Reference

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SlSiR 80-1976

Numerical Solution of Linear

Difference Equations

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Daniel W. Lozier

Mathematical Analysis Division

Center for Acplied Mathematics

National Engineering Laboratory

National Bureau of Standards

U.S. Department of Commerce

Washington, D C. 2C234

March 1980

Final Report

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U.S. DEPARTMENT OF COMMERCE

national SUR6AU OF STANOARDS

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80-1976

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NBSIR 80-1976

NUMERICAL SOLUTION OF LINEAR

DIFFERENCE EQUATIONS

O -

/Y 4 ^

Daniel W. Lozier

Mathematical Analysis Division

Center for Applied Mathematics

National Engineering Laboratory

National Bureau of Standards

U.S. Department of Commerce

Washington, D.C. 20234

March 1980

Final Report

U.S. DEPARTMENT OF COMMERCE, Philip M. Klutznick, Secretary

Luther H. Hodges, Jr., Deputy Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director
J . •* , i«!

V *

' r
1

ABSTRACT

Consider a given homogeneous or inhomogeneous linear difference

equation

Z.g-o “ g(r) where i ^2 and r = 0,1,2,... . Suppose y is a solution of this equation and u,v are solutions

of the homogeneous form of this equation such that u(r)/v(r) -> 0,

y(r)/v(r) -*• 0, u(r)/y(r) 0. It is knox^n that under these circumstances algorithms for the computation of y based on forward

recurrence or backward recurrence, such as the Miller algorithm,

are numerically unstable".

Stable algorithms, such as the method of Olver in the case 1=2,

have been based on approximating y(r) by the solutions of a certain sequence of boundary value problems. More specifically, Y^^^)

is a solution of the difference equation that coincides with y(r)

over some fixed initial range of r, say r = i, i+1, . . . , i+j -1 , and

satisfies Y^Cr) = 0 for r = n,n+l, , .

. ,n+£-j-l. Here j is an integer whose value depends on the asymptotic behavior of the chosen

solution y(r) and n is an arbitrary large integer. Boundary value

problems of this type are shown to be equivalent to tx<ro initial value

problems of order j and £-j by factorization of the linear difference operator. The solution of the problem of order j is obtained

by forward recurrence; the solution of the other problem is obtained

by backward recurrence.

The algorithm is specified completely for a broad class of linear

difference operators. This class includes, for example, every constant

coefficient operator. Convergence of v (r) to yCr) as n for

^ n fixed r is proved and an expansion of the truncation error is derived

Numerical stability is demonstrated under appropriate conditions. The

method is tested by numerical examples involving fourth-order equations

with variable coefficients.
PREFACE

This paper was submitted as a Ph.D. thesis in the Interdisciplinary Applied Mathematics Program of the University of Maryland

at College Park. The Ph.D. degree was awarded on December 18, 1979.

ACKNOWLEDGMENTS

I wish to acknowledge the aid and encouragement rendered by my

thesis professor, Dr. Frank W.J. Olver. His leadership was superb.

I also wish to express my appreciation to Dr. Daniel Shanks for

directing me to the literature on almost periodic functions, to

Donald J. Orser for helping with MACSYMA, to Richard N. Freemire

for producing the figures, to Katherine Mesztenyi for typing the

manuscript, and to my wife Elaine for her patience.

Computer support was provided by the Center for Applied

Mathematics, National Bureau of Standards, and the Mathlab Group,

Laboratory for Computer Science, Massachusetts Institute of

Technology
iii

CONTENTS

1. INTRODUCTION

1.1. Miller's and Olver's Algorithms 1

1.2. Factorization of Linear Difference Equations 7

1.3. A Preliminary Development of the Algorithm 22

2. CLASSIFICATION OF SEQUENCES

2.1. Constant-Coefficient Operators 36

2.2. Separable Operators '^2

2.3. The General Classification 51

3. THE GENERAL ALGORITHM

3.0. Preliminaries and Overview 55

3.1. Existence and Uniqueness 58

3.2. Convergence 63

3.3. Stability 70

4. NUMERICAL EXAMPLES

4.0. Introduction 74

4.1. Examples Involving the Homogeneous JYIK Equation .... 85

4.2. Examples Involving the Inhomogeneous JYIK Equation ... H7

4.3. Examples Involving the Homogeneous JYPQ Equation .... 121

4.4. Examples Involving the Inhomogeneous JYPQ Equation . . . 139

5. SUMMARY AND CONCLUSIONS 144

APPENDIX 154

REFERENCES 158
V

TABLES

1. Numerical coefficients, JYIK operator, x^=l,X2=10 90

2. (0, 1) “factorization, JYIK operator, x^=l,X2=10 93

3. (0,2) -factorization, JYIK operator, x^=l,X2=10 96

4. (0,3) -factorization, JYIK operator, x^=l,X2=10 99

5. J^(l), JYIK eq., x^=l,X2=10, i=0,m=100, j=l 102

6. 1^(10), JYIKeq., x^=l,x^=10, i=0,m=100, j=2 103

7. (-)^K^(IO), JYIKeq., x^=l,X2=10, i=0,m=100, j=3 104

8. Y^(l), JYIKeq., x^=l,X2=10, i=0,m=lQ0, j=4 105

9. 1^(1), JYIK eq., x^=10,X2=l, i=0,m=100, j=l 107

10. J^(IO), JYIKeq., x^=10,X2=l, i=0,m=100, j=2 108

11. Y^(IO), JYIKeq., x^=10,X2=l, i=0,m=100, j=3 109

12. (-)^K^(l), JYIK eq., x^=10,X2=l, i=0,m=100, j=4 110

13. A^(l), JYIKeq., x^=X

2 =l, i=0,m=100, j=2 112

14. J^(l), JYIKeq., x^=X

2 =l, i=0,m=100, j=2 113

15. 1^(1), JYIK eq., x^=X

2

=l, i=0,m=100, j=2 114
V

TABLES (Continued)

16. Y^(l), JYIK eq,, x^=X

2 =l, i=0,m=100, j=4 115

17. (-)^K^(l), JYIK eq., x^=X

2

=l , i=0,m=100, j=4 116

18. E^(l), JYIK eq., x^=x^=l, i=0,m=100, j=2 119

19. H^(O.l), JYIK eq., x^=X2=0.1, i=0,m=50, j=2 120

20. Numerical coefficients, JYPQ operator, y=0,x^=l , X2=0. 5 .... 123

21. (0 .1) -factorization, JYPQ operator, p=0,x^=l,X2=0.5 126

22. (0, 3) -factorization, JYPQ operator, y=0 ,x^=l ,X2=0 .5 129

23. J^(l), JYPQ eq., U=0,x^=l,X2=0.5, i=0,m=100, j=l 132

24. P^(0.5), JYPQ eq., U=0,x^=l , x^=0 . 5 , i=0 ,m=100, j=3 133

25. Q^(0.5), JYPQ eq., y=0,x^=l,X2=0.5,i=0,m=100, j=3 134

26. Y^(l), JYPQ eq., y=0,x^=l,X2=0 .5 , i=0,m=100, j=4 135

27. P^(0.5), JYPQ eq., y=l ,x^=l, X2=0. 5 , i=0,m=100 , j=3 137

28. q\o. 5), JYPQ eq., y=l,x^=l,X2=0.5,i=0,m=100, j = 3 138

29. E^(l), JYPQ eq . , y=l,x^=l,X2=0.5, i=0,m=100, j=3 142

H^(O.l), JYPQ eq . , y=l , x^=l , x^=0 .5 , i=0 ,m=100 , 30. j =1 143
^ri

FIGURES

1. The operator 9

2. The matrix 9

J

3. The matrix 10

J

4. The factorization (1.2.5) 16

5. The roots of (2.1.2) 39

6. Fragment of FORTRAN subroutine, JYIK operator 76

7. Fragment of FORTRAN subroutine, JYPQ operator 78
CHAPTER 1. INTRODUCTION

1 . 1 Miller's and Olveras Algorithms

Linear recurrence relations (linear difference equations) are satisfied by many of the higher transcendental functions of mathematics, and

they arise quite naturally in applications of mathematics. Discretization

of ordinary differential equations in numerical analysis is one important

area in which they occur. Because of their special form, linear recurrence

relations are especially well suited to the construction of computing

algorithms. Stability requirements in the case of second-order homogeneous

relations led to the well-known Miller backward recurrence algorithm [ 5 ]

which was developed for the computation of modified Bessel functions. This

reference was published in 1952. The review paper of Gautschi [9] provides a survey of algorithms and applications in the case of second-order

homogeneous equations. Nowadays, the name of Miller often attends when a

minimal solution (defined below) of a homogeneous equation of arbitrary

order is computed using backward recurrence [28]. These algorithms always

involve backward recurrence from assumed starting values (since these are

not known in general) followed by a normalization procedure, such as matching one value of the computed solution with the desired solution in order

to arrive at the proper scale factor which relates these two solutions.

A different class of algorithms becomes necessary when "intermediate"

solutions of a linear recurrence relation are to be computed. Intuitively

speaking, an intermediate solution is one for which two other solutions can

be found such that one dominates, and the other is dominated by, the intermediate solution. We are speaking here of dominance in the following sense:
if a = (a(0) , a(l) , . . . ) and b = (b(0) ,b(l) ,

. . .) are two infinite

sequences then b dominates a whenever a(r)/b(r) ->• 0 as r .

Neither forward recurrence nor the Miller algorithm is numerically

stable for an intermediate solution. This is because recurrence in either

direction, when carried out in finite precision, will be contaminated by

small components of the other solutions. An analysis of the propagation of these rounding errors appears in [18] . In the forward direction,

as the recurrence proceeds through more and more steps, the dominant solutions grow and eventually overwhelm the desired intermediate solution.

Since our concept of dominance is really dominance "at infinity", the situation for recurrence in the backward direction is not completely analogous.

However, if we start at a sufficiently large value of r and recur backward,

other solutions grow at a faster rate than the wanted intermediate solution

as r decreases.

In addition to intermediate solutions, a difference equation has minimal

(or "recessive") and maximal (or "dominant") solutions. Intuitively, a

minimal solution does not dominate, and a maximal solution is not dominated

by, any other solution of the difference equation as r . Ultimately,

forward recurrence is stable for maximal solutions since the error cannot

grow more rapidly than the solution. Similarly, backward recurrence is

stable for minimal solutions, at least in a region of sufficiently large r .

We shall regard the solutions of a linear difference equation, as well as the coefficients of the equation, as complex-valued functions of a discrete

real variable. We shall assume that this discrete real variable proceeds

in unit increments; thus the solutions, as well as the coefficients, of

linear difference equations are complex sequences. We shall be interested

only in those linear difference equations whose coefficient sequences are

of infinite length.
3

The lowest order of difference equation for which an intermediate solution can exist, and then only when the equation is inhomogeneous, is two.

Such an equation is of the form

(1.1.1) a(r)y(r-l)-b(r)y(r)+c(r)y(r+l) = d(r) , r = 1,2,... .

Since the general solution is

y(r) = au(r) + pv(r) + h(r) , r = 0,1,2,... ,

where a and p are constants, u= ( u(0) , u(l) , . . . ) and v= (v(0),v(l),

...) are linearly independent solutions of the homogeneous equation, and

h = (h(0) ,h(l) , . . .

)

is a particular solution of (1.1.1), there can be at

most three asymptotically distinct types of behavior of solutions as

r . Let us assume that v dominates u and that h is an intermediate solution. In addition, assume u(0) ^ 0 and v(r) 0 for all

sufficiently large r . Under these assumptions a stable algorithm for

computing intermediate solutions of (1.1.1) was presented by Olver in [21].

If y = (y (0) , y (1) , . .

.

)

is the desired intermediate solution. Giver's

algorithm approximates y by a sequence of solutions y = (y (0),y (1), n n n

...) such that y (0) = y(0) and y (n) = 0 where n is an appropriaten n ly chosen positive integer. Note that only one initial value of the wanted

intermediate solution is required to determine the sequence of approximations. The sequence is shown to converge pointwise to y as n .

Giver's algorithm efficiently produces the first n-1 unknown terms

of y as the solution of a tridiagonal linear system of order n-1 . The

n procedure used for solving these linear equations is equivalent to Gaussian

elimination without pivoting. It consists, in effect, of a forward elimination stage followed by a back substitution stage. The algorithm is constructed so that the numerical stability of this procedure may be assessed. The
fo-n^ard elimination stage is realized by two forward recurrences. One of

these uses the homogeneous form of (1.1.1) to generate a dominant solution

P ~ (p(0) >p(l) 5 •

•

•) . The initial values used for starting this recurrence

are p(0) = 0 and p(l) = 1 , and the solution p is asymptotically proportional to V . Therefore, the rounding errors ultimately propagate in

proportion to p , although they could propagate faster than p at first

(if p is essentially proportional to u for early values of r ,

corresponding to an excessively small value of u(0)). A combined algorithm

that employs both Miller's algorithm and Giver's algorithm in adjacent

ranges of r is proposed in [22].

The second forward recurrence in the forward elimination stage of

Giver's algorithm uses the inhomogeneous first-order equation

a(r)e(r-l) - c(r)e(r) = d(r)p(r) , r = 1,2,...,

where the solution e = (e (0) , e(l) , . .

.

)

satisfies e(0) = y(0) . Here

two distinct asymptotic forms of behavior may exist, but there is no reason

to expect that the calculation of e will be asymptotically unstable in

general. However, as was the case for the solution p , there may

be applications where rounding errors grow more rapidly then e(r) over

some finite initial range of values of r .

The back substitution consists of backward recurrence of the inhomogeneous first-order equation

p(r+l)y (r) - p(r)y (r+1) = e(r) , r = n-1 , n-2 , . . .

,

1

, n n

starting with ~ ^ stability of this backward recurrence is

guaranteed in the "asymptotic region" belonging to the chosen solution

y(r) , that is, the infinite range of r which is such that the asymptotic
5

behavior of y(r) relative to u(r) and v(r) is maintained.

Giver's method of solving the tridiagonal system leads to a very convenient representation of the truncation error. For each r and for all

n > r , the expansion

( 1 . 1 . 2 ) y(r) = y (r) + p(r)

n

y e(s)

^ p(s)p(s+i) s=n

is valid. Consequently, for each r and n > r and v > 1 we have

n+v

n+v-1

(r) - y^(r) = p(r) J e(s) s=n p(s)p(s+l)

This expansion is useful as a basis for deciding on an appropriate value of

n , i

e

. , for deciding when to stop the forward elimination and begin the

back substitution. Suppose that for particular values of e > 0 and

r , say r = m , it is desired that

ly(m) - y^(m) I < s .

A value of v(> 1) is selected and yfm) is approximated bv y , (m) . n+v The problem then is to determine the smallest value of n such that

< £ .

This is done by continuing the forward elimination until r = m at

which point calculation of the numbers

r\(r)

r+v-1 . !p(ni) I , T N I . r = m,m+l,...

p(s)p(s+l)'
is begun. Forward elimination is continued further, together with the

calculation of 'n(^) > until a value of r is reached such that Ti(r) < e .

This value of r is taken for n ; its adequacy may be checked by taking

additional terms in the infinite expansion of y(m) - Y^(^) » given by

(1.1.2) with r = m .

In addition to being appropriate for computing intermediate solutions

of (1.1.1), Giver’s algorithm is suited to computing the minimal solution

of a homogeneous equation. It can therefore be compared to Miller's

algorithm. The advantage of Giver's algorithm over Miller's algorithm

is that the latter does not have a criterion for determining automatically

an appropriate value for n . Gn the other hand, if an adequate value

of n is known from other considerations (as, for example, from an

asymptotic estimate), then Miller's algorithm may require less computational effort.

Zahar [27], Gliver [19,20], and Cash [ 6] have considered extensions of

Giver's algorithm to linear difference equations of higher order. In the

present work we shall construct and analyze an algorithm which has features

in common with these extensions. We shall also illustrate the algorithm

by means of numerical examples.
7

1.2 Factorization of Linear Difference Equations

Let F be the field of complex numbers and let S be the set of all

complex-valued infinite sequences x = (x(0) ,x(l) , . .

. )

. Clearly S is

an infinite-dimensional linear space over F . Let be the set of

all infinite upper- triangular band matrices of the form

dp(0) .... d^(0)

0

d„(l) .... d^(l)

dg(r) .... d^(r)

0

where d^,d^,...,d^ € S and d^Cr) ^ 0 , d^(r) 0 for infinitely

many values of r . We shall also use the concise notation [d^ j d^ , . .

. , d^]

to represent infinite matrices in the set . If we regard the infinite

sequences in 5 as infinite column vectors, then an infinite sequence

Dx € S is defined for each D ^ , x € S by ordinary matrix multiplication. In fact, the terms of Dx are given by

£

Dx(r) = y d (r)x(r+s) , r = 0,1,2,... . s=0

®

Clearly, each D € is a linear operator on S under this definition.

We shall say that is the class of linear difference operators of order

£ on S .

Note that is defined for each integer £ > 0 . We pointed out in

§1.1 that intermediate solutions of linear difference equations
can exist only if the order of the equation is at least two. But we are

going to factor linear difference operators into a product of two lowerorder operators. Therefore, operators of order one will arise. Furthermore, the formalism to be introduced will allow even the degenerate case

of operators of order zero.

If d^ and d^ have no terms equal to zero, we shall say the difference operator [d^ , d^ ,

. . . , d^] € is nonsingular ; compare [14, §12.0].

The terms of d^ and d^ will be called the leading and trailing coeffic -

ients of [d^,d^, . .

. ,d^] , respectively.

We shall use the notation to indicate a sequence from S whose

first term is x(i) . Thus

x^ = (x( i) , x( i+1 ),...) .

The point i will be called the initial point of the sequence. When we do

not indicate a superscript it is to be understood that zero is the initial

point of the sequence. Similarly, we shall use the notation x^’^ to

indicate the (finite) subsequence of x^ whose final term is x(n) . Thus

x^’^ = (x(i) ,x(i+l) , . . . ,x(n)

)

.

The point n will be called the terminal point of the subsequence.

Similarly, for linear difference operators from , we shall use the

notation or [d^jdj^, , . . ,d^] to indicate that i is the initial point

of each of the £ + 1 sequences involved in the definition of the operator. See Figure 1. When the superscript is omitted, it is understood

that these initial points are all zero unless it is clear from the context

(as in equation (1.2.1) below) what these initial points are.
9

Figure 1. The operator D

Figure 2. The matrix
Figure 3. The macrix
11

Two further notational conventions to be used in this thesis are

Dj , the infinite matrix obtained from by deleting the first j

columns (05j5-£.) ; and leading principal submatrix of order

n - i + 1 of Figures 2 and 3. Note that each diagonal of

below the principal diagonal is a subsequence of one of the sequences

d^jd^, . .

. ,d^_^ j the sequence of d^ is on the principal diagonal, and

the sequences d^_^^,...,d^ form the diagonals above the principal diagonal. The form of the finite matrix D^’'^ is similar, with the finite

subsequence d^’^ lying on the principal diagonal.

A linear difference equation is an equation of the form Dx = g ,

where g ^ S and D ^ are prescribed. Either x C S (or a subsequence of X of the form x^’'^) is to be determined. This equation

is equivalent to the infinite linear algebraic system

Dx(r) = g(r) , r = 0, 1, . .

.

.

Analogously to linear differential equations, auxiliary conditions are

required in order to specify a unique solution. In our investigations

these auxiliary conditions will consist of tv/o finite subsequences

i,i+j-l , n,n+k-l . ^ , . , . , X and X (or possibly only one of these; which are to be

specified in advance, where j + k = ^ and j > 0 , k > 0 . These are

called the initial and terminal conditions

,

respectively, when n > i + j

mine y

( 1 . 2 . 1 )

Let us consider the following finite boundary value problem:

i+j ,n-l

Detersuch that

Dy(r) = g(r) r = i, i+1, . . . ,n-j-l

together with initial conditions

v(i+r) = a , r = 0,l,...,j-l

r

( 1 . 2 . 2 )
and terminal conditions

(1.2.3) y(n+r) ~ ^ ~ 0,1,..., k-1

are satisfied, where DcP^,'2.>l,g^-S,jfc {0,1,...,-^} , k - > i>0,n>i+j and ^ F . In view of

(1.2.2) and (1.2.3), the finite subsequences ^ and

^

are known. Of course, if j = 0 then ^ is null. Similarly if

k = 0. These cases correspond to the absence of either (1.2.2) or (1.2.3).

The unknoxvn subsequence ^ satisfies a linear system of algebraic

equations which can be expressed in matrix terms using the notation we

introduced above. The system is

(1.2.4)

i,n-j-l i+j,n-l ^ i,n-j-l

j

/ S

provided that n - i - j > max(j,k) . The first partitioned column vector

on the right side of (1.2.4) has j entries in the upper part, and the

second partitioned column vector has k entries in the lower part. The

length of the other column vectors appearing in (1.2.4) is n - i - j.

This is what gives rise to the restriction n - i - j > max(j,k) . This

restriction is irrelevant for the applications we have in mind, since we

shall be considering sequences of finite boundary value problems as n -> «> .

Note chat the number of zeros represented by the symbol 0 in the partitioned column vectors is n - i - 2j in the first vector and n - i -

in the second vector. Obviously, the partitioned column vectors reflect

the influence of the boundary conditions (1.2.2) and (1.2.3).

Solution of the boundary value problem (1.2.1) to (1.2.3) is equivalent CO solution of the linear system (1.2.4). Equation (1.2.4) possesses

0

^ 0

0 ^n--d,n-i-l n,n+k-l

y
13

a unique solution if, and only if, the matrix ^ ^ is nonsingular.

If in addition we assume that the difference operator D is nonsingular,

then the solution ^ of (1.2.1) to (1.2.3) extends uniquely to

y € S by forward and backward recurrence. However, nonsingularity of D

is not necessary for this unique extension to exist; for example, some

of the leading and trailing coefficients of D which appear in the

• • * -r. ^ 5 nTM" T " 1 . , , finite matrix D. could be zero.

