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I . .- . 1-/ .'2r.. : ..·IC.,I .... 91 1 ., :f . 4'3"886-14327 051 ·. f 4I.

*VE .S·Repor*»No. 221

....

MASTER

'- 1 't: 40 .*4 ..-./....:

< bNUMERICAL INTEGRATION OF STIFF ORDINARY

B DIFFERENTIAL EQUA#IONS

by

C. W. Gear

January 20, 1967

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NUMERICAL INTEGRATION OF STIFF ORDINARYDIFFERENTIAL EQUATIONS*

by

C. W. Gear

: cr,EAR -79--- ...&.b i--0 Jzarl ABSTRI .%-

bit

January 20, 1967

Department of Computer ScienceUniversity of IllinoisUrbana, Illinois 61801

*This work was supported in part by AEC-(11-1) -1469 and in patt by theArgonne National Laboratory, Applied Mathematics Division.

PATENT R-v!:-W A:'P:NOVES RELEABE. PROCEDURESGOVERNING PATENT REVIEW AND RELEASE ARE ONRLE I.LY REQE,ivING SESTION,a OBMMUTION OF TRI9 OUCU (ERit 19

ONOWTEW

CONTENTS

ABSTRACT Page

1. The Physical Problem . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. The Numerical Problem . . . . . . . . . . . . . . . . . . . . . . . . 2

3. A Simple Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . 3

4. A Theorem of Dahlquist . . . . . . . . . . . . . . . . . . . . . . . . 5

5. The Region of Stability . . . . . . . . . . . . . . . . . . . . . . . . 5

6. Stiffly Stable Methods . . . . . . . . . . . . . . . . . . . . . . . . 7

7. Another Difficulty - Corrector Iteration . . . . . . . . . . . . . . .1 0

8. Practical Considerations - Starting and Step Changing . . . . . . . .1 4

9. Finding Stiffly Stable Methods . . . . . . . . . . . . . . . . . . . .1 6

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 8

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 9

LIST OF FIGURES

F i g u r e l. . . . . . . · · · · · · · · · · · · · · · · · · · · · · · · 1

Figure 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Figure 3. . . . . . . . . . . . . 6

Figure 4. . . . . . . . . . . . .1 1 1 1 1 1 1 1 1 1 1 1 1 1 1' . .8Figure 5. · · · · · · . . . . . . . . . . . . . . . . . . . . . . . . 9

ABSTRACT

Stiff equations are ones in which there are rapidly decaying com-

ponents which are of no interest because. of their small magnitude. Equations

of this form frequently arise in physical equilibrium problems such as chemical

or nuclear reactions where there are many components with a large range of time

constants. Although the rapidly decaying components are of small magnitude,

they cause severe numerical problems in most general purpose integration methods.

This note will examine the cause of these problems and propose some ways around

them. Some specific methods are developed, but the investigation of classes of

methods remains to be done.

r-i

1. The Physical Problem

Consider.the one parameter family of functions

Y(t) = ce + F(t) (1.1)-At

where X is fixed and c is the parameter. The family is the solution of

y'(i) = -1(y-F(t)) + m (t) (1.2)dt

If k (assumed real for the moment) is >>0, the family (1.1) all tend to F(t)

very rapidly as t increases.

Y(t) A other members of family

&10:-3-

F(t

l. . I

t

Figure 1

The initial Falue problem for (1.2) for increasing t is therefore very stable.Large initial errors are damped out very quickly. ,.

This simple example will· be treated in the sequel to clarify the pre-

sentation, but the methods to be discussed are equally applicable to systems of-At

non-linear equations as to single linear equations. The term ce can be

thought of as representing the local behavior of one of the error terms. 'The3fnumber -1 .is, of .course, .an eigen value of the matrix 3F.if f is a vector valued . ..

function of the n component vector y which represents the differential system.by

y' = f(t,y) (1.3)

Therefore k must be allowed to assume complex values in the discussion. The

time constants are + Re(x) · They are only defined for decaying solutions

-1-

such that 3-Re(A) > 0. We will assume that F(t) is a well behaved function with-At

a moderately sized derivative in relation to e

2. The Numerical Problem

dF(t)If F(t) is a smooth enough function such that 4 - could nor-dt dtmally be integrated with a step size h , one would hope that equation (1.2)

would have the same property. Unfortunately not: Most methods require that

h be chosen so that |hAI is bounded by a small number. The actual bound depends

on the method, 2 is not untypical. This means that h.has to be chosen to see-kt

the finest structure of the solution due to terms like ce , although these

components may be completely insignificant. (They will almost never.be zero

because of rounding and truncation errors. )

Thtse restrictions occur both in Runge-Kutta and multistep methods.

