Estimating and determining solutions of matrix vector recursions

Estimating and determining solutions of matrix vector

recursions

Citation for published version (APA):

Mattheij, R. M. M. (1977). Estimating and determining solutions of matrix vector recursions. Rijksuniversiteit

Utrecht.

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Published: 15/06/1977

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ESTIMATING AND DETERMINING SOLUTIONS

,•

OF MATRIX VECTOR RECURSIONS

determining soZutions of matrix vector recursions" door R.M.M.Mattheij.

I

Inzicht in het groeigedrag van oplossingen van recursies kan een nuttig hulpmiddel zijn om de stabiliteit van numerieke processen (zo deze zich recursief laten formuleren) te onder-zoeken.

II

De bewering van F.W.J.Olver dat een afrondfoutenanalyse voor zijn methode volgt uit hoofdstuk 4 van Wilkinson's "Algebraic Eigenvalue Problem" is aanvechtbaar.

Litt. F.W.J.Olver: Numerical soZution of second-order Zinear

difference equations, J.Res.N.B.S. 71 B (1967),pp.111-129.

III

In §2E van onderstaand boek wordt ten onrechte de stabiliteit van een tweepunts randwaarde probleem in verband gebracht met de perturbatiegevoeligheid van de oplossing t.a.v. de begin-waarde.

Litt. S.M.Roberts, J.S.Shipman: Two point boundary vaZue

probZems: Shooting methods, American Elsevier, New York (1972).

IV

Laat A een symmetrische matrix zijn als volgt

A B I 1 I I _I - - -I - - T - -I I B_{2 1} I

-+--'--I I B3 l?l

waarbij de Bi mde orde matrices zijn, zodanig dat voor i ~.2, Bi

=

ai B₁ + Si I Cai, Si E JR ) , Neem bovendien aan dat de eigenwaarden van de Bi in modulus ~ 2 zijn. Dan is A

non-singulier. Als men de operaties benodigd voor het diagonaliseren van B

1 niet meerekent, kan het stelsel Ax = b (voor zekere _{2 ..} vector b) opgelost worden ten koste van ongeveer m N vermenig-vuldigingen.

Laat A een tridiagonale matrix zijn van orde n. Laat D 0 een diagonaal matrix zijn die de diagonaal elementen van A bevat. Definieer B

0

=

A - D0. Laat alle elementen van D0 in

modulus

~

1 zijn en die van B₀ in modulus

~

t·

Dan kan men een rij {D.} ·;;;.1 van non-singuliere diagonaal-J ],,,,..

matrices en een rij {B.} ·;;;.1

J ],.- van matrices met diagonaal-element en gelijk aan nul, definieren door

-1

D. + B.

=

D.

1 - B. 1D. 1B. 1.

J J J- J- J-

J-Zij k het kleinste natuurlijke getal zodanig dat 2k+ 1

>

n. Dan is B.

=

O

J

lijk bestaat)

voor j ~ k. Voor de inverse van A (die

kenne-A-1 = D-1(I k geldt dan -1 - Bk-1Dk-1) VI

L aat A een n de or e matrix ZlJn. d . L aat µ een inwendig punt . . van het waardenveld van A zijn. Dan zijn er n onafhankelijke vectoren met Rayleigh quotient µ. Dit betekent dat A congruent is met een matrix B die als diagonaalelementen µ heeft.

VII

Laat A een normale nde orde matrix zijn met eigenwaarden A₁, . . . ,An. Definieer ~

matrix Q zodanig dat de

1 n

- n

L AJ.. Dan is er een unitaire j=1

diagonaalelementen van QHAQ alle

~

zijn. Het punt

s

is het enige punt uit het waardenveld van A met deze eigenschap.

VIII

De nauwe verwantschap die er blijkt te bestaan tussen wiskunde en theologie qua curriculum profiel, kan geen rechtvaardiging zijn voor onderstaande formulering:

( . . . ) "The inner product property extends to the spaces

W,

and also (thank God) to the strain-energies in linear elasticity and other applications."

Litt. G.Strang

&

G.J.Fix: An anaZysis of the finite eZement method, Prentice-Hall, Englewood Cliffs (1973), p.299.

Het verdient aanbeveling de verbetering van snelheidsrecords in de sport alleen dan te erkennen, als het een redelijke ver-betering in relatieve zin is.

x

Het gebruik van het Seys en Alcyone verhaal in Chaucer's "Book of the Duchess" kan als onvolledig plagiaat gekwalifi-ceerd worden.

XI

Het is niet terecht het werk van de jonge Mozart uitsluitend te beoordelen naar muzikale volwassenheid.

OF MATRIX VECTOR RECURSIONS

PROEFSCHRIFT

ter verkrijging van de graad van doctor in

de Wiskunde en

Natuurwetenschappen aan de

Rijksuniversiteit te Utrecht, op gezag van

de Rector Magnificus Prof .dr. A. Verhoeff,

volgens besluit vanhet College van Decanen

in het openbaar te

verdedigen op woensdag

15 juni 1977

des namiddags te 4.15 uur

door

ROBERT MARTINUS MARIA MATTHEIJ

geboren op 11 augustus 1947 te Sittard

I am indebted to all who have contributed to the completion of this thesis. I am particularly grateful to prof.A.van der Sluis for his support and wise counsel; I shall always remember with pleasure the friendly atmosphere - not only at the Mathematical Institute in which we worked. I owe a special debt to Mrs.Th.Breughel, who has most painstakingly prepared the typescript.

~

CHAPTER I 1

1 Introduction 2

2 Preliminary definitions and auxiliary properties 5 3 Outline of ideas; the reference case (RC) 13 4 Relating an inhomogeneous MVR to a homogeneous MVR 22 CHAPTER II 5 6 7 8 9 On the solutions of MVRs Triangularizing MVRs

Solution spaces; convergence Dominance and stability

Existence of k-dominant recursions and k-stable solutions; consistency 25 26 30 33 39 60 CHAPTER III 77

10 Directions and their recursions 78

11 Estimating directions 81

12 The directions of k-block UMVRs 89

13 Estimating solutions of UMVRs 102

14 Relating the normal form to Schur normal forms 107

CHAPTER IV 119

15 Finding a k-UMVR for stable approximation of a

solution 120

16 Error analysis of algorithm 8.21 134

17 Rounding errors in computed solutions 141

18 Computational aspects of algorithm 8.21 161

19 Scalar recursions 166

20 Finite boundary value problems 180

References 191

List of symbols 193

Index 197

Samenvatting 201

CHAPTER

§§1-4

This chapter is mainly introductory. In §1 we indicate the kind of problem that will be studied and give a brief account of several papers which are of interest for us. In §2 we summarize definitions and properties of a general kind. The framework of this thesis is sketched in §3.

In order to demonstrate the results we introduce a reference case for homogeneous recursions in §3 and in-homogeneous recursions in §4.

1. Introduction.

The history of recursions goes back to the times of Newton, when the differential calculus was being developed. After some years of diminished interest, Poincare, Perron and Birkhoff rediscovered the subject and extended the theory a great deal (for extensive lists of references one should consult [ 1,17] }. When fast computers came into use the study of recursions received new impulses, since many algorithms which are used in computer programs are recursive processes. The recursions which occur there may be divided into two classes. The first one springs from mathematical problems which are formulated in a recursive form themselves. Examples are the familiar three term scalar recursions for orthogonal functions, such as Bessel functions of the first and second kind. The second class consists of recursions, arising from mathematical problems, which have no recursive formulation. For this class one may think of a discretized different~al equation.

