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Numerical quadrature : a comparison of available proceduresat the Computing Centre of the Eindhoven University ofTechnologyCitation for published version (APA):Geurts, A. J., & Haan, den, P. J. (1981). Numerical quadrature : a comparison of available procedures at theComputing Centre of the Eindhoven University of Technology. (EUT report. WSK, Dept. of Mathematics andComputing Science; Vol. 81-WSK-04). Technische Hogeschool Eindhoven.

Document status and date:Published: 01/01/1981

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TECHNISCHE HOGESCHOOL EINDHOVEN TECHNOLOGICAL UNIVERSITY EINDHOVEN

NEDERLAND THE NETHERLANDS

ONDERAFDELING DER WISKUNDE EN INFORMATICA DEPARTMENT OF MATHEMATICS AND COMPUTING SCIENCE

N~i1[erical Quadrature

A comparison of available procedures at the

Computing Centre of the Eindhoven University of Technology

by

A.J. Geurts

p.J. den Haan

T.H.-Report 81-WSK-04

Augustus 1981

Summary. A comparison is made of the ALGOL procedures for numerical

quadrature, available for general use on the Burroughs B7700 at the

Computing Centre of the Eindhoven University of Technology. The pro

cedures are compared wit:i respect to the criteria: reliability and

efficiency.

The results may be of help for a user, dealing with an integral that

must be computed numerically, to select a procedure from the available

software.

AMS Subject Classification: 65D30, 68B99.

- 2 -

1. Introduction

Comparison of a number of individual items of numerical software, such

as programs or procedures/routines, all of which are intended to solve

the same group of probLems, may be done for several reasons, for in

stance:

- to show that, or under which conditions, a new developed procedure

is superior to a number of already existing ones;

- to select a subset out of a set of procedures, e.g. in constructing

a program library;

- to supply information with the help of which someone (a user) can

select for his problem an appropriate procedure from a collection

of available procedures (a program library).

As is well known, it is very difficult to set up a comparative test

such that the results are of practical use; the major difficulties

being:

simplification; the appreciation of a piece of software depends on

a large number of aspects. But only a few of them, often simplified,

are involved in the test, and these may even not be the relevant

ones;

extrapolation; a comparison is performed on a set of test problems,

the conclusions are of value only if they hold outside this set;

interpretation; a comparison yields a lot of data. These data must

be compressed into a readable form without loosing too much infor

mation. The information from which conclusions have to be drawn may

be indistinct, a concl~sion may be very personal.

- 3 -

In this report we describe a comparative test performed on the set of

procedures for automatic quadrature available in Algol for general use

on the B7700 at the Computing Centre of the Eindhoven University of

Technology. The:'goal ef the test is the third above mentioned, that is,

to provide information for the users of the computing centre which may

help them to make their choice out of the software on hand.

Numerical quadrature belongs to the class of so-called "unreliable

arithmetic algorithms", cf. Lyness [6J. This means that, in principle,

the algorithm does not solve the problem, but rather furnishe~ an

answer that might be an approximation of the required solution. In ac

tual practice, the situation with numerical quadrature is not as bad

as suggested; there are procedures which quite often give an approxi

mation within the required accuracy. Yet there is a fundamental uncer

tainty about the obtained result, unless extra information, eg., the

magnitude of some derivative, available. "So the question is not

whether the program is right or wrong, but rather how well it performs"

Rice [10, p.5J.

The choice of a procedure for a particular problem made by a user is

subjective and possibly based on both quantitative criteria (eg.,

CPU-time, storage requirement, complexity, failure probability) and

non-quantitative criteria (eg., readability of the documentation,

users convenience, confidence in the designer). A comparison test is

necessarily restricted to quantitative criteria, although non-quan

titative criteria might be more pertinent to the user.

- 4 -

The test presented in this report treats the following two criteria:

- reliability, measured by the fraction of successes within a test

set of integrals.

efficiency, measured ty the number of function evaluations ~n the

computation of an integral.

With respect to reliability we register not only whether a result is

correct or not, but also the indication given by the procedure (if any)

about the correctness of the result. If this indication is adequate, it

should be taken into account by the user, if not, the user should not

be troubled with it. We give our conclusion on the adequacy of the

considered indications as well.

With respect to efficiency we agree that the number of function eval

uations is only a substitute, since it is the total CPU-time needed

for the evaluation of the integral that counts. Furthermore, the use

of the former to measure the latter can be misleading (Lyness [6J,

Robinson [llJ). On the other hand, it is very difficult, if not im

possible, to measure the real CPU-time in a computing system with

time-sharing facilities, and measuring time by allowing the test pro

gram the sole and uninterrupted use of the computer (Robinson [11])

may bt misleading as well.

Another practical argument for taking the number of function evaluations

as a measure of efficiency is that it allows one to spread the compu

tations of the test in time, since it is to some extent insensitive to

modifications of the computing system, as long as the arithmetic re

mains unchanged. Even the addition of a new procedure afterwards can

- 5 -

easily be done.

