the nodes Xl' ••• , Xm and the linear differential operator E.
2.4 Some remarks on the rule for obtaining quadrature formulae 39
Such deductions substantially do not present anything new and generally are of no difficulty. 1)
On the contrary we may say that in the set of all quadrature formulae supplied by the above given systematic procedure, the classical ones are of no particular importance. We may also suggest that it is better to make the most of the procedure by getting, case by case, a quadrature formula
b
particularly suited to the calculation of a given integral J F(x) dx with a
F(x) E L[a, b] taking advantage, not only from the arbitrary choice of the nodes Xv ... , Xm and of the linear differential operator E, of order n, but also from the numberless ways we may write F(x) = g(x) u(x) with g(x) E L[a, b] and u(x) E A Cn - 1[a, b].
This point of view is not yet widespread and therefore, for the conveni ence of the reader, we shall gather in Chapter 4 several examples of de ductions of classical formulae, together with others that we consider new. For this purpose it will be necessary to state in Chapter 3 some notes on certain auxiliary elements (Bernoulli numbers and polynomials, orthogonal polynomials, etc.). In applying the general procedure it is useful to keep in mind some observations which can be of great utility, above all when we try to obtain quadrature formulae with certain pre-established quali fications (for instance that certain coefficients Aht be zero). We shall present these observations in the remainder of this § 2.4 and in the two successive ones.
The given rule for obtaining the formula
b n-l m J g(x) u(x) dx = 2: 2: Ahi u(h)(Xt) + R(u) , [E(u) = 0 ~ R(u) = 0] ,
a h=Oi=l (2.4.1)
with a, b finite or infinite, considers first the solutions Ifo(x), Ifl(X), ... , Ifm(x) of the differential equation E*(If) = g, then the computation of the co efficients A by means of
Ahi = [E:_ h _ 1 (lfi - lfi-l)J.=,."
and finally the expression of the remainder written
m XI+l
(2.4.2)
(2.4.3)
Sometimes it is more convenient to compute first the coefficients A hi ,
considering n linearly independent solutions u1(x), ... , un(x) of the linear differential.equation E(u) = 0 and writing the system of n linear equations
1) Some difficulties may arise when we try to deduce from the expression of the remainder R(u) formulae of majoration and evaluation of the remainder itself. In such operations it is often necessary to study the sign of the influence function 4>(x) then making use of very delicate technique.
in mn unknowns A hi ;
b n-I m J g(x) Uj(x) dx = L.; L.; Ahi U)h)(X) , a h~O i~1
(j = 1,2, ... , n) . (2.4.4)
From this system we may derive for instance the n coefficients A 01' An, ... , An-I, I in terms of the remaining (m - 1) n which can be fixed arbi. trarily (see § 2.1, Remark 2) or with any particular criteria. Furthermore we may apply the following theorem:
Theorem 2.4.1. Assuming that the coefficients Ahi of (2.4.1) are known, the functions To(x), Tl(X), ... , Tm(x) necessary to write (2.4.3), are given by the formula
x
Ti(X) = - IK(~, x) g(~) d~ + X: jfIAhj [:;', K(~, X)l~xj' (2.4.5)
a (i = 0, 1, ... , m)
where K(x, ~) is the resolvent kernel relative to the operator E [see (1.3.4)].
Proof. For i = 0 equation (2.4.5) is true since it coincides with the first equation of (2.1.9); for i = m it cannot immediately seen that (2.4.5) coincides with the second one of (2.1.9), that is that
a
(2.4.6)
But it can easily proved because if in (2.4.1) we denote by ~ the variable of integration and put u(~) = K(~, x) [solution of E(u) = 0, whence R(u) = 0], we obtain
b I g(~) K(~, x) d~ = :~: JI Ahi [:;h K(~, X)l~xi a
which is equivalent to (2.4.6). In order to finish the proof of (2.4.5) it will be sufficient to prove that
(2.4.2) is satisfied. For this purpose we observe that from (2.4.5) we may derive
and therefore for (1.3.4)
n-I n Ti(X) - Ti-l(X) = L.; Ahi L.; u~h)(Xi) Vj(X);
h~O j~l
it follows, using (1.3.9), that n~1 n
[E:~k~1 (rpi - rpi~1n"~Xi = L; Afti L; Ujh)(X;) [E:~k~1 (VI)]x~x;
which is (2.4.2) q.e.d.