We will restrict ourselves to multistep methods in this discussion after an

example of the common method, Euler's method. It illustrates the problem very

simply and pictorially.

y(t) AYn+2

F(t)

1 11111

Yn | | i l l€n 1

I l l1111 1Yn+1 1 1

'€n+2

l i l I1 Ii

tn tn+1 tn+2 1 1

.

tn+3

Figure.2

-2-

Suppose the numeriBal solution has reachbd (tn' yn) with an error En from.the

true solution F(t) shown in Figure 2. Euler's rule piojects the numeridal solu-

tion along the tangant, arriving at (tnfl' yn+1) and so oft to (tn+3' n+3 3

each error growing in magnitude. Mathematidally, we have

n+1 = n + h Y'(tn' yn)

Suppose F(t) = 0 so that

yl = -Xy

Then yntl = Yn - Ahyn = (1 - Ah)yE

Thus if 1-Ah < -1 or Ah > 2

| ntl | ) |yn|'

and the process is unstable.

Of course, nobody uses Euler' s inethod in oidinary differlntial equa-tions--but the problem will arise in iii'Bat methods. The reason that it alises

can be seen very easily in the above exam0le. The true relation of yn+1 to yn is

yA+1 = (1-Xh) yn

-AhThe accuracy with which 1-Ah approximated e determines the accuracy of the

method. For Ah < 0, the growth rate of the differential equation is larger than

that of the difference equation, for 0 < Ah < 2, the differential and the dif-

ference equations are stable, although the difference representation is "rela--Ah

tively unstable" (Hamming' s term) .when Ah > 1 + e , since then |1 - Ah| >|e-xh I . If kh > 2,. then |1 - kh| > 1 and the difference mathod is unstable.

The stability of the differdnce equation depends on Ah. The usual

asymptotic analysis considers only the case h + 0, but the problem is to per-

form the integration with a reasonably sized h.

3. A Simple Error Analysis

Multi-step methods are usually written in the form

rk kaO Vn+1 = - S ai yn+1-i + h s B y' . (3.1)

Li=1 i=0 1 n+1-id

-3-

where ao 4 0 and lak' + 'Bk| 0. If Bo 4 0,the method is called implicit

because, for the non-linear equation y' = f (x,y), n+l appears on the right

hand side in YA+1' otherwise it is called an explicit method. The coefficients

aI and Bi are chosen to make the method accurate in the sense the value of yn+11calculated from (3.1) does not differ from the exact value of y (t ) by v

eryn+1

much when the {y | i = 1, 2, .. k} are the exact values for a class ofn+1-i

functions y (t). This difference is called the truncation error, defined as

k

T (y, tn, h) = E (aiy (t ) + hBi y, (t .)) (3.2)n+1-i n+1-ii=0

with a = 1.0

Consider equation (1.2) with solution (1.1). Define the difference between the

calculated values yn and the true values y (tn = F (tn = Fn as En = yn - Fn'

By (3·1) a v = - Ea v -hI0 n+1 i -n+1-i YA+1-i

k k

0 (F+€ ) = - Ea. (F +€ ) -hE Bi (-A.(Fn+1-i + En+1-i - Fn+1-i)n+1 n+1 , n+1-i n+1-i1 + 0

+ F' .)n+1-i

k -

S (ai - hABi) En+1-i + T (F, tn, h) =0 (3.3)

0

Therefore the errors En obey an inhomogeneous k-step difference equation whose

inhomogeneous term is the truncation error for the exactsolution. Since F is

assumed to be well behaved, this can be assumed to be small for reasonable

methods, it is not influenced by the presence of rapidly decaying terms.

It is necessary to bound the solution of the inhomogeneous difference. .[2]

equation (3.3). This is done in all of the standard texts. (e.g. Henrici

Discrete Variable Methods in Differential Equations.) It depends on the sta-

bility of the corresponding homogeneous equation which arises when T is set to

zero. The difference equation is stable if and only if all roots of the poly-

nomial equation

(ai - hABi) Ek-i =0 (3.4)

0

are inside the unit circle or on the unit circle and simple.