An important tool for a test is the set of problems on which the test

is performed. Test prob:ems may be distinguished in "real life" and

"artificial" problems, the former generally hard to obtain and often

expensive in computer time, the latter probably too specific and con

sequently unrealistic. In any case the test set will depend on the goal

of the test. We think that for our goal "real life" problems would be

ideal, i.e., perfect, but not likely to be achieved. So we have chosen

for sets of artificial problems. And we hope they are not too far from

real life.

The test method used belongs to the category of the so-called battery

experiments, the pros and cons of which are discussed several times

in literature, cf. Lyness [7J, [8J, Robinson [IIJ. A major difficulty

with the battery experiment is the abundance of numbers the reader

confronted with, cf. Kahaner [5J, Robinson [11J. We use the idea of a

pairwise comparison matrix, cf. Saaty [12J, to compress the information

with respect to efficiency.

In section 2 a description is given of the procedures considered. Tn section

3 we describe how the population of test integrals is built up. The frame

work of the test method is explained in section 4. In section 5 and 6 we

give numQrical results with our conclusions.

Robinson [llJcontains a test set of integrals which is comparable with

the set used for our test. We used this set to check the validity of our

results. This check is reported in section 7.

We end up with some concluding remarks in section 8.

- 6 -

2. Description of the procedures

The set of procedures to be compared in this report consists of all

ALGOL procedures, available for general use on the B7700 at the Com-

puting Centre of the Findhoven University of Technology, that claim

to compute the integ .. :al

b

I := f f(x)dx

a

for finite a and b and a "general" integrand f, with a specified accu-

racy.

In this section a description of these procedures is given, consisting

of a global indication of the method used and furthermore answering

the following questions:

- is the algorithm adaptive *);

- does the procedure ask for an absolute or a relative tolerance. In

the latter case, is the tolerance relative to

b

f f(x)dxl

a

b

or to I If(x)ldx;

a

- does the procedure indicate whether the required accuracy has been

reached, and if it does, in which way;

- what is the maximum number of function evaluations allowed by the

procedure.

*'A procedure is called adaptive if the sequence of points for which the

integrand is evaluated, depends on the function values encountered.

- 7 -

The B7700 NUMERALS, Numerical Analysis Program Library contains the

following three procedures for automatic quadrature. For a complete

description and references, see [14].

ROMBERGINT.

Computes the integral by means of Romberg-extrapolation with polynomials,

combined with interval bisection. The algorithm is non-adaptive. The b

procedure asks for a relative tolerance with respect to f If(x)ldx. a

There is no indication on whether the required accuracy has been reached.

At most 15 subdivisions are applied, so the maximum number of function

1 . . 2 15 1 32769 eva uat10ns 1S + = •

BULIRSCHINT.

Computes the integral by means of Romberg-extrapolation with rational

functions and subdivisions according to the so-called Bulirsch sequence

(h := (b - a)/N, with N = 2,3,4,6, ••• ). The algorithm is non-adaptive.

The tolerance is the same as in ROMBERGINT. Whether the desired accu-

racy has been attained, might be concluded from the parameter ERR of

the procedure. This parameter contains, in case of convergence, a con-

servative error bound and with no convergence the value 10 (but this

is not mentioned in the documentation). The maximum number of subdivi-

sions is IS, so the maximum number of function evaluations is 3 x 27 + I = 385.

SIMPSONINT.

The procedure is based on the adaptive Simpson method of McKeeman, but

employs interval bisection in place of trisection. The tolerance is the

- 8 -

same as ~n ROMBERGINT. There is no indication on whether the required

accuracy has been reached. Since the depth of recursion is maximal

24J the number of function evaluations is bounded, but may be very

large (less than 226 + ~).

The local library of the university, called THE LIBRARY. contains four b

procedures for the computation of J f(x)dx with a prescribed accuracy. a

For a complete description and references, see [16J.

INTEGRAL TRAPEX.

The procedure is based on Romberg-extrapolation with rational functions

and interval subdivisions according to the Bulirsch sequence. So the

algorithm is similar to BULIRSCHINT. The procedure asks for an absolute

and a relative tolerance, AE and RE respectively, and the convergence

criterion then is

b

lOll ~ AE + RE x J f(x)dxl·

a

A Boolean parameter reveals whether the required accuracy has been

reached. The procedure asks for the maximum number of interval sub-

divisions, say M. The maximum number of function evaluations then is

ZM/2+1 + 1 if M is even and 3 x 2(M-l)/2 + 1 if M is odd.

INTEGRAL.

An adaptive procedure based on Simpson's rule with varying step size.

An absolute and a relative tolerance AE and RE, respectively, are

- 9 -

required. The convergence criterion in the n-th step then is

x n

lOIn' 5: AE + RE x J If(x)ldx.

Xn - 1

A minimal steplength HMIN must be supplied. The procedure returns the

number of times an interval of length HMIN is encountered which does

not satisfy the convergence criterion. This number is used as an indi-

cation with respect to the attained accuracy. On account of HMIN the

number of function evaluations is bounded, but may be very large.

