ima j appl math 1968 oliver 399 409

7/23/2019 Ima j Appl Math 1968 Oliver 399 409

1/11


2/11

4 0 0 J. OLIVER

2 .

Error Propagation

in

Initial Value Problem s

Consider first

the

system

y&)

=

Pr.i(x)

yi

(x) + ...+P

r

Jx)yJLx) +

RXx)

( r = l , . . . , n )

(la)

or, in the usual notation,

y'(x) = P(x)y(x)+R(x), (lb)

where we wish to evaluate some or all ofthecomponents

Y

r

(x)

ofaparticular solution

Y(x) over some range

[a,b],

Y(a) being specified (the theory may be extended to more

general initial conditions).

Suppose initiallyweemploy a linear one-step forward integration method to produce

a computed solution Y(x,) over a set of discrete pointsx,where

a=

* o < x i ,(*) {s=

1,2,...,/ ),

(4)

for some real Ai,...,/^ with A

t

^ ^2>. .. >

A*,

where the components (p,,i(x),...,

t

^(x)

of

,(jc)

are slowly varying functions of x.(In the special case of a constant matrix P

having distinct eigenvalues

Xi,...,

A, the,{x) are its eigenvectors, and thus independent

ofx.)In order to substitute fore

Jt

, in (2b) we denote byW

tir

(x)the cofactor of4>

lif

(x)

in

W{x) =

.. .

nt

i(x)

(5)

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3/11

INSTABILITY

OF

DIFFERENTIAL SYSTEMS

401

which leads

to

'

n

f

B

th Ix ~\ W (x

^

Yl x }

=

Q

E | E

W

(

X

J

Y ^ ) ^

X

J

)> X

exp{A,(x,-Xj)}

written

so as to

display bo th

the

local character

of

the errors ,

and the

manner

in

which

they are propagated.

The behaviour

of

the functions

which may in theory assume any values; we have written the result in this form,

however, since

in

many practical cases these quantities will

not

greatly exceed unity

in

magnitude. Similarly

the

initial effect

of the

propagated relative error

is

dependent

upon Yj(Xj)/YXxj),

and

upon

the

local truncation

and

rounding error. This latter

quantity depends, amongst other factors, uponthew ord-lengthof themachine: thus,

in the case of a binary machine having a mantissa in floating point mode of t bits,

we expect the quantity

Uxj)

=

\eJLxj)/YjLxj)\ (8)

to

be

either zero

or

else

of

order

2~ at

least.

So

long

as Y(*

y

) ~

Y(xj),

any

non-zero

tiXj) shouldbe at leastof this order, resultingin a certain minimum levelof relative

error, even

if the

local*truncation error

is

made very small.

The

effect

of a

relative

error

in one

component

Y,(xj) of the

required solution will

be

magnified

in

another

component

Y

r

(xj)in the

rat io

|Y&C j)IY

r

(xj)\ but

this will only

be

serious

if the

ratio

is larg e;wecommentonthis further inSection6.

Of greater concern, however, is the manner in which these local errors are pro-

pagated,andthisislargely de terminedby thebehaviourof

as

x

t

increases.It isimmed iately obvious that if, for example, some

Y

p

(x)

issuch th at

\exp(A,x)/Y

p

(x)\ increases rapidly withxfor one or moreA,,thenthe computed value

Y

p

(x)

will eventually

be

swamped

by

multiples

of

these complementary solutions.

N ow it is possible to derive a variety of elegant theoretical definitions of inherent

stability, based

on the

criterion that

the

propagated errors should

not

grow,

in

some

sense,asx increases. In practice, however, what matters is whether the propagated

relative errors grow

so

rapidly

as to

prevent at tainme nt

of the

required accuracy over

the particular range of interest in a given computational environment, and this is

necessarily

a

relative decision.

In a

similar manner,

we

shall speak

of one

solution

dominat ing another as meaning that their ratio increases appreciably in magnitude

as

x

increases, with ou t defining this precisely.

Thus the method will be ineffective for any particular solution

Y

p

(x)

which is

strongly dominatedby one or moreof exp(A,x),but it isimportant to realize that it

may neverthelessbeeffective for those

Y

r

{x)

whichare not dominated by any of the

complementary solutions.The disastrous error propagation in Y

p

(x) will eventually

cause

Y

p

(x)to

behave like

a

m ultiple

of its

dom inant complementary solution, rather

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4

J. OLIVER

than like

Y

p

(x),

thus altering the order of the rounding errorst]

p

{xj),but this will not

of itself induce instability in the other solutions.