3 Let us consider the nonsingular triangular cases of (1.2.4). The

first is given by j = -^ . Then (1.2.4) assumes the form

(1.2.4a) •ci-t-l i+£,n-l

y

i, n-£-l

g

^i,i+^-l i,i+Z-l

°0 ^

Writing out these equations in detail, we have

1-1

d.(i)y(i+£) = g(i) - \ d (i)y(i+s) ^ s=0

^

1-1

d^_^(i+l)y(i+^) + d^(i+l)y (i+^+1) = g(i+l) - \ d^ (i+l)y (i+s+1)

s=0

I

\ d (i+£)y(i+^+s) = g(i+^)

s=0

®

\ d (n--^-l)y(n-^-l+s) = g(n-£-l)

s=0

®

Since this system is lower triangular, it is solvable by determining

y{±+V) from the first equation, y(i+£+l) from the second equation, and

so on. This process is equivalent to developing y (i+^) , y (i+£+l) , . . . , y (n-1)

from (1.2.1) by forward recurrence starting with the given values (1.2.2).
Then (1.2.4) assumes

The second triangular case is given by j = 0 . the form

i,n-l i,n-l i,n-l

•o y = g

Writing out these equations in detail, we have

i

I d (i)y(i+s) = g(i)

s=0

®

n-£,n-l n,n+^-l

y

I

^ d ^ (n-£-l)y (n--£-l+s) = g(n-^-l)

s=0

^

I

dQ(n-2)y (n-2) -f- d^(n-2)y (n-1) = g(n-2) - I d^ (n-2)y (n-2+s)

s=2

£

d^Cn-DyCn-l) = g(n-l) - I d^ (n-l)y (n-l+s)

s=l

Since this system is upper triangular, it is solvable by determining

y(n-l) from the last equation, y(n-2) from the next-to-last equation,

and so on. This process is equivalent to developing y (n-1) , y (n-2) , . . .

y(i) from (1.2.1) by backward recurrence starting with the values (1.2.3).

The noncriangular cases of (1.2.4) in which 0 < j < ^ will be of

greater interest to us. Our primary goal is to find a way to solve these

problems by performing, in effect, a forward recurrence of order j followed by a backward recurrence of order £ - j

.

Definition 1.2.1 A nonsingular boundary value problem (1.2.4), or equivalently (1.2.1) to (1.2.3), having initial point i , terminal point n,

number of initial conditions j and difference operator

D = [d^,dj^ , . . . ,d^] , will be called factorizable provided there
15

exist difference operators A = [a^, . ,a^l of order i and

0 1 j

B

^ = [b^ ^

»

• • • ] of order k = I - j such that

(1.2.5) i,n-j-l ^ i,n-j-l i+j,n-l

If in addition the sequences and free of zeros and the

infinite matrix factorization

( 1 . 2 . 6 ) J J

is valid, then we shall say that the difference operator D is (i, j

-

factorizable .

Several remarks may be made here. Note that ^ ^ and are

: J band matrices each having total bandwidth t + 1 , lower bandwidth j ,

and upper bandwidth k . A nonsingular and factorizable boundary value

problem is one for which a finite matrix factorization of ^ ^

exists such that the left factor is lower triangular with total bandwidth

j + 1 and the right factor is upper triangular with total bandwidth

k + 1 . The difference operators A^ and may be arbitrary as long

as they satisfy the condition (1.2.5), since (1.2.5) involves only a finite

submatrix of each of A^ and . Equation (1.2.5) is illustrated

in Figure 4. Since ^ ^ is nonsingular, both A^’'^ ^ and

J J

Bq"*"^

’ ^ nonsingular. Thus the entries on the principal diagonals

of A^’'^ ^ ^ and ^ are all nonzero.

J 0

Noxa7 suppose the difference operator D € is (i,j) - factorizable;

it need not be a nonsingular operator. Because of the triangular forms

of Aj and B ^

, equation (1.2.6) implies equation (1.2.5) for every

n>i+j+l. In words, every leading principal submatrix of is

j

equal to the product of the corresponding leading principal submatrices of

A^ and . The assumption that the subsequences a^joand

are free of zeros is equivalent to assuming that the entries on the
Figure

4.

The

factorization

(1.2.5).
17

principal diagonals of the infinite matrices and are all nonzero. It follows that every leading principal submatrix of is

J nonsingular. Thus we have proved the following theorem:

Theorem 1.2.1 Let D ^ be(i,j)- factorizable . Then, for every

n > i + j + max(j,£-j) , the boundary value problem (1.2.4) having difference operator D, initial point i, number of initial conditions j and

terminal point n is nonsingular and factorizable. j^j

The next theorem shows it is possible to obtain a solution to a factorizable boundary value problem, in theory at least, by performing an

appropriate forward recurrence followed by an appropriate backward

recurrence

.

Theorem 1.2.2 Let (1.2.4) be a nonsingular and factorizable boundary

value problem with initial point i, number of initial conditions j, termi i inal point n and difference operator D € . If A and B are

difference operators of order j and k = -£-j , respectively, such that

(1.2.5) is valid, then the solution of (1.2.4) is identical to the solution of

(1.2.7)

i+j,n-l i+j,n-l _ i+j,n-l

y

~ ^ 0

(An-£,n-j-l -1 n--£,n-j-l n,n+k-l ) D

I

y

, i+j,n-l , , ... - where z is the solution of

( 1 . 2 . 8 )

i,n-j-l i+j,n-l _ i,n-j-l

Aj 2 - g

i.i+j-i 1,1+j-i

0

0
Proof The matrices are nonsingular because ^ ^ and ^ —— ^ j 0

i n*“ i “ 2- D^’ is nonsingular; see (1.2.5). Therefore, the diagonal elements

.i»ti-i-l , „i+i,n-l , , . ,

. -> . n or and (which are triangular matrices) are nonzero

and (1.2.7) and (1.2.8) are nonsingular linear systems. Furthermore,

n-c., n-j-1

A

J is nonsingular because it is a principal submatrix of

A^’^ ^ . Therefore the right side of (1.2.7) is unambiguously defined.

Now, premultiplying (1.2.7) by A^’'^ ^ ^ and using (1.2.5) and

(1.2.8), we obtain

j^i, n-j-1 i+j,n-l

j ^

_ 0

0 j , n-£, n-j-1 -1 n-£, n-j-1 n,n+k-l

Consider the last product on the right side of this equation. Its block

structure is

i,n-£-l

j

; 0 0

^ : ^n-£, n-j-1

‘ j

^^n-£ , n- j -1

^

’ n-j -l^n , n-k-1

^ J —

where 0 in the matrix denotes a zero submatrix with n - i - £ rows

and k(= £-j) columns, 0 in the vector denotes a zero subvector of

length n - i - £ , and * in the matrix denotes a submatrix which needs

no further identification here. Therefore, the product in question is

n-£, n-j-1 n,n+k-l

equal to the vector
Substitution into the equation and comparison with (1.2.4) completes

the proof,

An important remark is that if (1.2.6) is valid, i.e. if D is (i,j)-

factorizable with left factor and right factor , then

n-^,n-j-l _ n--C,n-j-l n-k,n-l

T .-u • r,n-k,n-l 1 r / A ~1

In this case may be written in place of (A )

j^n t,n i 1 (1.2.7). The truth of this remark is apparent from Figure 4 The next theorem gives necessary and sufficient conditions for a

difference operator to be (i,j) - f actorizable . The proof also shows

that (i,j) - factorizations may be constructed by Gaussian elimination in

natural order , i.e., without any row or column interchanges.

Theorem 1.2.3 Let D € . Then

i) A nonsingular boundary value problem having initial point i,

number of initial conditions j and terminal point n is factorizable if

every leading principal submatrix of ^ ^ is nonsingular.

ii) D is (i,j) - factorizable if every leading principal submatrix

of is nonsingular. Furthermore, this condition is necessary if D

is nonsingular.

Proof Our approach is to apply Gaussian elimination [10, §2.1] without any

row or column interchanges. Because Gaussian elimination is so familiar,

we merely sketch the proof. The proof of (i) is contained in the first

n - i - j stages of the proof of (ii); therefore we supply only the

proof of (ii). We begin by proving sufficiency of the condition.

The first row of is used to annihilate every nonzero element

a below the first element in the first colum.n. This is possible because

the pivot element, which is the leading principal minor of order one, is

nonzero. Because of the band structure, only the j rows immediately follow-
ing the first row are affected. Also (again because of the band structure)

the trailing coefficient in each of these rows is not changed. The

multipliers used in these annihilations go into positions 2 through j + 1

of the first column of the lower triangular factor . The first element is unity and every other element of the first column of A^ is zero.

j

This completes the first stage.

Now suppose the (r-l)st stage has been completed. Let the notation

6^ denote the transformation of the original resulting from the first

r - 1 stages. The leading principal submatrix of of order r is

upper triangular, having resulted in effect from the corresponding principal

submatrix of the original by Gaussian elimination in natural order.

Its diagonal elements are all nonzero, by hypothesis. Therefore the

pivotal element is nonzero and the r-th row of may be used to

annihilate every nonzero element below the r-th element in the r-th column.

Only the j rows immediately folloxd.ng the r-th row are affected, and the

trailing coefficient in each of these rows is not changed. The multipliers

used in this annihilation go into positions r + 1 through t + j of the

r-th column of A^ ; unity goes into the r-th position, and zeros go into

every other position. This completes the r-th stage.

Let be the infinite upper triangular matrix which results from

the full sequence of transformations on . By our construction, both

and B have the bandwidths required by (i,j) - f actorizability

.

Also by our construction, formally multiplying A^ on the right by

results in ; compare (1.2.6).

J In order to prove necessity of the condition in part (ii), note that

the factorization produced by the foregoing process is such chat leading

i i+i

and trailing coefficients of both A. and B are nonzero if D is
21

nonsingular. Therefore, no row or column interchanges could be made without xd-dening the bandwidth. Therefore, Gaussian elimination in natural

order is the only possibility for arriving at the factorization. But

Gaussian elimination in natural order is possible only if every leading

principal minor is nonzero. This completes the proof. |_|

Examination of the proof of Theorem 1.2.3 shows that a unique (i,j)-

f actorization does not exist, since a different scaling of the rows is

possible. This would result in the appearance of entries on the principal

diagonal of that are other than unity. In fact, the scaling used

in the (0, 1) -factorization in Giver's second-order algorithm put the

sequence (p ( 2) , p (3) , . . .

)

on the principal diagonal of the right factor;

compare §1.1. Thus the principal diagonal of the left factor will contain

entries other than unity, in general.
1 . 3 A Preliiainarv Development of the Algorithm

Let D € . It is clear that an extension of Diver's algorithm

for computing a particular solution y of a given difference equation

Dy = g should involve the posing of a boundary value problem whose solution y^ is an approximation of y over some finite subsequence y^’^ .

The boundary value problem will have some number j ^ 0 of initial

conditions (depending on the particular solution being sought) . These

will consist of a subsequence of j consecutive values of y . The

terminal conditions of the boundary value problem will be v (n) = y (n+1)

' n n

= • • • = y (n+k-1) = 0 , where k = Z - j

. n The first task is to decide how to choose j

. A necessary prerequisite is a classification of possible solution types when the operator

is of order higher than two. Oliver [19] provided an early extension of

Diver's algorithm. He also provided the following definition of dominance.

Suppose we are given two sequences a , b € S . Then b dominates a

if there exists some i > 0 such that

(1.3.1) |b(r+l)/b(r) I > | a (r+1) /a (r)

|

for all r > i . Oliver considered only those difference operators for

which a system of complementary solutions could be found such that the

complementary solutions are linearly ordered under this definition of

dominance. Furthermore, he considered only those particular solutions of

the given linear difference equation which could be compared to each of

the complementary solutions. But not all possible particular solutions

can be so compared; for example, no particular solution which has an

infinite subsequence of zeros can play the role of either a or b in

(1.3.1).
23

In §1.1, we adopted a different definition of dominance: b

dominates a if

(1.3.2)

lim a (r)

b ( r

This differs from Oliver’s definition (which is equivalent to requiring

that ultimately [a(r)/b(r)| be monotonically decreasing). Another

difference is that the limit of |a(r)/b(r)| under Oliver's definition

may be greater than zero, as the example a(r) = 1 + 1/r and b(r) =1

immediately shows.

Let D € . The set /((D) = {x€S|Dx=0} will be called the kernel

of D . Obviously, /((D) is a linear subspace of 5 . Let us suppose

that /((D) has dimension t , so that there exist t linearly independent sequences . . . x^ ( /((D) which span /((D) . A sufficient

condition for this to be the case is that D be nonsingular; compare

[L4,§12.1]. We shall say that the set x^,X

2

,...,x^ forms a basis

for D .

Definition 1.3.1 We shall say that D € is totally separable (as

r ^ «) provided there exists a basis y^>y2 >-*-»y^ D such that

y dominates y in the sense of (1.3.2) for each s , 1 < s < . s+1 s Any basis which satisfies the conditions of Definition 1.3.1 will be

called totally ranked . Such a basis is not unique because y^ , for example,

could be added to each of y^j.-.y^ and the resulting basis would also

be totally ranked.

For each s , 1 5 s < , let K denote the linear subspace of S

s which is spanned by

y]_’y2’'’’’^s * particular, = K(D) . It

seems obvious, and it will be proved in a more general setting in Chapter

2, that every totally ranked basis generates the same subspaces. The

subspaces satisfy
c ^2 c ... c = K(D) .

Each sec inclusion is proper, since the dimension of K is s . This

s ascending chain of subspaces induces an obvious classification of the

solutions of Dy = 0 into ^ distinct types. However, we are interested

in classifying the solutions of Dy = g for nonzero, and indeed, arbitrary

g

6 S . Therefore, we frame the definition so as to cover all cases:

Definition 1.3.2 Let D € be totally separable with totally ranked

basis • Then, for each y € S , we shall say that the

type of y (with respect to D) is s , where 0 < s < t , and we shall

write type(y) = s , provided that

(1.3.3)

lim y(r)

0

and

(1.3.4)

lim sup y(r)

y^(r) > 0

Furthermore, if (1.3.3) is true for s = 0 , we define type(y) = 0 ,

and if (1.3.4) is true for s = I , we define type(y) = Z .

The type of each y ^ S is the same regardless of which totally

ranked basis is used. This will be proved for a more general classification of sequences in Chapter 2. Thus we are justified in defining the

type with respect to D rather than with respect to a particular totally

ranked basis.

The conditions for Olver's algorithm [21] suggest that Definitions

1.3.1 and 1.3.2 are appropriate for an extension of the algorithm to

arbitrary order; see also Cash [ 6 ] . But it would be desirable to have a

classification scheme which is applicable, for example, to every difference
25

equation with constant coefficients. Not every constant-coefficient

operator is totally separable, however. In the next chapter we shall

introduce a more general class of difference operators ( separabl e

operators) which does include every constant-coefficient operator.

Incidentally, the class of difference equations considered by Oliver [19]

includes every constant-coefficient equation, since the standard basis

(see §2.1 below) is linearly ordered under his definition of dominance.

Before stating the next theorem, we explain what we mean by matrices

and determinants associated with the name Casorati. If x. ,x_,...,x € S 12 s are arbitrary sequences, then for each r > 0 the s x s matrix

X,12(r) X (r)

s X12(r+1) X (r+1) X (r+1)

s

X (r+s-1) X (r+s-1) .... x (r+s-1)

Z s

will be called a Casorati matrix. Let X = {x, ,x_,...,x }

. We shall use 12 s the concise notation [X] (r) to denote the Casorati matrix above. Furthermore, |x|(r) will denote the Casoratian (of X at r) , defined by

|xi(r) = det [X] (r) , r = 0,1,... .

If D € is nonsingular, Casorati 's theorem [14, §12. 11] states that

x^,x^, ...,x^ forms a basis for D if, and only if, |x|(r) ^ 0 for

all r .

Theorem 1.3.1 Let y be a solution of the difference equation Dy = g ,

where D $ has a totally ranked basis Y = {y^ jy

2

> • • • • Let
j = type(y) . Then y is uniquely determined by its values y(i),y(i+l),

...,y(i+j-l) provided that i > 0 is a point at which the leading principal minor of order j of |Yj(i) is nonzero.

Proof Assume j = 0 ; then the condition on the Casoratian is vacuous

and we must show y is uniquely determined without knowledge of any of

its values. Suppose z ^ y is another solution of type zero; then there

exist unique scalars (not all zero) such that

y(r) = z(r) + a^y^(r) + ••• + a^y^(r) , r = 0,1,,.. .

Let be the nonzero scalar of highest subscript. Dividing through by

y^(r) and allowing r , we see that = 0 , which is a contradiction.

Next, assume j > 0 . Suppose z ^ y is another solution of type j

such chat

z(i-l-r) = y(i+r) , r = 0,l,...,j-l .

There exist unique scalars . . . ,a^ 6 F (not all zero) such that

y(r) = z(r) + a^y^(r) + ... + a^y^(r) , r = 0,1

It is proved easily that each of

to that of the previous paragraph

by the linear system

The

..,a^ is

remaining

zero by a method similar

scalars are determined

aiyi

^ i » i+j -1 + a. y

J J 0

(where 0 stands for a column vector of j zeros and we have used the

notation developed in §1.2), This system is nonsingular by the assumption

on the Casoratian; therefore, = 0 , t = , which is a

contradiction,
27

Theorem 1.3.1 confirms the adequacy of the selection of j initial

conditions in the approximating boundary value problem, where j = type(y)

In ^3.3 we will demonstrate that stability considerations demand exactly

j initial conditions, in general, for a solution of type j.

Now let us suppose that a value of j has been selected, where

3 > type(y) . Since v = y and y

= 0 , the linear ' n n system to be solved is

(1.3.5) ^i,n-j-l i+j ,n-l ^ i,n-j-l

j ^

i.l+j-l

0 ^

0

compare (1.2.4). The value of n is to be found in the course of the

computation. In view of Theorem 1.3.1, y is the unique solution of the

infinite linear system

(1.3.6)

i+j i = g -

0

We assume that D is (i, j )-factorizable, i.e., that linear difference

operators $ P , and . exist such that . J 't-3 3 1 As shown in the proof of Theorem 1.2.3, successive rows of this factorization, and indeed of the triangularization of the infinite system (1.3.6),

may be generated indefinitely by the forw^ard stage of Gaussian elimination

without pivoting. The stability of this process, also, will be discussed

in §3.3.

In consequence of Theorems 1.2.1 and 1.2.2 and the terminal conditions

^n,n+k 1 _ Q ^ solution of (1.3.5) may be obtained as the solution of

i+j ,n-l i+j ,n-l ^l+j,n-l

(1.3.7)
where ^ is a finite subsequence of the solution of

(1.3.8)

. i i+j i = a —

i,i+j-i i.i+j-i

0 '

0

Comparison of (1.3.8) and (1.3.6) and reference to the identity

= A show that

j J

(1.3.9) T, i"*"! i+j

B y

= 2

Thus (1.3.7) represents the finite linear system that results from (1.3.9)

by truncation at the (n-i-j)th row and column. Since and z^"*"^

are generated by Gaussian elimination without pivoting, the solution of

(1.3.5) for any n is easily programmed for an automatic digital computer

The algorithm comprises two stages: ( i) forward elimination (solution of

(1.3.8)); (ii) back substitution (solution of (1.3.7)). The forward

elimination can be continued arbitrarily far. The back substitution is

numerically equivalent to back^^^ard recurrence, and proceeds ontil we

return to the initial point i .

The question of the stability of the back substitution is easily

settled. For totally separable operators we have the following theorem:

Theorem 1.3.2 Let D z have a totally ranked basis Y = {y^ , .

. . .

. y ^ Suppose i > 0 and j £ {0, 1, . . . ,£-l} are such that D is (i,j) -

factorizable and the leading principal minor of order j of |Y|(i) is

nonzero. If = A^’B^'*’^ is an (i,j) - factorization of D , then

J J is totally separable. Furthermore, if z € S has type j or less with

respect to D , then z^"^^ has type zero with respect to .

Proof Suppose j = 0 . Then A^ is a diagonal matrix with nonzero elements on the diagonal; see Definition 1.2.1. For each y ^Y we have
29

= 0 , i.e.

I

I a^(r)b (r)y(r+s) = 0 , r = i,i+l,... . s=0

Therefore,

I

I b (r)y(r-i-s) = 0 , r = i,i+l, . . . s=0

®

which proves {y^|y ^ Y} is a totally ranked basis for . It follows

that has type zero with respect to B^ whenever z has type zero with respect to D .

Suppose 0< j< Z . Let U = {y^>y2

» • • • »Yj

}

• Since |u|(i) ^ 0

it is possible to find a linear combination x of y^yV

2 ’‘’'yy^ such

that x(i) ,x(i+l) , . . . ,x(i+j-l) take on any prescribed set of values.

Therefore, appropriate linear combinations x^ , s = j+1 , j + 2, . . . ,

can be found such that y (r) + x (r) vanish for r = i, i+1 , . , . . i+j -1 .

^s-j

” ^s ’ ^ ~ j+1 , j + 2, . . . . It is easy to verify that

{Yj^j • • • jYj > ,

. . . , v^} is a totally ranked basis for D . Now let

^ ^ > "'^ 2

’

' ’ ' ’ • In view of Theorems 1.2.1 and 1.2.2, the remark

following Theorem 1.2.2, and the facts that Dv = 0 and v(i+r) = 0

for r = 0,1,..., j-1 , we conclude that ^ satisfies

i+j,n-l i+j,n-l

Bq V

0

„n-k,n-l n,n+k-l \

for every n > i + j + max(j,k) ; compare equation (1.2.7). On letting

n -> oo this proves B^'*'^v^'''^ = 0 . Thus {vj"'*”'^ , v^"^^ , . . . , v^"^^

} is a

totally ranked basis for B^~^^ . Finally, let z be of type j or less

with respect to D . Let s £ {l,2,...,k} . Then
—V oo

1

Z (r) ^ z(r)

V (r) y (r){l+o(l)}

s j+s

as

i'^i T> ^”^3 which shows chat z has type zero with respect to 3 . Lj

j

I In consequence of Theorem 1.3.2 and our remarks in §1.1, when

j > type(y) the back substitution stage of the algorithm is stable, at

least for all r in a sufficiently large range.

It remains to determine the appropriate value of the truncation

parameter n . It is necessary to introduce Green ' s formula for linear

difference operators ; see, e.g.. Miller [13]. We apply Green's formula

not CO the operator D but to Che operator . Our development is

essentially the same as that of Cash [ 6 ]

.

Let B = [b-,b ,...,b ] i V . The infinite upper triangular matrix | 0 1 k k jj B = ^ which is obtained from B by deleting the first

|

k columns and transposing the result, i.e. whose entries are defined by

(1.3.10) b (r) = b (r+s) , s k-s

/s is a linear difference operator of order k . We shall call B the

operator which is adjoint to ^ , or more briefly, the adjoint operator

of ^ . An operator and its adjoint satisfy a number of elementary

properties which are easily verified from the definition. Among these are

(1.3.11) bP .