QADRAT •.

An adaptive procedure based on Gaussian quadrature formulae of high

order and a recursive interval bisection strategy. Tolerances and error

bound are the same as those of INTEGRAL.

The procedure uses an internal minimal steplength HMIN which is pro-

portional to RE, viz. HMIN = 32 x RE x (b - a). Similar to INTEGRAL

the procedure returns the number of subintervals of length HMIN for

which the required accuracy is not reached. So with respect to the

accuracy indication and the number of function evaluations, see INTE-

GRAL.

INTEGRML.

An adaptive procedure based on a 5-point Gauss-Legendre quadrature

formula and a varying step size strategy. The procedure asks for an

- 10 -

*) absolute tolerance. The procedure returns a realistic approximation

to the error in the computed value, which indicates whether the de-

sired accuracy has been obtained. The procedure uses a fixed internal

minimal steplength FJ:.llN = 10-4 x Ib - al. The maximum number of func

tion evaluations therefore is approx. 105

•

The NAG LIBRARY Mark 7 (ALGOL 60) contains the following four proce-

dures for automatic numerical quadrature. For a complete description

and references, see [15J.

D01AAA.

Computes the integral by means of the Clenshaw-Curtis method with the

O'Hara-Smith error estimate. This method is based on finite Chebyshev

series approximations, and is non-adaptive. The procedure requires a

tolerance relative to the value of the integral. A parameter IFAIL in-

dicates whether the accuracy has been reached. The maximum number of

function evaluations is M + 1 where M. the maximum number of subintervals.

which should be a power of 2 not greater than 128, must be supplied.

Remark. The weights and abscissae used in the Chebyshev series are cal

culated before the evaluation of the integral. Subsequent calls to the

procedure bypass this calculation. Therefore the first call of the

procedure is much more expensive than subsequent calls.

*) This is a quantity that is generally of the same order of magnitude.

It needs not to be an upper bound, and it may occasionally be a bad

approximation, especially when the accuracy is not reached.

- II -

DOIABA.

The procedure is based on Romberg-extrapolation with polynomials and

interval bisection,cf. ROMBERGINT. For tolerance and accuracy indica

tion, see DOlAAA.

The maximum number of function evaluations is M + 1 analogous to

DOlAAA. In this case M may not exceed 1024.

DOIACA.

Computes the integral by means of a method described by Patterson.

This method uses a family of quadrature formulae of increasing order

over the whole interval. so it is non-adaptive. The procedure requires

an absolute and a relative (to the integral) accuracy and will termi

nate whenever either accuracy has been reached.

The accuracy indication is realized by the parameter IFAIL. The pro

cedure supplies a realistic approximation to the error in the computed

integral as well. The maximum number of function evaluations in 255.

Remark. The procedure makes no use of the values of the integrand at

the endpoints of the interval.

DOIAGA.

This procedure evaluates the integral by the Clenshaw-Curtis method

and a double-adaptive interval subdivision strategy. An absolute tole

rance must be supplied. The accuracy indication is realized by the

parameter lEAIL, furthermore a realistic approximation to the error in

- 12 -

the computed value is returned.

The procedure uses a minimum steplength HMIN = Ib - al 2-m

, where m

must be supplied by the user. So the number of function evaluations

is bounded, but may bp very large.

The description given are summarized in the following table.

Method adap- re re acc. error max tive ae f(x) If(x)1 indic. estim. fu.ev.

ROMB ERG INT Romberg + 32769

BULIRSCHINT Romberg-Bulirsch + + + 385

SIMPSONINT Simpson + + < 00

INT. TRAP EX Romberg-Bulirsch + + + 2 !11+ 1 + 1

INTEGRAL Simpson + + + + < 00

QUADRAT Gauss + + + + < 00

INTEGRAAL Gauss + + + + < 00

DOlAAA Clenshaw-Curtis + + M+I

DOIABA Romberg + + M+I

DOIACA Patterson + + + + 255

DOIAGA Clenshaw-Curtis + + + + < 00

- 13 -

3. Description of the population of test functions

As already has been pointed out in the introduction, we have used

"artificial" problems. We gathered these problems from test sets of

others, from textbooks dnd tables, etc. We have chosen functions which

are finite in the ivterval of integration, easy to evaluate and of

which the integral is exactly known, or at least to machine precision.

Like others (Robinson [11J, Lyness [7J) we believe that a population

of all sorts of integrable functions is too broad and too various to

be used as a population for the comparison of quadrature procedures.

So we distinguished the following groups and performed comparisons

on each group separately (cf. Robinson [lIJ).