il~O j~1
We next note that in (2.4.1) the condition
E(u) = O~ R(u) = 0
E(u) = E(ii) ~ R(u) = R(i/,) ,
41
(2.4.7)
(2.4.8)
where u(x) denotes another function satisfying the same hypotheses of u(x). Finally note that (2.4.7) requires that the quadrature formula (2.4.1) be
exact for the functions u(x) belonging to a certain n-dimensional linear space. Instead of defining the space by means of the operator E, we can define it by its linearly independent n points; namely (2.4.7) can be replaced by the condition that the quadrature formula be exact for n assigned functions ul(x), ... , u,,(x) having Wronskian U(x) '* 0 in the interval of integration. We then have the result
u u' ... ,
(2.4.9)
2.5. Gauss problem.!) Let us consider again the elementary quadrature formula
b n~1 m f g(x) u(x) dx = L; L; Ahi U(h)(Xi) + R(u) , a h~O i=1
[E(u) = O~ R(u) = 0]. (2.5.1)
In its right member we find the values, at the nodes Xl' X 2' ••• , x m , of the function u and of its derivatives up to order n - 1; we know (§ 2.1, Remark 2) that in the coefficients Aili there are altogether (m - 1) n arbitrary para meters. Therefore the question is whether, having fixed an integer p (with 1 <p < n - 1), it is possible to make use of the arbitrary nature of these parameters to drop from the formula the values U(h)(Xi) of the derivatives of order higher than n - p - 1, that is whether there can exist a formula of the type
b n~p~1 m
f g(x) u(x) dx = L; L; Ahi U(hl(X;) + R(u) , a h=O i~l
') See A. GHIZZETTI [I].
[E(u) = O~ R(u) = 0].
If (UI , UZ' ••• , un) are a fundamental system of solutions of E(u) = 0 it is clear that a necessary and sufficient condition that (2.5.2) hold is that the m (n - p) coefficients Ahi satisfy the following system of n linear equations
n-p-l m b
L: L: Ani UY')(Xi) = J g(x) Uj(X) dx , h~O i~l a
(j = 1, 2, ... , n) . (2.5.3)
To discuss this system of equations, let us consider the transposed homo geneous system which, denoting by CI , C2, ••• , Cn the relevant unknowns, may be written
n :E U~h)(xtl cj = 0 , j~l
(h=O,I, ... ,n-p-l; i=I,2, ... ,m). (2.5.4)
If the rank of the matrix (ujh)(Xi)) with m (n - p) rows and n columns, is equal to n, 1) the system (2.5.4) has no non-trivial solutions and therefore the system (2.5.3) is certainly consistent and has oom(n-p)-n solutions.
Let us now assume that the matrix has rank n - q (with q ~ 1).2) In this case the system (2.5.4) has q linearly independent non-trivial solutions (cl1 , C2 ,' ••• , cn,)' (r = 1, 2, ... , q) and by a well-known criterion the system (2.5.3) is consistent if and only if the q conditions
n b
L: cjr J g(x) Uj(x) dx = 0 , (1" = 1,2, ... , q); (2.5.5) j~l a
are satisfied. In this case (2.5.3) has oom (n-p)-(n-q) solutions. Now we shall try
to present these results in a more illuminating form. To this purpose we shall consider the following problem: determine a solution of the differential equation E(u) = 0 which together with its derivatives up to the order n - p - 1 is zero at each of the points Xl' X2, ••. , X",.
This is an homogeneous boundary problem expressed by the formula
E(u) = 0; u(h)(xtl = 0, (h = 0, 1, ... , n - p - 1; (2.5.6)
i = 1,2, ... , m) . n
Since the general solution of E(u) = 0 is given by u(x) = L: cj Uj(x), with j~l
Cj arbitrary constants, the boundary conditions are expressed by the fol lowing system of m (n - p) equations in n unknowns Cv ... , Cn:
n :E Cj U)h)(Xi) = 0, (h = 0, 1, ... , n - p - 1; i = 1, 2, ... , m) , j~l
which coincides with the system (2.5.4).
