Table Of ContentTHEORETICAL
NUMERICAL
ANALYSIS
BURTON WENDROFF
Los Alamos Scientific Laboratory
Los Alamos, New Mexico
ACADEMIC PRESS New York and London
COPYRIGHT © 1966 BY ACADEMIC PRESS INC.
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To My Mother
Preface
My purpose in writing this book is to present numerical analysis as a
legitimate branch of mathematics, deserving attention from mature mathe
maticians and students alike. In keeping with this theme the reader will
find himself traveling a narrow, often deep path through five basic fields
of numerical analysis: interpolation, approximation, numerical solution
of ordinary and partial differential equations, and numerical solution of
systems of equations. The direction and depth of the path, while largely
a matter of my own taste, are constrained when feasible so as to lead to a
consideration of good computing technique (large scale digital).
Most of the book can be read by anyone with a good understanding of
advanced calculus and linear algebra. There are, however, occasional
references to ideas usually covered in a course on methods of applied
mathematics, such as Green's functions, maximum minimum principles,
self-adjoint boundary value problems, Fourier analysis, and notions from
ordinary and partial differential equations. These items are not explained
in the text. This fact, together with the complexity of some of the argu
ments, makes this volume primarily a graduate text.
If it were just a matter of proving convergence theorems or finding error
estimates, numerical analysis would be a sterile subject indeed. What gives
it life is the fact that numerical procedures must face up to experimental
test, a point at which many an elegant method has had to be abandoned. In
spite of this I have not included any numerical examples, because I feel
that numbers have a much greater impact when produced by the student
in his own programs. At the end of most chapters a few computer problems
are suggested, and I strongly urge that these be worked out, if the proper
equipment is available. Some exercises are also provided at the end of each
chapter, with hints for their solution where I felt such were necessary.
vu
vin PREFACE
Many results stated herein are not credited to a specific person because it
would have been impossible to trace the priority through the literature.
Most of this book was written while I was Visiting Associate Professor
at Brown University. I am indebted to the Division of Applied Mathe
matics ofthat university and to the U.S. Office of Naval Research for their
support during the academic year 1963-1964.1 am particularly grateful to
Philip Davis, who arranged my stay at Brown and who gave me much
encouragement and inspiration. Bruce Chartres, also of Brown, read the
preliminary manuscript and suggested many corrections and improve
ments. Two colleagues at Los Alamos, William Beyer and Blair Swartz,
read parts of the manuscript. The entire final version was read by Herbert
Kranzer of Adelphi University. The first draft was typed by Ezoura
Fonseca, whose accomplishment can only be appreciated by those who
have seen my unimaginably bad handwriting.
Los Alamos, New Mexico BURTON WENDROFF
Preliminaries
A reference in Chapter m to Eq. (n) means the nth numbered equation
of Chapter m. A reference in some other chapter to (m-n) means the
same. Theorems, lemmas, and corollaries are identified by the chapter in
which they occur, e.g., Th. m.n. is the wth theorem in Chapter m.
The symbol [x , ..., x ] signifies the smallest closed interval containing
x k
the points x ..., x .
l9 k
Following a suggestion of B. Chartres we have replaced statements such
as
■t,1 f = l,2, ..., n
r=l J — A> z» ···» *
by
For i = 1, 2, ..., n
For j = l,2,...,i
fl«7 = Σ air + !
I I r=i
in order to show precisely the order in which the operations are to be
performed.
CHAPTER 1
Interpolation and Quadrature
Interpolation is probably the most important mathematical tool used in
numerical analysis. In abstract terms, interpolation involves representing
functions in some class by those of another class, presumably with a
simpler structure. The representative can be used to approximate the
original in the calculation of integrals or derivatives or in other operations.
The most useful interpolating class is the class of polynomials, and it is
this which we shall consider in this chapter.
1.1. HERMITE INTERPOLATION
The most commonly used interpolating polynomial is that of Lagrange,
namely, the polynomial of degree n — 1 which agrees with a given function
f(x) at n distinct points. A more general procedure is that of Hermite, in
which certain derivatives of the interpolating polynomial are required to
agree with corresponding derivatives off(x).
To be precise, let there be given real distinct points x, ..., x and integers
l k
m , ..., m such that
i k
mi > 1, i = 1, 2, ..., k
m + m + ··· + m = n
x 2 k
1
2 1. INTERPOLATION AND QUADRATURE
The Hermite interpolation problem is to find the polynomial P(x) of degree
n — 1 such thatf(x) — P(x) has zeros at least of order m at the x.
t t
We always assume that/(x) has enough derivatives so that this definition
makes sense.