I. Smooth or well-behaved functions.

2. Functions with one or more sharp peaks within the interval.

3. Functions with a singularity in (one of) the endpoints.

4. Functions wi th a singularity wi thin the open interval.

S. Rapidly oscillating functions.

During the computations we encountered a number of functions for which

nearly all the procedures failed. These were unbounded functions (that

is why the functions in all five groups are finite) and a number of

functions we have gathered in a separate class of miscellaneous func-

tions. We called class

6. Pest functions.

In order to enlarge the number of functions in a simple way and to

avoid to some extent partiality in the choice of test functions we have

- 14 -

used the following transformation, due to O'Hara and Smith, cf [9J,

(3.1 ) := (qb - a)x - (g - I)ab

y (q - I)x - ag + b

This transformation ~ps, for any positive value of the parameter g,

the interval [a,bJ onto itself, and if the function g(q,x) is defined

by

(3.2)

then

a

g(q,x)

b

.- fey) .:!l dx

2 q(b - a)

«q - l)x - aq + b)2

b

J f(x)dx = J g(q,x)dx

a

f (~b - a)x - (q - l)ab) \ q - 1) x - aq + b '

For the effect of the transformation on the integrand, see Van der

Vorst [l3J.

Remark. From the detai.led results of the tests it follows that integrals

corresponding to the same basic function and different values of q

usually behave similar, but there are cases with considerable differences.

So in our opinion the use of the transformation (3.1) makes sense.

b Since the relative tolerance is related either to I f f(x)dxl or to

b a f If(x)ldx we used only positive functions (if necessary, we have made

a an integrand positive by adding a simple function). From (3.2) we see

- IS -

that if f is positive, then g is positive as well. Another argument

for the choice of positive functions is that the integral of a posi-

tive function is well-conditioned with respect to perturbations in

the function values.

A'list of all test functions, arranged 1n the various groups, 1S given

in Appendix A.

4. Framework of the test

In order to obtain a fair comparison the parameters of the va~ious

procedures must be chosen with care. The specific choices we have made

for that reason are the following:

- All integrands are positive functions.

If possible, then a relative tolerance £ is chosen; if not, then

the absolute tolerance £ fbf(x)dx is used. a

- Some procedures ask for the maximum number m of interval subdivisions,

others ask for a minimal steplength h . • Sometimes the documentation m1n

gives restrictions, sometimes recommendations, for the values of these

parame ters •

documentation

recommends/demands we took

INT. TRAPEX m::; 10 20

INTEGRAL h min ~ 10-5

Ib - al 10-6 Ib - al

DOlAAA 8 ~ m 128 128

DOIABA 4 ::; m ::; 1024 1024

2-m Ib - al

2 DOIAGA h min = m = - log(£)

m s 50

- 16 -

A test set consists of a group of Nintegrals, and one tolerance. In a

test run all integrals are evaluated with tolerance E by all (eleven)

procedures. This yields the following 11 x N test results:

{I, ~I, IND, NFE}.

I the computed approximation of the integral I

t,1 the relative error, viz. &1 := 1([ - 1)/11

IND the accuracy indication, default value is true

NFE the number of function evaluations.

From these results we first derive a reliability score. We distinguish

the following four cases, coded from 0 to 3:

code accuracy obtained

o

2

3

The reliabili ty score (in %) is

table.

Example. Group 4, £ = J 0-9 , N

Method

3 SIMPSONINT

4 INT. TRAPEX

5 INTEGRAL

6 QADRAT

7 INTEGRAAL

I 1 DOIAGA

true

true

false

false

registered

200.

0

95

I 1 3

45

51 46

18 41

68

indication

true

false

false

true

in a so-called reliability

2 3

5

82 4

37 18

3

41

4 27

- 17 -

From a reliability table we may derive conclusions about the reliability

of the procedure in solving the quadrature problem (columns (0 + 1)

versus columns (2 + 3»; and about the reliability of the accuracy

indicator (columns· (0 + 2) versus columns (1 + 3».

With respect to effLciency we dislike the presentation of a bulk of

numbers, from which everyone can, and indeed has to, make his own com-

parison, cf. Kahaner [5J, Robinson [IIJ. We would rather express

the relative efficiency of one procedure with respect to another by a

single value. To this end we proceed as follows.

Let Pk and P 9, be two procedures to be compared. Let J := {Ii 11 sis N}

be a set of integrals correctly computed by both Pk and P~. Let Fmi

be the number of function evaluations used by P , m = k,9" for the m

computation of Ii' and finally let Qk9,i := Fki/F2i , lsi s N. Then

the relative efficiency of Pk with respect to P2 may be expressed by

the geometric mean

(4.1)

Now let vm' m k,!(', be defined by

N 1

(4.2) v := (i~l F . \N" m mL )

then

(4.3) Ek2 vk/v9,'

So we may assign to each procedure the corresponding value (4.2) as an

efficiency number, of which only the relative magnitude counts.

- 18 -

Now if the procedures P , m = 1,2, ••• ,M are to be compared and we use m

one and the same test set J, then for any pair :k and P£ the above

statements hold. This implies, in particular, that we may use the num-

bers v , m = 1,2, ••• ,M as an efficiency score to compare the procedures m

mutually.