-4-

Asymptotic theory considers the ease hA = 0 only. If stiff equations

are to be integrated with large h, then large values of hA must not make (3.4)

unstable. Since there may also be componerits.with large time constants (small

X), small values of hk must not make (3.4) unstable either. One root of:(3.4)hk

must approximate e for values of hk for which accuracy is required,.the other

roots must be small

4. A Theorem of Dahlquist

The order p of. a multiatep method is defined as the highest degree

of polynomial functions y for which T (y, t, h) vanishes identically in t and

h. Since the truncation error is then of the form

h P+1 k (Yb t, h)

where k=0 (1) as h- *O,i t i s Usually desirable to make p large. Dahlquist[1]

has shown that if (3.4) is to be stable for all X such that Re (hk) 0 (all

stable equations) then P < 2. He also showed that the "best" method is thetfapezoidal rule (with p = 2). This is a negative result that appears dis-

couraging. However.,. let us examine the requirement Re (hk·) ) 0.

5. The Region of Stability

The roots of (3.4) are to be small, except that one root must be close-hA

to e for those values of hA for which accuracy is necessary. However, they

cannot be bounded·everywhere except in trivial cases, in which case the

method cannot be accurate. Therefore we must ask "for what regions of the hk

plane is stability necessary and for what regions is accuracy necessary]"

Consider first the region where Re (hk) )D>0. In this region-At -Ah,

the term ce is reduced by |e I in magnitude in one step of size h. Now

|e-Ah I S e-D, so if D is chosen so that e- is insignificant at the accuracywe are interested in, then we are not interested in accurate representdtion of

this term, only in stability of the difference equation. For Re (hk) < D we

may also be interested in accuracy. Suppose that hA =x+ iy. In one step-At -iythe term ce dhanges by e-X e . Thus, in addition to a magnitude change

of e x,i t oscillates Z- complete cycles. For x

values at points with a spacing of h. It is impossible to represent a band

limited function with less than two samples per cycle of the maximum frequency

present, and in practice, 6 to 10 samples per cycle are needed for any sort of

accuracy. Therefore, if x < D, we must arrange that L < 8 where e is smaller2Kif higher accuracy is required. Typically e will be around 1/5. If the solu-tion is growing (x < 0), then the step size must be chosen small enough to pro-

vide sufficient points on the curve, so that x must be limited below by some

number -a, where e' is the maximum ·amount of growth to be allowed in one step,-hk

Inside the region D 2 x 2 -a, e 2 y 2 -8, accurate representation of e by one

root· of (3.4) is required, while other roots of (3.4) must not interfere withthe solution. (This could be satisfied by relative stability, stability or a

, combination, depending on the problem. )

In the remainder of the half plane Re (hk) < D, we do not care what

happens.

This discussion can be summarized graphically.

f AhA=-h -planey iJj 11

STABLE

8 \/ /

CCU RATE

\\\\1 -'ArJD ;. '.. '0 4 .... /.

-,«A0 LE « »-0

1 1.17 10 liFigure 3

6-

A method which satisfies these properties will be ealled "stiffly stable with

respect to D, 8, and a".

Roughly'speaking, if mgre accuracy is required, then larger Dand the smaller e and a should be used. Note that, for a given method, if hk.is in the left half plane, then h must be chosen so that hk lies inside the regionof accuracy. This is a reasonable thing to 40; it represents a reduction in h

to achieve the desired accuracy. If hA is in the right half plane (not includ-

ing the imaginary axis) then either h can be made small enough to bring hk inside

the region of accuracy or else it can be made large gnough to put Ah ip the

"don't-care" region of stability, depending on whether the solution for that

component is of interest or not.

6. Stiffly Stable Methods

The existence of @tiffly stable methods depends on the parameters D,

8, and a and on the definition of acegracy. The usual definition of accuracy

is that of order. Dahlquist has shown thAt if D is zero then the order p cannot

exceed 2. The only positive result so far is that methods of higher order than

2 do exist if D > 0. To be precise, methods of order as high as 6 have been

obtained and shcwn to be stiffly stable for suitable parameters D, 8, and e.These methods were found by observing that as hk changes from 0 to

plus infinity the roots of (3.4) move from the roots of the polynomialk

k-i k-ik

p (E) = E ai E to those of the polynomigl G (t) = E Bi g · From this1=0 i=0

it is obvious that G (E) must itself have stable roots, and must have degree k.kThe most stable possibility is a (g) = & which has zero roots. If the unique

method of degree k is calculated for this 0 (g), it can be shown to be stablefor k

F .