In the present investigation we shall always assume that a recursion has been given in an explicit form, irrespective of the problem i t originates from. This means that errors will only be related to solutions of this recursion. A sub-sequent analysis which relates the solutions of the recursion to those of the original problem, as may be necessary for the second class above, is outside the scope of this thesis.

We shall restrict ourself to situations where the re~ cursion is linear, i.e. we shall consider matrix vector recursions

( 1.1) i ;;;. 0,

where x

0,x1, ... - to be called a solution - and r0,r1, ... are vectors and A

0,A1, ... are matrices. The following aspects will be studied:

In the first place we shall examine the space of solutions satisfying (1.1), for given sets of {A.} and {r.}. We shall

J_ J_

show that in many situations the solutions can be classified according to their growth character. In order to estimate solutions of a certain class we shall develop a practicable technique for performing this estimation. This then can be regarded as the second aspect of our thesis. In the third place we shall consider stable methods for approximating solutions of a recursion. From a numerical point of view this is by no means a trivial problem. Although we have an eXJ?licit relation for the sequence {xi}, rounding errors

may

blur the result. We shall therefore formulate alternative ways for

approximating {xi}' which are believed to be new (cf.[ 9,p.125] ). A number of papers having some relation to our investigation are the following: A survey of problems and algorithms en-countered in three term scalar recursions is given by Gautschi [7]; i t mainly deals with the so-called minimal solution, i.e. a solution outgrown by any other solution, which is not a multiple of the former solution. In [ 19] Olver describes an algorithm for computing a solution of an inhomogeneous scalar recursion, which is outgrown by one type of solution of the homogeneous part on the one hand, and which is outgrowing an other type of such a solution on the other hand. In [ 18] Oliver gives an extension of Olver's method for n-th order scalar recursions. Unfortunately, the

algorithmic approach of the last two authors does not allow for a straightforward matrix vector generalization.

Finally, estimates for solutions of linear recursions can be found in [11,23,25], though the first paper only deals with second order recursions.

2. Preliminary definitions and auxiliary properties.

2.1. Space and mappings.

Let R denote either one of the linear spaces JRn and a:n. We shall always use complex notations such as y x H

the inner product of a vector x and a vector y since for

they induce the real ones. If no dimension is specified i t will always be assumed to be equal to n.

Let s₁,s₂ be finite dimensional linear spaces over the same field as R. For a linear mapping A: s1 ~ _{s2, define}

( 2. 1) ker(A) = {x E s1 Ax = 0}' the kernel of A, ( 2. 2) ran(A) = {y E s2 3_xES

1Y = Ax}'

the range of A. We shall always identify a linear mapping with its matrix.

Let x

1, . . . ,xk ER, then (2.3) span(x

1\ ... \xk) =the linear subspace of R spanned by

Let (2.4)(a) (2.4)(b) {A.} J q II j =p q L j =p

be a set of square matrices,

rq···Ap

'

if q ;;;;, p A. = J _{I i f q <p} { A + ••• +A A.

=

p q J 0 ' if q ;;;;, p if q

<

p 2.2. Eigenvalues. then

If A denotes a matrix, then we shall always suppose that the eigenvalues A

1, . . . ,An are ordered such that \ A j

I ;;;;, \

A _{j + 1}

I

for j

=

1 , ••• , n-1.

In the sequel we shall frequently encounter matrices Ai. The eigenvalues of Ai. will then be denoted by A.

1, . . . ,A. ,

l in

where \A·₁

J;;. ...

;;.\A·

I·

Definition 2.5. Let A be a matrix with eigenvalues A

1, ... ,An· Then the eigenvalues are said to be k-separated if

(therefore !Aj!

>

IA

1!, j=1, ... ,k; 1=k+1, ... ,n).

2.3. Partitioning of vectors and matrices.

By

grouping the elements of a square matrix A into blocks such that the diagonal blocks are square we obtain

~ partitioning for the matrix as follows

( 2. 6)

A~

(L ....

IJ

For such a partitioned matrix we shall use the following notations: 0 A11 0 ' 0 21'

_'

A ' ' LA = ' _' DA = ' 0 ' '-

_'

'

AR-1 ... .'.fi/l-1 ₀ ' 'AH

( 2. 7)

12

A, •••

' 0

The case where the integer 1 equals 2 (in (2.6)) is of special importance:

Let k be a fixed integer (1 ~ k

<

n). For a vector

T

Clearly x1 is a vector of a k-dimensional space, to be called R1, which is induced in a natural way, by the partitioning. Similarly x 2 can be considered as a vector of an

(n-k)-dimensional space R2 , say.

For this partitioning the relation y = Ax then reads:

(2.9)

c:)

0

t~~:+~~~)

( ::)

If A21 = O, then A is called a k-block upper matrix. If more-over A11 is uppertriangular, then A is called a k-upper matrix.

An nxk-matrix will simply be called a k-matrix.

If a 1 , ... ,ak ER, then Ca11 • • • !ak) denotes the k-matrix

with a₁, ... ,ak as columns respectively.

2.4. Norms.

The space R will be provided with a Holder norm

II

·II p, p ;;;;. 1. This norm is monotonic, i.e._ if x ER and ·y E R, and Ix! .;;;; !YI, then llxllp.;;;; llyllp (cf. [3] ). Also the spaces JRk (or ttk),for k=1, ... ,n-1,will be provided with this norm. _{This implies llx}1_{11p .;;;; llxll p and llx}2_11P .;;;; _llxllp

for any x E R.

Let

s

1

,s

2 be linear spaces over the same field as R of dimension k₁ and k₂ respectively Ck₁,k₂ .;;;; n). Then the norm ll·llp induces an associated operator norm (also called least

upper bound) and a greatest lower bound for A:

s

₁ ~

s

₂ as

(2.10) (2.11) lub CA) p glb (A) p

=

-D llAllP _D II Axil min

l!Xlfl?.

xESi p x~O

!I

Axil

max~

xESi x p x~O

Definition 2.12. A vector x E

s

₁ is said to be a maximizing

veator for A (cf.(2.10)) i f llAxllp

=

lubp(A)llxllp. Similarly

x i s said to be a minimizing veator of A (cf.(2.11)) if llAxllp

=

glbp(A)llxllp (see [24,p.14] ).

Remark 2.13. The indices p for the norms will often be omitted.

Apart from the associated norm (2.10) we shall sometimes use other matrix norms, such as II ·II F, the Frobenius norm (also called the Euclidean-, Schur norm), see [27,p.57].

For matrix norms we have the following important concept: Definition 2.14. A matrix norm

II ·II

(defined for matrices of all possible dimensions) is said to be partiaZZy monotonic if for any matrix A and any submatrix A, derived from A by

striking off one or more rows and/or one or more columns,

llAll .;;;;

llAll.

Property 2.15. The associated Holder norm II ·II and the

p

Frobenius norm ll·llF are partially monotonic. Property 2.16. Let A be a matrix and let

u

1 and

u

2 be unitary matrices of proper dimensions, then:

lub

2CU1AU2)

=

lub2CA) glb

2CU1AU2)

=

glb2CA) llU 1AU 211F

=

llAllF

For products of matrices we have the following useful inequalities.