1) This is possible only if n :::;; m (n - p). 2) \Ve must have n - q :::;; m (n - p). Moreover the matrix considered contains
the m matrices (n - p) X n formed by the first n - p rows of the W'ronskian of u1' ••• , Un
computed at the points XI (i = 1,2, ... , m); these matrices have rank n - p and therefore we certainly have n - q ;;:; n - p. In summary, we may write
n - p :::;; n - q :::;; m (n - p) .
2.6 Tchebychef problem 43
Therefore, saying that the system (2.5.4) has no non-trivial solutions is equivalent to saying that the problem (2.5.6) admits only the solution u(x) = 0; similarly, saying that the system (2.5.4) has q linearly independent solutions (cIr , C2" ••• , cn,) is equivalent to saying that the problem (2.5.6) has q linearly independent solutions
n U,(x) = L: c,• u,(x) ,
j=1 (r = 1,2, ... , q) . (2.5.7)
In the latter case, the necessary and sufficient conditions (2.5.5) that the quadrature formula (2.5.2) exist, can be written as
b
J g(x) U,(x) dx = 0 , (r = 1, 2, ... , q) . (2.5.8) a
Therefore we may state the following theorem:
Theorem 2.5.1. Given the nodes Xl' X2, ••• , X"" the linear differential operator E of order n and the positive integer p < n - 1, consider the homogeneous differential problem (2.5.6). If this problem has no non-trivial solutions [whence n < m (n - p)] it is possible to write a quadrature formula of the type (2.5.2) in oom(n-p)-n different ways. If on the other hand the problem (2.5.6) has q linearly independent solutions U,(x) [r = 1, 2, ... , q with n - m (n - p) < q < p] then the formula (2.5.2) may apply only if the q conditions (2.5.8) are satisfied; if so there are oom(n-p)-n+q possible formulae of form (2.5.2).
In the conditions stated in this theorem, the equation (2.5.8) may appear as conditions for the weight g(x) (see § 2.7, Problem 4); but this point of view has little importance since generally the weight g(x) is given.
However we may fix only the number m of the nodes and consider their abscissae Xl' x2, ••• , xm, all or in part, as undetermined parameters. In this case, (2.5.8) appear as conditions for the nodes; they show us how we have to choose these points to make it possible to get the quadrature formula (2.5.2) required. I) This problem, presented in these terms, will be called Gaus8 problem since it is a wide generalization of a particular problem studied by Gauss which originated the quadrature formulae having his name (see § 4.6).
2.6. Tchebychef problem.2) Now let us consider another problem regarding quadrature formulae which, in certain particular cases, have been studied by Tchebychef. In analogy with Gauss problem [see (2.5.2)] we try to
1) It may be that there many groups of nodes (Xl"'" Xm) satisfying (2.5.8). In that case it is obvious that the latter statement of theorem 2.5.1 (existence of oom(n-p)-n+q quadrature formulae) is to be applied to each of such groups (see § 2.7, Problems 2,3).
a) See A. GHIZZETTI [1].