Taking one and the same problem set J for all procedures would quite

probably imply a considerable loss of information, since this set,

being the set of integrals correctly computed by all procedures, will

often be much smaller than any of the sets correctly computed by a pair of

procedures. This loss of information may be avoided in the following way.

Let J be a given set of N problems. Let Jk~ be the subset of J consis

ting of all integrals correctly computed by both Pk and P2 and let Nkt

be the number of problems in J k£* Let Ek£ be defined by

(4.4)

where the product is taken over all i with Ii E J k£. The numbers Ekt ,

I ~ k,~ ~ M constitute a positive matrix

(4.5)

In Appendix B we show that the positive eigenvector, w say, correspon-

ding to the eigenvalue p(E) m~y be used as a replacement for the vector

v defined by (4.2), and that

(4.6) fl : = LP~( E~):.-...-....,...;;.:;M M-]

- 19 -

called consistency, is a workable measure for the quality of this

replacement.

As a result, the efficiency score of a test set consists of the eigen-

vector w corresponding to the eigenvalue p(E) of the matrix E, and

the number ~. We record the results in a so-called efficiency table.

The lower triangle ~ ~ < k ~ M contains the fractions 100 x Nkt/N,

the upper triangle ~ k < £ ~ M contains the matrix elements Ekt ,

which describes the entire matrix, since Ek£ == E~~ for all k.L The

( M ) third last column contains the mean values of the rows, i.e., \ I Ek£ 1M t= I

for the k-th row, as an initial quess for the eigenvector w. The

second last column contains w. These two columns are normalized such

that the minimum value equals 1. The difference between the mean

value sum and w is also an indication for the consistency. The last

column contains 2log w. The value of p is given separately.

Example:

EFFICIENCY TABLE INPUT IDENTIFICATION ! GI~OUl;:' 4

EPS=@'-9 j.94 NUMBER OF SELECTED INTEGRALS:

METHOD 3 4 5 7

F<EL • EFF • NUMBI::F:S MEAN EIG. LOG. ROW VEC. SCAt SUM

--------------------------------------------------------------_._--3: SIMPSONINT 0.2 0.5 :L.O 0.5 1.6 1.6 1.6 0.69

4: INT.TRAPEX 14 '') .! ..: .. + \J 7.5 2.9 8.5 8.5 8.4 3.06

5: INTEGRAL 46 13 2.4 1.0 2.9 3.1 3.1 1.61

6: OArtFMT 98 14 46 0.4 1.5 1.5 1.4 0.46

7: INTEGF~AAL 60 :1.3 40 60 2. <J 3.1 3.0 1. 60

11: D01AGA 70 11 :35 70 45 1.0 1.0 0.00

CONSISTENCY: 6.163E··-03

- 20 -

In establishing the reliability and efficiency we assumed the computed

integral to be correct if the actual error is less than the given

tolerance 8. Many users are less rigorous, that is they rather accept

a result if the error is not much larger than the tolerance. To get

an idea of the infl~~nce a less stringent acceptance criterion may

have on the results of the test we investigated how often the error

in the computed value is only a little larger than E. To this end we

have considered, for every problem group and every tolerance, the set

of integrals which are evaluated with an error greater than E, and

we determined the percentage of integrals in this set with an error

less than 2E. These percentages are given 1n the following table.

group

e = 10-3 25

10-6 19

10-9 46

2

6

3

5

3

42

18

7

4

28

9

5

5 all

9 18

8 9

5 7

The overall percentage amounts to 10%. So we see that errors larger

than the tolerance are mostly considerably larger, and the results of

the test will hardly be influenced by a small enlargement of the accep

tance of the computed result.

5. Reliability and efficiency, numerical results

In this section we give the numerical results with respect to the

cri,teria reliability and efficiency, We remind that there are five

- 21 -

groups of integrals, namely:

1. Smooth or well-behaved functions.

2. Functions with one or more sharp peaks within the interval.

3. Functions with a sLngularity in (one of) the endpoints.

4. Functions with a singularity within the open interval.

5. Rapidly oscillating functions.

For every group we have approximately 20 basic integrals; these in-

tegrals are listed in Appendix A. From every basic integral we have

derived 10 integrals by means of transformation (3.1) with the follow-

ing values of q : 0.33, 0.50, 0.75, 0.78, 1.00, 1.05, 1.25, 1.50,

1.63 and 2.14. So every problem group contains approximately 200 in-

tegrals.

-3 -6 -9 We have used three tolerances: £ = 10 ,10 and 10 ,the last one

being about the smallest possible, since the machine-precision of the

Burroughs B7700 equals ~ x 8- 12 ~ 0.7 x 10- 11 .

On each of the pages 23-27 the results concerning reliability

and efficiency of one group are given. The upper half of a page con-

tains the reliability tables of all three tolerances. For convenience

of the reader we repeat that the columns of a table contain the

following scores, in percentages:

Column 0 integrals correctly computed, agreed by the indication used;

Column integral correctly computed, disagreed by the indication used;

Column 2 integral incorrectly computed, failure indicated by the pro-

cedure;

Column 3 integral incorrectly computed, failure not indicated by the

procedure.