(1.3.12)

(1.3.13)

k

i
31

(1.3.14) _ .pP+k.q+k Tr

®k - ’

valid for q 2: p and p = 0,1,2,... .

Now take any u,v € S and B $ P , and assume k > 1 . Green's

K. formula is the identity

q-1

(1.3.15) I {v(r)Bu(r) - u(r+k) Bv(r)

}

r=p

p,p+k-l Tr p,p+k-l p,p+k-l

= (^V ) ^ ^

q,q+k-l Tr q,q+k-l q,q+k-l

— (^v ) n u

valid for q>p+k, p>0 . Green's fonnula may be verified by expressing the left side of (1.3.15) in the form

and reducing this expression with the aid of the definition (1.3.10) and

the identities (1.3.11) - (1.3.14).

Theorem 1.3.3 Let be an (i , j

) -factorization of D ^ , J 1

"4 where D is totally separable and j < ^ . Let y be a solution of

Dy = g , where type(y) 5 1 and g € S . Let ^ and be

n

the corresponding solutions of (1.3.5) and (1.3.8). Also, let s be a

s4“X fixed integer satisfying i+j +k-l<s<n and w = (w (s+1)

,

s s w (s+2),...) be the solution of

s

(1.3.16)

;>s-k+l s+1

B, w

k s

^s-k+i,s s-k+1,3 w

where k = -£ - j and
(1.3.17) W, = (0,0,. ..,0,1) .

Then

(1.3.18)

n-1

y^(s) = 5^ w^(r)z(r) 78^(3) ,

where ^^(s) is the (s-i-j+l)st leading coefficient of .

Remark: Equations (1.3.16), (1.3.17) are equivalent to the initial value

problem

k

I b (r)w (r+t) = 0 , r = s-k+1 , s-k+2 , . .

.

t=0 ®

with initial conditions w (s-k+1) = • • • = w (s-1) = 0 , w (s) = 1

• s s s compare (1. 2.1)-(1. 2.3) and (1.2.4a).

Proof: Let p=s-k+l,q=n,v=w and u = y in (1.3.15) . By

s n virtue of (1.3.7), (1.3.16) and (1.3.17), the left side of (1.3.15)

becomes

n-1

I w (r)z(r) . s r=s

. q,q+k-l n,n+k-l , , , . , - , ^ Since u = V = 0 , the second term on the right side of

' n (1.3.15) is zero. Thus, expanding the right side and using (1.3.17),

we derive
33

, P+k-l^Tr^p , p+k-l^p , p+k-1

(0,0,...,0,1) t>o(s-k+l) ... b^_^(s-k+l) y (s-k+1)

n

1

O' o

cn 1

yjs)

= bg(s)y^(s) .

Equating the left and right sides thus arrived at, we obtain (1.3.18).

In §3.2 with appropriate conditions it will be proved that

lim y (r) = y(r) n-^ n

for each fixed r > i + j

. Thus, allowing n « in (1.3.18), we have

(1.3.19) y(s) = ^ w (r)z(r)/b (s)

s 0 r=s

for each s>i+j+k-l. Similarly, if

Ti (s ) = y(s ) - y (s

)

n n

is the truncation error incurred in accepting the approximation y (s

)

n

for y(s) , we have

(1.3.20) T] (s) = I w (r)z(r)/b (s) r=n

for each s>i+j+k-l .

Let us compare (1.3.20) with (1.1.2), the truncation error expansion

for the original algorithm of Olver. Suppose 1=2 and j = k = 1 .
The difference operator which is implicit in (1.1.1) is D = [dQ,d^,d

2

] where

dg(r) = a(r+l) , d^(r) = - b(r+l) , d^(r) = c(r+l)

for r = 0,1,2,... . Similarly, the right side of (1.1.1) is, in our

present notation,

g(r) = d(r+l) , r = 0,1.2,... .

The rov;s of the (0 , 1) -factorization are scaled so that

bg(r) = p(r+l) , b^(r) = - p(r)

for r = 1,2,..., where = [tig>b^] and p = (p (0) , p (1) , . . . ) is the

sequence defined in §1.1. The solution z^ of the forward elimination

stage of the algorithm is given by

z (r) = e(r) , r = 1 , 2, . . . ;

compare (1.3.8). Finally, it may be verified readily that the solution

of the adjoint equation Bw (r) = 0 , r = s,s+l,... such that w (s) =

-j

- s =

is given by

w (r)

s p(s)p(s+l)

p(r)p(r+l)

s , s+1 , . .

.

Making these substitutions into (1.3.20) and exchanging the roles of r

and s , we obtain (1.1.2) .

The simple form of the equation for w (r) depends solely on the

s particular scaling of the ( 0, 1) -factor iza tion used by Olver and the fact

that k = 1 . A similar simplification can be obtained for difference

equations of arbitrary order provided only that k = 1 . It stems from
35

the fact that the order of the adjoint equation is one, hence all solutions can be expressed as multiples of a single particular solution.

These conclusions were also arrived at in Cash’s paper [6].

The expansions (1.3.19) and (1.3.20) may be employed in the estimation

of the optimal value of n such that a given termination criterion is met.

Typical termination criteria would be to require one or more values of

y^(r) in the range i+j<r5m to approximate y(r) to a specified

absolute or relative precision. Since the back substitition stage is

stable, controlling the absolute or relative error only at the point r = m

is often sufficient. In practice equations (1.3.19) and (1.3.20) are

employed by replacing the upper limit by n + v , where v ^ 1 is a

parameter chosen to suit the problem at hand.

The solution of (1.3.16), (1.3.17) is obtained by for\^7ard recurrence.

The stability of this process will depend on the behavior of for

large r relative to the other solutions of the adjoint equation. In

general there is no reason to expect that w (r) will not contain a coms ponent of a maximal solution. Thus the solution of (1.3.16) will normally

be stable, although there could be an initial range where some unstable

rounding error propagation takes place.
CHAPTER 2. CLASSIFICATION OF SEQUENCES

2.1 Constant-Coefficient Operators

Let D = . . .d^] - be a constant- coefficient operator . By

this we mean that there exist 6 ,6 ,...,S„ ^ F such that 6 0 ,

5, 7^ 0 and d (r) = 6 for all r . We assume without loss of

L s s generality that 5^

= 1 in every constant-coefficient operator. Since

every constant-coefficient operator is nonsingular, by our definitions, the

kernel K(D) has dimension Z ; see discussion preceding Definition 1.3.1.

A particular basis, which we shall call the standard basis , exists

for each constant-coefficient operator; see [ 11 ,§§1 3 .0-13 . 1] . The polynomial

I 1-1

P(C) = r + + ••• +

is known as the characteristic polynomial for D . The standard basis is

the set of sequences

1 = {(r°^\^) € S I P(\) =0 and m 6 {0, 1, . . . ,M.(X)-1}}

where M-(X) denotes the multiplicity of the root X . We are using the

notation (r'^^X^) here as an alternative way of denoting the sequence

(p ,X, 2 X ,3 X , , . .

)

m

where p„ = 1 and p =0 when m > 0 . 0 ra The standard basis of a constant-coefficient operator is not necessarily

totally ranked; recall Definition 1.3.1. For example, if X^ and X^

are two distinct zeros of the characteristic polynomial such that

|X^! = jX^I then |X^| = | X^

|

for all r . Neither solution dominates

the other.
37

Let us x^rite a < b whenever a,b c S are such that a is dominated by b . We shall say that a and b are separated (at infinity) if

a<b or b<a. A totally ranked basis is linearly ordered by <

but in general the standard basis is not, as we have just seen. However,

we can introduce a linearly ordered partition of the standard basis . We

shall proceed to do this.

m, m„

1 r 2 r It is apparent that (r < (r if, and only if, either (i)

|l-ll < (ii) = 1 ^

2

! TM1 ^ denote the

number of roots of P(^) =0 of distinct absolute value, and let

X , X , ...,\ be representative roots of P(^) = 0 such that |X | < |k j 1 2 O’ s s+

i

for each s . Define u. = max{!-i.(X)! |Xl = !X 1} . We define the sets

' s s

= {(r'^x’') t a| |Xl = |Xj} , s = l,2,...,c

and, for each s , the sets

A = {(r^X^) € A |p.(X) > m} , m = 0,1, . . . ,p -1 . S 5

Til S 3

It may be verified readilv that the sets A are disjoint and nonempty

s , m

and

u -1

O’ ' s (2.1.1) A = U U A

1 n s,m

s=l m=0

The sets A are linearly ordered according to the definition:

S ) HI

XI < A , , if, and only if, either (i) s < s’ or (ii) s = s' and

s , m s , m

m < m ' .

An illustrative example is afforded by the constant-coefficient equation
y(r+8) - (4+4V'2)y(r+7) + (15+16v'I)y (r+6) - (48+12V2)y (r+5)

- 12y(r+4) + (192+64v'2)y (r+3) - ( 208+256Vl)y (r+2) + ( 256+19 2V2)y (r+1)

- 192y(r) = 0 . Its characteristic polynomial, in real factored form, is

(2.1.2) (4-1)(C-2)(C+2)(C^-2\/2?+4)^(5-3) = 0 .

The roots of (2.1.2) include the complex numbers V2(l+i) and V2(l-i) ,

each of multiplicity two. The subsets of the linearly ordered partition

in this example are

‘ 1,0

A

2,0

^ 2,1

3,0

= {2^(-2)^V2U-i)^V2"(l+i)^}

{rV2'"(l-i)’',rV^’'(l+i)’'}

{ 3 ^}

Note that if a 6 ^ and ^^^"2

1 ia(r)/b(r)| = r -1

V7hen r > 1

In contrast, if a 6 A and b 6 A , where s < s’ then !a(r)/b(r)|

s s contains a decreasing exponential factor.

In general, if a,b 6 A are such that a < b , then the separation

ratios ja(r)/b(r)| decrease as the product of a power of r

^ and an

exponential function in r . If the exponential function is identically

equal to one, as will be the case whenever a and b are standard basis

solutions which correspond to characteristic roots of equal magnitude,

then we shall say that a and b are algebraically separated . Otherwise,

we shall say that a and b are exponentially separated .

Figure 5 is obtained by plotting the roots of (2.1.2) in the complex

plane and drawing circles through them, centered at the origin. There are

three circles, corresponding to the distinct absolute values of the roots.
39

The double roots are indicated on the circle of radius two; it is helpful to imagine them duplicated on a second circle of radius two that is

raised slightly out of the plane. Let us number the four circles, thus

arrived at, hy increasing three-dimensional distance from the origin.

This determines geometrically the proper linear ordering of the four subsets A of the partition. The ordering of elements within any one of

s > m

the subsets is unimportant for our purposes.

Clearly, this geometric indication of the linear ordering of the subsets of the partition can be extended to any constant-coefficient operator.

Figure 5. The roots of (2.1.2). The double roots ''/2(l±i) are

denoted X, \ .
. r.e ere- _ .1 Let (2.1.1) denote the linearly ordered partition of the

standard basis for a given constant-coefficient operator. Then:

i) If a ^ and b £ '•

, , , then a < b if, and only if,

^ s , n

ii) If

iii) If

a,b c ' , then a(r) ' = ib(r)- for all r . 3 ,ni

a, , a , . . . , a , b € A

1 2 q ; where the a's are distinct, and

7

’

‘ ..,c € F are all nonzero, then

(2.1.3) lin sun T t a (r)/b(r) > 0 . P P r-^ p=l

Proof: Parts (i) and ( ii) are trivial consequences of our definitions.

Turning to oart (iii), we note that the sequence c a (r)/b(r) is p=l p p not identically zero (since the a's are linearly independent). Let

,\ be the c'naracteristic roots associated with the set A

1 ^ q s,E

such that

O D p = 1 , 2 , . . . , q

n, r and b(r) = r X , Since the X's lie on the same circle in the complex

plane, there exist real constants lying in the halfopen real interval [-"',“) such that

a^(r) = , p

= 1,2, ...,q

and b(r) = r^ \,"^exp(irp) , i - -1 . Thus

I t a (r)/b(r) = i z p=l P P p=l P

where ' = '; - 3 • The v's are distinct real numbers (modulo 2n)

P P
41

because the sequences a are assumed distinct.

1 Z q

The right member of the last equation represents a so-called almost

periodic function of r when r is regarded as a continuous real variable. To complete the proof we appeal to the following lemma, the proof

of which is given in the Appendix.

Lemma 2.1.1 Let a, be nonzero complex numbers and r, 1 2 q 12

distinct real numbers (modulo 2tt) . Define the infinite sequence

ir"v* (2.1.4) x(r) = y a e P

, r = 0,1,2,... .

1 P p=l

Then lim sup^_^^j x(r) | > 0 .

An equivalent statement to inequality (2.1.3) is that there exists an

infinite subsequence of the sequence

q

I a a (r)/b(r) , r = 0,1,2, .. . p=l P P

which is bounded away from zero. Intuitively, this means that it is

impossible to create a new sequence with qualitatively different asymptotic

behavior merely by forming a linear combination of standard basis solutions

of similar asymptotic behavior. Note, however, that unlike the original

sequences infinitely many zeros may occur in the new sequence. For example,

r r if 2 and (-2) are sequences in a standard basis, then every other

r r term of the sequence 2 + (-2) is zero.

We relegate the proof of Lemma 2.1.1 to the Appendix because the theory

of almost periodic functions is not germane to the rest of this thesis.
2.2 Separable Operators

In this section we introduce a subclass of the general set of infinite

detail the general structure of the kernel of an operator in this subclass. This will prepare the way for the presentation, in the next

section, of a general classification of all sequences in S with respect

to a given linear difference operator. In Theorem 2.1.1 we presented

three properties of the standard basis of a constant-coefficient operator.

These properties suggest the following extension of Definition 1.3.1:

Definition 2.2.1 Let D i V

be such that fC(D) has dimension t .

Then D will be said to be separable if there exists a basis X for D

which can be partitioned into nonempty disjoint subsets in

such a way that the following three conditions are all satisfied:

i) If X € X and y € X , where s < t , then lim x(r)/y(r) = 0

S t r^

ii) If x,y 6 X for some s , then 0 < lim inf lx(r)/y(r)|

s r->" and lim sup^_^^| x (r) /y (r) | < °°

;

iii) If X is any linear combination of sequences from X^ , other

than X = 0 , and y £ X^ , then lim sup^^_^ ! x(r) /y (r) | > 0 .

Let D be a separable operator. A basis X for D which is such

that

where the X satisfy the conditions of Definition 2.2.1 will be said to

s be optimally ranked . Let

CT

( 2 . 2 . 1 ) X = U X ,

1 ^ s=l

( 2 . 2 . 2 )

where the subscript n is positive (because X

s s is nonempty)

.
43

Obviously, we have

(2.2.3) t- I s=l

n

s

In Lemma 2.2.1 below we will show that two distinct optimally ranked bases

for a given separable operator have the same number (cr) of disjoint subsets

and the same number (n^) of sequences in corresponding subsets.

An optimally ranked basis exhibits the full range of possible

asymptotic behavior of solutions to the corresponding homogeneous difference equation. Conditions (i) and (ii) of Definition 2.2.1 establish the

ranking of sequences within the basis. If (2.2.1) is a particular optimally

ranked basis, we shall say that x € X is a sequence of type s provided

that X € . More generally, we shall say that x € K(D) is a sequence

of type ^ s orovided that x 0 and there exist scalars a € F such that

‘ p»q

n

(2.2.4) X

sI P=1

a X

p.q p>q

where a ^0 for some q , 1 5 q 5 n . Such a representation of x s,q

^ ^ s exists and is unique for every nonzero x € K(D) . Again, in consequence

of Lemma 2.2.1 below, if we change to a different optimally ranked basis,

the type of every sequence in /((D) remains invariant. Condition (iii)

of Definition 2.2.1 presages this; it requires that no nontrivial linear

combination of basis sequences of the same type may be dominated by a

basis sequence of that type.

Let L(Y) denote the linear subspace of S which is spanned by Y

where Y c S may be any finite subset of sequences. If (2.2.1) is an

optimally ranked basis for a separable operator D, then we shall say that

s the subspace I ( U X ) is the subdominant subspace of type s for D . - p p=i
These subspaces are independent of the basis; again, see Lemma 2.2.1

s below. If i denotes the dimension of L( U X ) , then

" P=i p

(2.2.5) ' I p=l

compare (2.2.2) and (2.2.3). In addition, we shall say that {0} , the

linear subspace of dimension zero (whose only sequence is the zero sequence)

is the subdominant subspace of type 0.

Let U be one of the subdominant subspaces for D . The complementary

subspace of (J in f((D) , i.e., the subspace 1/ of K(D) such that

( 2 . 2 . 6 ) K(D) = a © 1/ ,

will be called the corresponding dominant subspace. The symbol ® on the

right side of (2.2.6) means that K(D) is the direct sum of the subspaces

U and 1/ ; that is, every nonzero sequence x 6 /((D) is uniquely

expressible in one of the forms x = u , or x = v , or x = u+v ,

where u € U and v 6 1/ . Clearly, if U is the subdominant subspace

a

of type s, 0 5 s < a , then 1/ = L( U X ) ; and if U is the subp=s+l P dominant subspace of type a, then 1/ = {0} .

Before stating and proving Lemm.a 2.2.1, we note that

s L( U X ) = L(X. ) © L(X ) a • • • ® L(X ) p=i p

^ ^

Thus, an alternative characterization of a sequence of type s in f((D) is

the following: x € K(D) is of type s, 1 5 s < a , provided that

s

r X

p

6 L(X ) P X 0 ; s

compare (2.2.4)
45

Lemma 2.2.1 Let

and

CT O’ X = U X = U {x ,x ^

, . . . ,x }

, s , s,l s,2 s,m

S=1 s=l s

T T

Y = U Y = U {y ,y , . .

. ,y } .s ,s,ls,2 s,n

S=1 s=l s

be distinct optimally ranked bases for a separable operator D ^ .

Then cr = t and m = n for all s . Furthermore,

s s

s s L(UX)=L(UY), s = 1,2, . . ., <j , , s , s P=1 P=1

and the type of every nonzero sequence z € K(D) is the same with respect

to X as it is with respect to Y ,

Proof Let u S and v € X^ , so that relative to X the types of u

and V are s and t . Let the types of u and v relative to Y

be s' and t' . Then there exist unique representations

and

s

u = } u. , u.€L(Y.), u,^0

jii j : j

t' V = 1 V. , V. 6 L(Y.) , V , 7* 0 . jil 3 J :

Furthermore, by condition (i) of Definition 2.2.1, we have, as r ,

(2.2.7) u(r) = u^,(r) + o{y(r)> , y ^ ,

and
( 2 . 2 . 8 ) v(r) = v^,(r) + o{y(r)} , y 6 , .

i) First we prove that s'=t' if s=t,i.e.,if u and v are of

the same type with respect to X , then they are also of the same type

with respect to Y .

Since s = t , the sequence u(r)/v(r) is asymptotically bounded

away from zero; compare part (ii) of Definition 2.2.1. Suppose that

s' < t' . We shall obtain a contradiction by showing that, under this

assumption, u(r)/v(r) has an infinite subsequence which converges to zero. Choose any y ^ Y^ ,

. Then we have

lim u(r) ^ Q

r-x» y(r)

From (2.2.8), we have

v(r)

y(r)

v^,(r)

y(r) + 0 ( 1 )

and, using the triangle inequality, an elementary property of the limit

superior, and condition (iii) of Definition 2.2.1, we obtain

lim sup

j V (r

)

|y(r)

Thus there exists an infinite subsequence of v(r)/y(r) that is bounded

away from zero, and therefore also a subsequence of

lim sup , Vi y r^ y(r)

u(r) ^ u(r) /y(r)

v(r) v(r)/y(r)

that converges to zero. This supplies the needed contradiction.
47

A similar contradiction is obtained if we suppose that s' > t' . Hence

s' = t' .

ii) Next we prove that s' < t' if s < t .

Since s < t , the sequence u(r)/v(r) converges to zero. Suppose

that s' > t' . We shall derive a contradiction by showing that, under

this assumption, u(r)/v(r) has a subsequence which does not converge to

zero. Choose any y € , . First, by using (2.2.7) and an argument

similar to that used in part (i) of this proof, we have

lim sup

>00

u(r)

y(r) > 0 .

Thus there exists an infinite subsequence of u(r)/y(r) that is bounded

away from zero.

Assume now that s' > t' . Then

lim v(r) ^ Q

r-»oo y(r)

In this case it follows that there exists a subsequence of

u(r) _ u(r) /y (r)

v(r) v(r)/y(r)

that diverges to « , which supplies the needed contradiction.

Alternatively, assume that s' = t' . Using (2.2.8), we have for

sufficiently large r

v(r)

Vj., (r)

y(r) y(r)

Furthermore, using condition (ii) of Definition 2.2.1, we see that there

exist r >0 and S > 0 such that

o
,v^,(r)

.

'_J:: i <: ? r > r

' y(r) j ' ^ ’ O

Tnerefore, for sufficiently large r , we have

| V(r)

'y(r) ? +

1

and

i u(r)

v(r) - (1+p)

-l| u(r)

'y(r)

It follows that u(r)/v(r) has a subsequence that is bounded away from

zero, which again supplies the needed contradiction.

iii) Next, we prove that i = z and m = n for all s , and also

s s s s that L( j X ) = L( U Y ) for all s . p-1 p=l

Part (i) of this proof implies that for each s , 1 5 s < a , the type

relative to Y of every sequence in is the same, say t^ . Using

part (ii), we have

(2.2.9) 1 5 t < t <

••' 12 < t 5 T . O’

Furthermore, that s < t for all s , 1 5 s 5 o , is easily proved bv

s induction. In particular, o < t and, using (2.2.9), we see that o 5 T

o

Inverting the roles of X and Y and repeating the argument, we see

that T S T . Thus o = t , and (2.2.9) implies t^ = s for all s .

For each s , it follows that

L(X,J---UXJ IS c L(Y isIJ---UY ) ,

and by symmetry.
Thus

L(X,U* • -U'X ) = L(Y.U- • -UY ) 1 s 1 s

for each s . In particular, l(X^) = L(Y^) . Since X^ and Y^ are

bases of the same linear space, we have ~ easy induction

completes the proof of this part.

iv) To complete the proof, let z 6 K(D) be a nonzero sequence such that

z = 1

^ u r 0 , u € i.(X ) , p=l P s P P

and

t

z = 1 ^ V # 0 , V 6 L(Y )

. p=l P

’ t ? P

Assume s < t , and let V € Y . Then

t

(-* <

rt lim s

V u ( r ) -V

P ? (r) t-1 V (r) — n

r-*-» y(r) L _p=i y(r) p=Li

— U .