get, for a fixed p (0 < P < n - 1), a quadrature formula of the type b n-p-1 m J g(x) u(x) dx = 2: Ah 2: U(h)(X;} + R(u) , [E(u) = 0 ~ R(u) = 0] ,
a h~O i~l
(2.6.1)
that is, one in which the coefficients of the values U(h)(Xi), of the derivatives
of the same order of the function u(x), are always equal to each other. Reasoning as in the preceding § 2.5 we find for the n - p coefficients Ah the following system of n linear equations
n-p-1 m b L Ah 2: U~h)(Xi) = J g(x) Uj(x) dx , (j = 1,2, ... , n) , h~O i~l a
(2.6.2)
whose homogeneous transpose can be written n m
2: ci 2: U~h)(X;} = 0, (h = 0, 1, ... , n - p - 1) . (2.6.3) j~l i~l
If the rank of the matrix (~u}h)(Xd) (with n - p rows and n columns)
comes out to be equal to n (this is possible only if p = 0), the system (2.6.3) has no non-trivial solutions, therefore the system (2.6.2) is consistent and admits of one and only one solution. Instead let us assume that the matrix has rank n - q (with q > 1, 0 < n - q ~ n - p). In this case the aystem (2.6.3) has q linearly independent solutions (cl " c2 " .•• , cnr) (r = 1, 2, ... , q) and the system (2.6.2) is consistent if and only if the q conditions
n b
2: ciT J g(x) Uj(X) dx = 0, (r = 1,2, ... , q) , (2.6.4) j~l a
are satisfied, resulting oo(n-p)-(n-q) solutions of (2.6.1). These results can be presented in a different way, considering the following homogeneous boundary problem:
E(u) = 0; m
(h = 0, 1, ... , n - p - 1). (2.6.5)
n Since the general solution of E(u) = 0 is u(x) = 2: Cj Uj(x) assuming the
j~l
boundary conditions satisfied, we get for the arbitrary constants ci precisely the system (2.6.3). Therefore when we say that (2.6.3) has no non-trivial solutions we mean that (2.6.5) admits only the solution u(x) = 0; to say that (2.6.3) has q linearly independent solutions (cIr , c2" ••• , cnr) means that the problem (2.6.5) has q linearly independent solutions
n UT(x) = E Cjr Uj(x) ,
j~l
(r = 1,2, ... , q) , (2.6.6)
2.7 Problems 45
We have then proved the following theorem:
Theorem 2.6.1. Given the nodes Xl> X 2, ••• , X m, the linear differential operator E of order n and the non-negative integer p < n - 1, if the problem (2.6.5) has no non-trivial solutions (this is possible only if p = 0) we may get a quadrature formula of the type (2.6.1) in only one way.
If on the other hand problem (2.6.5) has q linearly independent solutions Ur(x), (r = 1,2, ... , q with q 2: 1, P < q < n) 1) then it is possible to con struct a formula (2.6.1) (in ooq-p different ways) if and only if the q conditions (2.6.7) are satisfied.
We may repeat here the remarks stated in the preceding § 2.5: (2.6.7) may be considered as conditions on the weight g (but this has a little im portance) or as conditions on the choice of the nodes.
2.7. Problems.
1. Examine the possibility of devise a quadrature formula of the type b
J u(x) dx = AOl u(x1 ) + A02 u(x2 ) + Aoa u(xa) + R(u) (1) a
[with a < Xl < x2 < xa < b] which must be exact when u(x) is a polynomial of degree :s 3.
2. The Gauss problem of § 2.5 can be generalized by substituting (2.5.2) by the formula
b m n-P/-l J g(x) u(x) dx = L: L: AM U(h)(Xi) + R(u) , (1) a i~l h~O
E(u) = 0 =:> R(u) = 0,
where the maximum order n - Pi - 1 of the derivatives of the argument function u(x), which must be considered in the node Xi, depends on the node Xi' Prove the following theorem (which is an extension of theor. 2.5.1): given the nodes Xl> x2' ••. , X m, the linear diffential operator E of order nand the integers PI> P2' .•. , Pm (with 0 <Pi < n - 1), consider the boundary problem:
E(u) = 0; U(h)(Xi) = 0, (i = 1,2, ... , m; h = 0, 1, ... , n - Pi - 1). (2) m
If this problem has only the trivial solution u(x) - 0 [whence n < m - L: pd
m mn- Epi-n
it is possible to write a quadrature formula of the type (1) in 00 i~l
1) It may happen that q = n, namely that all the solutions of E(u) = ° satisfy m
the boundary conditions 2,' u(h)(Xi) = 0, (h = 0, 1, ... , n - p - 1). Example: n = 2, i~l
d2
m = 2, P = 0, E = dX2 + 1, Xl = 0, x2 = n.
different ways. If on the other hand the problem (2) has q linearly independent m
solutions Ur(x) [r = 1, 2, ... , q with n - m n + 2: Pi < q < min Pi], i=1 - -.
then the formula (1) is possible if and only if these q conditions are satisfied: b
J g(x) Ur(x) dx = 0 , a
(r = 1,2, ... , q);
m mn-I;PI-n+q
(3)
in such a case this formula is possible in 00 i=1 different ways.