Property 2.17. Let A

1 and A2 be matrices, then (a) lub(A 1A2) .;;;; lubCA1 )lubCA2) (b) lubCA 1A2) ;;;, glb CA1 Hub CA2) (c) glbCA 1A2) ;;;, glbCA1)glbCA2) (d) glb(A 1A2) .;;;; lubCA1)glbCA2) If A 2 is square, then ( e) lubCA 1A2)

;;;,

lubCA1)glbCA2) (f) glbCA 1A2) .;;;; glb CA1 )lubCA2)

Definition 2.18. Let X be a set of nonsingular matrices. If

-1

3KE:ffi+ \fTEX max(llTll ,llT

II) .;;;;

K, then

X

is said to be

invertibZy bounded.

Remark 2.19. Instead of Holder norms one may as well use more generally monotonic norms on R. This will then induce norms on R1 and R2 and operator norms for the blocks of a partitioned matrix and the norm of such a block will then at most equal the norm of the whole matrix (cf.2.14). However, we shall not only consider matrices which originate from operators on a linear space, but also matrices which are just constructed as rows of vectors in R (or R1 or R2) see e.g. §8.1 - and for which no norm is induced naturally. For such matrices a norm and glb may (and should) be defined then separately. However, in order not to confuse the reader unduly, we have opted for the Holder norm, but the reader may use the more general one if he wishes.

2.5. Recursions and solutions. For q

>

p ~ 0 let A , ... ,A

1 be matrices and

p

q-rp• · · · ,rq_₁ be vectors. Let the vectors xp, ... ,xq satisfy

p

<

i

<

q.

{ } q-1 { }

Then the sequence A.;r. A ,r ,A +

1,r +1, ... ,A 1,r 1

l l i=p D p p p p q- q-is said to define a matrix vector recursion (MVR) for

q

{x.} and

l i=p

The sequence

q

{x.} is then called a solution of that MVR.

l .

l=p

{A. ;r.}q will be called an MVR for short.

l l .

i=p

If ~iri = O, then the MVR is called homogeneous; in all other cases the MVR is called inhomogeneous.

The MVR {A.;r.}00 ({A.;0}00 ) will also be denoted by 1 1

i=O 1 i=O

[A,r] ([A]); a solution will then be denoted by x.

q-1 q'

If for a given MVR {A.;r.} the sequence {x.} with

l l i=p l i=p'

p

<

p'

<

q' ~ q, obeys the relation (2.20), then

q'

{x.} will be called a partial solution of that MVR. l i=p'

If we use the relation (2.20) for determining xi+₁ from

x. (p

<

i

<

q), then this is called forward recursion or

l

recurring in forward direction.If we use the relation for

determining xi from xi+

1 Ci~ p

<

q), then this is called

backward recursion or recurring in backward direction.

The set of solutions of the MVR[A,r] ([Al) is denoted by IR(A,r) (tR(A)).

Definition 2.21. Fork~ 1, let x(1), . . . ,x(k) be linearly independent solutions of an MVR [Al . Then

(x(1) ! ... !x(k)) D {(x

0(1)j ... !x0(k)),(x1(1)! . . . !x1(k)), ... } is called a k-(matrix) solution of [A].

Property 2.22. If Xis a matrix solution of an MVR [A] then

Definition 2.23. If x(1), •.. ,x(n) are linearly independent solutions of a homogeneous MVR [A] then the n-solution (x(1) . . . x(n)) is called a fundamental system of [A]

(cf. [15 ,p. 39]).

The definitions 2.21 and q-1

for the MVRs {A. ;r.} and

l l .

2.23 have obvious analogues q-1

{A.; O} •

l .

i=p l=p

For a sequence of k-matrices X

D

{X

0

,x

1, . . . } denote MXN

=

{MX

0N ,MX1 N, ... } , if M and N are matrices of proper dimensions.

Let x(1), . . . ,x(k) be sequences of vectors, then span(x(1) ! ... !x(k)) denotes the linear space generated by x(1), ... ,x(k).

Definition 2.24. [A,r] will be called

(a) a UMVR if V. A. is an uppertriangular matrix,

l l

(b) a k-block UMVR if Vi Ai is a k-block upper matrix,

(c) a k-UMVR if

v.

A. is a k-upper matrix.

2.6. Equivalence of MVRs.

Definition 2.25. An MVR[A,r] is called equivalent to an

MVR [A,r] if there exists a set of nonsingular transformations {P.}.~

₀

, such that

l l""'

and

Property 2.26. The relation "equivalent'' is an equivalence relation.

2. 7. The sets ID

p

The following notation for subsets of ID will be used. (2.27) ID

=

{n E ID

I

n ;;;. p}.

3. Outline of ideas; the reference case (RC).

It is the object of this section to sketch the ideas which underlie this thesis. In order to demonstrate them we shall introduce an MVR, with some known relevant properties, which will be called a reference case. In this section and later on we shall often resort to this case.

A most simple MVR is the following homogeneous recursion [A], where

i ;;;;. 0, and the eigenvalues A

01, ... A01 of A0 are such that !A01i>···>IAon1. Denote the corresponding eigenvectors by e₀₁, ... ,e

0n respect-ively. Then, defining

( 3. 2) x.(j) = CA

0.)ie0. j=1, ... ,n,

l J J

we see that (x(1)

I··.

!xCn)) is a fundamental system of [A], and that

( 3. 3) _'ef. llxi+ 1 Cj)ll ₍

i

II

xi j

)II

j=1, ... ,n.

II x . +

1 C j ) II

We shall refer to a quantity like

11

~.(j)!I in (3.3) as a

l

growth factor of xi(j) (cf. [11,§2.1] ). By the growth of a solution we shall then mean the behaviour of the sequence of

growth factors; we thus say e.g. that a solution is slowly growing if the growth factors are about 1. As for the recursion above we see that the solutions x(1), ... ,x(j) are outgrowing x(j+1), ... ,x(n) (1 ~ j

<

n-1), i.e. they

dominate x(j+1), ... ,x(n).

Now suppose that {Ai} is a sequence of nonsingular matrices such that for all i:

(3.4)(a) (3.4)(b) (3.4)(c) II _{Ai - Ai+ 1}11 is small, the eigenvalues Ai 1, ... ,Ain of Ai are 1-, 2-, ... , (n-1)-separated, such that

llA.-A. 111

v.

l i+ is not large, J

IA.. \-IA·. 11

l ] l , ] + for any A ..

l ] (j=1, ... ,n) and corresponding eigen-vector e .. , the angle between e .. and the subspace

l ] l ]

of R spanned by eigenvectors belonging to the other eigenvalues is not small.

It has been shown in (25] that if the matrices of an MVR[A] satisfy conditions like (3.4), then there exist solutions x( 1), ... ,x (n) of [A] , for which xi ( j) is direc-tionally close to the eigenvector belonging to A·· and

more-l ]

over xl.(j) has a growth factor of about A··· This can be con-lJ

sidered as a generalization of the result for a constant MVR as the one above. We shall refer to this situation as the

PefePenae aase (RC). In the context of the RC, x(1), •.. ,x(n)

will always denote solutions having properties just described.