We have thus far analysed error propaga tion in terms of

the

fundamental solutions

of

the

original differential system, whereas a practical one-step method approximates

it by a system of linear difference equations having a particular solution Y*(x,) and

complementary solutions vW*(x,). One desirable feature of any method is, however,

that the local trunca tion errors in each of the required solutions

Y

r

(x )

should be small,

and achieving this will usually ensure that any dominant complementary solution of

the differential equations has a corresponding dom inant solution in the approximating

system. Consequently any inherent instability in the original problem almost always

leads to similar error propagation in any discrete system which closely approximates it.

The same is true both of linear multi-step methods, with the added complication of

the spurious complementary solutions which the replacement system of difference

equations possesses, and also of global methods of solution such as Chebyshev series

(although their theory is not yet well developed).

It may of course be possible to represent the differential system by a discrete one

whose particular solutions closely approximate the ones required, but in which the

dominant complementary solutions are replaced by ones with a milder form of

behaviour. An example of this, involving Chebyshev collocation applied to a single

fourth order equation, is cited by Oliver (1968a). However, this remedy will not

usually be feasible, and so an alternative method of solving an inherently unstable

initial value problem must be found: we return to this point jn Section 5.

3 .

Error Propagation in Boundary Value Problems

The stability of the two-point boundary value problem may be analysed similarly.

For simplicity, we take the following type of conditions:

y

r

(a)

=

Y

r

(a \

0

= \,...,n-m)

yp(r)(b) = YpO)(b),(r =

n-m+l),...,n)

where

1

-x )} exp

{^(x -b)}

,.Z

+l

W(a,b)

with the stability characteristics largely dependent upon the terms

We conclude, therefore, that inherent instability in Y

p

(x)with respect to the boundary

conditions at x = b will result if and only if exp (A,^) is sufficiently dominated by

Y

p

(x).

Finally, to investigate the effect of errors introduced during the computations,

we suppose that a linear one-step formula, of the form

y(xd = A(xOy(*,-i)+*(*


6/11

4 0 4 J. OLIVER

Under the above assumption regarding the complementary solutions, we may write

**/)= t cJ'Xxd+S'Mxj)0 =0,1,...,;)

1 = 1

t j \ .,N ) (17)

where the constants are determined by (16) and by equating the representations (17)

at their common pointXj.This gives

0

0

Ija) 0

0 t > -

m +

0 >( )

0

.)(*)

dt

An,

0

.(18)

Using similar algebraic manipulation to tha t given earlier, substitution for

c

r

and

d

r

in (17) shows that each component

i,(

x

j)

contributes an am ount e^ x , ) to (*,) where

(19)

{

Xj

)

V(xJ)

the determinants F having been norm alized as previously. Examination of the

terms which govern propagation, namely

indicates that perturbations will propagate unstably in Y

p

(x ) if and only if Y

p

(x ) is

dominated byexp

( *+i*)

o r

itself dominatesexp

{X^pc),

agreeing with the conclusions

already reached in regard to changes in the boundary conditions. The perturbation

{xj) obviously encompasses local truncation error, and rounding errors can also be

regarded inthislightsee, for example, an analogous treatment of recurrence relations

by Oliver (1968b)though it is more difficult to assess their probable order of

magnitude than in the initial value case.

The comments m ade in Section 2 regarding the relationship between the comple-

mentary solutions of the differential system and those of the discrete approximating

system apply equally well here.

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INSTABILITY OF DIFFERENTIAL SYSTEMS 4 0 5

4. Choice of Boundary Conditions

It

is obvious from the foregoing analysis that for a differential system having

exponential-type complementary solutions and particular solutions of a somewhat

similar nature, there is an appropriate choice of boundary conditions which lead to

a well-posed problem for each Y

r

(x). More precisely, if YXx) is dominated by

exp(A,x)fors= 1,..., m but not for s=m

+

1,...,n, then boundary conditions of the

form (9) are most effective. In particular, forward integration of an initial value

problem should be employed when Y

r

(x ) is not dominated by any of the comple-

mentary solutions, and of course integration in the reverse direction if it does not

dominate any of them.

Although a particular boundary value problem may be inherently stable with

respect to the required solutions, it suffers from the major disadvantage that jury

methods of solution entail considerable computation and storage. Shooting methods

will normally be impracticable, due to inherent instability, although Conte (1966)

suggests that this instability may

be

avoided

by

orthogonalizing

the

computed solutions

as the forward integration proceeds. We stress, however, that the boundary value

problem itself m ust be well-posed if

this

approach is to succeed.

If the available boundary conditions specify an inherently unstable system for one

or more of the required solutions, and no further conditions are available, then no

improvement

is

possibleindeed, the effect of the instability cannot

even be

recognized

in one computed solution. Although this may be true of a mathematical system,

however, in physical problems some additional information is usually available,

perhaps in the form of vaguely-stated conditions of boundedness. For example,

Midgeley (1966) cites the case of the linearized equation for vertical wave propagation

in a vertically inhomogeneous gas, for which the relative behaviour of the various

complementary solutions is known from physical considerations. This mathematically

redundant information may indicate that the problem as specified is ill-posed, and

tha t boundary conditions of a different nature should, if possible,

be

obtained and used.