A

4

STABLE

2 REGION

6 (D95

m.-6 -4 -2 2 4

-2

-4

.

./ „...·

k=5 k=6

A hX plane

k= 4

5

Maximum values in4 real positive direction

'.1-Jp k=3 k Max (approx)20

w 3 3 .1UNSTABLE SIDE 4 .7

k=2 5 2.46 6.1

2

This is a lowerbound on D.

1STABLE SIDE

1 2 3 4 5 6

circle.

This boundary can be found in the hk plane as the locus of -2 11aCE) for'

iet=e , 6+ - [0, 2*]. This locus can be plotted in the hk plane. It is known

that at hk. = + infinity, the method is stable so that any region connected to

that point will be stable by a continuity argument. These plots are shown for

k=2 (1) 6 i n Figure 4 and 5. It can be seen that up to k=6 methods are

stiffly stable for. D > 6.1, e < .5 and for some a that has not been determined.

These methods have been used with success on a set of three equations arising

in a. chemistry reaction where the time constants are 1/.01 (1 a small amount)

and 1/3500 respectively.

7, Another Difficulty - Corrector Iteration

An implicit corrector formula can not, in general, be solved explicitly

for y Usually a first guess (the predicted value) is used to start an inter-n+1'

active solution, usually of the successive substitution form. Specifically, if '

the corrector formula isk ' . +

GO n+1 = -hBO y' (xn+1' Yn+1) - E .(aj n+1-j + hBj YA-j+1)1-1.

(0) 'and the predicted value of y is yn+1' the corrector iteration is usuallyn+1

(m+1) (m)) -200 yn+1 = -h BO y' (xn+1' Vn+1 ' ""

This will converge if |to *1

is less than 1 in a region encompassing a

large enough neighborhood of the solution. Since this depends on h, it is

usual to say that for sufficiently small h the bound is satisfied. .Unfor-By'tunately we are interested in large values of· h 39". Instead, let us

consider .·

the corrector iteration

a (m+1) 4. hBQ I)(m) Y(m+1) =0 n+1 .n+10

- h (y'(xh+1, n+1 D n+1) L. . . .(m) _ (m) (m) _ s(7.1)

(m) (m)where B is a matrix in case we are talking of a system of equations. D

-10-

is not specified at this time except that I + -, D is to be non-singularhOO _(m) .a

so that (701) can be solved for.y .. . Note that if D is - (y ) then(mfl) 0 (m) . By' (m)n+1 oy n+1(7.1) is the Newton iteration to solve the corrector formula.

.If a sequence D(m) can be chosen to .make (7,1) converge, then it

converges to the solution of the corrector· formula. Therefore, if an approxi-A,·,

mation to - can be calculated, it can be used for D(11). This need not becalculated for every iterate, nor yet for every step if - changes slawly.

It is convenient to re-express (7.1) to simplify the arithmetic.Subtract (711) with m-1 replacing m from (7.1) to get

I,. :P DC.).11 'Cm.1) - 1 I + BO DC.-1)-1 'Cm) -0

- hBO 1, (y(m)) - D(m)y(m) _ y,Cy(m-·1)) + D(m-1.) y(m=l) (762)0

(Note that the subscript 'n+1' has been dropped and. that y' (y (m)) hasbeen abbreviated to y' (y(m)).)

n+1' n+1

Rearranging (7.2), we get

I + BO D(m) (yimfl) _ 3(m)) = _ BO .Y, (1(m)) _ Cy, Cy(m·-1)) + D(m-1) Cy(m)·

ao L

- y(m.-1) 3 (7.3)

Define the variable d(Ill) = hy' (y(m-1.) ) 1. hD(Ill-.1) (y(m) _ (m-1) n+1 n+1 n+1 n+1