- 22 -

The lower half of the page contains histograms of the last columns of

2 the efficiency tables, i.e. log w, with the methods arranged in de-

creasing order of efficiency. If for some method the percentage of

correctly computed inte;rals is too low (actually less than 5%), then

this method is excluded from the histogram. We repeat that for a pair-

wise comparison only the difference of the corresponding values of

2 log w counts: a difference of I cm in height of two bars in a histo-

gram corresponds to a factor 1:2 in efficiency, in favour of the method

corresponding to the lower bar.

With these results it LS easy to compare the performances of all

methods together with respect to one problem group.

On the next pages we discuss the performances of the methods on account

of these results. The discussion is far from exhaustive; the recommen-

dation of some procedures for each group, on account of both reliability

and efficiency, is rather personal.

Finally we give the complete efficiency tables of every test set of

integrals.

- 23 -

Group 1: Smooth or well-behaved functions

METHOD

1:

2:

3:

4:

5:

6:

7:

8:

9:

10:

11:

ROMBERGINT

BULIRSCHINT

SIMPSONINT

INT.TRAPEX

INTEGRAL

QADRAT

INTEGRAAL

D01AAA

DOl ABA

n01ACA

D01AGA

E= -3 10

0

100

49

100

99

100

100

100

100

99

100

100

2810411693715

~ ~

Reliability tables .~ ---- -

e-3 e-6 1 2 3 0 1 2

100

27 24 10 48 42

100

1 100

100

100

94

100

:I. 100

100

100

Efficiency histogra~s

2861110479315

e-9 3 0 1 2 3

l.OO

7 59 34

100

100

97 3

100

6 9"'" ,J 5

100

99 1

100

100

£:= 10-9

r- - 4

- 3

- 2

,.....""- _ 1

8 6 11 2 10 4 7 9 1 3 5

- 24 -

Group 2: Functions with one ore more sharp peaks, within the interval

METHOD

1:

2:

3:

4:

5:

6:

7:

8:

9:

10:

11:

ROMBERGINT

BULIRSCHINT

SIMF'SONINT

INT.TRAPEX

INTEGRAL

QADRAT

INTEGRAAL

D01AAA

D01ABA

D01ACA

D01AGA

E;= -3 10

-

--

386115107491

0

98

1

99

54

leO

23

66

13

49

28

73

(!-3 1 2 3 0 1

2 80

1 90 8 6

1 100

2 35 9 48 3

100

19 47 11 100

12 22 78 20

1 3 83 7 2

4 46 1 17 21

21 51 12 17

11 16 99

Reliability tables

£= -6 10

--

r--

..-.--

~

r--

,..-

-

ll6810375491

Efficiency histograms

(!-6 2 3

20

94

48 1

2

86 5

62

'71

1

. . ~ .. , ._-

0

78

96

30

34

87

46

6

7

84

s= -9 10

r--

-

1

1

14

4

8

13

8

-

(!-9 2 3

22

99

4

56

8 10

c' >J

4~S 3

93 1

87

8~j

16

- - 5

-4

_ 3

-2

- 1

- 25 -

Group 3: Functions with a singularity 1n (one of) the endpoints

METHOD

1 :

2:

3:

4:

5:

6:

7:

8:

9:

10:

11:

ROMBERGINT

BULIRSCHINT

SIMPSONINT

INT.TRAPEX

INTEGRAL

QADRAT

INTEGRAAL

D01AAA

D01ABA

D01ACA

D01AGA

E;:: -3 10

0

100

14

:1.00

'72

94

67

100

98

98

100

99

10 8 6 11 2 4- 3 9 1 7 5

Reliability tables

~-'3 ~--6

1 2 3 0 1 2

95

9 73 4 1 1 97

1.00

28 55 29

6 46 40 9

7 26 44 56

70 29

2 FJ6 5

l 1 42 10 48

100

1 97

E,fficiency histograms

Q r-

.----

r-

,--'--

-,-

-

I 8 10113 6 7 5 4- 9 1

~-9

3 0 1 2

5 C::'") >J"-

1 1 99

95

16 lO 2 86

... >J 2~~~ 7 46

~56 44

1 12 56 3") .:..

9 34 1.4 c'''' >J.,.

7 2 9l

71 5 24

3 98

s= 10-9

Q ,....