Furthermore, if an exact well-posed problem cannot be formulated, a knowledge of

the general relative behaviour of the various solutions can itself lead to a stable

system.

5. Replacement of

n

Hi-posed Initial Value Problem

To illustrate this, we return to the initial value system in which Y(a) is specified,

and suppose tha t

a

particular solutionY

p

(x )

is

required which

is

dominated

by exp

(A,*),

(s

= l,...,m ), to such an extent that forward integration methods are ineffective.

Assume also that no other conditions are available, except the knowledge that Y

p

(x )

is dominated by exp(b,

and consider the solution

z(x)

specified by the

boundary conditions

zj(a)= YJta) (s = l , . . . ,n-m )

Note that the detailed choice of boundary conditions is unimportant; what matters

is tha t onlyn mof

the

exact ones atx = ashould be used, and zero values of Y^x)

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4 0 6

J.

OLIVER

substituted nearx = x

M

for the remainder. z(x) existsjf

# 0, (21)

and this we assume to be true. Following the usual procedure of introducing

normalized determ inants in Cramer's rule, the relative difference betweenz(x)and

is found to be of the form

(22)

and thus the more exp(k^x)dominates Y

p

(x), the closer will z

p

(x,) approximate to

YJxd over[a, b].

A similar analysis to tha t performed in Section 3 shows that this differential system

for z

p

{x^ is well-conditioned to truncation and rounding errors, so that this is an

effective method for evaluating an approximation to the required solution. The

parameter x

M

is taken sufficiently large so that the required accuracy is obtained,

though as always in such algorithms the optimum choice ofx

u

is a non-trivial decision

for an analogous problem in recurrence relations see Olver (1964, 1967).

Midgeley (1966) has suggested a method using step-by-step methods and quadrature

to evaluate each of the complementary solutions in turn in order of decreasing dom in-

ance. This avoids the necessity for solving a boundary value problem, although he

effectively solves one for each complementary solution, using an appropriate number

of arbitrary initial conditions with zero values at the opposite end of the range. The

method is riot, however, immediately applicable to non-homogeneous equations.

Finally we note that even when the nature of the complementary solutions is not

apparent from physical considerations, their behaviour can be determined by a process

such as Midgeley's, or equivalently by solving directly a series of boundary value

problems for the homogeneous form of the differential equations using m arbitrary

conditions at some point to the left of x = a,and

nm

zero conditions at x

u

> b,

for m = n,n1,...,0. Provided the complementary solutions are of exponential type,

this will indicate their behaviour (including those with nearly equal dominance), and

if the nature of the required solution is known then a well-conditioned boundary

value problem can often be constructed.

6. Numerical Examples

We take first an example based on that of Conte (1966):

y'^x) = -

1-8368>'

1

(X)-6-2976>'

2

(X)-1008}'

4

(X)

+

10-08(1

+ x

2

/2) ,

y'

2

(x)=

-6-2976>>

1

(x)

+ l-8368.y

2

(x)-10-08.K3(x)-49-875 exp(20x), (23)

y'

3

(x )

= -10-08 j>

2

(x)+3-4832y

3

(x)-ll-9424>>

4

(x)-

37-40625

exp (20x)

+11

-9424(1

+

x

2

/2),

y't(x) = -1008j '

1

(x)- l l -9424y3(x)-3-4832^(x) +

x +3-4832(1+x

2

/2).

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INSTABILITY

OF DIFFERENTIAL SYSTEMS 4 0 7

This is, conveniently, the special case of a constant matrix P, and the complement-

ary solutions vary exactly like the exponentials

exp

(20*),

exp(*) , ex p (- *) and

exp] (- 2 0 * ). Using the appropriate initial values to specify the particular solution

Yi(x)= 0-76506 exp (20*)-2

exp

(*)/3-2exp( - * ) / 3 ,

Y

2

(x) = -

2-65283

exp (20*) -

exp

(x)/2+8 exp ( - x)/9,

F

3

(*) = - 0-64575 exp

(20*)

+

3 exp

(*)/8 +

2

exp ( - *)/3,

(

'

F

4

(*) = l+*2/2+sinh(*) ,

we note tha t Y

4

(x)is strongly dominated by exp (20*), whereas the othe r solutions are

not. Consequently we expect that forward integration is inherently stable for all

solutions except

YA(X).