Hence (703) can be written a.s

y I+ -D' hy, (y ) - (7.4)(m+1) = y(m) _ BO 130 (m) 1-1 (m) \ d(m) 10We .also need· a ,recurrence "relation for d(m)

d

= hy, Cy(m)) _ FI + BO D(m) FI + hOO D(m) - -1 hy, Cy(m)) - d.(m) L ao L ao

+ FI + hBO D(m). -1 hy, Cy(m)) - d(m) L ao

Therefore

d.(m+1) = d.(m) + I + BO D(m) -1 hy, (y(m)) - d(m) (7.5)L ao

We need to define d It should be chosen to make (7.4) correct for m = 0.fo)

From (7.1)

F hBoII +

- I)(O) Cy(1) _ y(0)) =L ao

hB- 5.-2 (y, (y(0)) - D(0)y(0)) - - (aj yn-j+1 + B hy, )0 j=1 0

j n-j+1

- I S. hOOIDCO) yCO)0

Bo i(0) ) FOOy (0)

1= - -- 1 hy, (ywo

C

+ [Br + 1'1 BO (al n-j+1 + Bjhy.'.j+1)]

a y(0) kHence d(0) =- 0 n+1 - E 1 (a + he y' ) (7.6)n+1

BO B j n-j+1 tjn-j+1j=1 0

(0)Since yn+l, the predicted value is a linear combination of earlier function values

and derivatives, so is d(O . Note that if y is the value of y which satis-n+1- n+1 n+1

fies the corrector equation, then

-k kd(0) = -1|Z a v +h Z B yn+1 Bolj=o m «n-j+1 j=l j n-j+11

1= -h@ vi = hy,

BO 0n+1 n+1

-12-

(0)Therefore, we identify d as a "preditted" vallie Af hy' When (7.5) (6h*n+1 n+1

verges, d = hy, (Y ) so that d .converges td hy'(m) (m)· (m)

n+1 n+1

The pradictor and cbtrector formdiae can noW be rewritten in bhe[3]form giveh in Gear. Defint the vector

Z(m) i y(m), dCm), yn-1, ya-2, 1-n · ' i i hyj-i, hyn-2 1 ' 0 '

The predictor forHula. and (7.5) can be ·ekpressed as

Z

n

A is the Pascal ttiangle matrix

1 1 1 1 1 ... 1

1234 k13614

1.1 k

1

and £ is determined by the method. The equivalent methods to those discussed

in Section 6 are given by the following f.

234 5 680 2/3 6/11 24/50 120/274 720

/1764 :

1 3/3 11/11 50/50 274/274 1764/1764

f2 1/3 6/11 35/50 225/274 1624/1764

g O 2/11 10/50 85/274 735/17643

E4 0 0 1/50 15/274 175/1764 i,

E 0 0 0 1/274 21/17645 "

..:0

£6 0 0 0 0 1/1764

8. Practical Considerations -·Starting and Step Changing

To write a useful code, a number of practical problems must be

solved. These include:

(a) How to start

(b) How to choose.the step size

(c) How to decide when to re-evaluate an approximation to

I, + 2 Dll-'(d) What order method to use.

-14-

Insufficient numerical work has been done so far to substantiate the suggestions

below, but preliminary tests suggest that the following provides a practical

solution to these problems. (Parenthetically it should be' noted .that the fol-lowing remarks also apply to other than stiff equations, which also raises the

question of when to use stiffly.-stable formulae and evaluate an approximation to

'- hBO 1-1

1. I tar D -1and when not to bother. If stiffly stable methods with reason-

able error terms can be found, then it may be worth always using them. The fol-

lowing method is based on this assumption.)

The method consists of the following steps. Initially it is assumed

that some initial values exist for the function values, their derivatives and

possibly higher derivatives. All unknown derivatives are set to zero. The

order p of the method first selected has been set so that a sufficient number

of derivatives are known. (If only the function values and first derivatives

are known, then p is 1.) The formulae used must be such that the predictor and

the corrector have the same order so that the difference between the predictor

and the corrector can be used to estimate the first ignored derivative, and

hence the truncation error.

Step 1 Predict the value at the next point by a =A a-n+1 -n

Step 2 Iterate the corrector without changing the value of D until it con-

verges or until.too many iterations have been used (5?).

Step 3A If too many iterations were used, and this is the first attempt of

step 2, re-evaluate D and repeat step 2. If then is the second

attempt, halve the step size.and return to step 1.