But this result contradicts condition (iii) of Definition 2.2.1. Hence

must have s > t . A similar contradiction is obtained if we assume

s > t . Therefore s = t and the proof of Lemma 2.2.1 is complete. !_

In general, the actual determination of optimally ranked bases for

variable coefficient difference operators is a difficult problem. Some

results have been given by Wimp [26]. This reference extends earlier

work by Birkhoff and Trjitzinsky [2,3] on the analytic theory of singular

difference equations whose coefficients possess asymptotic expansions
of a orescribed form. All equations with coefficients rational

in r are included, for example. Wimp's analysis provides a

means of constructing a basis, which he calls a canonical set :

see [26, Theorem 3.3 and Definition 3.6]. The canonical set is

analogous to our optimally ranked basis. More specifically, there

exists a basis {Yt jY-j • • • > 7 ;?} such that, for each s = , J- z ^

y (r) = c M (r){l+o(l)} as r ^ “

s s s

where c 7^ 0 and

s

Q (r) 6 M (r) = e r ^(£n r)

s

Here 9 is a comnlex number and p is a positive integer. Q is of

s

‘ s s the foirm

Q^(r) = a^r In r + u^r + + ••• +

where 0 is an integer, p > 1 , and p ,u.., ,

. . . are complex numbers.

U -L p

The rich variety of possible separation ratios of solutions as r

-» «>

,

compared with the algebraic or exponential separation of solutions in the

constant-coefficient case, described in § 2.1 above, is evident.
51

2.3 The General Classification

In §2.2 we showed that every nonzero sequence in the kernel of a

separable operator has a unique t^/pe with respect to that particular

operator. In this section we extend this classification to every nonzero

sequence in S . The following definition should be compared with

Definition 1.3.2.

Definition 2.3.1 Let D € be separable with optimally ranked basis

a

X = U X . Then, for each y € 5 , we sav that tvpe (y) = s , or more

3 = 1

^ fully, the type of y with respect to ^ is s , provided that 0 < s < c

and the following two conditions are satisfied:

i) If X € X

s+1

then

lim y (r)

r-*«> x(r)

0 ;

ii) If X T X then = > lim sup I y (r)

r-*oo * X ( r

)

> 0

If (i) is satisfied for s = 0 , then (ii) does not apply, and we define

type(y) = 0 . Similarly, if (ii) is satisfied for s = cr , (i) does not

apply, and we define type(y) = ct .

The classification of sequences introduced in Definition 2.3.1 allows

us to treat both homogeneous and inhomogeneous difference equations in a

uniform manner when the difference operator is separable. The kernel of

the operator contains every solution of the homogeneous equation, by definition. But solutions of the inhomogeneous equation with arbitary right-hand

side may lie anywhere in S . This is why a classification of all sequences

in S is desirable.

Theorem 2.3.1 Every sequence in 5 possesses an unambiguous type with

respect to each separable operator in V

^

.

Proof (i) First we show that the type with respect to ^ fixed optimally
ranked basis is determined uniquely by Definition 2.3.1 for every nonzero

sequence in S .

Let z € S . Assuming that

(2.3.1)

lim sup z(r)

x(r) > 0

for at least one sequence X ^ X , let X € X

type such that (2.3.1) is satisfied. Let s

is satisfied for every x 6 X

s

, since

z(r) z(r) x(r)

x(r) x(r) x(r)

be a sequence of largest

type (ic) . Then (2.3.1)

and lim inf

^ ^^ jx(r) /x(r)

|

>0 as a consequence of condition (ii) of

Definition 2.2.1.

If s = CT , then type(z) = a according to Definition 2.3.1. Therefore suppose that s < ct . By the maximality of x we see that

lim sup^_^ j z (r) /x (r)

I

= 0 whenever type (x) > s or, equivalently,

(2.3.2) = 0 whenever type (x) > s . r-^ X ( r

)

Accordingly, type (z) = s .

To conclude the proof of part (i), let us suppose that (2.3.1) is

not satisfied for any x ^ X . Then (2.3.2) is satisfied for every x 6 X

and in particular for every x € X^ . Then type (z) = 0 . a a

(ii) In this part we prove that if X = U X and X' = U X[

s=l ^ s=l

^ are two distinct optimally ranked bases for D then the type of every

sequence in S is the same with respect to X and X'.

Choose any s , 1 < s 5 a , and choose x ^ X , x' <: X' . There

s s
exist X

P

Furthermore

X , 1 < p 5 s , such that

P we have

x,+*--+x ; see Lemma 2.2.1

i s

x^(r)

x(r)

= 0 ( 1 ) +

x (r)

sx(r)

as r ^ .

Since x^ is a linear combination of sequences of type s, a brief compute

tion and application of condition (ii) of Definition 2.2.1 yields

lim sup

X (r)

sx(r) < oo

It follows that

(2.3.3)

lim sup X ' (r)

r-»-“ x(r)

and

lim inf xlrl

0 whenever type (x') = type (x) x’(r)

Since the roles of x and x' are interchangeable, we also have

x(r)

(2.3.4)

lim sup

and

<

x' (r)

1 im inf x*(rl

x(r) > 0 whenever type (x') = type (x)

Let z € S and suppose type (z) = 0 with respect to X . Thus

z(r)/x(r) -*• 0 for every x ^ X . Choose x' € X' . Let s = type (x')

and choose x C X . Then

s

z(r) z(r) i x(r)

x'(r) x(r) |x' (r) ^ 0 as r

by virtue of (2.3.4). This proves type (z) = 0 with respect to X' .
Let z t S and suppose type (z) = <j with respect to X . Thus

lit! sup lz(r)/x(r)| > 0 for every x c X . Choose x' ^ X' . Let >oo

s = type (x') and choose x 6 X^ . Then the sequence

z(r)

x'(r)

I z(r)

|x(r)

x(r)

x'(r)|

satisfies lim sup |z(r)/x'(r)| >0 by virtue of (2.3.3). This proves

type (z) = 0

" with respect to X' .

Finally, let z C S and suppose type (z) = s , 0 < s < ct . Thus

z(r)/x(r) 0 for every x $ X ^ IJ*‘*UX and lim sup |z(r)/x(r)| > 0

s ' 1 CT

>*oo for every x € X^ . It can be verified that z(r)/x'(r) -> 0 for every

x' € X' ,,U--‘UX' and that lim sup lz(r)/x’(r)| > 0 for every

x' ' using arguments similar to the ones used in the preceding two

paragraphs . j^‘|
CHAPTER 3. THE GENERAL ALGORITHM

3 . 0 Preliminaries and Overview

In this chapter we fix our attention on a particular linear difference equation

(3.0.1) Dy(r) = g(r) , r = 0,1,2...

assuming that D t P is separable and g $ 5 . More specifically,

Z are interested in computing an approximation y of a particular solution

n

y of (3.0.1) which is valid over some finite subsequence y^’^ . i and

m given.

Let CT be the number of distinct types in any optimally ranked basis

for D . We assume that y is known to be a sequence of type t with

respect to D , where 0 5 t < a . We also assume that initial values

y(i) syCi"*"!) ,• • • jyCi'fj”!) are known, where j is the dimension of the

subdominant subspace of type t for D . The extension of Theorem 1.3.1,

from totally separable to separable operators, is given in §3.1. Accordingly, these initial values (in the presence of one additional condition)

suffice to determine the solution uniquely.

Let k = -^ - j . The approximating sequence y^ is defined as the

solution of (3.0.1) which satisfies the conditions

(3.0.2) y^(i+r) = y(i+r) , r = 0,1, . .

. , j-1

and

(3.0.3) y (n+r) = 0 , r = 0,l,...,k-l . n

r-> • T . T ^ x.Tn Since y is to be approximated by y , it is clear that the value
of n tnusc exceed m . The existence and uniqueness of the approximating

sequences y , for all sufficiently large n , and the convergence of n the sequence of aporoximate values v (r) to y(r) for each value of r n in the range i + j < r < ” , are proved under appropriate conditions

in §3.2.

The general algorithm, which is valid for any separable operator, reduces to the algorithm presented in §1.3 for totally separable operators.

We again assume that the separable operator is (i,j) - factorizable, just

as we did for totally separable operators. The linear system (1.3.5) is

solved by the factorization method by means of a forward elimination

stage followed by a back substitution stage; see §1.3. The expansions

(1.3.19) and (1.3.20), which are used to estimate the optimal value of

n , are unchanged for separable operators. In §3.3 we discuss the stability

of the forward elimination stage, which was omitted in §1.3. We also

extend Theorem 1.3.2 to separable operators, showing that the back substitution is stable. Finally, we note that our remarks on the stability of

the forward recurrence solution of the adjoint equation, whose solution

enters in the truncation error expansion of §1.3, apply to separable

operators as well as totally separable operators.

We conclude this introductory section by fixing some notation for

later use in this chapter. Let

a

(3.0.4) X = U X

s=l ®

be an optimally ranked basis for D , where

(3.0.5) X ={x ,,x ,..,,x }, s = l,2,...,o-.

s s , 1 s , 2 s , n

s

Then the dimension of the subdorainant subspace of type t for D is
t (3.0.6) j = I n s=l ^

Define the set

(3.0.7)

t U = {u ,u , . . . ,u . } = U X 12 1 s=l

"

by means of the correspondence

X

p,q

When t = 0 it is understood that j = 0 and U is empty,

define the set

(3.0.8)

9

’

' U X : s=t+l ^

Similarly

,

when t = a it is understood that V is empty.
3 . 1 Existence and Uniqueness

Let y be a solution of (3.0.1) of type t such that

(3.1.1) y(i+r) = > r = 0 . 1 , . . . , j -1

for some i > 0 , where j is given by (3.0.6). Using the notation |u|(

for the Casoratian of U at i , we also suppose that

(3.1.2) |U| (i) ^ 0 ;

see (3.0.7). Note that for every i (3.1.2) is satisfied vacuously if

t = 0 , and also (by virtue of Casorati's theorem [14,§12. 11]) if t = c

and D is nonsingular. The following theorem generalizes Theorem 1.3.1.

Theorem 3.1.1 If (3.1.2) is valid then there exists a unique solution of

(3.0.1) of type t such that (3.1.1) is satisfied.

Proof: Suppose t = 0 . Then j = 0 and we must prove that without any

specified initial values y is uniquely determined. Let v 6 S be a

solution of (3.0.1), other than y , such that type (v) = 0 . Since

V - y € f^(D) , there exist scalars a $ F such that

^ p>q

n

CT p y-v=y Va X ; pil q=l P’P

see (3.0.5). At least one of the a is nonzero; let the largest

p.q

value of p such that a 0 for some q, l5q5n,be p = s.

P.q p Then

lim y (r) -V ( r)

r-rco X (r)

s , 1

because type (y) = type (v) = 0 . By part (i) of Definition 2.2.1, we
59

have

^ . s-1 n a X (r) lim r P,q P»q

p=l q=l s,l

= 0

It follows from the above three equations that

n

lim

> oo

s a X (r)

I = 0

X, 1 (r) q-1 s ,

1

But this result contradicts part (iii) of Definition 2.2.1; therefore we

conclude that y

= v .

Alternatively suppose t > 0 . Let v ^ S be a sequence of type t

which satisfies (3.0.1) and is coincident with y(i+r) for r = 0,1,...,

j-1 . Then there exist scalars a 6 F such that

p>q

n V = y + I I a „

• p=l q=i P'1 P'<!

Let the largest value of p in this sum such that at least one of the

scalars a is nonzero be p = s . Then s ^ t , because s > t p,q

would imply

lim v(r)-y (r)

r-«° x , (r)

S j 1

0

(since type (v) = type (y) = t) and

lim

>oo

S*1

I p=l

pa X (r)

y p>q^>q 0

(by part (i) of Definition 2.2.1). This gives
lim

= 0 ,

s a X (r)

y s.q s,q

q-1 s,l

which is a contradiction.

Thus, a =0 whenever p > s . The a having p 5 s satisfy

p»q p,q

the system of linear equations

s n

y y a X (i+r) = 0 , r = 0,1,..., j-1 . pil qii p.q p,q

This system is nonsingular by the condition ju|(i) 7^ 0 , so we have

G =0 for all p and q . Thus v(r) = y(r) for all r and the

p,q

theorem is proved. [TT]

Theorem 3.1.2 If D is nonsingular then, for each i > 0 , !u|(i+r) ^ 0

for at least one value of r in the range 0 5 r 5 -2 - j

.

Proof: Since D is nonsingular, Casorati's theorem is valid, i.e.,

iX'(i) f 0 for all i . Theorem 3.1.2 is then an immediate consequence

of Laplace's general theorem on the expansion of determinants [15, §93]. Q

Theorem 3.1.2 implies there is no shortage of points i which satisfy

condition (3.1.2) when the difference operator is nonsingular. Our next

goal is to show that any optimally ranked basis may be used to locate an

admissible value of i .

Theorem 3.1.3 If |uj(i) r 0 for some fixed i > 0 , then the leading

principal minor of order j of the Casoratian evaluated at i of every

optimally ranked basis for D is nonzero.

Proof: If j = 0 then there is nothing to prove, so let us assume j > 0 .

Also, let us assume temporarily that ] < I . Let X' be an optimally

ranked basis distinct from X , where

X’ o

U X’

1 s s=l
61

and

XL = {x' ,x’ , S s, 1 s,2

,x j- , s ,n

s

s 1,2,.. .jO” \

compare (3.0.4), (3.0.5). Define the sets

U' = {u' ,u ' . .

. ,u' } = U X'

^ 3=1

and

V = {v’ v’ = u X' . ^

^ s=t+l =

There exist scalars a € F such that

p.q

P qil P>P P ,=5+1 P-'l P-2

and

j v'.= ya u + / a V.

P-3 qii P,q q P,q q-3 p = j-l-1, j+2, . . . ,^ .

The matrix of coefficients (a ) is nonsingular because both X and X'

P» q are bases of the finite-dimensional linear space K (D) . Furthermore,

from the definition of an optimally ranked basis it is easy to show that

(a ) is block lower triangular with the s-th diagonal block having order

p>q

n . Because of the block triangular structure, the determinant of (a ) s p,q

is equal to the product of the determinants of the diagonal blocks. And

because the determinant of (a ) is nonzero, so is the determinant of

P.q

every diagonal block. Therefore, the leading principal minor of order j

of the determinant of (a ) is nonzero. Let A. denote this minor.

P.q 3
Next, consider the Casorati matrix [X](i). It can be written in

block form as

[X](i) = [U](i)

[V] (i+j)

where the actual form of the blocks indicated by asterisks is unimportant

Similarly, we write

[X'](i) = [U'](i)

[V'](i+j)

If

(a ) =

P i 1

is the corresponding block form of (a ) , we verify readily that

P ,q

A 0

11

A , A „ 21 22

[U](i)

[X'](i) = [V](i+j)

The leading principal minor of order j of |X'j(i) is evidently given by

^11 •A^n

. 21

0

• Tr

:^22

U'|(i) = |U|(i) -det(A^p .

Since |Uj(i) ^ 0 , by hypothesis, and det(A^^) = 0 we have

!U'|(i) r 0 . This completes the proof for j < -3 .

Finally, we note that the proof in the case j = ^ follows by an

argument analogous to Chat just used for j < £ , except that there is

no partitioning of matrices since V and V are empty. j'_J
63

3 . 2 Convergenc e

y Assume

(3.2.1) iu| (i) 0

and let y be a solution of type t of (3.0.1), where 0 5 t 5 ct . For

each n > m + 1 such that a solution y of (3.0.1) exists satisfying

n both (3.0.2) and (3.0.3), there exist scalars (n) ,U

2 (n) ,

. . . ,a^ (n) and

j3^(n) ,p^(n) ,

. . . ,p^(n) such that

j k

(3.2.2) ^ ^ ^ ^

' S=1 S=1

In the case t = a , we have

y = y+ V a(n)u

^-i s s s=l

Then we prove readily, using (3.0.2) and (3.2.1), that ~ ^

s = 1,2,...,^ . Thus for each n , v (r) = v(r) for r = i.i+1

n

"

Therefore we restrict the ensuing discussion to the cases in which

0 5 t < c .

Consider the set C of all subsets of k distinct sequences from

X . We have k > 0 by our assumption that t < c . Obviously,

V = {v ,v , . .

. ,v } € C . We shall say that the optimally ranked basis X 12 k

is j -normal provided that |v| (r) ^ 0 for all sufficiently large r , and

(3.2.3) |v|(r) =o{|v|(r)} as r -> «>

for every V 6 C - {V} . When t = 0 , C = {V} and X is automatically

^The reader is referred to §3.0 for notation and underlying assumptions.
j-normal (0-normal) , When t > 0 the sets V contain at least one

sequence from U = {u^ , u^, . . .

, u^

}

. In this case (3.2.3) expresses the

quite reasonable condition that whenever one or more of the "dominant"

solutions V,12k ,v^,...,v, is replaced in the Casoratian |vl I by a "sub- I y

dominant" solution, then the resulting Casoratian is dominated by |v|(r)

as r .

We now state and prove the convergence theorem. A similar result,

but only for homogeneous linear difference equations, is given by Zahar

[27 , Th. 5.1]

.

Theorem 3-2.1 Assume X is a j-normal optimally ranked basis such that

(3.2.1) is satisfied and y(r) is a solution of (3.0.1) of type t. Also

assume

(3.2.4)

1 (y) I (r) lim ' s Vi (r) =0, s=l,2,...,k

where V (y) = {v , .,,,v ,y,v ,...,v }

. Then for sufficiently large

S J- S ' i. n there exists a unique solution y of (3.0.1) which satisfies (3.0.2)

n and (3.0.3). Furthermore,

(3.2.5)

lim

n-v<» y (r) n y(r)

for each fixed value of r > i + j

.

Proof: From (3.2.2), (3.0.2) and (3.0.3) we derive the linear system

U^(i) ... (i) v^(i) ... V^(l) a^(n) “

0

u^(i+j-l). . . u . (i+j-1) (i+j-1).. . Vj^(i+j-L)

.bl"?.

0

Uj^(.n) u . (n)

J

_ v^(n) ... v^(n) P^(n) -y(n)

u ( n+k- 1 )

. . . u^ (n+k-1) v^ (n+k-l)

.

.Vj^( n+k-1) CO. c 1

-y ( n+k-1)
Although the block structure indicated in this system disappears when

t = 0 , this modification^ causes no difficulty in the following proof.

Suppose V were such that the upper right block of the matrix is

zero (again, this is satisfied vacuously if t = 0) . Then the linear

system becomes

U^(i) ... (i) 0 0 a^(n) 0

u^(i+j-l) . . . u . (i+j-1) 0 0 a. (n) 0

u^(n) Uj (n) v^(n) Vk(n) P^(n) -y(n)

u^(n+k-l)

.u . (n+k-1)

J v^ (n-hk-1) . . .V (n-hk-1)

k J

-y ( n-hk- 1

)

The determinant of this system is | U j (i) *| V | (n) , and it is nonzero for

all sufficiently large n , in view of our assumptions. Using Cramer's

rule and Laplace's general theorem on the expansion of determinants

[15, §93], we then find that

^iCn) = a^(n) = • • • = (n) = 0

and

P^(n)

Vg(y) I (n)

jvj (n)

s = 1,-2, ,k .

Therefore, for all r and for all sufficiently large n , we have

(3.2.6) y^(r) = y(r)

k

r

L s=l

I V^(y) I (n)

V (r) .

The other case in which the block structure disappears, which is

t = c , was disposed of earlier.
Then (3.2.5) follows from (3.2.4).

In order to complete the proof, let us (3.2.7) v' = V + r u, + + Y .u P P P,1 1 P,J j

introduce the sequences

, p = 1,2, ... ,k ,

where the y are to be chosen in such a way that

p.q

^

v^(i+r) =0 , r = 0,1,..., j-1 .

Thus the y have to satisfy the linear systems

p>q

u (i) u.(i) y , -V (i)

1

u^(i+j-l) .

.

J

_

.u. (i+j-1)

P,1

=

P

-V (i+j-1)

J J ^ P’jJ L P J

each of which is nonsingular because |u|(i) 0 , by hypothesis. Let

V' = {v' , v' , . .

.

, v'

}

. Clearly it suffices to show that U U V is a 12 k

j -normal optimally ranked basis for D such that

(3.2.8)

r->-« I V I ( r

)

= 0 s = 1,2, ,k .

Since v dominates every sequence in U , we have from (3.2.7)

P

v’(r)

^

.

-^1 as r ->- «> V (r)

P

for each p , 1 < p < k . Therefore v' has the same type as v

P P Furthermore, we may verify the three conditions of Definition 2.2.1,

as follows:

i)

•‘(f) ^ u(r) ^ v'(r) v(r) v'(r)

P P P for every u € U , and

0 as r oo
67

v'(r) v'(r) V (r) v (r) —2_ = —2— . —2_ . _2_ ->Q og 1- v'(r) v(r) v(r) v'(r)

a p q q

whenever type (v') < type (v');

P q

ii)

v'(r) v'(r) V (r)

_2 2 . _2

V ' (r) V (r) V (r)

q p q v'(r) V (r)

q q

as r “

whenever type (v^) = type (v');

(iii) Let v',v' ,...,v' 6 V be all the sequences of one type,

P P+1 q

and let v' be any fixed one of these. Let the corresponding unprimed

sequences be v,v ,v v 6 V . Take 5 ,5 S € F , not all

P P+1 q P P+1 q zero. For each s = p,p+l,...,q we have

5 v' 5 (v +Y ,u +---+V .u.) 6 V

s s _ s s s, 1 1 3,1 1 _ s s ... V

(since are of lower type than v’ = v{l+o(l)}) . Hence

Furthermore

6v’+***+5 v' S V +---+6 V

= . V V

5 v'+*’’+6 v' S V +‘*-+6 V

P .P - P. P- q q V

{ 1+0 ( 1 )} + 0 ( 1 )

from which it is clear that

lim sup

J-'—woo I 6 v'(r)/v'(r)

s s s=p > 0

Thus, U U V is an optimally ranked basis.
Next we prove that |v'j(r) ^ 0 for all sufficiently large r .

We recall that a determinant is a multilinear function of its columns.