To illustrate this without introducing the complication of parasitic solutions, we

approximate the differential system at equally spaced points *

;

by

- i )} . (25)

Taking an interval h of 2~

7

, and using (25) recursively over [0,2] on KDF9 (which

has a 40-bit mantissa) with the initial values appropriate to the differential system

produces a reasonab ly accurate approximation to Y\(x),as Table

1

shows, and simil-

arly for Yz{x)and Yi(x),whereas the computed values of

Y*(x)

are rapidly swamped

by the propagated error. Employing a smaller interval would of course improve the

accuracy of the first three computed solutions, but would not remove the inherent

instability for YA{X).

TABLE 1

Forward integration

for

first differential system

*

0-25

0-5

0-75

1

1-25

1-5

1-75

2

Computed

1 1 3 x 1 0 2

1-71x104

2-55x106

3-80x108

5-67x1010

8-46x1012

1-26x1013

1-89x1017

Tito

Relative er ror

6-89x10-3

1-24x10-2

1-79x10-2

2-36x10-2

2-93x 10-2

3-51 x 10-2

4 0 9 x 1 0 - 2

4-68x 10-2

Y4.x)

Computed

+ 6 -9 5 x 1 0 -1

- 2 1 0 x 1 0 2

- 5 0 1 x 1 0 4

- 1 0 2 x l 0

7

- 1 - 9 4 x 1 0 9

-3-50x1011

- 6 1 4 x 1 0 1 3

- 1 0 5 x 1 0 1 6

Exact

1-28

1-65

2 1 0

2-68

3-38

4-25

5-32

6-63

Since Y-t(*) is dominated by only one of the complementary solutions, it would

seem at first sight that a boundary value problem in which three conditions were

specified at * = 0, and a further one for some * > 0, would be well-conditioned. This

is indeed the case, yet computed values of Y*(x)again exhibit catastrophic error

accum ulation. This is not due to growth of local errors, however, but rathe r to relative

errors in Yi(*,),

Y

2

(xj)

and

Y

3

(xj)

producing greatly magnified errors in Y^Xj),

particularly for large *,. Referring to (19), the relevant term is Y&Cj)IY

r

(xj), with

equivalent ones in (12) and (13). Consequently the problem is well-conditioned in one

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4 8

J. OLIVER

sense, yet not in another, and there seems little remedy in such a situation (except

perhapstoproduce a fourth order differential equation for

YA(X)),

though itisthought,

fortunately, to be of infrequent occurrence in practice.

A more common problem is that typified by the same system (23), but with the

terms inexp(20*) omitted from the equations for

y

2

(x)

and

y

3

(x).\With

appropriate

initial conditions, the solution is

Yi(pc) = - 2 exp (*)/3 -

2

exp (- x)/3,

Y

2

(x) = - exp(x)/2+8 exp (- x)/9,

Y

3

(x) = 3exp(x)/8+2exp(-x)/3,

Y

A

{x ) =

(26)

and

the

initial value prob lem

is

ill-conditioned with respect

to all the

particular

solutions

Y

r

(x),

asillustratedbythe resultsofTable 2, obtained w hentheapproxima-

ting system (25)

was

solved recursively over [0,2] w ith

an

interval

of

2~

7

.

TABLE 2

Second differential system: relative errorsincomputedsolutions

X

0

0-25

0-5

0-75

1 0

1-25

1-5

1-75

2 0

Forward

Y i x )

0

3-12x10-7

1-31x10-6

2-02x10-3

2-24x10-3

2-74x10-1

3-29x101

3-92x103

4-63x103

integration

Y*x)

0

1-02x10-6

1-91x10-

218x10-5

2-29x10-3

2-72x10-1

3-24x101

3-88x103

4-67x105

Boundary

Yi(x)

0

4-98X10-6

1-88x10-3

3-88x10-3

6-20x10-5

8-63x10-5

111x10-4

1-34x10-4

108x10-4

value method

Y4.x)

-4 -7 3 X10-11

1-63x10-3

2-79x10-5

3-76x10-5

4-69

x 10-5

5-68x10-3

6-75x10-3

7-90x10-3

4-31x

10-s

If,

however,

one of

the initial conditions

a tx =0 is

discarded

in

favour

of

o ne

for

some positivex,and the approximating system solvedby a boundary value method,

then

as

expected the re

is no

evidence

of

inherent instability. Fu rtherm ore, we may use

the methodofSection5, withm = 1inthis case,togood effect. Forexample, solving

the system (25) subject

to

*,(0)= YJ[0) (s = 1,2,3)

24(2-5)= 0,

{27)

with

a

fairly large interval

of

2~

3

produces

the

approximations

to

Yi{x)

and

YA(X)

shown in Table 2. More accurate approximations can be obtained with a smaller

interval,or byapplication ofdeferred correction proce dures .

7. Conclusion

W e have investigated the inhe rent stability of systems of linear differential equ ation s

having complementary solutionsofexponential type,and shown under a numberof

reasonable assumptions that

to

any particular solution

of

exponential type there will,

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ima j appl math 1968 oliver 399 409

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