A basic concept in this thesis will be stability. We shall call a process stable if small perturbations, wherever they might be introduced, will not have large effects. We

shall demonstrate this for a solution x of a reference case MVR to be determined by using the recursion in forward direction. Thus let x obey

x

0 a given initial vector. Let ox₀ be a perturbation of x

0 and let x denote the solution of [A] with x

0 = x0 + ox0, then

from which we see that the error

{x. -

x.}00 is a

l l i=O

solution of [A] . Due to linearity there exist scalars a.₁,. .. , an such that

(3.7)(a) ox

0 = a1x0(1) + ••• + anx0Cn),

(3.7)(b) _{xi - xi= a 1xi(1)} + .•• + anxi(n), i ~ 1. We see from (3.7) that the effect of the perturbation ox

0 on xi, viz.

x. -

x., may become arbitrarily large (in norm)

l l

with respect to xi as i 4

oo,

unless x has a nontrivial

com-ponent of x(1). Especially, if x = x(1) we have x.

l (i 4 00 ) . Investigation of the sensibility

of a solution to perturbations at other stages (which can now be done in an obvious way) will yield a similar result.

Summarizing we can say that forward recursion is unstable (in a relative sense) if the desired solution x is composed of x(2), ... ,x(n) only. For stability i t is desirable that per-turbations will have an overall small effect. In our RC this is likely to be fulfilled if x contains a component of x(1).

If the matrices A. are nonsingular, as in the RC, and if

l

for a certain solution x an approximation of XN' xN(N) say, has been given, then the recursion can be used in backward direction as follows:

( 3. 8) i

=

N-1, ... ,0.

If the error xN(N) - xN is small, then its propagated effect on x. Ci 4 0) will not be large if x E span(x(n)); in fact,

l

xi(N) xi will become almost proportional to xi as i 4 0

(with a proportionality constant close to 1). This is a con-sequence of the assertion above for the forward recursion. In a similar way, it can be seen that the effect 0£ small

pertur-bations made at other stages will not be large. Therefore backward recursion will be stable for x(n).

We even see that x.(N) becomes almost proportional to

l

xi (as i ' 0) for arbitrary choice of xN(N), if only xN(N) is directionally close to xN. In practice we just choose some starting vector, vN(N) say, and compute VN-1(N), ... ,vo(N) in backward direction, after which this sequence is normalized (e.g. by comparing v

0CN) with a given vector x0), yielding a sequence, {x.(N)}N say. This process is

calle~

MiZZer's

aZgo-i

rithm [14]. A detailed error analysis has been given in (12] .

We shall say that {x(N)} converges to x if lim x.(N) ' X ·

l l

N+oo

for all i. From the preceding analysis it follows that in the RC we expect convergence under very mild conditions. We also refer to the MVR with constant coefficients, which was intro-duced at the beginning of this section: Miller's algorithm then is equivalent to inverse iteration (cf.[ 27,p.619] ).

Suppose we have an RC recursion(AJ. Let [A,r] be a

corresponding inhomogeneous MVR and x a solution which is

out-grown by the solutions x(1), ... ,x(n) of [A]. We shall show that Miller's algorithm converges, if we choose (approximate

solutions) x(N) with XN(N)

= o,

i.e. lim x.(N)

=

x. for any i.

N+oo l l

(The final normalization step is superfluous in the inhomo-geneous case). Define ( 3. 9)

x.

l (3.10) Gi

=

(xi(1)1 ... Jxi(n)), -1 -1

=

X. diag(IJx.(1)11 , ... ,llx.(n)ll ). l l l

Since x(N) is a solution of the inhomogeneous MVR [A,r] , we have for some b(N) E R:

(3.11) xi(N) =xi+ Xib(N). The error is given by: (3.12)

We find: (3.13)

=

X.b(N).

l

which is estimated by:

1 [ llxi(j)llJ (3.14) llti(N)ll .o;;;llGillllG~

II

._max llx (j)ll llxNll

J-1, ... ,n N -1

From ( 3. 4 )( c) we see that llGillll GN

II

will not be large with respect to 1; since we have assumed that x was outgrown by x(1), . . . ,x(n), we can therefore deduce from (3.14) that 1 im t i ( N ) = 0 .

N+oo

For the solutions x(2), •.. ,x(n-1) in the RC neither forward nor backward recursion is stable as shown above. We shall now indicate how a combination of forward and backward recursion for a suitably reformulated problem may yield a stable way to determine them. The basic idea is to uncouple the recursion in such a way that we have lower order MVRs so that the solutions can be determined in parts. This will be done by transforming the MVR into an equivalent UMVR. We shall demonstrate this for the RC in a somewhat heuristic and qualitative way.

Let

z

0 = Cz0C1)1 ... !z0(n)) be a matrix such that for

j

=

1, ... ,n the vector zo(j) is close to eoj (i.e. the

eigen-vector of

A

0 belonging to AOj) and, therefore, close to x0(j). Define a matrix solution Z by

Denote the columns of Zi as in: (3.17) zi = (zi(1)J ... lzi(n)).

will be close to Then span(zi(1)J ... Jzi(j))

span(xi (1)

I •••

J xi (j)) and therefore close to span(e.

1J ... Je .. )

i i ]

for j=1, ... ,n and all i.

We factorize Zi as follows:

(3.18) Zi

=

QiRi' Qi unitary and Ri uppertriangular. (Qi is unique but for postmultiplication by a unitary diagonal matrix, cf. [27,p.241] ).

The first j columns of Qi span the same subspace of R as the first j columns of Zi and this subspace was close to

span(ei

1J ... leij). Since the directions of ei1, ... ,eij were sufficiently separated, Qi will be close to the unitary matrix Si obtained by Gram-Schmidt orthogonalization from

(e.

1J ... Je. ), again but for postmultiplication by a unitary

i i n

diagonal matrix. Note that

S~AiSi

is an uppertriangular matrix with

A.

₁, ...

,A.

as diagonal elements, the so-called

i i n

Schur normal, form of A. (cf. [26 ,p. 17] ) . Moreover, on ·account

l

of (3.4) _{we can choose Qi+ 1 so that Qi+1} ~Qi. We shall assume that Qi+

1 has been chosen that way. Defining

(3.19) see (3.16),(3.18))

we therefore see that

v.

(being triangular) is close to a

l

Schur normal form of A. with ordered diagonal.

i

Suppose we wish to approximate the solution x(k+1) (of [A]) in a stable way.

Partition V. as: i

( B. : C.)

(3.20)

vi=

\/-:-/-i B. k-th order . i

For any solution x of [A] denote (3.21) i ;;;;., 0.

Let the superscripts 1 and 2 indicate a partitioning for vectors corresponding to the partitioning of Vi. Hence

[V] is a UMVR which is equivalent to [A] and a solution x of [A] corresponds to a solution x of [V] , where:

(3.22)(a) -1 _{xi+1 = B.x. + Ci xi'} -1 -2

l l

-2 -2

xi+1 = E.x. l l

(3.22)(b) i ;;;;., 0.

We notice the following:

(3.23)(a) The left upper blocks Bi of Vi define an MVR ~], having

<z

1(1)i ...

lz

1(k)) as fundamental system. Since x(1), ... ,x(k) and thus z(1), ... ,z(k) are out-growing x(k+1), we obtain that z(1), ... ,z(k) are outgrowing x(k+1);

-1

-1 -1

hence z (1), ..• ,z (k) are out-growing x (k+1) (note that

and i= 0, 1, ... ) .