If we express

V'(r) IV

1

’

’ V 2’ * ’ •’ k

(r)

where v'12v' ...,v' temporarily denote the successive columns of | V | (r) , ic then we derive

|V'|(r) = lv^,v^,...,v^|(r) + 'f'l ,q

i ""q

q=l

= V ,v ,v' . . . ,vj (r) + I r ju ,v' . . . ,vj (r)

1’ 2’ 3 a=l

l,q' q’ 2-

and hence

|v'^(r) = |v!(r) + I T, | u , v

' . . ., v

'

| ( r)

q=l l,q' q’ 2

q=l

+ • • •

q=l h.q'''r''2

If we continue this construction until every reference to a primed v is

removed, then it is clear that |v'|(r) is equal to the sum of |v|(r)

plus a linear combination of determinants formed from | V j ( r) by replace-
ment of at least one v by a u . By our assumption that X is j -normal,

we conclude

(3.2.9) iV'l(r) = |vj (r){l+o(l)} as r .

Since !vj(r) ^ 0 for sufficiently larger, we also have |V | (r) ^ 0

for sufficiently large r.

If one or more of the solutions replaced in

I V j (r) by subdominant solutions from U , then a construction like the

one preceding shows chat the modified Casoratian |v'j(r) is dominated by

|v|(r) as r ® , since the original optimally ranked basis is j-normal.

Consequently, using (3.2.9), we conclude U € V is j-normal. A similar

argument shows that (3.2.8) is satisfied, and the theorem is proved,
3.3 Stability

Let us consider solutions of (3.0.1) of type t, where 0 5 t < ex .

'

Let j be the dimension of the subdominant subspace of type t for D .

Let i be a point at which |u!(i) 0 ; see (3.0.7). Suppose D is

(i, j)-factorizable; see Definition 1.2.1. Let = [a^, a^, . . .a^] be a

lert factor and B

^ = [b^ ^

,

. . . , b^^ '^ ] the corresponding right

factor of , so that ^ is an (i, i)-factorization of D . 3

' 3 3 The next theorem provides a generalization of Theorem 1.3.2, from

totally separable linear difference equations to equations that are merely

separable. The remark following Theorem. 1.3.2 applies here as well. That

is, the back substitution stage of the algorithm is stable, at least for

sufficiently large r.

Theorem 3.3.1 If lu|(i) 7^ 0 and = A^B^"*"^ , then the difference

operator is separable and any solution y of (3.0.1) of type t or

less is a sequence of type zero with respect to .

Proof: Suppose 3 = 0 , i.e., type (y) = 0 with respect to D. Then

A^ = [a^] and B^ = [b^ , b^ , . .

. , b^] and we have = A^B^ . 0 0 1 V Now let X € X , where X is the optimally ranked basis (3.0.4). Then

Dx = 0 ; hence D^x^ = A^B^x^ = 0 . This last equation is equivalent to

ap,(r) y b (r)x(r+s) = 0 . r = i,i+l,... . 0 'n s

s =0

Since a^ is free of zeros by the definition of (i,j) - factorizability

,

we have = 0 . Thus {x^|x i X} is an optimally ranked basis for

and it follows im.mediately that type (y) = 0 with respect to B

The case t = o corresponds to a maximal solution of (3.0.1). Since th

general algorithm reduces to pure forward recurrence in this case, it is

stable ; see ^1 . 1 .
71

Next suppose j satisfies 0 < j < -S . Let v 6 V , where V is

given by (3.0.8). In view of our assumption that ju|(i) 0 , we may

assume that

v(i+r) =0, r=0,l,...,j-l;

compare the proof of Theorem 3.2.1. Since v £ /((D) we have Dv = 0 .

It follows that for each n > i + j + max(j,k) the subsequence ^

satisfies the finite boundary value problem

pi,n-j-l^i+j ,n-l

3

p

^n-'^ ,n-j-l^n, n+k-1

compare (1.2.4). For each n, these boundary value problems are nonsingular

and factorizable as shox^m by Theorem 1,2.1.

Therefore, Theorem 1.2.2 is applicable and we see that ^ satisfies

i+j,n-l i+j,n-l _

»0

for each n > i + j + max(j,k) ; compare (1.2.7) and the remark following

the proof of Theorem 1.2.2. The last equation is equivalent to

k

I b (r)v(r+s) = 0 , r = i+j . i+j+l , . . . .n-1 . s=0

®

0

„n-k,n-l n,n+k-l

B, V

Since n may be arbitarily large, we have

k

I b (r)v(r+s) = 0 , r = i+j , i+j+1 , . .

.

s=0

^

or, equivalently, = 0 . Therefore, {v^"*"^|v 6 V} is an optimally

ranked basis for B^"^^ . Since y is dominated by every sequence in V ,
ic follows that type (y) = 0 with respect to . | | Now let us turn to the for^^7ard elimination stage of the algorithm.

This consists of applying a finite number of steps of Gaussian elimination, without pivoting, to the infinite linear system

(3.3.1)

i i+i i

D.y ^ = g

-

y

compare (1.3.6). Rounding errors introduced in this process are equivalent

to perturbations in the original problem. Thus, instead of satisfying

(3.3.1) the computed solution y^ ^

, say, is an exact solution of a

system of the form

(3.3.2) (D^ + =

g"* -

•^i,i+j-y,i+j-i

b + e'

where the terms and e^ represent the perturbation.

The effect of introducing a single element of e(r) , say at r = s ,

is to perturb the true solution y^"^^ by a linear combination of the

solutions of the corresponding homogeneous equation. Because of the

boundary conditions we have imposed "at infinity", only the solutions

u^.u^j.-.u^ enter from the optimally ranked basis {u^.u^ ,

• • . , u^ , v^ , v^

,

...,v, } ; compare equation (3.0.3) and Theorem 3.2.1. Subsequently,

eC we have only to consider what happens when r > s , and here it will be

the multiple of u^ that will be the fastest growing.

The effect of , also, can be allowed for by making a perturbation

3 of the right side. To terras of the first order, we have
73

Then the arguments of the preceding paragraph again apply.

Since the wanted solution y is of type t , we conclude that the

process of forward elimination is stable in the sense that each perturbation subsequently grows at a rate that does not exceed that of che wanted

solution. Of course, there will be a loss of accuracy when an element of

(or e^) is large compared with the rounding error in the stored value

of the corresponding element of (or of the right side of (3.3.1)).

This may occur, for example, when there is heavy cancellation in the

formation of an element that is subsequently used as a pivot. Hoi>;ever,

the important point is that the effect of each such loss is not magnified

in subsequent steps of the algorithm.

An extension of the foregoing discussion shows that if we select a

value of j (that is, the number of prescribed initial conditions) that is

less than the dimension of the subdominant subspace of the wanted solution,

then the forward elimination remains stable. However, the back-substitution

is now unstable; compare the proof of Theorem 3-3.1. Similarly if j is

too large then the forward elimination is unstable, and the back-substitution is stable. Nevertheless if, in practice, the actual instabilities are

weak owing to weak separation of solutions of consecutive types as r ,

then it may be advantageous to select a value of j that differs from

the dimension of the subdominant subspace of the wanted solution. This is

because the loss of accuracy caused by the instability is offset by a

substantial reduction in the value of the terminal point n owing to

increased convergence. This modification is illustrated by numerical

examples in Chapter 4.
CHAPTER 4. NUMERICAL EXAMPLES

i . 0 Introduction

Several numerical examples will be given to illustrate the general

algorithm. The examples all involve fourth-order linear difference

operators with nonconstant coefficients. Both homogeneous and inhomogeneous difference equations are treated.

The fourth-order operators were produced by using the method described

in [14, §12.22], starting with two second-order recurrence relations which

have as solutions Bessel functions, modified Bessel functions or associated

Legendre functions. The inhomogeneous equations have as particular

solutions Anger-Weber functions or Struve functions. The right sides of

these equations were produced as a by-product of the method used to produce the fourth-order operators.

The production of the fourth-order difference operators from the

corresponding pairs of second-order operators requires only elementary

algebraic manipulations. These calculations are rather lengthy, however,

and would be extremely difficult (as well as tedious) to complete accurately

by hand. Instead, the MACSYMA symbol manipulation system, described in

[12], was employed . This procedure had the added advantage of storing

the required formulas for the coefficients directly in the computer, ready

for numerical evaluation, thereby further reducing the possibility of

human error. Similarly, MACSYMA was used to advantage in the production

of the right sides for the inhomogeneous examples.

Development of which is currently supported, in part, by the United States

Department of Energy under Contract Number DE-AC02-79ER10357 and by the

National Aeronautics and Space Administration under Contract Number NSG1321.
In the case of the Bessel functions J (x) , Y (x) and modified

r r Bessel functions I (x) , K (x) we restrict ourselves to integer order

r r

r > 0 and real argument x > 0 . Relevant properties of these functions

are given in [16, Chapter 9]. For example, for fixed x the functions

J (x) and Y (x) satisfy the linear recurrence relation

r - ^

(4.0.1) y(r-l) - — y(r) + y(r+l) = 0

Similarly, I^(x) and (-) K^(x) satisfy

(4.0.2) y(r-l) - — y(r) - y(r+l) = 0

The Anger-Weber functions E^(x) satisfy

(4.0.3) y(r-l) - ^ y(r) + y(r+l) = 2{l-(-l) }

and the Struve functions H (x) satisfy

r

(4.0.4) y(r-l) - ^ y(r) + y(r+l) =

,1 . r x)

VnF(r+ |)

compare (4.0.1) in both cases.. Relevant properties of the E^(x) and

H^(x) appear in Giver's paper [21] where they were used as examples for

second-order inhomogeneous equations.

The coefficients of the fourth-order homogeneous recurrence relation

obtained from (4.0.1) with x = x^ and (4.0.2) with x = x^ are shown

in Figure 6 in the form of FORTRAN statements produced by MACSYMA. In

addition. Figure 6 shows the right-hand sides corresponding to E^(x^)

and H^(x^) . We designate the difference operator D defined by these
-X- fa * f

-'- + i] — X fa

Cl pc > Cl i Df fa -Xt Cl -X- •*r Cl w D4 Cl VT X Cl X-

'P— Cl -X- -X -Xt •Ml

X tC Cl -Xt PM P'1 -X- .My

-:• p^ -X- C] p-p yMy Cl Cl Cl X CD

-;:- *r TT Cl Cl •;> rM

fr Cl -X- X- -X- fa

X DC

-It C| tXX fa -X .'•V -Xt + -It ?i Pm fa. Cl KM o Cl

Cl -X Pn “r Cl -X- *r s w

fa Cl cv •fr Cl Cl \.

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i\ cJ X- o /My Cl VC X a KS -X- o CD

fa -It - V

£'J

X f !

fa

w

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ci sJ Cl -X -Xr -XV X, 1 1 -X- Pm ! ir fa fa fa Cl Cl vl .V Cl r’i fa- ©

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r** T

t} iTC -X- X- -Xt tj;

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fa

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Cl -X-

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VC Cl ii »-‘S -X- Cl -X Cl -.i fa fa

d:-V • -X •Xt

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fa «M Tc ?1X £1

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w Cl t' -*• Cl -Xt 1 -X -X- Cl -X- •X '1=14 w C-;

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— fa fa VG— 1M4 + —«

PC Cl Ps •M Ps < CJ ClX

w

Cl rC Xt fa -Xt < fa CJ Cm

fa fa X- ''1

w •X

X -Xt vD ko* fa •X: •X- X '•1 DC X g o fa

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Cl •X c T? o t- fa PC TT O -X- o MU pl» vr -X- •—

-X- -X 'J XI <! o < Cl c fa -X O -X- DC «- s CD © X c < •X •r VC Xt o < -M -'V » fa My < -X- Cl :.'l o < .‘V p'l 6 Cl i •Ml fa imJ 1 Cl i p'C i < o 1 Cl -X <x o \ 7r

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II •,l II DD •I Cl i> CJ 11 il :i 1» •M» Ml fa PCX

/»s Cl VC o Cl Ml VM o •«M :i CJ Pm

«» 1 -X- -X- -X- . y-p Cl ; o • Pm Sy^ fa.

;:- Cl -X- <* 10 <f.T o M. -,;• v»* Cl i Ml VM| fa fa

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CJ fa C-/ fai CJ fa fa fa fa CJ Ul.

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cc

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y

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G

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dJ rC < 3

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01

G J3 x:

G

a fa fa fa c fa fa

0) •fa

GG

G

u ja fa o rfa u fa G •fa 3 3 G G fa G o 3 •fa 01 c G rfa fa u G G o fa G G

fa

G T—\(a- o •fa CN faG G

CO o fa ffa •M < 0 3 G G G G

0) rfa •fa •H * A fa 60 a fa o fa G

*H 0 II

G •fa fa fa s^ a >

fa

GG

0) fa pi

G rfa o G 3 a fa cn G G o M G O

60 •fa c 60 fa •H M « G G w fa G •fa 3 3 n fa > CT

(faG

G

G e G fa J3 0 JZ G G o fa e c 3 G o 0

fa 60 fa •faG o c G fa G

fa •fa

a

G

G > 3 60

01 ffa fa cr

O 3 o 3 G g

•H > 3- 0

fa

G

fp

G rG 3 •fa •fa 3 C o O •fa fa G G

ja

r—\ fa 3 G • 3 fa G G rC

CO s./ 60 fa

r 60 Pi O X s s fa r—r < II 'T3 O o ! 3 ja aa H r—V G G A fa G •fa /-N. 3 o -t- CN rH o

fa fa

X

G •fa

rr

A s.^ fa fa fa fa fa G 0 T-S X w 3

fa

A 3

fa Sfa

G c Cfa

G

G • O o

0) T3 fa •fa •faG e CM o fa fa >

60 1 faG

G 3 G II CJ 3 3 fa faCN W G CT 3 fa —J

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OJM3

GO

fa
77

coefficients the JYIK operator . Since the functions ’

^r^^2^ , (-) K^Cx^) are linearly independent, they form a fundamental

set of solutions of Dy = 0 ,

Similarly in the case of the associated Legendre functions P’j^(x) ,

Q^(x) we restrict consideration to integer degree r > 0 , integer order

u > 0 , and real argument x > 0 . Relevant properties of these functions

may be found in [16, Chapter 8]. We use the r-wise linear recurrence

relation

(4.0.5) (r+u)y(r-l) - (2r+l)xy(r) + (r-u+l)y (r+1) = 0 ,

valid for fixed x and u . The coefficients of the fourth-order difference operator D, shown in Figure 7, were produced (using MACSYMA) from

(4.0.1) with X = x^ and (4.0.5) with x = x^ . Figure 7 also shows the

right sides corresponding to E^(x^) and H^(x^) . In this case the

functions J (x^ ) , Y (x^ ) , p'^(x.) , Q^(x„) form a fundamental set of

r 1 r 1 r 2 r 2 solutions of Dy = 0 . Accordingly, we designate D the JYPQ operator .

A FORTRAN program has been written to implement the general algorithm.

The user may select either the JYIK operator or the JYPQ operator. He may

also elect to solve the homogeneous equation, the inhomogeneous equation

having the Anger-Weber function as a solution, or the inhomogeneous

equation having the Struve function as a solution. Alternatively, he may

provide a subroutine which generates the coefficients of an arbitrary

linear difference equation of any order.

The user supplies the program with the starting point i and the terminal

point m of the desired subsequence of the solution to be computed, the

number j of initial values, the subsequence y^’^***^ ^ of initial values,

and the parameters v and e for use in the convergence test (to be described
UJ

c: C3 1 ff * c* Cl 1:3 c; Cl if -X Cl if 1C T" .•If if if

a C3 ys.

w

— i c? cs ff if 1 —

y

y-yy 4, ! Cl if CC

a ! -r- ! 1— ifa + if V if «T* if if Cl -1, '? f'3 Cl y-s if 1 if O CD Cl X Cl j.

p] tI- if ?i fl e: ZX if cp

a

— if f a If

V y-y y— if y—.

y-iy

—

vtV J, ?]

V ff if if

a “V C3 if !-C if

a -:-

G s.^ if Syy if ! -I'.

a e I if 1 I y-N

a ff if S •e if s zX

a

—

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G 1 •w f? ff C'l w c s if a o if i- "4

if •f?

a If if

a o if 'C -f if Cl

-f

w C3 if -f » i" :3 if T-y <- ei £3

a if .-s if yy-s if Cl is- -;- f“ ff '•3 JL 12 Vyy C3 y^ Cl if yyy if a C3 Cl a if If Cl if a Cl

C'3

— ^<1 + if IS if -'-

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‘4

if if :3 cc y^ c; Cl

a Cl y*y

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a if

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fr

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a

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<- if if if if if s_y

a

T Cl if <<

•{ i'- S Ci C1 C'3 if if C3 if

a y-s Xa y-s V* -f if

a if <-

if if cc ff if if if i'f n5 zX 1 O -fr cx if CD y-s if

a -> Cl f if 1 72

}'r C3

V ‘ Cl •? y-s if it if y*y Cl CO if y— 1 o y-s ;I;

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— zX

a

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— 1 ! ff 1 W ! T -f if if •f >c if 1-^ 1 I a — y— y-s if -If cC

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<- s if Cl f- 1 zZ y«y

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c.^ •L

a i',- % •f if •;:• iT if if •2 if -f — yy»y if ! a if 6i s.y a CD .'1 C3 If 72 if 'C' CX <- — if — if Cl Cl o yyN Cl n. 3C if C3 x« ( if a if Cl 1 1 j. if r* — Kn Tf <- cc if y-y 1 Tyy if C3 if Cl s_y /L a -s CD i t-iy 1 <' —y

s«y

n

a ii *T

w 1 ff y-N Cl -i- if 1 a 1 y»y y',i Syyy 1^-4 if if Syy i f S_y y-y 1 o

ff CO ;i ! if

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if if y-s

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a :-j Cl

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iT.4 vva

if

g

1'^-. S e O © a a

CJ j C.J CJ C.' r.; n; a ua e a CJ

-- c: c:

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— Cl

aa o —< Cl CD <<

44 3 /•-V cn

-y 3 3 rH

C

a

O XX

X

O 3 3 u -s-y •H -H 3 a U 3 14 a

CO 3 C50 o U O 3

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a

a

a

a

01 3 3 o 33 *H v-y 3 c 3 a 3

QJ 3 3 a

a

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a

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a

iJ

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CSC o o 3 3 3 XX a a

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+J

a 3 3 3 D-i XX 3 •a a a 44 O a e •H a c 3 3 a 3 u a U 3 a a 3 a 3 o 3 o a a 3

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79 IIII

vr vr ?! 44 ?! ?! 44

a

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V' + ?<

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74 ?! r? ?] ?? ?!

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a 2 1 44 t:? 77 v4 - — 44 2 -4 ?! 2

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a

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X

<- 2 * f! -{- 44 44 2 X 4- -’- 44 44 1 « 0

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•!• *'%* ?! 2 W 44 2 1 2 ' 1 4 7 li

a

!7 77 ?! 44 ?! 2 ?! + V7 2 44 44 0

f.- 44 2 44

X 44 !-=* 44 44 i’J < 1 -::• - i 44 44 44 2 44 2 -:4 -r /-S 2 44 Ci ?-! ?! '! ; ?! r 77 ?! ?! — * 2 1 77 V- ir 44 ?! ?! 44 77 ?! 1 ! — 2

c! ?! -;r 44 ?3 44

X 44 77 44 2 0 \ —

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11 7! i7 77 11 1 ?'!

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k«

N 44 2 :5 2 44 44 44 2 44 44 -2 -- --=> 0 » U ?!

-::- ?- ^ 44 44 44 44 ?! 0 -r ?! 0 W :i Tra v,^ * %c 7*! ?! 44 ?! ^7 ?! ?! ?i 44 4? — : T— a ^ C! -- 1*. ?! w. ; i >• J ; .) 0 ;s t. •!; IJ 44 ?! - •"7 -=« a

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•Z.^ '.f “’ r,

a “T yc V- 2 t? 0 2

CO ^ 77 ;4 :d ?! •7 77 44 i r*^ 3 2 s 0 e 2 2 a V 2 CJ a 2 a X 2 < t'4 < 2 2 22s 2 2

?! ?: - ?! ?: <- 2 N ?! ::

w- ?! C! c 0 0

OJ T3Oeou

(U

!-i 3

&0

•H
below) . The program solves the proposed problem by executing three phases

In the initial phase the first m+l+v-i-j rows of the infinite

linear system (1.3.6) are stored. Forward elimination without pivoting

is performed on these rows. This results in the storing of the first

m+l+v-i-j rows of the left and right factors of the (i,j)~

factorization ^

, and also the subsequence

J J solution of (1.3.8). The elimination procedure is carried out by modifica

tions, described below, of the subroutines DGBFA and DGBSL which are

available in the mathematical software package known as LINPACK; see [ 8 ]

.

The unmodified DGBFA performs forward elimination with partial

pivoting on a finite band matrix, ignoring the right side. It produces

an upper triangular matrix, a lower triangular matrix and a vector of

pivoting information. These results may then be used by the unmodified

DGBSL to solve the corresponding linear system with specified right side.

Alternatively, the user can direct that DGBSL solve a linear system having

as its matrix of coefficients the transpose of the matrix input to DGBFA.

Provision has been made in DGBFA to suppress the partial pivoting. When

this is done the vector of pivoting information is superfluous and (except

for rounding errors) the matrix obtained by premultiplying the upper

triangular matrix by the lower triangular matrix is the original band

matrix. This factorization is therefore of exactly the kind required by

Definition 1.2.1; see equation (1.2.5).

When partial pivoting has been suppressed in DGBFA, DGBSL proceeds,

in effect, by forward elimination without pivoting followed by back

substitution. Provision has been made in DGBSL to exit after the forward

elimination stage and to re-enter at the start of the back substitution

stage. Thus the user has the option of examining the result of the
81

forward elimination stage and perhaps electing to continue it without

having to do any back substitution. All of the above modifications of

DGBFA and DGBSL, to provide just the tools that are needed in our

algorithm, were easily achieved because of the clarity and flexibility of

the coding and documentation of LINPACK.

We return to the description of the initial phase. The next computation is the subsequence . compare equations (1.3.16) and

m

(1.3.17) with s = m . Since w^ satisfies the adjoint equation, it

m

is produced using the forward elimination stage of the (modified) subroutine DGBSL in which the input matrix to DGBFA is automatically transposed. Finally, the initial phase computes

r-1

(4.0.6) y (m) = V w (s)z(s)/b (m)

r ^ m o s=m

for r = m+1 ,m+2 , . .