3. Examine the possibility of devise a quadrature formula of the type 1
J U(x) dx = AOI U(X1) + An U'(Xl) + A21 U"(Xl) + AOIl U(xlI ) + R(u) , (1) -1
which must be exact when u(x) is a polynomial of degree < 5.
4. Determine the weight functions g(x) in correspondence of which it is possible to devise a quadrature formula of the type
1 .
J g(x) u(x) dx = AOI u(- 1) + AOIl u(O) + A03 u(l) + R(u) , (1) -1
which most be exact when u(x) is a polynomial of degree < 5.
CHAPTER 3
Special functions
a.l. llernouilli polynomials and numbers. The Bemouilli polynomials Bn(x) are defined by the following expansion (valid for \z\ < 2 n):
eZ _ 1 = 1: Bn(x) n! ,1) z eX Z 00 ztl I B 1(x) = X _ ~n~o B 2(x) = X2 _ X + ~ , ... ,]
(3.1.1 )
therefore we have z 00 zn 1 00 zn
-ez---l - 1: B - = 1 - --- z + 1: B - n=O n n! 2 n=2 n n!
(3.1.3)
. z 1 z cosh z/2 and smce .-1 - 1 + 2 z = 2 . h /2 - 1 is an even function, we deduce that e - sm z
Ba = B5 = B7 = ... = 0 , (3.l.4)
while the other Bernouilli numbers B 2, B 4 , B 6 , ••• are all non zero. Now note that from the identity
we may deduce, expanding every term in power series of z:
00 n+l DC nOOn
1: xn :,-- = 1: Bn(x + 1):' - 1: Bn(x):' n=O n. n=O n. n=O n.
or also 00 nOOn
1: n xn- 1 :, = 1: [Bn (x + 1) - Bn(x)] ~,- ; n=l n. n=l n.
therefore we have
Bn (x + 1) - Bn(x) = n xn- 1 , (n = 1,2, ... ) (3.l.5)
and particularly, putting x = 0 and using (3.l.2)
Bn(1) = B n , (n = 2, 3, ... ) . (3.l.6)
1) That Bn(x) come out to be effectively a polynomial of degree n, it immediately follows from the formulae given furtherwards [for instance from (3.1.8)].
48 3. Special functions
From equation (3.1.1) there follows, differentiating with respect to x
Z2 eX Z 00 I zn
eZ _ 1 = L Bn(x) ;I ; n~l .
(3.1.7)
on the other part, always for (3.1.1), the function occurring III the first co zn+l co ~n
member has value "B (x)-- = L n B (x) ~ so that from the n~ n n! n~l n -1 n!'
comparison with (3.1.7) we deduce
B~(x) = n Bn_1(x) , (n = 1,2, ... ) . (3.1.8)
From this equation with further differentiations we obtain B~(x) = n X B~_l(X) = n (n - 1) B n_ 2(x), B~'(x) = n (n - 1) B~_2(X) = n (n - 1) (n - 2) X Bn _ 3(x), ... and in general
(k) __ ~ Bn (x) - (n _ k)! Bn_k(x) , (k = 0, 1, 2, ... , n) . (3.1.9)
Supposing k < n - 2, that is n - k > 2, if we put x = ° or x = 1 in (3.1.9), using (3.1.2) and (3.1.6), we obtain
nlk) _ (k) _ n! D" (0) - Bn (1) - (n _ k)! B n- k , (k = 0, 1, ... , n - 2), (3.1.10)
while in the case of k = n - 1, since (3.1.9) becomes B~n-l)(x) = n! (x - {-),
we obtain 1 1 B~n-l)(O) = - "2 n! , B~n-1)(1) = "2 n! . (3.1.11)
We get another interesting consequence of (3.1.8) integrating it between ° and x; then we have
x
Bn(x) - Bn = n J Bn-M) d~ , (n = 1,2, ... ) . (3.1.12) o
From it, changing n by n + 1 and putting x = 1, using (3.1.6) and supposing n ~ 1 we get:
1
J Bn(x) dx = ° , (n = 1, 2, ... ) . (3.1.13) o
Let us now deduce a property of symmetry or semisymmetry of the poly-
nomials Bn(x) with respect to the point x = ~ . We write again (3.1.1),
a first time changing x by 1 - x and a second time changing z by - z, thus obtaining the two formulae
z e(1-"')z co zn Z e-XZ co, zn eZ _ 1 = nfo Bn (1 - x) n! ' - e-Z _ 1 = n~ (- 1)n Bn(x) n! ;
it is easy to see that the first members are equal to each other and therefore we have the formula
Bn (1 - x) = (- 1)n Bn(x) , (n = 0, 1,2, ... ) (3.1.14)
which expresses the wanted property.