,--

,--'----

,--

-

--

810116735491

~~

48

5

2

'")I:!' ~ .. .J

2

4

_ 3

- 2

_ 1

- 0

- 26 -

Group 4: Functions with a singularity within the open interval

Reliability tables

e-3 e-6 e-9 METHOD 0 1 2 3 0 1 2 3 0 1 2 3

1: ROMBERGINT 97 3 94 6 50 50

2: BULIRSCHINT 12 9 75 4 2 2 96 1 1 98

3: SIMPSONINT 100 100 95 0::' .J

4: INT.TRAPEX 73 27 33 10 38 19 11 3 82 4

5: INTEGRAL 96 4 70 14 16 45 37 1.8

6: QADRAT 48 12 1 39 43 53 2 2 51 46 3

7: INTEGRAAL 89 11 46 43 6 5 18 41 4:1,

8: D01AAA 34 66 2 8 90 1 87 12

9: D01ABA 93 1 6 34 20 45 1 5 7 88

10: D01ACA 82 1 4 13 15 7 76 2 1 99

11: D01AGA 73 1 26 67 2 3 28 68 1 4 27


E= 10-6

o r-

- 4

-r- _ 3

r-

-r-r--

- 2

r-r- r--

- 1 r-

r-

[ I - 0 8362J1Jt91D175 3J16571D941 n 6 3 7 5 9 4 1

- 27 -

Group 5: Rapidly oscillating functions

METHOD 0

1: ROMBERGINT 96

2: BULIRSCHINT 7

3: SIMPSONINT 87

4: INT.TRAPEX 79

5: INTEGRAL 96

6: QADRAT 66

7: INTEGRAAL 90

8: D01AAA 73

9: D01ABA 82

10: D01ACA 100

11: [l01AGA 91

r--

_r-"

....-,.....-

r-

,......--

-

8 2 10 6 3 4- Jl 7 9 5 1

Reliability tables

(!-3 (!-6 1 2 3 0 1

4 87

15 75 3 4 26

13 94

21 83 1

2 2 87

16 4 14 87 13

10 84 7

2 25 80 2

6 12 67 10

88 8

7 2 9:3


E: ::: 10-6

,..-r-

...-

....-

r--

r-,......

r-

.--

I"

2

70

14 ,

13

8

15

23

3

7

(!-9 3 0 1 2 3

13 81 19

7 36 57

6 86 14

2 71 5 20 4

48 39

85 10 5

1 74 5 20 1

3 72 2 21 5

42 15 43

1 76 1 22 1

95 1 4

Q -

r-"---

,..-

_ 3

,..- 2 .--

...--,.....--

1

-r o -8 10 6 2 11 7 4- 3 9 10 5 8 10 2 6 11 7 4- 9 1 2 5

- 28 -

Discussion

General remarks.

I. For the sake of limitation of the CPU-time, procedures are not

allowed to use more than 35000 function evaluations per integral.

If this maximum has been reached, then the computation of the inte-

gral is aborted and no results are registered. Consequently, the

Sum of the reliability scores may be less than 100%, see for in

-9 stance: group 2, ~ = 10 ,INTEGRAL or QADRAT.

2. The procedure BULIRSCHINT is useless because of its bad reliability

score for all groups. Therefore BULIRSCHINT will be left out of

the discussion in the sequel.

3. The behaviour of QADRAT fore= 10-3 is remarkably bad in all groups

except group 1. It turns out that, if the required accuracy has

not been achieved, then the procedure has used only a few function

values. This suggests that the termination criterion is not ade-

quate for low accuracy.

4. The first call of DOlAAA requires a lot of extra computations (see

page ]0). Therefore, in the discussion of the individual groups

this procedure is not recommended in the cases when its efficiency

score belongs to the best.

5. If the eigenvector w has a large component, corresponding to one

procedure (e.g. ROMBERGINT), then the consistency may improve con-

siderably if this procedure is omitted from the efficiency table.

On the other hand, this omission hardly influences the other compo-

nents and has therefore no effect on the assessment of the procedures.

- 29 -

2 Example. The consistency and the components of log w 1n group 3

with E = 10-9, with ROMBERGINT (see pag40, last column) and with-

out ROMBERGINT are, respectively,

consistency I 3 4 5 6 7 8 9 10 I 1

6.51 10-2 6.61 2.26 3.31 3.24 0.98 1. 91 0.00 4.01 0.02 0.90

1.1310-2 ---- 2.23 2.97 3.18 1.00 1.90 0.00 3.50 0.03 0.89

Remark::: on the individual groups.

Group I: Smooth or well-behaved functions.

I. All procedures behave very reliably.

The procedures based upon Romberg-extrapolation with polynomials

or Simpson's rule are significantly less efficient than the other

ones.

2. Recommended procedures.

QADRAT, DOIACA, DOIAGA.

Group 2: Functions with one or more sharp peaks within the interval.

I. SIMPSONINT is very reliable, followed by ROMBERGINT and DOIAGA.

QPDRAT and DOIAGA are the most efficient procedures.

The non-adaptive procedures are evidently not suited for this type

of integrals.


DOIAGA; SIMPSONINT if reliability is highly wanted; QADRAT for high

accuracy only.

- 30 -

Group 3: Functions with a singularity in (one of) the endpoints.

1. SIMPSONINT and DOIAGA are highly reliable, DOIACA is only for

-9 E = 10 less reliable, yet very faithful, i.e., indicates properly

that the result is incorrect.

The procedures based upon Romberg-extrapolation are not suited for

this type of integrals.