. ,m+v ,m+v+l and

r+v-1

(4.0.7) r\ (m) = 7 w (s)z(s)/b (m)

r,v ^ m o s=r

for r = m+1 . Equation (4.0.6) is the same as (1.3.18) apart from a

change of symbols. Equation (4.0.7) is the approximation to the truncation

error

Tj^(m) = y(m) - y^(m)

which is obtained when y(m) is replaced by y , (m) : compare (1.3.20).

r+'j

After completion of the initial phase, the program proceeds to the

iteration phase . One additional row of the infinite linear system (1.3.6)

is stored. Since the previously determined part of the ( i, j

) -factorization

has replaced the corresponding rows of the original matrix in the computer
storage, these rows are no longer available to extend the forward elimination stage. Therefore, extension of the forward elimination is accomplished by an alternative procedure that requires the solutions of two triangular linear systems of orders j and k . This is achieved using DTRSL, a UNPACK subroutine that solves triangular systems. Next, the formerly

obtained subsequences of the solutions and wTM"*"^ are extended to m

include the next higher term. Then the series (4.0.6) and (4.0.7) are

summed with the new value of r , and the following convergence test is

applied: If

(4.0.8)

then the program proceeds to the third and final phase; otherwise, the

iteration phase is repeated and the convergence test re-applied.

The final phase of the program is the back substitution stage of the

algorithm. Let n denote the value of r which satisfies (4.0.8). The

back substitution is begun at this point, using the (modified) subroutine

DGBSL. This yields the computed solution ^ (1.3.7) and also

the desired approximation of y(r) over the range r = i+j , i+j+1 , . . . ,m .

In all of the examples presented in this thesis the chosen termination

criterion corresponds to an estimated relative error of 10 in the final

term y (m) of the requested sequence of values of the function. Thus

n

t = 10 in equation (4.0.8). For the values of x^ and x^ selected

in the examples the solutions are reasonably well-separated as r ,

hence we take the "testing parameter" v in equation (4.0.7) to be 1

throughout. Initial values were entered to 10 significant figures in

every case. All computations were performed internally to 18 figures
and the output printed to 10 figures. For each example we present (i)

the solution z (i+j ) , z (i+j-fl) ,

. . . ,z (n) of equation (1.3.8) (the forward

elimination stage); (ii) the solution w (m+1) (m+2) , mm . . . ,w (n) of m

(1.3.16), (1.3.17) (the forward recurrence of the adjoint equation for use

in the termination criterion); (iii) the sequence y ^^(m),y .(m),..., m+1 m+2

y

(m) of approximants to the value y(m) (see equation (4.0.6));

n“K;

(iv) the sequence ri (m) ,"n , » , (m) , . . . ,ri (m) of estimates of the m+1 , 1 m+ 2,1 n ,

truncation error incurred in accepting the terminal points of m+1 ,m+2, . . . ,

respectively (see equation (4.0.7)); (v) the sequence y^(n-l) , y^(n-2) , . .

.

,

y^(i+j) derived from equation (1.3.7) (the back substitution stage).

The initial values that were used in the boundary value problem actually solved in each example are shox«m in parentheses in the tables; see,

for example. Table 5. The terminal values (which are all zero) are also

shown. Forward recurrence of the j-th order difference equation A^z^ = g^

starting from the given initial values produces a solution of (1.3.8) (the

algorithm produces this solution by forward elimination) . Sim.ilarly, backward recurrence of the k-th order difference equation ' n starting from the terminal values (which are all zero) produces the corresponding solution of the boundary value problem; this is the back substitution stage. Data is given in the tables only for selected values of r .

All of the values of r that were used beyond r = m are included, however. These values illustrate the termination procedure.

It should be noted that when j = 4 the method as implemented

by the FORTRAN program is equivalent to, but not identical with,

forward recurrence of the given difference equation. See, for

example, Table 8. This is because DGBFA, the UNPACK matrix

factorization subroutine, scales the linear system in such a way
chat the principal diagonal of the lower triangular factor is a sequence

of ones. Therefore, the result of the forward elimination stage differs

from the result of the back substitution stage (unless the trailing coefficients of the original linear difference operator are all equal to one)

.

Since there is no question of truncation error when j = 4 , the sequences

w (r) , y (m) and ri (m) , r = m+l,m+2,..., are not defined, m r ^ > i

It has been observed that Giver's original second-order method is

likely to cause overflow in most computers. This same likelihood of overflow exists for the extended algorithm. The examples to be presented in

this chapter were computed using double precision on a Univac 1108. Double

precision was used not for higher precision but for wider exponent range.

308

The double precision range extends to 10 whereas the largest number

200

generated in the examples was somewhat greater than 10 . Van der Cruyssen

[25] gives a rescaled version of the second-order algorithm, based on LU

decomposition, that greatly reduces the incidence of overflow or underflow.

An equivalent procedure for overcoming this difficulty, based directly on

the use of ratios of consecutive terms in each solution, is described by

Aggarwal and Burgmeier [1]. In the present writer's view, however, it is

preferable (if possible) to avoid mathematical modifications of algorithms

simply to suit arbitrary limitations of existing computers. A software

alternative to rescaling is proposed in [24], in whicn a full integer

storage location is allocated to the exponent of each floating-point

number. A FORTRAN implementation of this proposal is available; see [11].
85

4 . 1 Examples Involving the Homogeneous JYIK Equation

From Che asymptotic forms

(4.1.1) J^(x)

(2Trr)

1/2 \(x) -

and

(4.1.2) I^(x) (-)’^K^(x) - (-)

which are valid for fixed x and large r, we obtain

(4.1.3) < I^(Xj) < (-)'^K^(x,) < Y^(x^)

when x^ < x^ and

(4.1.4) I^(xp < J^(x^) < Y^(x^) < (-)''k^(x2)

^jhen x^ < , where in (4.1.3) and (4.1.4) the symbol < is used to indicate

the order of dominance of the functions as r . Thus when x^ ^ x^

the JYIK operator is totally separable. As with any totally separable

operator of order 4, there are four distinct types of solution and the dimension of the subdominant subspace of type s is s . This means that j = s

is the proper number of initial values to specify for a solution of type s .

When all four Bessel functions have the same argument, from (4.1.1)

and (4.1.2) we have

(4.1.5) J^(x) ~ I^(x) , (-)"K^(x) -- (-)^^^ ^trY^(x) , r -> » .

The first of these relations suggests that we define the further solution
Then it may be verified that

(4.1.7) A^(x)

1 x~

2 r+1

I^(x)

J^(x)

^ 00

where either function in the braces on the right may be chosen. Furthermore, since there is no nonzero linear combination of (-) K (x) and

r Y^(x) that reduces to a solution of lower type, we conclude that there

are exactly three distinct types of solution. Symbolically, we have

(4.1.8)

r ^

I (x)

^ r

(-) K (x) r

< >• <

r > J (x) Y (x)

L r J L ^ J

where in each pair of braces either function may be chosen. Thus when

x^ = x^ the JYIK operator is separable but not totally separable. The

dimensions of the subdominant subspaces of types 1,2 and 3 are 1,2 and

4 respectively.

Although A^(x) has type 1, it is separated only by a relative

factor of 1/r from solutions of type 2. Let v (r) denote the approxima-

' n tion to A^(x) that is obtained from the algorithm with terminal point n.

Then, suppressing the arguments of A^(x) and all the Bessel functions

and using equation (3.2.6), we have'

y^(r) ~ A^ - ?^(n)I^ - P

2 ('^)Y^ ~ , n - »

where

Since I^(x) , Y^(x) and K^(x) are not normalized as required for the

validity of (3.2.6), we have only the weaker statement given here.
87

A

n

Y n

/

I n Y

n

=

"^n+1 "^n+1 Vi Vi

^n+2 / n+2 ''n+2

I n A

n {-)\

/

I n

Y

n (-)\

"^n+1 ^nh2 Vi

^n+2 ^n+2 / ^n+2

and

I n

Y

n A

n /

I n Y

n (-)\

p3(n) =

"^n+1 /

T""n+l

^n+2 V2 \+2 / ^n+2 V2

In consequence of (4.1.1), (4.1.2), and (4.1.7) we see that

2 -1

P^(n) -X ( 2n) for large n , whereas ^2^^^ ^3^^^ approach zero

much more rapidly. Thus we expect that the rate of convergence of the

algorithm will be approximately as 1/n . This has been confirmed in

numerical tests. Since the separation of A^(x) from solutions of type

two is so weak, it is usually advantageous to regard A^(x) as belonging

to the same type as d^(x) and I^(x) ; this means that we specify two

initial conditions rather than one. This modification improves convergence;

in consequence the value of the terminal point n is reduced very considerably. This gain is offset to a minor extent by some loss in accuracy in

the values of A^(x) for higher values of r ; the extent of this loss is

no more than the cancellation that takes place between the corresponding

values of I (x) and J (x) in Eq. (4.1,5).

r r
1

Example . 1 .In this example we compute the solutions J^(l) , 1^(10) ,

(-)'K^(IO) and Y^(l) of the homogeneous equation defined in Figure 6,

with x^ = 1 , x^ = 10 , for r = 0 , 1 , 2 , . .

. , 100 . Since each of these

solutions is monotonic (in magnitude), the algorithm will be numerically

stable provided that the proper number j of initial values is used in

each case. This number is j = 1 for J^(l) , j = 2 for 1^(10) ,

j = 3 for (-)^K^(IO) and j = 4 for Y^(l) ; compare (4.1.3).

Table 1 gives the numerical coefficients of the JYIK operator with

X., = 1 and = 10 for r = 0 , 1 , 2 ,

. . . , 109 . Tables 2, 3 and 4 give the

numerical coefficients of the (0 ,j ) -factorizations of this JYIK operator

for j =1,2, and 3. (We do not present the ( 0, 4) -factorization . ) Since

X X x+i

the (i, j)-factorization = A^B produced by the test program has

the principal diagonal of equal to (1,1,1,...) , we do not include

values of a^ (r) .

Tables 5 through 8 give the results of the computation of the four '

desired solutions J^(l) , 1^(10) , (-)^K^(IO) and Y^(l) for selected

values of r ; see §4.0 for a full description of the tables. Initial

%»alues were taken from Tables 9.4 and 9.11 in [16]; they are given in

parentheses. These tables also supply values of the four Bessel functions

for certain other values of r in the range 0 5 r 5 100 . These entries

all agree with our computed results, except for the tenth significant

figure in some instances. This slight discrepancy is explained by the

fact that the starting values that were used inevitably have an error of

several units in the eleventh significant figure. Incidentally, because in

this (and other examples) we have not attempted rigorous control of I

I

)

(

individual rounding errors, a guaranteed error bound cannot be given.
Nevertheless, the stability of the algorithm for all solutions in this

example is firmly demonstrated.
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ttie

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with

x

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and
1

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c; ;i '-' i' I'- V 3 u-i L-i !? T2 ifi i>i '} :: c3 ts :<• « r- v o 3 ^ .5 - 3 - S' 3 I- :•! i i - C-: 3 3 <- - \? T? f] S' S' V 3 C-? 3 r? c? Li: o r; ;2 3 ?i l- 3 C5 3 - 3 r. r- jo :5 3 3 3 o 3

‘ ‘ - • - - ci 3 - N cj C-: 3 - 3 -> v 3 3 r- - ;•: 3

"i ?] « '? 13 -3 s 3 C] ':• ?5s|i

- T-3

I I

<- n - - 3

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+ + + + + + + + + + + +

'5' L3 3 f1

••:' r- 3 '? Li N N 03 N 3 Li 01 3 3 “ 01 0] N 3 - 01 S' N ti 01 3 Li — 3 CJ 3 3 — 3 3 Li 3 - 01 3 3 N e V •“ 3 C? c:i o: Li - 01 3 r? Cl 3

C? - 3 V r- ^* zi 3 ci 3 3 CO O <' 3 3 C3 3 Li 01 N

3 C-2 ti t 'i o: 3 3 N 01 S3 C3 - si Ol Vi -•? V? V 3

- 3 S' 3 0- Si c O'! Jv 3 3 N O 3 C-3 '? S3 CO O 3 -

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01 01 01 8888888888888 I I I I I i I I I I I I I — O'.-'> — ::-- 3 —— i-1 - Li Li 013 3... o —

— 3 3 3 '? 3 3 Li S' Li 3 CJ SI r«- 3 3 SJ S' 3 3 3 i- 3 O I- Li to 3

— Li 1I •. 013 V V i- —— '?V V O O i'*' —

3 C3 Li 01 C l 3 - 3 Li Li - OK

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.i 3 1 - 3 Li Li

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t

01 01 01 Cl 01 3 0 0 0 0

0 3

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cj 1'- 3

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Vj) Li Li

n 01 01 01 Ci 01 01 01 Ci 01 Cl oooooooo 00000333 I I I I I I I 1 3 O 3 Cl S' Cl Li

3 3 0

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C5 3

Li 3 b -.v V : ..j Ci Ci - 3 v5 S' S' V

3 01 1 - — 330 V

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01-

CS 3 3 SI C q 3 V 3 S' o: c; 3

3 1 - S' '? S' 3 <r? “ 3 1 - Li V ci — 3 Ci Ci Cl Ci 01 01 01

I 1

C - Cl 3 V Li 3 r- 3 3 O - Cl 3 '? Li 3 1 - C3 3 O - 01 3 S' Li -i N 3 3 C - 01 3 S' Li v9 - 01 01 Ci 01 01 C! Ci 03 i; 01 Ci Ci 3 Ci Ci 3 C 5

Table

3.

Numerical

coefficients

of

the

(0,

2)-factorization

of

the

JYIK

operator

with

x

=1

and
97

N N N N N N

O S S ^ S ^ OO'SO^OO^ ©SG>eoss30oos3 0^^Y’'7 99?^®‘?’'?9^'S9'

c©o©o©©o©©c5©os©'i?©oo©©^©©©9oo©^ooo© 9 © o

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CNI —• V © r- 3 © f] © © tJ> V © © "^ V © © C^l 3 © N 3 ^ ‘ © V — I - -© © N © © T? 5^ ©

n © © C3 © © ?3 © CS © ?J N V C* ^' N ?3 © © © — © © 33 © © © N © © © V © © © C3 © ©

© 33 © o; N © V 73 © 31 33 33 33 v3 33 3J V © N 7^ O 33 V © © — V N © v C3 — © © -^7^ ^-^ V" © © N N S © 3 — 33 © -^ LO © 3'* iC © © — 33 © © © N 33 ss — A] ^ N, jv g; Q-\-j — -.I -.I c3 © © © © •© © © ^? ’? v "C" v V © © © ©

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I I I

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Q o o o © s s s © s o s s s s Cy o o s s o s o s s

+ + -f- + + + + + -*- + -i- + 4. + + 4- + + + + + X X + X X X X -}- X X X X

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+ + + + X + + + + 4* + + + + + + + + + -1-

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Cl o a mm G PA IT? — G vS 31 A*- o Cv s 7C N. '»C ts* © LC V •* •>« N 33 y-> TA o cc •-N '.V a a © © 1'* © C' 31 © W •—

LC \* 31 '•S 31 1? 'vS N a 33 7C o 33 *<• V —V~» © C' «»A AO © 31 G VV T? N •V © © V- C! © N a o 31 W 'iS Ij* •A © 35 a -A »rX '^ N w N ;N* :*. c> AT N 7$ —

— N •—« w 7’$ N w' V 37 T4 o AT. Q Cj Cj Cl TV L"$ vC vS 3l LC AT

cc © L75 — \5 Ci © —

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33 Cl 33 33 31 33 33 33 33 33 33 Cl 33 33 33 33 c$ a a a C5 a a aa

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1 1 1 1 1 j s O o O © a rA C' $• Cl © —

33 a Cl G Jsj C5 i-* G u$ C' CC Zt r T

‘-T> L-$ c5 Cl s—

N Cl Cl L*^ A ^ o N Cl 33 C*" 7?- r:'-* o Cl N» f-X o vS o Cl vS V G

V lA LQ C' mm V“ LC LC vS Cl << © w C3 G O C$ Co Cl 7> s s ;2'. LC; T> o cc N '•? Cl V.Z- VJ

CO N N Ci C3 o o o V* iiC © — a © C'l C' O v' N — S vS V A". ih N V A a

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La i.C o L"C 17$ A a vu aA o G s5 o a vS V ^•* C-C cl

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v-^ o «Al o Cl z$ Q 33 V c$ L*3 © V Cl •s» Q C' T? *?y» C' C-5 — C3 Cl AT © vS a mm c$ Cj CC lc cc V N —

V C-j 0 -A Q X? G G C7$ V c$ mm

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A a *« c*$ C$ C7' © N '•S iTD T? cc Cl Cl •A —

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(r+ 2)

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Table

4.

Numerical

coefficients

of

tlie

(0,

3)-factorization

of

the

JYIK

operator

with

x

=1

and

x

-

10.
(r)

b

(r+3)

b,(r»-3)

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Table

4

(Continued)
2003+001

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101

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6.

Computation

ofI

(10)

from

the

JYIK

equation

with x

-1,x

-10,

1-0,

m-100
;'JT0062:'-Li-004)

1

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Table

7.

Computation

of

(-) K

(10)

from

the

JYIK

equation

with

x

=1,

x

=10,

1=0,

m=100.
fi32r>696420-OO

1

)

73H2l232K:+ft«())

iGuc^inafiov+ooi)

105

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03
I

Examp 1 e 4.1.2 In this example we compute the solutions 1^(1) , J (10) .

Y^(IO) and (-) K^(l) of the homogeneous equation defined in Figure 6,

with x^ = 10 , x^ = 1 , for r = 0 , 1 , 2 ,

. . . , 100 . Both 1^.(1)

K^(l) are monotonic over this range of r . The solutions J (10)

and Y^(IO) are monotonic only for 10 < r 5 100 ; for lower

values they oscillate with approximately the same amplitude. However,

since there is no "crossing over" of growth behavior in the four solutions

in the nonasymptotic region 0 < r < 10 , we expect the algorithm to be

stable with the following numbers of initial conditions: j = 1 for 1^(1),

j = 2 for J^(IO) , j = 3 for Y^(IO) and j = 4 for (-)’^K^(l) ;

compare (4.1.4). Tables 9 through 12 give the results of the computation

of each of these solutions for selected values of r ; see §4.0 for

a full description of these tables. Again, the stability of the algorithm

for each solution is confirmed by comparison with Tables 9.4 and 9.11 of

[ 16 ].
107

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Table

9.

Computation

ofI

(1)

from

the

JYIK

equation

with

x

=10,

x

=1,

i=0,

m=100,
o o c o

n N C"

C3

S' — ^ O O t'» Crf *^* V “ S' <? <? vSJ N — N

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C-] ;- - o L". 3 N

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Table

10.

Computation

ofJ

(10)

from

the

JYIK

equation

with

x

=10,

x„=l,

1=0,

m=100
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-33H
Ill

Example 4.1.3 In this example we compute the solutions^ A^(l) , J^(l) ,

1^(1) , Y^(l) and (-) K^(l) of the homogeneous equation defined in

Figure 6, with x^ = 1 , x^ = 1 , for r = 0 , 1 , 2, . . . , 100 . Since each

solution is monotonic (in magnitude) except possibly at the very beginning

of the range of r , the algorithm is expected to be stable provided that

the proper number of initial values is taken in each case. This number

is j = 1 for A^(l) , j = 2 for J^(l) and ^ ~ ^ f°^

X

Y^(l) and (-) K^(l) • However, in the case of A^(l) , for reasons given

earlier in this section, we take j = 2 instead of j = 1 , thereby

exchanging mild instability for a much improved rate of convergence.

Because of this mild instability, each individual rounding error,

introduced at say r = s, is amplified in direct proportion to the number

of steps taken beyond the point s , since in (4.1,7) A^/

0(l/r) .

Thus, after 100 steps, approximately two decimal places of precision will

be lost. On the other hand, if we were to take the number of initial

conditions theoretically required (in this case it is J~=l,- since

is dominated by all of the other homogeneous solutions), then the converg gence would be extremely slow; A terminal point of the order of 10

would be required to obtain 8 significant figures of accuracy in A^^^d) .

Tables 13 through 17 give the results of these computations for selected

values of r ; see §4.0 for a full description of these tables. Again,

these results agree with the entries in Table 9.4 and 9.11 of [16]. In

the case of A^(l) , the method we have used is equivalent to computing

J (1) and I (1) as type 2 solutions and then forming their difference.

The function A (1) is defined by A (1) = J (1) - I (1) ; compare

(4.1.6).

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Table

14.

Computation

ofJ

(1)

from

the

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equation

with

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=x

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i-0,

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Table

15.

Computation

ofI

(1)

from

the

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equation

with

x

=x

=1,

i=0,

m=100
{?.ri2r»696420-00

1

)

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115

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4 . 2 Examples Involving the Inhomogeneous JYIK Equation

For fixed x and large r the Anger-Weber function has the

asymptotic forms

( 4 . 2 . 1 ) EzrCx)

2x

(4r -1)TT

compare [21]. Comparison of (4.2.1) with (4.1.1) and (4.1.2) shows that

'lyxz)'

1 (4.2.2) . ycxy

J

< E^(x) < .< vyxi)

for any given set of values of x^ , x^ , x . Thus E^(x) is a solution

of type 2 of the first inhomogeneous equation defined in Figure 6 and the

correct number of initial values of E^(x) to be used in every case is

j = 2 .

The asymptotic form of the Struve function, also taken from [21], is

(4.2.3)

y 2

' ' rrr

for fixed x and large r . Comparison of (4.2.3) with (4.1.1) and (4.1.2)

shows that

(4.2.4)

ry(x)'i

H (x)

Vm

> , r

L 'J^(x) J

Thus H^(x) is a solution of type 1 of the second inhomogeneous equation

defined in Figure 6 when = x^ ; compare (4.1.8). However, since

the separation of H^(x) from solutions of type 2 is even weaker than
chac of the solution A^(x) defined by (4.1.6) it is again advantageous

in most applications to specify the number of initial values appropriate

to solutions of type 2, i.e., j = 2 instead of j = 1 ; compare

Example 4.1.3.

Example 4.2.1 For the inhomogeneous fourth-order difference equations

defined in Figure 6, we compute E^(l) with x^ = x^ = 1 for r = 0,1,2,

...,100 ; and H^(O.l) with ~ ^2 ~ ^ ~ 0,1 , 2, . . . , 50 .