-2 .

z.(J) = 0 for j=1, ... ,k

l

(3.23)(b) Since Vi is close to a Schur normal form of Ai' with ordered diagonal elements and moreover since

v.

is

l

expected to have no large off diagonal elements (i.e. the anormality is not large, due to (3.4)(c)), we see that the MVR[E] has solutions with growth factors

~

I

\.,k+l

I .

(3.23)(c) Since the direction of xi(k+1) is sufficiently well separated from the directions of xi(1), ..• ,xi(k), 11xf Ck+1)1! cannot be small with respect to llx1Ck+1)1!;

-2

-hence llxi(k+1)1! ""'llxi(k+1)1!.

From the observations (a), (b) and (c) above i t follows that we expect forward recursion with (3.22)(b) to be stable for

x2Ck+1) (assuming

x~(k+1)

has been given). If for some N the partial solution

{x~(k+1)}~=0

(of

~])

has been found, then (3.22)(a) can be- used in backward direction if we have

-1

some approximation for xN. In fact this backward recursion can be considered as Miller's algorithm in the k-th order

in--1

homogeneous case. Choosing 0 as approximant of xN we there-fore expect convergence and stability.

The approximation for x.(k+1) (0 ~ i ~ N) can then be

l

found by backtransformation using Qi (cf.(3.21)). Of course, (N-i) has to be sufficiently large for a satisfactory approxi-mation.

In the remainder this idea will be worked out for more general MVRs. Special attention will be paid to problems such as the convergence (the choice of N included) and the stability of this approximation method for subdominant solutions. Also the question of finding suitable transformation matrices, as Qi above, and the stability of the triangularization process will be studied. An interesting application of the method will be dealt with in §20, viz. for the approximate solution of a differential equation obeying linear boundary conditions.

Beside these algorithmic aspects, we shall also use the triangularized recursion for classifying solutions according to their growth and give estimates for their growth factors

(cf.(3.23)(a) and (3.23)(b)). This way of estimating has a certain resemblance to the method used in [25], as far as both methods take advantage of the nicer form of the matrices of a certain equivalent MVR.

The remaining three chapters may be classified as follows:

Chapter II is devoted to characterizations of MVRs and their solutions,

Chapter III deals with estimates for solutions, Chapter IV contains the numerical aspects.

4. Relating an inhomogeneous MVR to a homogeneous MVR.

Let [A,r] be an inhomogeneous MVR. It will be useful to relate [A,r] to an (n+1)-th order homogeneous MVR, especially if one is interested in estimates for the solutions (cf.

Ch. III).

In order to get some idea about the behaviour of the solutions of inhomogeneous MVRs consider the following

Example 4.1. Let [A] be the MVR with constant coefficients as defined in (3.1). Let {r.} be a sequence of vectors such that

l

for some

p,o

with 0

<

p,o

<

1 and all i

1 II r. 111

~1Ao,k+11

<

11~:11

<al Aoki (1) Define T

D (

e

01

I .. ·I

e On) ( e Oj being an eigenvector belonging to

. -1 -1

A

0j ) ' D = diagCA01, ... ,A0n) = T AT and vi ri = T r. Let D be partitioned as (D1

0 )

D1 of k-th order and

0 D2

let a corresponding partitioning be defined for vectors

00 1 i-,Q,-1

(cf. §2.3). From (1) i t follows that I.

[DJ

r,Q,_

1 exists for all i. We now define x. = l Then x = {Tx.} 00 D l i=O 00

-I

t=i+1 l 2 i-,Q, 2 I [D ] r,Q,_₁ t=1 is a solution of

p and a are not close to 1, then xi

t=i

[A,r], and clearly, if

""' T (-

[D~

-1

ri )

Hence ri-1

i f the directions of the r. are not too close to

l

-2 spanCe

01

j ...

leok2~ i.e. ifthe llrill have the same order of magni-tude as the II r .11 , then { x.} 00 has the same growth character as

{r. 1}

00

(If, in an extreme situation, the r. should lie

i - i=1 l

in span(e

01J ... Je0k)' then {xi}:=l has the same growth character as {r.}00 , which virtually is similar to the

l i=1

above one, due to (1)). ~

A

most simple related MVR for [A,r] and a solution x of [A,r] would be defined by the relation

We see that the matrix in (4.2) has the eigenvalues

A

₀₁, ..•

,AOk

and 1. On account of the RC of

§3,

on the other hand, we hope that these eigenvalues reflect the growth factors of certain solutions. Considering example 4.1 we see that the growth factor of xi Cx being the particular solution there) only corresponds to an eigenvalue if llr.11/llr.

111 ,.,, 1. We thus

l i

-conclude that the choice (4.2) is undesirable if llr.11/llr. 111

l i

-is not almost 1. The following related MVR may therefore be more successful: Let \j. r. 1- 0. Define l l ( 4. 3)

_no

= llr 011 2 llr 111

(::)

AO

llr 111 - - 2 ro (4.4)(a) = _{~ r_ol~ ___}

-

-

-

-

-0 llr₁11

11r

_{0 11} r.

A.

l

c+1)

l _!lri-111 I (4.4)(b)

_{- -, - - fr}

~II

-11i+1 - ₀ r l 11ri-1ll Note that lli

=

!Ir. ₁11.

i

-C)

If we apply (4.3) and (4.4) to example 4.1, then the matrix in (4.4)(b) will have eigenvalues that correspond to the growth of the solutions x(1), ... ,x(n) of the homogeneous

[Al (cf.(3.2)) and the partial solution x of the inhomo-geneous [A,rl. If in addition to requirement (1) of 4.1 the directions of the r. are only slowly varying, then the matrix

l

in (4.4)(b), too, is only slowly varying.

If, more generally, [A] is an RC recursion and for some p

,a

E JR+ with 0

<

p

,a

<

1

1 llri+111

<4

.s)

Vi

~1Ai,k+11

<

Urill

<

aJAikl

and, moreover, tte directions of the ri are slowly varying, then the corresponding homogeneous MVR as constructed in (4.3) and (4.4) will be an RC recursion also. Therefore, such an inhomogeneous recursion [A,r] will be called a reference

CHAPTER II

§§5-9

Chapter II deals with characterizations of solutions of an MVR. Special attention will be paid to k-block UMVRs; properties of their solutions will be related to properties of the recursion. The dominance, stability and convergence concepts, which were introduced in §3 for the RC will be considered in detail for general MVRs. Finally, a useful relation will be given for MVRs with certain stability and convergence properties.

5. On the solutions of MVRs.

In this section we shall give a number of definitions describing relations of solutions with respect to each other. Moreover we shall investigate the relation of a solution with respect to initial and boundary conditions.

q-1 If an MVR {A. ;r.}

l l i=p

and a matrix X have been given,

p

then a matrix solution {X.} q

l i=p

of that recursion is com-pletely determined. The problem of computing this solution will be called an initial value problem. We remark that q

allowed to be 00 and {X.} q to be a sequence of vectors. l i=p

We introduce the following concepts:

Definition 5 .1. Let [ A,r] be an inhomogeneous MVR and let is

y E ~(A,r). Then any nontrivial matrix solution of the homo-geneous MVR[A] is called a complementary matrix solution for y (of [A,r] ).