Step 3B If the corrector converges, examine the difference between the pre-

dicted and corrected values of the functions. This is an estimate

of k hP+1 y(P+1) where k depends on the method. Since the leadingP+1 (P+1)term of the error has the form C h y where C is also known,

an estimate of the error can be made. This should be compared with

Eh where E is a measure of the error to be permitted on the unit in-

terval.. (If the equations are known to be very stable or unstable,

then .E should be changed throughout the interval.) If the estimated

error is too large the step must be rejected; if it is too small, it

-15-

can be accepted, but in either case, the next step should be an

order determining type (Step #4). If the error is within bounds,

return to Step 1 for the next integration.

Step 4 If the error is too large or too small the step size must be adjusted

(by a factor of 2 down or up on a binary machine). However, do not

increase the step until at least p steps have been made since the

last change. (From a multistep point of view, this would amount to ·

using points further back in history than the last step change, where

presumably, things were less well behaved.)

Having adjusted the step, perform a test for order as follows.

An estimate of the scaled (p+1)st derivative is available from the

last successful step (except in the case of the first step). Use

this and perform an integration with the abope technique of order

p+1. Do not check the error; instead use the predictor corrector

difference to estimate the error, and compare this with the estimated

error for methods of order p and p-1 which can be obtained using theP+1 (P+1) p (p)available values of the scaled derivatives h y and h y(P+1). P:

Choose the order method which gives the smallest error (in relation

to the components of E if a system is b eing integrated) . Repeat the

last step if necessary.

9. Finding Stiffly Stable Methods

At present, no technique is known for finding stiffly stable methods

except that of using common sense and investigating'the properties of the guess.

A possible approach might be the following (although it has not been successful

yet).

P(E)- should approximate log & for an appropriate region of the E-a(E)

plane corresponding to the accurate region of Figure 3. The part of |E| > 1

that does not correspond to the accurate region should map into the don' t care

region while the part of | E| < 1 th at does not correspond to the accurate region

may map inte. anything. p(E) is a rational fraction, so we ask the question="how is a ratioial fraction chosen to satisfy the aims above?" Such a method

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method may not be consistent, but that does not matter since we do not want con-vergence as h + 0, the point of the work is to handle large h.

\

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BIBLIOGRAPHY

[1] Dahlquist, G. G., "A Special Stability Problem for Linear Multistep Methods,"BIT 3 (1963) pp. 27-43.

[2] Henrici, P., Discrete Variable Methods for Ordinary Differential Equations,Wiley, New York.

[3] Gear, C. W., "Numerical Solution of Ordinary Differential Equations ofVarious Orders," Argonne National Laboratory Report, ANL 7126, Argonne,Illinois, 1966.

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L

.

APPENDIX

kDetermining p(&) when a(E) =g.·

[2]· It is known (Henrici ) that if the method is of degree p, then

I. P(E) - logE 0(E) - c(&-1)P+1 as & = 1 (Al.10(This is the statement that the root of p(E) - log(B) 0(E) = 0 is close to B.)

k (E)If we set p=k and 0(&) =E, (Al.1) determines p in the following way.Set E-1 = x

p(x+1) = log (x+1) . (1+x)k truncated to terms in xk and lower.

If we represent p(E) by the column vector

2 = Iak' (Xk-1' ..., 00-1

kk-jwhere p(E) = E a. ' g , then f can be expressed as

0 J

1-1 1-1. . . (-l)k -1 00 1 -2 3 (-l)k-1 k kl +10 0 1 -3 (_l,k-2 (]A 2j /14 k 1 -1/2

2 = 1 (2//k) /k) k 1

01/3

t t 00 1 1k (:) . . ' 8}(:j k l -(l)k/k

The last column consists of the coefficient of the logarithm series for (1+x).kThe hecond matrix corresponds to multiplying by (1+x) and ignoring terms abbvek

x . The first matrix arises from the conversion back te E = 1+x.

If this is done for k = 2(1)6, we get the examples:

2+1 = 'l n + a2 n-1 + ' ' ' + akYn-k+1 + hBO n+1

where {ai J and BO are given by:

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-

k BO al a a '4 a2 3 5 a62 ·213 4/3 -1/3 0 0 0 0

3 6/11 18/11 -9/11 2/11 0 0 0

4 12/25 48/25 -36/25 16/25 -3/25 0 0

5 60/137 300/137 -300/137 200/137 -75/137 12/137 0

6 60/147 360/147 -450/147 400/147 -225/147 72/147 -10/147

4

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