We now have
Theorem 1.1. The Hermite interpolation problem has a unique solution.
Proof Let
ίθ if ι=0
(1) KO =
[m, + ··· + m-, if />0
(I if (=0
(2) π(χ, (') =
|(x-x,)mi ···(*-- χ,Γ if />0
To be complete we should write n(x i,m, ..., /w,·) for π(χ, /), but this would
9 x
be cumbersome.
Consider a polynomial of the form
k - 1 m, + i — 1
(3) P(x)= Σπ(χ,ί) £ a j(x-x y
r(i)+ i+i
i' = 0 j = 0
If we could find coefficients a, / = 0, 1, ..., n — 1 such that
t
for i = 1, 2, ..., k
for a =0, 1, ..., m — 1
f
(4) L L f{a\xi)=p{a\xi)
then P(x) would solve the problem. To show that (4) uniquely determines
the coefficients of P, associate with each equation (4) the unique integer
s = r(i — 1) 4- a. Inspection of P(x) shows that for any relevant / and a
P(x) = A(x) + π(χ, i - l)(x - Xi)"a + B(x)
s
where A(x) is a linear combination of the coefficients α ,α, ...,a_
0 χ s l
and B(x) is a linear combination of a , ...,a _. Since, using Leibnitz'
s+l n x
rule,
Ρ(α)0,.) = A*\xd + 7t(jf„ i - \)oi\a
s
1.1. HERMITE INTERPOLATION 3
we see that (4) is a triangular system ; that is, the 5th equation involves only
a ,a, ..., a. Having found α ,α, ..., a_, then
0 l s 0 { s i
(5) a* = —7 '■—7T7~
π(χ·, / — 1)α!
(
Thus, the algorithm (4) defines a ,a, ..., α _, in that order.
0 i η ί
To establish the uniqueness of P(x) suppose some other polynomial
Q(x) of degree n — 1 solves the Hermite problem. Then P(x) — Q(x) is a
polynomial of degree n— 1 having zeros of total order n\ therefore
P(x) = Q(x), finishing the theorem.
The usefulness of the Hermite interpolating polynomial P(x) as a re
placement ïorf{x) is limited by the amount of information available about
the behavior of the error, f(x) — P(x), when x is not a point of interpola
tion. Let us write this error in the form
(6) f(x) - P(x) = n(x)G(x)
where π(χ) = π(χ, k). Then we have
Theorem 1.2. Let m = max <,·<*. m and let J be some interval containing
x t
the Xi. If fix) has m continuous derivatives on J then G(x) is continuous
onJ.
Proof. If x is not a point of interpolation, then, since π(χ) Φ 0, the
relation
fix) - Pix)
il) Gix) =J-
π(χ)
defines G as a continuous function at such an x. Now for 0 < a < w we
f
have/^fo·) - Pia\xi) = 0 and π^Ο,) = 0, whereas nmix^ φ 0, therefore
if we define G(x) as
f
, Uimi)(Xi) - P^Kxdl
r(
πκ l,iXi)
then by L'Hospital's rule
G(x) = lim Gix)
f
X-*Xi
proving the theorem.
4 1. INTERPOLATION AND QUADRATURE
With a little more effort we can prove
Theorem 1.3. With m and Jas defined in Th. 1.2, iff has m + 1 continuous
derivatives on J then G' is continuous on J.
Proof Differentiating (7) we have
4r-n-<v-n
π
defining G as continuous at a noninterpolating x. Let
π' d
h(x) =(x — Xi) — =(x — Xi) — In π
π dx
J = 1 (X-Xj)
Then
, (x - x,W - P') - *(xX/ - P)
(9) Gr = Ϊ^Φ
Since all derivatives of the denominator of order m or less vanish at x but
i {
the (m + l)st derivative does not, we need only establish that all deriva
x
tives of the numerator of (9) of order wi,· or less also vanish at x , for then
{
it follows that Ηιη _ G\x) exists. It follows readily from the continuity of
χ Χι
G at Xi that
lim G'(x) = G'(x)
t
X-*Xi
The numerator itself clearly vanishes at x. By Leibnitz' rule we have
t
C( )(/ F)] (/ p)< +1_r,(x x r
έ * - *·· ' - - - Lt(") - ° - ' L·
(0 if a < m
f
and
,
έ««-«ΐ..·|Σ0(ί-*-* ] .
ι
(0 if a < m,
Λ(χ,Χ/- PA"-! if « = m,