'

The appropriate number of initial values in these cases is j = 2 for

E (1) and j = 1 for (0.1) . However, for reasons given above we

actually use j = 2 for H^(O.l). Initial values for E^(l) and H^(O.l)

were computed from series expansions given in [21] and checked against value

found in [21] and [23]. Tables 18 and 19 give the results of these computations for selected values of r ; see §4.0 for a full description of

these tables. They agree with the 10-figure values of E^(l) and H^(O.l)

r = 2,3,4, 5 , given in [23]; the 7 through 9-figure values of E^(l) ,

r = 2, 3,..., 10 , given in [21]; and the 9-figure values of H^(O.l) ,

r = 1,2,..., 13 , given in [21]. In addition, they agree with series calculation of the functions to 10 significant figures in the case of E^(l) ,

r = 2, 3,..., 100 , and to 9 significant figures in the case of H^(O.l) ,

r = 2, 3,..., 50 . The loss of precision in the Struve function computation IS due to mild instability which is acceptable in lieu of impossibly

slow convergence.

We restrict the sequence of H (0.1) to r < 50 in order to avoid underflow in the computer. As suggested at the end of §4.0, this difficulty

could be overcome by using the software described in [11].
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121

4 . 3 Examples Involving the HomogeneotiS JYPQ Equation

For large values of r and fixed values of p and x , the associated

Legendre functions have the asymptotic forms

(4.3.1) P^(x) = ^ ^^^cos(r0 + ^ - ^

+ + O(^)

and

(4.3.2) Q!^(x) r(r^+l)

r(r+3/2)

(

TT_ 2 sin G

1/2

) cos (r 0 + —{ + ^

+ iin) + 0(i)

where x = cos 0 with 0 < 0 < rr

; see [16, Chapter 8] . Consequently,

for values of x less than 1 in magnitude, neither associated Legendre

function dominates the other. Comparing (4.3.1) and (4.3.2) with (4.1.1)

we obtain

"P^(X2)"

(4.3.3) Jr(Xi) <

Q^(k2)

< » r -> «

for all admissible values of [j, , x^ , X

2

• Thus the JYPQ equation has

three distinct types of solution, and the dimensions of the subdominant

subspaces of types 1, 2 and 3 are 1 , 3 and 4 respectively; this

contrasts with (4.1.3), (4.1.4), and (4.1.8).

Example 4.3.1 In this example we compute the solutions 3^(1) ? P^(0.5) ,

Q^(0.5) and Y^(l) of the homogeneous equation defined in Figure 7,

with u=0 , x^ = l , X

2 = 0.5 , for r = 0 , 1 , 2 , . . . , 100 . Since these

solutions exhibit their characteristic asymptotic separation, described by

(4.3.3), from the very beginning of this range of r , the algorithm

"'"in accordance x^ith custom, when the order of the associated Legendre

functions is zero we drop the superscript in the notation.
will be stable provided the proper number of initial values is used in

each case. This number is j = 1 for d^(l) , j = 3 for P (0.5) and

Q^(0.5) , and j = 4 for Y^(l) .

Table 20 gives the numerical coefficients of the JYPQ operator with

u = 0 , = 1 and = 0.5 for r = 0 , 1 , 2 , . . . , 104 . Tables 21 and 22

give the numerical coefficients of the (0 , 1) -factorization and the (0,3)-

factorization . Since the (i, j )-factorization produced by

the test program has the principal diagonal of equal to (1,1,1,...),

we do not include values of a^ (r) .

Tables 23 through 26 give the results of the computation of the four

desired solutions for selected values of r ; see §4.0 for a full description of these tables. Initial values were obtained from Table 9.4 of [16]

in the case of the Bessel functions and by hand calculation of the explicit

elementary forms given in [16], Section 8.4 in the case of the Legendre

functions. The computed Bessel function values agree satisfactorily with

the values given in [16] . The Legendre function values were compared with

data available for r < 10 in [16] and (in the case of the P^(0.5)) [17].

Agreonent is satisfactory to the precision available in these references;

this ranges from 6 to 8 significant figures. In addition, all of the

computed Legendre function values agree satisfactorily with double-precision values computed by an unpublished program of Dr. J. M. Smith of the

National Bureau of Standards. This is sufficient to demonstrate stability.
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Table

20.

Numerical

coefficients

of

the

JYPQ

operator

with

y=0,

x

=1,

and

x

=0.5.
d.

(r)

do(r)

d..(r)

I I I I I I I ! I I I I I I I i I i I I I I I I I I I I I I I I I I I I I

— — — — — — — — — 01 01 Cj Cl Cl 01 01 01 01 Cl 03 01 Cj Cl 01 01 01 01 01 01 01 01 01 CO Cl Cl G G

G c o G O o G o o o G G -G o G G o G <> G o o G o G G o O o

+ ** -r + *r + -T“ i” •r S- *r *r + -r + + *r + 4* 4“ T 4* 4* -{- 4* 4- »•- *r 4- -J- -V o •— 01 N* i-m N. Ifi N c-0 -—

•

o G •— r^o G C3 •--( G o L3 CO N N Cl A' Ci N. — CJ C^ V CO Cl G Oi 4> V* i-N 01 01 G O C5 LO G

G Q LO V L*^ •CO N CO c> Cl -* V V — c;J 01 O) C3 G 'G O \5 V5 O fA a* a. C ^•o ct CO C) cc

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\5 N N ci O' f-<fs 01 CO cO Na ci V G 01 LI — Cl o V C-3 N* i\a M G G 01

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—

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Table

20.

(Continued)
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21.

Numerical

coefficients

of

the

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actorizatlon

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the

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with

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and
127

+

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N G ‘0 G — Cl CO S' 13 0 C* 43 0 O Cl G S' 13 0 N G 0 O — Cl G 'iC 13 0 N 'G 0 O

G G G S’ S' S' S' S' S S' S' S' S' 13 13 G 13 13 13 13 13 13 13 0 0 0 0 0 0

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Table

21.

(Continued)
10

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.

iri-IOSiU

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I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

11

y C3 CD G CD G G G G G G C5 C5 G CO CO G CO CO CO G CO O' 'O •<? G G o o o o G o o c o G G G /*> O G O o G :-; o o o G c o

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C'J N. G N. -— G tM N3 G r-s > fOM (.3 CD 'V o CD 01 CO N b > » G <* CD ei N G Cl G N N V* G G b LD iO N CO

t'

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C'; G *«»r 'C '‘D o LD N V? G o CD ei 01 C-J N» G Vw G

C‘2 w w N \V 'D G fl n 1-M 01 G N G G w-l *—

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'• J CJ CJ :> G G CD V’ G LD G N N C3 C3 G G —

CD CD CD CD CD CD CO CO CD CO CO G C y CO CO G CO CO CD '<• y CO CO CO CO CO CO T? w «aa aa OM wa -=•( *sm '*=1 -sa. oa Bt T=* T*1 !^ -=< •:=» G 9 O C G Q G G c o G o G G G G G o G o G G o o G O Q Q © G

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03 o c /M 03 f'. N LfD £3 '.5> N G £3 ^ A SV CD O C3 Vi (» "N CD 05 G G O C-5 CD a.4 o o N 03 N 03 G G G C^ 01 -a.

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21.

(Concluded)

.
129

+

c-i: Q c-3 c3 K C3 :; S' s' cn o-;- 9-;- o-.. o-.- 9 o o-- 9-c- 9-;- ©

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+ + + + + © © © © — V © N

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G G G G C G G

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S' © G © S' VI a N Vi CJ CJ S' S' -> N © N VI © G VI Vj V CJ <' O C3 ©

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e — VI © •«? © G N C3 G G — VI © V' © G r- S3 G G — ?i © <' © G iV © G G Cl © <' © G

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Table

22,

Numerical

coefficients

of

the

(0,

3)-factorization

of

the

JYPQ

operator

with

p=0,

x

-1

and
g s s X 5 5^ s e c 6 s c b c -3

S C-^ Cl -3 v5 ;;-< © L"; ,3 \3 -? 13 3 ® >3 3 3 -.3 i> !3 — S' © S3 3 © N o 3 S' :: © © — 13 — — c; ?3 © ;5 O — vS a - v3 © 33 © © © £S C'l N :i N f 1 C-5 3“ ?3 :] C l c-3 S3 :o ;•; © ;o -? 'T- 13 13 13 © > n ©: ©3 S'

i I I i I I I I I i I ! I I I I j i ; I ; I ! I I I I I 1

+

u

— S3 S3 S3 C-3 S3 S-3 S3 S3 S3 S3 S3 S3 S3 S3 S3 S3 S3 S3 S3 S3 S3 S3 CS CS C3 CS SS G®©C3CG©0GG©©0©©03©S©©G00©0©0©CG0©©00

^ - VSP- 'fc.' .•« t'' V4/ •. J J v' ^ w' ’V^ C3 S' S3 <• S3 S3 13 N S3 c; N S3 S3 G S3 N — N C-3 s: N G - 13 3'> G L3 N N O N V ;-3 S3 O S3 N S3 S3 O S3 N S3 O O - 13 G S3 G S3 S3 S3 - N > - O S3 - 13 '3 S3 i'3 ©- - G N > N G - S3 '•• G Cl s: S3 S3 V 13 13 O N S3 G S3 S

G G O .- 13 S3 S3 S3 S3 S3 'C- =V N - G G 13 - N S3 G N 13 O - S3 r) G

S3 S3 <• 13 13 'i G N C3 S3 G

CN

^:3

§g-.- + G S3 id

iiG G

G G G G G G

-i- -i- T - S3 N” V S3 V

13 i'. tv.

-.3 G

13 SC. G

G ^3 —

13 13 13

G G G G G 3 G O G G C

*r -r ---r ^ V S-* S3 13 G

G S3 SI S3 G

G - S3 N S3 r; — S3 13 13 13 !' ssicsir

G G

G G G O GGGOGGGGGGOOGG OGGGGGGGGGOGGGGGG -r -r *'" •>* " -r -r *i- -> -f -r -r -r -i- -> i- vr •> S3 S3 — =N S3 N S3 — 13 S3 SI S3 S3 S3 >• — G S3 — S3 T -r G V

S3 IS J-- G S3 — <' Cl G " O c; \? S3 G S3 S3 13 G G .'.i S3 G S3 G G O S3 s'' -C‘ S3 S3 S3 S' S3 S3 G C3 S3 'S S3 iv S3 © 13 S3 > Si' G G G S3 G — 13 G S' Q V G G S3 L3 13 S3 S3 G G -< S3 G S3 G 13 S3 pi

G O G G G S3 L3 13 S3 S3 G G -< S3 G S3 G 13 G G 13 S3 C3 O G 13 C3 G S3 G V S 3 ?'> G S3 LS G G S3 G

13 S3 © S' S3 G G — S3 G G S3 S3 S3 S3 V O S' <?' S3 G N

tS G V' — G i-" 13 >1’ S3 G S' S- 10 <3' S3 — O S-3 N G << S3 — — — O O G G ©3 o S' G G -C.' G G G :3 c-S S3 S3 S3 O O O 3 O O G O C O O G

I I I I I I I I I I I I I I I I I I I ; I I I I ! i I I I ! I I I I 1 I

G O © O G

G O

o o OOOOOGOOOOCGOGOOCGOOG OOOOGOGOOGOOOOOCOOGGO -f S S -f + S -r T -; -r -r S -r S -i- -r -!- -r -r -r + -^ S -

1- S + -r -!- -f -> S -> -I- N Ct S'. 1-3 G S3 G G S3 — O L-3 S3 S3 S3 G D" S3 — S3 -e C G O O G O V S3 O - S3 G N S3 N

S3 V S3 V? S3 S3 G S3 N S3 'O N 13 10 V G S3 G '5 'G 10 N S3 G N G SJ 10 13 - '0 V S3 G - - 'G - - - - - - - - N S3 10 N s.. - i> N. 03 G 10 G G — S3 10 - 13 G G N - N S3 N — S3 O N G S3 '5 S3

S3 S3 S3 S3 S3 Cl S3 88 S 835 s 8 S 8 S 8838 S§S 388 S 8 §Sg 8 S 88888888 S

c '3 o - 6 N - c G — 13 'S - N 13 S3 b Cl o c'c 10 fo © :•; b n o cb ib ~ b 6 g c

'3 < V G G :. G G 10 V c:

:• c 8 13 v o s' e> 10

g o - o siN g g ©* ~ ci S3 o s> 10 si CD C' C N N i-. << -c- O - G G S' G C3 V S3 C' G '<5 C3 t" S' S3 S' '? S' O Iv S' SI G N SI 50 — 10 S’ I» V 10 - G 10 G S3 G N S3 — C3 S' V G G S' 1‘ 13 SI '0 CD O G SI 10 G C3 S- C3 G 10 S3 CD ©- C3 G vl C; C' i. CC G CD N N N ©' -0 SI SI G 13 O 13 CD G 10 G CD O O S' CD O -

Ci l: N C; :> .. '> — G i3 i; ;; c,-; O lO G il il v S' i' - CD S' cl C3 li - G — 1^, o' ./ :.} C 51 10 C . G G 10 O CD O C G C S3 CD G SJ Si S3 lO CD C3 S' 13 C3 - - ~ CD CD C S' C5 .'3 G ?- G

N G .1 1* S' C3 c'. c; g g - cc o i- g - cd 13 c: o ci g g si o si .0 vo cD si g .© g ~ C S' G s:. i'. 'G G 10 10 G G 13 C3 CD '1 Cl Cl “ ~ - O O C C CD O G Cl - G G G LO

f.j “ C- S'. S' S' G CD S. CD CD

I I I ; I I I I I I I I I I I I I I I I I I I ! I I I I I i I I I

i- :: G G — CI r: .0 g c- sd g g - si sc g i-- vo i. so g o - ci © o' -.o g n sD g c - si c;

r. CO C3 V G V G G G 'l- 10 10 13 IT 13 10 10 13 lO 10 '3 G G G G G G G '0 'D ;• O- i'*

Table

22.

(Continued)
131

+

»-4

0 O O S O O O O O O O O ^ O O O O O O O O -t=f o o

r “=“ i* -T* > 4- -1- — -.-- •- — -- -- -|- .- ~r -,• *j' *-- !- C-: o c r: C-: :ri n — o :-j c-: v cc •r^^ z-l > :4) n n o ^? n

S' - ~ T? S' Lf; -.? 5 N K N ;-3 N ^ - c-3 L-? L-:: = -r c; .=: s' -t? v :s

V? V S' c; LS '0 — v ® L? --^ N t7 © S' N t3 O sp 'C“ ’= N S S C; s- s

Li S' V -5 - -:•: V -s ei S' S' o - C5 -? ©; '> S' o n ? :s :3 •“ s: -- f:

'5 -,i S': - S; V T? N. s: s N S3 S3 G s; 3 .2 ;s c-' “ s S' r: v g V S' G S' ^ C5 G v5 CQ N -5 Sp <- N S3 G ri S3 '5 S' 3s C G S3 S3 S3 N

c: — V S' S' G SI V — ;; ® •:: ® v3 G v ss - v o ® ;; e: iS :: -? s~ v

— :; S3 SI c S' S N N 3> ©= S' o cf G S' C3 :? S' ss ?3 — G ® ® - -G s c G - S3 s: s: V sj ® n k s' - m s; •g g n s' g c: 'J" g cc s' 33 ^ g c3 ^ ^ ^ ;'7 ;-.j vl ,;.] ;; v?. t? <:*

1 I I I I I I I I I I I I I I I I !! I I I I I ! I I I I

m

+

!_I

c: c; G G G C3 ' G G G G G G

+ + + + + -!- N — O G G G G S3 N G ?3 :'l

•<? S3 LO S3 N C-3 G S3 53 G N vS N S3 V G N -3 G N G G e N

C N G N — S3 G G ?3 G C3 G

G G 33 G

CGGGGOGGGGGGGGGGG GGGGGGGGGGGGGGGGG

-rT--r'rS'T*r"r*r"r“T"T‘r"3 -r~r^

G 3^ G G G G G L3 L-3 G '5 V V G S' G V

V- G G f-3 G N S3 G G G G G © G 13 G G

S' f3 N © — S3 G v3 ?3 ~ S3 — ^s r-3 v v’ V S3 G G G 3 i-- G G — ^ fj p? L3 G L-3 :M ^ S3 S3 G G S3 G i> V G V N S3 G S3 G S3 G LS <- G S3 S3 G G V G G i'- G G S3 ' N

S: — © G G N G ss V G S3 - G G G S3 N G S3 G 3-' 'v. N>i N. Pv, 's. .5 v5 \5 GGGGOGGGGOGGGGGOG

O

•r *r

G iS3 G S3 G 'G G G

G G

G C

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SI: G N.

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G S3 G — -p," i> V 33 S3

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I I I I I I I I ! I I I I I I I I I I I I I I I I I I

:§i^- C G

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-r -r -:- LG G S3 G LS N N LG G G S3 S3 G G C3 N V S3 G N S' ~

- '3 G G G G S3 G S3 N S3 —

— G > G P? 53 — G X.

<- G G S3 G 5-3 G S3 S3 S3 S3 S3 - -

S3 S3 S3 S3 S3 S3 S3 S3 S3 S3 S3 G G C G G G G G G © G OCOOOG5POGGGS3 S3 S3 S3 S3 S3 S3 S3 S3 S3 S3 S3 S3 S3 S3 S3 S' GOGGCGGsGOG' ‘ ~ “ OOOOOOOOG 53

<1 i i j >1 j . ; 8888 i 8

3 G G

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T3

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oo

CM

CM

^ LS G N t; G G = S3 <- l-: G 5? G G

N N N N N N cS 53 S G G G e; SS :3 G

- S3 G -f? LS G r- G G G — S3

t' S-- (i' S' S- C' S' t'' G O O

JZ

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c o - M c: V N

588SS888S3

:: o z2 -r: 6 c i c

L-^ C ^ ''j -•^

S O C; ^3 b Q b b :'j v3 c i' v3 -3 N :; C'' ; : S' o

O V L3 S' S' S S'l V* fJ V — S' V T? G L3 <-> S'] '3 V - - .=•: J] ;'] - S' -s c]

"• '> S' '> “• S]

Ls S' •=: '7-‘ 0 0=>”-

; : . : , I ! G V - C" N G G O S' G 'V O] V

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S3 CD S' O ~ Cv S] G -T] N '3 C-3 G 0] N C3 O O C ? V S3 CS ^ S> G G " G G ;] = N -

S3 > S' S'l V G :3 ’ G S3 C4 G '3 G N- N N ::. CD CD CJ

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G ^ : o G ~ C3 G N G G — S] si iv CD G G S' CD c: C: c: S' S' S3 G N G C3 C? \P -

S3 N — G S3 G G S' :3 Cj C- V G G N G S' C'3 S3 C3 G V' S3 S'] £5 'S' SI -= G G “> CD

G S3 S3 N

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I I I I

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C3 CD G G G '3

frj CD C3 CD CJ CJ

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S' CD N I''* 3'' i''» iiiiii

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t-? - V « V ^

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S3 S' -GDI - S C3 C3 S' S' S' N C-3

01 S' Op G - CO V.-. CD G 03 p S3 13 l3 S3 'G 'G 'G i'p >,'* ip

I I I I ; : i I I I S' *- c “ G •- S3 u3 S' . p S' CO S3 S' '3 G ~ 03 CD

S3 S3 O CO CO CO O O'] ~ CO '•' S3 o S' S' : ' - o S3 CO C3 '1 o CO CO C3 jp ~ •;•] Ip s- oi b f'i j.i b \ o CO Jp ' J

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— 01 G S' S3 G !p O O O O O G O

r

n

Table

23.

Computation

ofJ

(1)

from

the

JYPQ

equation

with

p=0,

x

=1,

x

=0.5,

i=0,

m=100
133

unc

0 0 — 00

o o o o o O O Q O O

*!* I 'f' O O C

o o o 1; rj

Si© f|i G

G

© ©

G G s ; ©© iiiSS © G

9 ©

©

o

©

G iSS i I i -i“ 01 G 01 N. © ©

! ***1 S3 -G

I N

1 N ©: ; i i I 03 ^ Cl 01 SO

: i N C'

i ! •GI 1 I S3 .0 01 G

© © N © V- 01 Cl G — C'l — '=1' S3 V SO — © © ©

01 N Cl S3 N

so N N © ’'o G S3 O — — S3 © © N © 01 03 S3 © S3 © O

S3 S3 SO

S3 S3 V

© 03 © — © © £3 S3 '=: Cl <- © Cl S' O' 1^ S3 © © > O' N. -> S3 -3 O'

*

C3 '3 G C' S3 — © ©

S3 © Cl cQ -e © © © 0- SC © S3 L3 -:-3 01 CO si CO 0- S3 N -C S3 CO ©

G O'* S3 S3 01 S3 V N 'C SO Cl 7© 01 — © S3 © — G S3 S3 G O

© © V G '5 01 '1? S3 •e © G '0 G 01 '5' 01 O S3 © N G Cl ©

« G S3 S3 V © O 01 3? f*-J 03 © L3 c-1 Cl 'O' - 01 © 1—

t

- © - S

0- C l - V — © — N C'l S3 G Cl S3 'G Cl C3 G 01 G Cl S3 © Cl ©

I I i I I I I I ! ! I I I

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n L-? ca o « ©GO — - ©GOO© I I I i I S' c: C3 c-i o

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1 I I I : I

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C^ © © C-5 CC © O © — N

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<?* 01 N ©

© 03 N 01 01

N- © V O S3 O © — '? ©

N — C O —

Tf C'l — V Cl

01 01 O Cl « N

L, ^ O G v3 O 01 Cl © 01 © G © O © O - Cl V5 O © V1;; © © G N 01 :c o © N cc ©

G O S3 -3 O Cl © O © Cl " ' - S3 © - 01 O Cl N r^->. ^-'1 o © ;.3 Cl CO © o N N o ©

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24.

Computation

ofP

(0.5)

from

the

JYPQ

equation

witli

p=0,

x^=l,

x

=0.5,

1=0,

m=100
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25.

Computation

ofQ

(0.5)

from

the

JYPQ

equation

with

p=0,

x

=1,

x„=0.5,

1=0,

m=100
L35

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Example 4.3.2 In this example we compute the solutions P^(0.5) and

Q^(0.5) of the homogeneous equation defined in Figure 7, with la = 1 ,

x^ = 1 , = 0.5 , for r = 0 , 1 , 2 ,

. . .

, 100 . The algorithm will be stable

provided that we take three initial values for each of these functions;

compare (4.3.3). Tables 27 and 28 give the results of these computations

for selected values of r ; see §4.0 for a full description of these

tables. Initial values were obtained by hand calculation of the explicit

elementary forms given in Tables 8.2 and 8.4 of [16]. The computed function

values agreed satisfactorily for r 5 10 with the limited data available

in [16] and [17]

.

In addition, satisfactory agreement was obtained with

double-precision values produced by an unpublished program of Dr. J. M.