Definition 5.2. Let [A] be a homogeneous MVR and let Y be a matrix solution of [A). Then any nontrivial matrix solution X, which is such that span(X

0) n spanCY0) =

0

is called a

complementary matrix solution for Y (of [A] ) •

Remark 5.3. It should be noted that a complementary solution of an inhomogeneous recursion [A,r] (cf.5.1) is no solution of the recursion at all and therefore its name is a misnomer, which is, however, commonly used in literature (cf.[5,p.123]). Definition 5.4. Let [A,r] be an inhomogeneous MVR and let

y E ~(A,r). Then any nontrivial matrix solution of an MVR

be-longing to the set {{A.;0}00 \t=0,1, . . . } is called a parasitic

l i=t

Definition 5.5. Let [A] be a homogeneous MVR and let Y be a matrix solution of [A] Then for any t, any nontrivial matrix solution {Z.}00 of {A.;0}00 , for which

l i=t l i=t

n spanCY

2)

=

0

is called a parasitic matrix

solution for Y (of [A]).

Remark 5.6. If

V.

A· is nonsingular then any parasitic solution

l l

can be extended to a complementary solution using the re-cursion. However, if any of the Ai is singular this is not always possible.

A parasitic solution {Z.}00 will always be identified

l i=t

with the sequence {O, ••• ,O

,z

₂ ,Zt+l' · · ·} ·

Remark 5.7. Analogues to the definitions 5.1, 5.2, 5.4 and 5.5 will be used for partial solutions, {Y.}N say, as well;

l i=t

the range of indices for complementary or parasitic solutions is then restricted to [ t,N].

As for situations encountered in the theory of differential equations we can also impose some boundary rather than initial conditions on a solution of an MVR

(cf. [26,p.152) ): If a recursion q-1 (5.8) {A.;r.} l l . i=p

and a linear relation between two matrices

x

2 and XN Cq>N>t)

(5.9) M₁Xp + M₂XN

=

M₃

have been given, then the problem of computing a matrix solution {X.}q of (5.8) and (5.9) will be called a

l •

i=p

We are especially interested in the existence and uniqueness of the solution of such a boundary value problem in case {X.}q l i=p Define (5.10) F(p,N) = is a vector solution {x.}q

N-1

II j=p A .• J l i=p

Theorem 5.11. There exists a solution {x.}q of (5.8) and

l i=p q -1 (of{A.;r.} ) l l i=p (note that M 3 is a vector).

Moreover, this solution is unique iff (M

2F(p,N)+M1) is nonsingular.

Proof: (a) Existence; if: Let {y.}q

- l . be a solution of l=p q-1 {A. ;r.} l l . l=p , then M 3-M1yp -M2yN can be written as M1 zp + M2zN

for some solution {z.}q of {A.;O}q-1 By defining

l . l .

l=p l=P

V.-,,.,... x. = y.+z., we find that the solution satisfies l7J,-' l l l

the boundary condition.

Only if: For any particular solution {y. }q there exists a

l .

l=p

such that V.~ _i,,,.p Y· =

l

ran(M

2F(p,N)+M1).

x.-z ..

l l

(b) Uniqueness. Any two solutions of the problem differ by a complementary solution {zi}: satisfying M

1zp+M2zN = O. l=p

The assertion is now trivial. ~

Corollary 5.1~. Let Vi ri = 0 in (5.8). Then there exists a solution {xi}: of (5.8) and (5.9) iff M

3 E ran(M2F(p,N)+M1). i=p

This solution is unique iff (M

q-1

Corollary 5.13. Let {A.;r.} beak-block UMVR

(homo-l l .

l=p

geneous or inhomogeneous). Let R1 Ck-dimensional) and R2 ((n-k)-dimensional) be the linear spaces as defined in §2.2.

1 2

be given. Then xN and xp

L t e xN 1 E R1 and x _p 2 E R2 define

a unique solution {x.}q of { A. ;r.} q-1 iff the k-th

l .

i=p order leading principal minor nonzero.

l l .

i=p

of F(p,N) (see (5.10)) is

Definition 5.14. If there exists a unique solution of (5.8) and (5.9) then the boundary value problem is said to be

we Z Z posed.

6. Triangularizing MVRs.

In §3 we saw how useful it could be to transform a recursion into an equivalent one in which the matrices are uppertriangular.

In the present section we consider such transformations more generally.

The methods which are given below, show great similarity with Bauer's 'Treppeniteration' (staircase iteration, cf. [2) ), in which only one matrix A is involved, whereas we use a new matrix Ai at each step.

Let [A,r) be an MVR.

Let T₀ and Ti be such (nonsingular) matrices, that (6.1)

is an uppertriangular matrix.

Define sequences of nonsingular matrices {Ti} and upper-triangular matrices {Vi} recursively by:

i=1,2, ... Moreover defining:

( 6. 3)

we have found a UMVR[V,s), which is equivalent to [A,r). Each step in this triangularization process thus consists of post-multiplying A. by T. and of decomposing this product into a

l l

matrix Ti+i and an uppertriangular matrix Vi. Two familiar types of such triangularizations are

(a) L-U decomposition, i.e. the product of a lowertriangular

matrix and an uppertriangular matrix,

(b) Q-U decomposition, i.e. the product of a unitary matrix

(For numerical aspects of these methods, see [27,Ch4] .)

For our purposes, Q-U decomposition should be preferred, mainly because of the better numerical stability of unitary transformations.

For the matrices Ti and Vi obtained by Q-U decomposition we have the following relation with the matrices Qi and Vi encountered in §3:

Let Z, as in §3, be an n-solution of [A] (cf.(3.16)), let Qi correspond to Zi as in §3 and take T

0 = Q0. Then: (6.4) zo = TORO = OoRo

i-1 Since z. = (IT A.)z

0, we obtain from (6.1) and (6.2): i j=O J

(6.5) zi = TiVi-1 ... VoRo.

In (3.8) we also had Zi = QiRi. Since the unitary matrix in the Q-U decomposition of a nonsingular matrix is unique but for postmultiplication by a unitary diagonal matrix, there exists a sequence of unitary diagonal matrices {Pi} such that (6.6) Ti= QiPi

Denote for any solution x (of [A] )

( 6. 7) x i = Tixi = PiQixi = Pixi (cf.(3.21)),

H

H H

H-

i ;;;;., 0.

It will be clear now that the observations (3.23) also apply to the UMVR[V] of this section if the - notation is replaced by by a ~ notation; the only exception is that Vi now is close to a Schur normal form of Ai' but for pre- and postmultipli-cation with unitary diagonal matrices.

As will turn out in §9.1 it may be known beforehand which partitioning of the matrices of the equivalent UMVR is required in order to enable us to approximate solutions like

x(2), ... ,x(n-1) (in the RC) in a stable way; in other words the order k of the left upper block of the matrices Vi may be known. In such situations we may restrict ourself to

determining an equivalent

k-UMVR

rather than a

UMVR,

which will reduce the amount of computation. Indeed, if we use e.g. Householder's method for Q-U decomposition (p.148), we only have to determine k elementary Her~itians at ~ach step, obt~ining a decomposition of the form

( 6. 8)

(T. unitary).

l

If the matrices T

0 and T1 are unitary and equal, then

v

0 (see (6.1)) is a Schur normal form of A

0. For an MVR with constant coefficients we shall then have a kind of optimal choice for an equivalent UMVR if moreover the diagonal elements of

v

0 are the eigenvalues A01, ... ,Aon of A0 in this order. This motivates the following:

Definition 6.9. Let [A,r] be an MVR. Let {Ti} be a set of unitary matrices, {Vi} be a set of uppertriangular matrices and {si} be a set of vectors which satisfy (6.1), (6.2) and

(6.3). Finally, let the diagonal elements of

v

0 be the eigen-values A

01, ... ,Aon of A0 in this order. Then [V,s]

a normal form

of [A,r].