3.1 Bernouilli polynomials and numbers 49
Finally let us consider the polynomial
P 2n(x) = B 2n(x) - B 2n , (n = I, 2, ... ) , (3.1.15)
on the interval [0, I]. Using (3.1.2), (3.1.4), (3.1.6), (3.1.10) we see that
P 2n(0) = P 2n(l) = 0,
p~2:-1)(0) = ~2:-1)(1) = 0 ,
(k = 1,2, ... , n);
wherefrom and from the Rolle theorem it is easy to deduce that P2n(x) does not vanish for 0 < x < 1. In fact if P2n(x) vanishes (besides at the points x = 0, x = 1 also) at a point internal in [0, I], its.derivative p; .. (x) should vanish (besides at the points x = 0, x = I) at least at two internal points, p~ .. (x) at least at three internal points, P~~(x) (besides at x = 0, x = I) at least at two internal points and so on until the absurd conclusion that p~2nn-2)(x) (polynomial of second degree) should vanish at least at three points internal in [0, I].
This reasoning proves also that P;n(x) vanishes only at one point internal 1
in [0, I], which point has necessarily abscissa 2in virtue of the symmetry
property expressed by (3.1.14). Therefore P 2n(x) is always poeitive (or nega-
tive) for 0 < x < 1 and gets its maximum (or minimum)
For computing Pn(~) we write (3.1.1) with x = I:
z eZ 00 zn eZ _ 1 = 2: Bn(l) n! ;
.. =0
(3.1.17)
we observe then that the first member may also be written
1 2 z e 2 2 z e2z ( ~2z )
2 e2z - 1 + e2z - 1
1 so that, applying again (3.1.1) with x = 2 or x = 1 and 2 z instead of z, we obtain
(3.1.18)
From the comparison between (3.1.17) and (3.1.18) obviously follows that
Bn(l) = 2n- 1 l Bn (~) + Bn(I)] , whence we have Bn (~) = - (I - 2nl_1)
XBn(l) and finally, using (3.1.6):
Bn(~)= - (1-2nl_1)Bn, (n=2,3, ... ). (3.1.19)
From equations (3.1.15) and (3.1.19) then follows that the above mentioned maximum value (or minimum) of P2n(x) in [0, I] is given by
P 2n (~ ) = - 2 (I - 2:" ) B2t1 , (n = I, 2, ... ) . (3.1.20)
We may speak of maximum if B2n < 0, of minimum if B2n > 0. In the first case the polynomial P2n(X) hasin [0,1] the minimum (equal to zero) at the extreme points x = ° and x = 1; from it and from the fact for (3.1.10) we have P~n(O) = P~n(l) = 2 n (2 n - 1) B2n - 2 follows B2n - 2 > 0. Ana logously in the other case we obtain B2n - 2 < 0. We conclude that B2n - 2
and B 2n have always opposite signs and therefore, being B2 > 0, that B 2n
has the sign of (- 1)"-1; there follows that P2n(x) has for ° < x < 1 the sign of (- It·
3.2. Euler functions. The Euler function of second kind or gamma function is defined for iX > ° by the following formula
+00 r(iX) = J e-X X,,-1 dx,
o (iX> 0) ,
r(I) = 1 ,
Then we have +00 +00
r(iX + 1) = J e-X x" dx = [- x" e-X]:oo + iX J e-X X,,-l dx o 0
that is r(iX + 1) = iX r(iX) .