Smith; compare Example 4.3.1. Consequently, the stability of the method

is confirmed for this example.
137

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Table

27.

Computation

ofP

(0.5)

from

the

JYPQ

equation

with

p=l,

x

=1,

x^=0.5,

i=0,

m=100
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28.

Computation

of

Q^(0.5)

from

the

JYPQ

equation

with
139

4 . 4 Examples Involving the Inhomogeneous JYPQ Equation

Using Stirling's formula, we find

(4.4.1) r(r+p+l)

r(r+3/2)

U-1/2

for fixed values of u = 0,1,2,... . Comparison of (4.2.1) with (4.3.1),

(4.3.2) and (4.4.1) shows that

(4.4.2) Er(xi) < ^ Q^(x2)

for fixed [X , x^ and x^ such that x^ > 0 and ^9 < ^ • Accordingly,

the Anger-Weber function is a solution of type 1 and asymptotically the

correct number of initial values to be used in every case is j = 1 ;

compared (4.3.3). However, the separation of E^(x^) from the solutions

- 1/2

of type 2 is weak when u = 0 (being of relative order r at the odd

-3/2

terms of the sequence and r at the even terms of the sequence) , The

-3/2 separation is also weak when u- = 1 (being of relative order r at

-5/2

the odd terms of the sequence and r at the even terms of the sequence)

Accordingly, we treat E^(l) as a solution of type 2 in both cases, even

though when u, = 1 greater numerical instability is incurred than in any

of the other examples. This is especially true at the even terms of the

sequence, which are separated from the Legendre functions by the relative

a P -5/2 order of r

From (4.2.4) and (4.3.3) it is seen that the type of the Struve

function H (x^) with respect to the JYPQ operator is 0 for all

admissible values of x^ and x^ . But again because of the weakness

of the separation, we regard H^(x^) as belonging to type 1 .
Example 4.4,1 For the inhomogeneous fourth-order difference equations

defined in Figure 7, we compute E^(l) with u = 1 , = 1 and x^ = 0.5 ,

for r = 0,1,2, ... ,100; and H^(O.l) with u. = 1 , x^ = 0.1 and x^ = 0.5

for r = 0,1,2, ... ,50. (We restrict the sequence of H^(O.l) to r S 50

for the same reasons that were described in Example 4.2.1.) For reasons

given above, we use 3 initial values (rather than 1) for E^(l) and 1

initial value (rather than 0) for H (0.1) . Initial values for E (1) r r and H^(O.l) were computed from series expansions given in [21] and

checked against values found in [21] and [23].

Tables 29 and 30 give extracts from the results of these computations

for selected values of r ; see §4.0 for a full description of these

tables. Comparison of the function values produced by the algorithm with

values given in [21] and [23] show good agreement over the available

ranges, which, however, do not extend beyond r = 13 . Comparison of the

values of E^(l) in Table 2.9 with values computed to 10 significant

figures from the series expansion shows a steady deterioration of precision

at the even values of r , culminating in a relative error of 0.5(10)

^

at r = 100 . At the odd values of r the precision also deteriorates

steadily, but at a slower rate, culminating in a relative error of

0.8(10) ^ at r = 99. These rates of loss of precision are consistent

-5/2 -3/2 with the separation ratios of r and r which govern the growth

of the error in this example.

-10

That is, an individual relative rounding error of, say, 10 in one

of the stored initial values of E (1) will be amplified by a factor of

S/75 3/23 approximately 100 = 10 at E _(1) and by 100 = 10 at E (1).

i uu y y (As noted in Eq. (4.2.1), the function is in behavior different at even and

odd values of r .) These amplifications after 100 steps are greater than

the factor of 100 that was predicted and observed in the computation of
141

A (1) in Example 4.1.3 and Table 13, where A (1) was computed as if

r r

it were a function of the next higher type than it actually is.

Finally, comparison of the values of H^(O.l) in Table 30 with

values computed to 10 significant figures from the series expansion

shows agreement to 9 or more significant figures at all values of r .

The loss of precision in the Struve function computation is slight, due

- 1/2

to the separation ratio of only r , but it offsets impossibly slow

convergence which would be evidenced if the theoretically correct number

of initial values were to be taken.
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I
CHAPTER 5. SUMMARY AND CONCLUSIONS

This thesis is concerned with the numerical solution of a broad class

of linear difference equations of arbitrary order, and of either homogeneous

or inhomogeneous form. An algorithm is presented for the stable computation of any solution of such an equation. The results apply directly to

scalar linear difference equations but could be adapted to similar problems

posed in vector form.

Historically, the algorithm is an outgrowth of algorithms of Miller

and Olver. Tne underlying idea of Miller's algorithm is the following:

If a solution y(r) has growth behavior, as r

-> ®, that is less than or

equal to the growth behavior of all independent solutions, then in the

reverse direction its growth behavior is greater than or equal to the

growth behavior of all independent solutions. Consequently, because of

the superposition principle, errors introduced in computing y(r) by

backward recurrence cannot grow faster than y(r) itself, and in this sense

the backward recurrence process is stable. However, this is true only in

a region in which the solutions actually maintain the growth behavior

indicated by their asymptotic forms as r For this reason we call

this the asymptotic region; evidently it depends on the asymptotic structure of the given difference equation for large r.

Miller's algorithm starts the backward recurrence process at a

value of r that is larger than any r for which a valid approximation

of y(r) is wanted. Arbitrary starting values are assumed. The computed
145

solution is essentially a scalar multiple of y(r) for early values of r

when (i) the difference equation is homogeneous; (ii) y(r) is subdominant

compared to every independent solution of the equation. Under these conditions the algorithm is stable and y(r) is obtained from the computed

solution for early values of r by a normalization procedure, such as

matching the value of the computed solution at r = 0 with y(0) in order

to determine the correct scale factor. Modified normalization procedures

are required when the asymptotic behaviors of more than one fundamental

solution are similar.

Miller's algorithm involves no forward recurrence, and is stable

only for solutions whose growth behavior is less than or equal to the

growth behavior of all independent solutions. Giver's algorithm was

applied originally to second-order inhomogeneous difference equations,

in cases where a particular solution y(r) has growth behavior as r

that lies between that of a pair of linearly/ independent solutions of

the homogeneous equation. To construct a stable algor ithm, Giver posed

a boundary value problem having one condition at each end. The initial

condition is the value of y(r) at a suitable point r = i. The terminal

condition is zero, it being anticipated that isolated values of y(r)

for large r will not be easily obtained. The resulting boundary value

problem is a tri-diagonal linear system of algebraic equations. Giver

specified a method for solving this linear system that involves a

forward recurrence stage and a bacfavard recurrence stage. The method

is equivalent to Gaussian elimination without pivoting, scaled in a
particular way. Analysis of the method shows that (i) the forward and

backward recurrence stages are stable, at least in ranges beginning at

a sufficiently large value of r; (ii) the second-order linear difference

operator that corresponds to the difference equation is, in effect,

factored into the product of two first-order operators. Furthermore

Olver derived an infinite series expansion of the truncation error, valid

at any point r, that is incurred by choosing a particular terminal

point for the boundary value problem. This enabled the optimal terminal

point to be calculated automatically, in contrast to Miller's algorithm

in which the terminal point is guessed and subsequently tested.

Extensions of Giver's algorithm to difference equations of higher

order were proposed by Zahar, Oliver and Cash. Criteria for selecting

the correct nimnber of initial conditions (and therefore the correct

number of terminal conditions, since the total number of conditions

must equal the order of the difference equation) were given independently

by Zahar and Oliver. Zahar proved convergence of the algorithm under

assumed conditions on the solutions of the adjoint equation. Oliver

investigated stability by analyzing the factorization of the finite

boundary value problem. Cash provided an extension of the series

expansion of the truncation error. The work of Zahar, Oliver and Cash,

taken together, contains the essential elements of a practicable

generalization of Giver's algorithm.

In this thesis Giver's algorithm has been extended and analyzed

afresh. The view was taken that a linear difference operator is an
147

infinite upper-triangular band matrix. For a specified initial point i

and number of initial conditions j, an (i, j )-factorization of a linear

difference operator D is said to exist if Gaussian elimination without

pivoting about the j-th super-diagonal of D, starting at the point i

on this diagonal, can be carried out indefinitely. This process produces

two lower-order difference operators (infinite band matrices) . Their

product is, of course, the original operator, and the sum of their

orders is the order of the original operator. Furthermore, the solution

of any finite boundary-value problem having initial point i, number of

initial points j and terminal point n may be found by forward recursion

with one of these lower-order operators followed backward recursion with

the other (Theorem 1.2.2). Procedurally, a finite boundary value problem

is solved by the algebraic method of forward elimination without pivoting

followed by back substitution; this is accompanied by an analysis of

the stability of this process in terms of linear difference operators.

Determination of the correct number of initial conditions for

stability requires a knowledge of the relative rates of asymptotic

growth of solutions of the homogeneous equation as r ^ . Since the

number of linearly independent homogeneous solutions is equal to the

order il of the linear difference operator, there cannot be more than

I distinct growth rates as r Cases in which there are exactly

I distinct growth rates are the easiest to treat. In these cases a

basis exists that is linearly ordered by growth rate, and we call the

operator totally separable. However, there exist operators of practical
interest, including constant-coefficient operators for which one or more

of the distinct characteristic roots are equal in magnitude, that are

not totally separable. Accordingly, a more general class of operators,

called separable operators, is introduced (Definition 2.2.1). The solutions of a homogeneous equation that has a separable, but not totally

separable, operator exhibit less than Z distinct rates of growth as r ^ “

Separable operators are characterized by the existence of certain bases,

called optimally ranked. An optimally ranked basis includes a ^ I

disjoint subsets such that (i) neither of two distinct solutions in

the same subset dominates the other as r (ii) any two solutions

taken from distinct subsets are such that one dominates the other;

(iii) no nonzero linear combination of solutions from one subset is

dominated by any solution in the subset. These conditions ensure

that separable operators are unambiguously defined in the sense that

any two distinct optimally ranked bases have similar structure

(Lemma 2.2.1)

.

The disjoint subsets of an optimally ranked basis are arranged

in linear order by increasing rate of growth. All possible solutions

of an inhomogeneous difference equation with arbitrary right side are

classifiable by comparison with the solutions in an optimally ranked

basis of the corresponding homogeneous form. Thus a particular

solution y(r) is said to be of type s if (i) y(r) is not dominated as

r -r CO by any solution in the first s subsets of an optimally ranked

basis; (ii) y(r) is dominated by every solution in the remaining

a-s subsets. The extreme cases s=a and s=0 correspond to particular
149

solutions for which forward recurrence and backward recurrence of the

given difference equation are stable, respectively. For the intermediate cases 0 < s < cr, let j be the number of solutions in the first

s subsets. If in addition to being separable the difference operator is

(i, j

) -factorizable for some initial point i, analysis of the boundary

value problem shows that the equivalent pair of forward and backward

recurrence problems (Theorem 1.2.2) are stable in the sense that rounding

errors are not propagated more rapidly than the solutions, at least in

the asymptotic region belonging to the given problem. Accordingly, the

solution y(r) of the boundary value problem is obtained in a stable

manner by forward elimination without pivoting followed by back substitution (§3.3 and Theorem 3.3.1). Furthermore, under appropriate

conditions it follows that (i) y(r) is uniquely determined by j initial

values (Theorem 3.1.1); (ii) the algorithm converges pointwise as the

terminal point proceeds to infinity (Theorem 3.2.1).

The numerical examples included in this thesis were chosen so

that known asymptotic results could be used in conjunction with the

new classification theory for separable operators. This enabled the

correct number of initial conditions for any chosen solution to be

ascertained immediately. The examples also show that it is sometimes

desirable to associate a solution with solutions of the next higher

theoretical type. This artificial raising of the type is manifested in

the algorithm by the specification of the number of initial conditions

appropriate for the higher type. This artifice preserves unique determination
of the solution and also convergence of the algorithm, but introduces

instability. It is employed only when the asymptotic separation between

two t>n3es is weak, consequently the loss of precision resulting from the

instability is mild. The compensating advantage is that the rate of convergence is increased very substantially, enabling a much smaller terminal

point to be used.

The trade-off between numerical instability and slow convergence can

be regarded heuristically in the following way. First, suppose u(r) ~ e

and v(r) e are two fundamental solutions of a second-order equation.

Then one initial value of u(r) determines the function uniquely and the

algorithm is both convergent and stable. Furthermore, the rate of convergence

is very rapid. This can be seen from the equation

-n

u (r) = u(r) - -— • v(r){l+o(l)} , n , n n

e

where solution of the approximating boundary value problem

having n as its terminal point (compare Eq . 3.2.6). The rapid convergence

of as n ^ for fixed r is readily apparent. Thus the

algorithm performs very well with regard to both numerical stability and

rapidity of convergence when the separation of solutions is strong as in

this example of exponential separation.

-1 - 1/2

Next, suppose that u(r) r and v(r) ~ r . Here again, one

initial value of u(r) determines the function uniquely and the algorithm

is both convergent and stable. But in this case

u (r

)

n

-1

u(r) - - 1/2

• v(r){l+o(l)} , n -f 00
151

and it is apparent that the rate of convergence will be governed by the very

- 1/2 slowly decreasing function n . Suppose, however, that we give an

additional initial value of u(r) . With two initial values our algorithm

is equivalent to forward recurrence of the given difference equation and

all questions of convergence disappear. This forward recurrence is, of

course, unstable for the desired solution u(r) but the instability is

very weak; any small component of the more dominant solution v(r) can

1/2

grow no faster than r relative to u(r) . Although this rate of growth

(or any other positive rate of growth) is ultimately sufficient to obliterate

the desired solution when computed in finite precision arithmetic, subsequences of the solution of considerable length can be computed before this

happens. Thus in many practical situations where the separation of solutions

T IS weak, a minor modification of the algorithm performs acceptably well

with regard to both stability and convergence.

Finally, suppose that the relative separation of u(r) and v(r) is

-3/2 -5/2

like r or r (compare Example 4.4.1 which presents the computation

of a function exhibiting both forms of behavior, r -3/2

on the subsequence

-5/2

of odd terms and r on the subsequence of even terms). These separation ratios are too weak to allow rapid convergence of the unmodified

algorithm, and at the same time too strong for the numerical instability in

the modified algorithm to be truly insignificant. Computation of cases such

as this are the most difficult for the algorithm to perform well, and performance can be improved only by recourse to extended-precision computation.

In general the modified algorithm will require a back substitution as well

as a forward elimination stage. Nevertheless, similar considerations govern

the trade-off between numerical instability and slow convergence.
Perhaps it: should be stressed that a complete knowledge of the asymptotic behavior of all of the solutions of the homogeneous equation is

essential to determine the correct number of initial conditions. This information may be developed for a broad class of linear difference equations

using known analytical results (§2.2). An alternative would be to attempt

to construct the essential classification parameters - the number of distinct types of solution, and the number of linearly independent solutions

of each type - by numerical experimentation. Under this approach, the

algorithm presented in this thesis would be applied with chosen initial conditions appropriate to different possible realizations of the classification theory, that is, different possible forms of the optimally ranked basis.

Presumably, incorrect choices would be revealed by discernible numerical

instability. The effectiveness of such an approach has not be assessed,

however

In conclusion, the results presented in this thesis are distinguished

from the work of earlier authors in the following ways. First, the interplav between linear difference operators and infinite band matrices achieves

a fuller blending of analytic and algebraic ideas. This leads to a clearer

understanding of stability, including the tradeoff choices that exist

between slow convergence and mild instability when the asymptotic separation of adjacent solutions is weak. Secondly, the classification theory

for separable operators is believed to be new. It enables the proper

number of initial conditions that are prescribed by the convergence

theorem to be determined directly from the asymptotic behavior of fundamental solutions of the given difference equation. In contrast, Zahar employs

fundamental solutions of the adjoint equation and Oliver applies 3
153

point-by-point growth criterion between pairs of solutions. Finally, the

numerical examples presented here have been chosen in such a way that the

solutions exhibit a much richer variety of growth rates than in any previous investigation.
APPENDIX

Proof of Lemma 2.1.1

The proof consists of ( i) identifying the sequence defined by

equation (2.1.4) as an almost periodic sequence, and (ii) proving that

evem,’ ncr-zero almost periodic sequence has an infinite subsequence which is

bounded away from zero. L'e begin by presenting the essential elements of

the theoiq/ of almost periodic functions.

First we give the characterization of almost periodic functions that

is taken as the definition by Corduneanu [7 ]. A function of the form

T(x) = I c e , - » < X < “ , k=l

where the c, are complex numbers, the \ are real numbers and

.< k,

i = V-1 , is called a complex trigonometric polynomial . A function f (x)

taking complex values and having all real numbers as domain is an almost

periodic function provided that to each s > 0 there corresponds some

trigonometric polynomial T_ (x) such that

if(x) - T^(x) 1 < e

uniformly on - » < x < » . Clearly, every complex trigonometric polynomial is itself an almost periodic function.

An alternative characterization is the definition of H. Bohr [4 ]:

A continuous complex valued function f(x) defined on the real line is

almost periodic if to each z > 0 there corresponds a positive real

number ^ such that every open interval of the real line of length

contains at least one point ^ for which

I f (x+O - f (x) I < e
155

uniformly for every x . Any number ^ which satisfies this condition

is called a translation number of f corresponding to s , or more briefly, an 5 -translation number of ^ ,

An t-translation number of f is "almost" a period of f . Every

periodic function is almost periodic as well. If p is the fundamental

period of a periodic function f , then rp for r = 0,+l,+2,... are

translation numbers of f for each s . Bohr's definition generalizes

the fact that the periods of a periodic function form an arithmetic

progression.

Bohr's definition and Corduneanu's definition define exactly the

same class of functions. This is the "fundamental theorem" of the theory

of almost periodic functions. See either [4 ] or [7 ] for a full account.

Turning now to sequences, we have the following definition presented

in [ 7 ,§1.6]: A function f(t) , taking complex values and having the set

of integers as domain, is an almost period sequence provided that to

each e > 0 there corresponds some positive integer N = N(s) such that,

within any set of N consecutive integers, there exists an

integer p for which

, f (r+p) - f (r) ! < £ .

uniformly over all integers r . Corduneanu proves "A necessary and

sufficient condition for a sequence almost periodic is the

existence of an almost periodic function f(x) , - =» < x < “= , such that

= f(r) , r = 0 , j;^l ,+2, . . . " ; see [7, Theorem 1.27] .

VTe are now ready to prove Lemma 2.1.1. If r were allowed to assume

all real values, instead of being restricted to r = 0,1,2,..., then

equation (2.1.4) defines a trigcnometric polynomial. By Corduneanu's

definition, every trigonometric polynomial is an almost periodic function.
By the theorem quoted above, the sequence x(r) , r = 0,+l,+2,...

defined by equation (2.1.4) is an almost periodic sequence. The only

thing remaining is to prove the following assertion:

If an almost periodic sequence {f(r)} is not identically zero,

then there exists an infinite subsequence of {f(^)} which is bounded

away from zero.

Proof Assume {f(r)} is not identically zero. Let be such that

fCr^) ^ 0 and put A = |f(rQ)| . Put e = A/(2+A) . Then e $ (0,1) ,

and by the definition of an almost periodic sequence there exists a

positive integer , say, such that for some integer p^ c {1,2,...,N^}

the inequality

f (^o+Pi)

- f (^q) I < e

is satisfied. Put r^ = • Next, there exists a positive integer

such that for some integer p^ - { 1 , 2 ,...,N

2

} the inequality

f (ti+Pi) - f (t^) I < £

is satisfied. Put ^^2 ~ ^1 ^2 ' Continuing in this manner, we see that

for each k = 3,4,5,... we can find an integer ^ ]_ such that

1,2,3,... we have

f(r^) - f(r^j)| S |f(r.) -

r j £ 1 . < 4

& < irr < T

^

j=i

For each k =
Since A = ! f (i^g) I . we

!f(r.

have

^)| = |f(rg) + - f(rg)|

2 A - |f(r^) - f(rg)| >

which finishes the proof of the assertion • m

N)

IH"
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U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBSIR 80-1976

Accessioa No. 5. Rociptent's Accession No.

4. TITLE AND SUBTITLE

Numerical Solution of Linear Difference Equations

5. Publication Date MARCH, 1980

$.^,BsrfofTOBg;Ofganfzati Code

7. AUTHOR(S) 8. Performing Organ. Report No. Daniel W. Lozier

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

lit Pro|ect/TjB^^p)* wltN >

11. Contract/Grant No.

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS rs free f. City, State. ZIP)

Same as Item 9

13. Type of Report & Period Covered

Final

Ag^cy Co^ ^

15. SUPPLEMENTARY NOTES

I

! Document describes a computer program; SF-185, FIPS' Software Summary, is attached.

16. ABSTRACT (A 200-ivord or less (actual summary ot most significant information. If document includes a significant bibliography or

literature survey, mention it here.) ^ Consider a given linear difference equation d^(r)y(r+s) = g(r) where S, ^ 2 and

r = 0,1,2,... . Suppose y is a solution of this equation and u,v are solutions of the

homogeneous form of this equation such that u(r)/v(r) -> 0, y(r)/v(r) -> 0, u(r)/y(r) -> 0.

Under these circumstances algorithms for the computation of y based on forward or backward recurrence, such as the Miller algorithm, are numerically unstable.

Stable algorithms, such as the method of Olver, have been based on approximating

y(r) by the solutions of a certain sequence of boundary value problems. Specifically,

y (r) is a solution that coincides with y(r) over some initial range of r, say r=i,i+l,

...,i+j-l, and satisfies y (r) = 0 for r=n,n+l,..., n+H-j-1. Here j is an integer whose

value depends on the asymptotic behavior of the chosen solution y(r) and n is an arbitrary large integer. Boundary value problems of this type are shown to be equivalent to

two initial value problems of order j and il-j by factorization of the difference operator. The solution of the problem of order j is obtained by forward recurrence; the solution of the other problem is obtained by backward recurrence.

The algorithm is specified completely for a broad class of operators including, for

example, every constant-coefficient operator. Convergence of y (r) to y(r) as n " for

fixed r is proved and an expansion of the truncation error is derived. Numerical stability is demonstrated. The method is tested by numerical examples involving fourthorder equations with variable coefficients.

17. KEY WORDS Csix to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name;

separated by semicolons)

Factorization methods; linear difference equations; linear recurrence relations;

Miller's algorithm; Giver's algorithm; numerical stability.

18. AVAILABILITY Unlimited

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UNCLASSIFIED

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21. NO. OF

PRINTED PAGES

166

22. Price

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