7. Solution spaces; convergence.

In §3 we gave a method for approximating the solutions x(2), ... ,x(n-1) of the RC by solutions of suitable boundary value problems. This was performed by using an equivalent UMVR. For an N

>

0 we defined an approximate solution of x(k+1), ~(k+1;N) say, for which the last (n-k) coordinates of x

0Ck+1;N) were equal to those of xo(k+1) and the first k coordinates of xN(k+1;N) were chosen somehow. Especially, if the xN(k+1;N) were chosen equal to zero, we showed that xi(k+1;N) converged to xi(k+1), as N ~ 00_{, i fixed.}

We shall now formulate a condition which makes this con-vergence explicit for more general recursions. It will give rise to a splitting up of the solution space. We shall assume that the recursion already has k-block uppertriangular form; for a general MVR such a k-block UMVR can be found using the method given in §6. We shall distinguish between homogeneous and inhomogeneous recursions.

7 .1. The homogeneous case.

Let [VJ b.e a k-block UMVR, where ( 7. 1)

v.

(-"-i_~S-)

i ;;;. 0

l

"1 : E.

l

Assumption 7.2. Bi is nonsingular for all i.

Assumption 7.2 will hold throughout this section. For a solution x of [VJ , we obtain:

2 i-1 2

(7.3)(a) x. = ( II Ej )x

0, i ;;;. 0.

l

j=O

1 i-1 1 i-1 i-1 t-1 2

(7.3)(b) x. = ( II _{Bj)x 0 +} { L: ( II B.) Ct ( II Ej)}x

0, i ;;;. 0.

l

It will be convenient to introduce the following operator: ( 7. 4) q-1 £ -1 £-1 I (II B.) C£( II E.) £=p j=p J j=p J (note that nq p

Remark 7.5. In order to get an idea of nq consider a second p'

order RC, where we obtain: q-1 £-1 A.2 1 nq

=

I < II

~) A-=-:-

c£.

p £=p j=p j1 £1

(1)

If {C.} is only slowly varying, we find from (1) that

l nq cP

P """

Xp1-Xp2 ( 7. 6) In ( 7. 7) 1 i-1 1 . 2 x. = ( . II B. ) {x 0 + rl~x

0

}, i J =O J

a similar way we have:

1 N-1 -1 1 i-1 x. = ( II B.) _~

-

n~

( II l j=i J l j =O i ;;;;: 0. 2 Ej)x₀, N ;;;;: i.

An obvious class of solutions obeying initial conditions is given by: 2 Definition 7.8. £k = {x E ~(V)

I

x 0 = O}. Property 7.9. \;f

.x?

=

0. l l

Moreover £k is a k dimensional subspace of ~(V).

For any (n-k) dimensional vector a, there exists

a unique solution y(N;a) of the boundary value problem with boundary conditions

y~(N;a)

=

a and yN(N;a) 1

=

O, due to 7.2 (see 5.13). A second class of solutions of

IV]

will now con-sist of those X E ~(V) for which _{XO = a} 2 and where _XO 1 is a limit point of {y1 N=1,2, ... } arbitrary.

0(N;a)

_'

a From

(7. 7) i t can be seen that this will only make sense if {rlN} 0 is bounded; this is especially so if lim nN exists. (For

the example in remark 7.5 we deduce from (1) that N CO

lim

n

~

)

N+oo O ~ A01-\02 .

Definition 7.10. If 7.2 holds and

{n~}

is bounded, then [VJ is called weakly k-convergent.

Property 7 .1 ~· If [VJ is weakly k-convergent, then there exists at least one infinite set I

0 c JN exists.

This limit will be denoted by

n

0<I0).

h h 1. N

sue t at lm

no

NEio

N+oo

Definition 7.12. If 7.2 holds and lim

nN

exists, then N+oo O

[V] is called k-convergent. Denote

n

0<JN0) by

n

0 for short.

(We remind the reader that JN - {n E JN

p D n ~ p} (see 2.27)). Examples of k-convergent k-block UMVRs can be found in §8.3.

We now introduce

Definition 7.13. Let [VJ be weakly k-convergent. Then Aen _ k ( I

0 )

D {

x E tit<

v )

I

x

~

= - n

0 <I 0 ) x

~

} . Denote .4Cn-k ( IN

0) (if i t exists) by .At n-k for short.

Remark 7.14.

A

classification which is similar to 7.8 and 7.13 is given in [25]; the sets L

1 [25,§3.6] and L2 [25,§9] resemble the sets £k and u ...f(n-k(I₀) respectively.

IO

Property 7.15. Let [V] be weakly k-convergent. Let I

0 be a subset of JN₀ such that ..kn-k(I

0) is well defined. Then ..4fn-k(I

0) is an (n-k) dimensional subspace of dt(V) and moreover £k $.Mn-k(IO)

=

dt(V).

Remark 7.16. We shall call {V.;0}00 weakly k-convergent if

l i=1

{

n~

I

1 fixed' N = 1+ 1'1+ 2,. .. } is bounded. In such a situation we shall denote by I

1 a subset of

:w

1

limn~

D

n 1CI1) exists. NEI1 N-+oo such that

In a similar way we can introduce Jtn-kCI

1) (cf.7.13). From (7.4) we obtain the relation:

i-1 i N N i-1

(TI BJ.)(S11-n1) = -n.c TI E.),

j=1 l j=1 J

(7.17)(a)

If ni(Ii) exists for some i, we see from (7.17)(a) that n 1Cii) exists for 1

~

i, and is a mapping defined on R2.

We have therefore

i-1 . i-1

(7.17)(b) (TI B.)(n~-n

0

(I.))=-n.(I.)( TI E.), j=1 J ,, ,, l l l j=1 J

If, on the other hand, n

1cI1) exists for some 1, then i t can be seen

i-1

from (7.17)(a) that for i

>

1 the restriction of

n~

l

to ( TI Ej)R2 has a limit as N

~

00 , NE I

1. We shall denote j

=

1

this limit by niCI₁;1). We thus have: (7.17)(c) ( TI i-1 BJ.)(ni 1-n1CI1)) = j=1 i-1

-n.

CI 1;1)( TI E.), l j

=

1 J i

>

1. Remark 7.18. If n

1cI1) exists for some I1 and if for an

i

>

1 the matrices E

1, ... ,Ei_1 are nonsingular, then i t follows that niCI

1;1) is defined on R 2

, and we have nicI 1 ,1) = nicI 1 ).

We have the following consequence of 7.13: Property 7.19. Let [VJ be weakly k-convergent. Then x E,Mn-k(I 0 ) iff Vi xt = -S1i(I 0 ;0)xf

Proof. The if is trivial. Only if : Let x E .ACn-kCI 0)

1 2

then x

1 i-1 _{i 2} x. = c IT BJ.)cn 0-n0CI0))x0 = i j =O i-1 2 -Q.(Io;O)( IT EJ.)xo l j =O

Remark 7.20. In the sequel we shall mainly deal with

k-block UMVRs [VJ , where for all 2 { V. ; 0} 00 is k-convergent,

l

i=R-exists for all 2.