For n positive integer, a repeated application of this formula yields
(3.2.1)
(3.2.2)
(3.2.3)
(3.2.4)
r(iX + n) = (iX + n - 1) (iX + n - 2) ... (iX + 1) iX r(iX) , (3.2.5)
I and particularly putting iX = 1 or iX = -2- and using (3.2.2) and (3.2.3):
r (n + 1) = n! (3.2.6)
r(n+ ~)=(n- ~)(n- :) ... ~-. ~V;=2~nn~:V;. (3.2.7)
From equation (3.2.4) follows too
r(iX + 1) = iX (iX - 1) (iX - 2) ... (iX - n + 1) r (iX - n + 1), (iX> n -1)
whence the binomial coefficients (:) may be expressed as follows
( ",) 1 r(ex + 1) n = n! r(ex _ n +1)' (iX> n - 1) . (3.2.8)
The Euler function of first kind or beta function is defined by the formula 1
B(iX, fJ) = J X,,-1 (1 - x)fJ- 1 dx , o
(iX > 0, fJ > 0) . (3.2.9)
3.2 Euler functions 51
Now we prove that this function is connected to the gamma function by the following relation
In fact for (3.2.1) we have
r(ex) r({3) B(CI., fJ) = r (ex + (3)
+00 +00
(3.2.10)
T(CI.) T(fJ) = J e- X X",-1 dx . J e-Y yf3- 1 dy = = J J e-(x+y) X",-1 yf3- 1 dx dy , o 0 Q
where Q denotes the quadrant x 2: 0, y > ° of the plane x y. Operating in this double integral the changing of variables x = u v, y = (1 - 1t) v, which transforms the quadrant Q into the strip S defined by ° < u < 1, v > 0, we get
T(CI.) T(fJ) = J J e-V U",-1 v"'+f3-1 (1 - u)f3-1 du dv s 1 +00
= J U",-1 (1 - u)f3-1 du . J e- v 1,,,,+f3-1 dv; o 0
but for (3.2.1) and (3.2.9) the latter expression has value B(CI., fJ) T(CI. + fJ) and (3.2.10) follows.
Finally we prove, for the gamma function, the following Legendre duplication formula
r(2 CI.) = V~ 22",-1 T(CI.) T(CI. + ~). (3.2.11)
From (3.2.9) follows 1 1/2
B(CI., CI.) = J X",-1 (1 - X)",-1 dx = 2 J (x - X2)",-1 dx o 0
and operating on this integral the substitution x = (1 - Y-l='y)/2, which implies y = 4 (x - X2):
J1 ( y)"'-1 dy 1 J1 - ~ B(CI., CI.) = 2 4 4 VI _ Y = 22",-1 y",-1 (1 - y) dy,
o 0
that is for (3.2.9)
B(CI., CI.) = 22:-1 B (CI.,~-) . For (3.2.lO) this formula may be transformed into
[r(ex)]2 1 r(ex) r ( ~ ) r(2 ex) = 22 ",-1 ( 1 )
rex+ 2
4·
3.3. Orthogonal polynomials connected to a given weight function. Let a weight function p(x), measurable and non negative, be defined on an interval [a, b], finite or infinite, being p(x) > ° in a set of positive measure and p(x) xn E L[a, b] for n = 0, 1,2, ... ;1) therefore there exist finite all the moments
b
fln = J p(x) xn dx , (n = 0, 1, 2, ... ) , (3.3.1 ) a
of p(x) and we have flo > 0. Let then a sequence {an}n~o, 1, 2, ... of numbers all different from zero be assigned.
Then the following theorem holds:
Theorem 3.3.1. It is possible to construct in one and only one way a sequence {Pn(X)}n~o,I,2, ... with Pn(x) polynomial of n degree having an as coefficients of x n, in such a way that
b
J p(x) Pm (x) Pn(x) dx = ° if m 0:/= n, a
or, under an equivalent form b
J p(x) IIn~I(x) P n(X) dx = ° , a
(n = 1,2, ... ) ,
where IIn~I(X) denotes an arbitrary polynomial of degree < n - 1.
(3.3.2)
(3.3.3)
Proof. For the polynomial Po(x

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