In such a situation we have n. (JN.)

l l

=

Q.(JNO) l "'2

=

For the sake of convenience we shall then denote their common value by ni.

7.2. The inhomogeneous case.

The definitions of the preceding subsection have

analogues for the inhomogeneous case. We shall indicate them briefly.

Let [V,s] beak-block UMVR, with Vi as in (7.1). Assume 7.2.

For x E ~CV;s) we obtain:

i-1 i-1 i-1

2 2 2

(7.21)(a) x. = ( IT _{Ej )xO +} L: ( II E.) s (cf.(7.3)(a))

l _{J m} 1 (7.21)(b) x. l where ni 0 is (7.22) j=O m=O j=m+1 i-1 _{2 i 2 i}

=

( IT Bj){x 0 + noxo + 1T 0} j=O

defined by (7.4-) and _1T i ₀ by: q-1 Q, -1 1 2-1 2-1 L: ( II BJ.) {s 2 +

c

2 L: ( IT R-=p j=p m=O j=m+1 (cf. (7. 6)) 2 E.) s } . J m

'

Remark 7.23. In order to get an idea of nq, consider a second

p

order inhomogeneous RC (cf.§4-), where the particular solution x is "second in growth'': Let \i

1 and \i2 be the eigenvalues

II Si+111

of

v.

and let II II _{=µ .. Assume l\- 1}

!

_{>µi > l\i 2}

1·

If the

s~

have the same order of magnitude as the s.

l l

(cf.example 4.1) then we obtain

1

1T q ""'

~-s..._p_·

-P A _p1 -µ _p

This is an analogue to the result for nq as was found in 7.5.

p We also find (cf. ( 7 . 7) ) : 1 N-1 _{B. )-1} 1

_n~c

i-1 2 N (7.24) x. = ( II _XN

-

II Ej)x₀ 1T . l j=i J l j =O l Definition 7.25. £k

=

{x E ~(V) (cf.(7.8)).

Definition 7.26. If (7.1) holds and the set of matrices

{Cn~

I

1T~)}

is bounded, then [ V,s] is called weakly

k-con-vergent. If

limCn~

J

7T~)

exists, i.e. if both lim nN and

N N+oo N+oo 0

lim 1TO exist, then [V,s] is called k-convergent.

N+oo

In an obvious way we can introduce cniCI2)

I

1Ti(I2)) and (n.

17f.),

i ~ O, if they exist (cf.7.16).

l l

1 2

Definition 7.27. Mn-k(IO)

D

{x E ~CV,s) Jx

0 = -n0CI0)x0 -1T0CI0)} and in a similar way.M

11_k (cf.7.13) and.Mn-kCI2) (cf.7.16). From (7.28)(a) (7.22) we obtain: i-1 i N ( . II B. ) ( 1T 2 -1T 2) = J =2 J N~i~L Hence, if 7Ti(I) exists for some i, then 1T

2(I) exists for all 2. We find:

(7.28)(b)

Property 7.29. Let x E .Mn-kCI 0 ) iff vi

Remark 7. 30. I f [VJ

[V,s] be weakly k-convergent. Then

1 i-1 2

x. = -n.CIO;O)( II EJ.)xo - 1Ti(IO).

i i j =O

is k-convergent, then we define: 1T.

=

8. Dominance and stability.

In §3 we saw that the effect of possible perturbations of the recursion was not goodnatured (i.e. the problem was unstable) if we could indicate parasitic solutions which were outgrowing the desired solution. Therefore we shall first investigate this growth behaviour of solutions with respect to each other more precisely. This will lead to several dominance concepts, enabling us to give a rigorous definition for the stability of the computation of a

solution of an (approximating) boundary value problem.

8.1. Dominant and dominated solutions.

Let {xi} and {yi} be sequences of vectors, such that {x.} is outgrowing {yl.}, i.e. ~im lly.IJ/llx.11

=

0. I t is usual

l l~OO l l

then to say that {xi} dominates {yi} (cf.[ 8,p.25;25,§3.9] ). This concept can be generalized to sequences of matrices X = {Xi} and Y = {XiJ in various ways. Defining span(X) as we did in §2.5, one such generalization might be that any nontrivial x E span(X) dominates any y E span(Y). However, as the following example shows, this dominance concept would still allow a certain unpleasant situation to take place: Example 8.1. Let

[VJ

be an

MVR,

where

vi-1 = Define X.

=

l . 1 2 (~) l 0 0 ( (i+:1) 2 -(i+1) i i+1 0

~)

i+1 0

i+1-~-L

i iz 1 i+1 1 i ~ 1 i ~ 0

Then X = {X.} and y = {y.} are a 2-solution and a solution

D i D l

of [VJ respectively and any nontrivial x E span(X) dominates y. However, let for j=0,1, ... the solution x(i) be defined by

x(Cl

~

x

(~:i)

=

.Q.+1.

Hence

₌

Example 8.1 shows that for one and the same recursion i t may last arbitrarily' long before a certain dominant solution starts to outgrow a certain dominated one, and in the meantime this dominated solution may have grown arbitrar.ily much more than the dominant one. This explains our definition

8. 2:

Definition 8. 2. Let

x

= {Xo,X1,···} be a sequence of k-matrices such that x E span(X), x f. 0 ~ \;f.

l x. l f.

let y _{= {YO,Y1, ... } be a sequence of m-matrices.}

(i) If lim sup

_<

00 '

then we (ii) I f

i+oo xEspan(X),xf.O yEspan(Y),y

0t.o shall say that Y is

lim sup

i+oo xEspan(X) ,xf.O

yEspan(Y) ,y 0t.o

weakly dominated

llyill

I

~xi~

TIYaT

rx;;n-

=

then we shall say that Y is dominated by X. by 0.

0'

x.

and

In addition to this we shall also say that the sequence {O,O, ... } is dominated, or weakly dominated,by X.

We shall sometimes use an even more restrictive dominance concept:

Definition 8.3. Let X = {X

0

,x

1, ... } be a sequence of k-matrices and let y = {y

0,y1, ... } be a sequence of vectors. If y is (weakly) dominated ::JK '\J.R, sup xEspan(X),x#O i;;;>.R. llyill lly.R,11 moreover

then we shall say that y

is uniformly (weakly) dominated by

X. In addition to this we shall also say that the

sequence {O,O, .•• } is uniformly weakly dominated or uniformly dominated by X.

Remark 8.4. Dominance implies weak dominance.

Remark 8.5. If X dominates y, then X does not necessarily dominate y uniformly as can be seen by taking X = {1,1, . . . }

1 1 2 1 1 2 and y = {1,1,

2 ,( 2 ) ,3

,c

3 ) , ... }.

For the solutions x(1), ..• ,x(n) in the RC we expect that x(j) uniformly dominates x(j+1), . . . ,x(n) for j=1, ... ,n-1.

We have the following properties:

Property 8.6. Let X and Y be sequences of k-matrices and m-matrices respectively and let glb(X

0),glb(Y0) t 0. Then Y is (weakly) dominated by X iff

lub(Y.)

~im

l b ( / )

=

0 i-+00 g l i-+oo (lim lub(Yi) glb(X.) l

<

oo) •

Proof. If: Let x E span(X) and y E span(Y). Then there exist vectors c and d (of appropriate dimensions) such that x = Xe and y = Yd. We have then:

lub(Yi) glb(Y

0) lldll

l1Cilf

assertions can be deduced. lub(X

0) llcll