The most efficient procedure 1S DOIACA. Remind that this procedure

makes no use of the endpoints of the interval.


SIHPSONINT, DOIACA, DOIAGA; QADRAT for high accuracy only.

Group 4: Functions with a singularity within the open interval.

I. SIHPSONINT is the only procedure with high reliability for all

tolerances. The rather bad reliability of DOIAGA for all tolerances

is remarkable. This bad performance turns out to be neither re-

stricted to some basic problems of the group, nor to specific values

of q in the transformation (3.1). The incorrectly computed integrals

with indication IFAIL = 0 (column (3) of the reliability table) are

more or less homogeneously distributed over the group.

The non-adaptive procedures are not suited for this type of il1te-

grals.

SIMPSONINT, QADRAT and DOIAGA are significantly more efficient than

the other procedures.


SIMPSONINT; QADRAT for high accuracy only.

- 31 -

5: Rapidly oscillating functions. ----~~

I. The procedures ROMBERGINT, SIMPSONINT, INTEGRAAL, DOIACA and DOIAGA

are reliable for all tolerances.

The procedures based upon Romberg-extrapolation with polynomials

or Simpson's rule are the least efficient. The most efficient of

the reliable procedures is DOIACA.


DOIACA; QADRAT and DOIAGA for high accuracy only.

EFFICIENCY TABLE INPUT IDENTIFICATION : GROUP 1

EPS:::e-3 NUMBER OF SELECTED INTEGRALS: 220

REL .. EFF.NUMBERS MEAN EIG. LOG. ROW VEC. SCA.

METHOD 1 2 3 4 5 6 7 8 9 10 11 SUM -------------------------------------------------------------------------------------------

1: ROMBERGINT 2.1 1.3 1.9 0.9 1.8 1.0 2.1 1.4 2.0 1.9 2.:' 2.2 1.13

2: BULIRSCHINT 76 0.5 0.9 0.4 0.8 0.5 0.9 0.7 0.9 0.9 1.0 1.0 0.00

3: SIMPSONINT 100 76 1.5 0.7 1.4 0.8 1.7 1.1 1.6 1.5 1.8 1.7 0.81

4: INT.TRAPEX 99 75 99 0.5 1.0 0.5 1.1 0.7 1.0 1.0 1.1 1.1 0.20 IoU

5: INTEGRAL 100 76 100 99 2.0 1.1 2.3 1.5 2.1 2.0 2.4 2.4 1.25 N

6: QADRAT 100 75 100 98 100 0.6 1.2 0.8 1.1 1.0 1.2 1.2 0.29

7:: INTEGRAAL 100 76 100 99 100 100 2.1 1.4 1.9 1.8 2.1 2.1 1.10

8: D01AAA 100 76 100 99 100 100 100 0.7 0.9 0.9 1..0 1.0 0.04

9: [l01ABA 99 76 99 97 99 98 99 99 1.4 1.3 1.6 1.6 0.64

10: D01ACA 100 76 100 99 100 100 100 100 99 1.0 1.1 1.1 0.16

11: J)olAGA 100 76 100 99 100 100 100 100 99 100 1.2 1.2 0.22

CONSISTENCY: 1.827£-04


EF'S=e-6 NUMBER OF SELECTED INTEGRALS: 220

REL.EFF.NUMBERS MEAN EIG. LOG. ROW VEC. SCA ..

METHOD 1 2 3 4 5 6 7 8 9 10 11 SUM

-------------------------------------------------------------------------------------------1: ROMBERGINT 3.0 1.0 2.1 0.7 3.2 1.9 3.3 1 .. 1 2.7 2 .. 9 3 .. 3 3.2 1 .. 69

2: BULIRSCHINT 58 0.3 On8 0 .. 2 0 .. 9 0.6 0 .. 8 0.4 0.8 0.9 1 .. 0 1.0 0 .. 00

3: SIMF'SONINT 100 58 2.1 0.6 3.1 1.8 3.2 1.1 2.6 2 .. 9 3.3 3 .. 2 1.68

4: INT.TRAF'EX 100 58 100 0 .. 3 1 .. 5 0.9 1.5 0.5 1.2 1.4 1.5 1.S 0.S8 w

5: INTEGRAL 100 S8 100 100 4 .. 9 2 .. 9 5.0 1.7 4.1 4.5 5.1 5.0 2 .. 33 w

6: QADRAT 100 S8 100 99 100 0.6 1.0 0.3 0.8 0.9 1.0 1.0 0.03

7: INTEGRAAL 94 55 94 93 94 93 1.7 0.6 1.4 1.6 1.7 1 .. 7 0.78

8: D01AAA 100 58 100 100 100 100 94 0.3 0 .. 8 0.9 1.0 1.0 0.02

9: [l01ABA 100 58 100 100 100 100 94 100 2.4 206 3.0 2 .. 9 l .. SS

10: [l01ACA 100 58 100 100 100 100 94 100 100 1.1 1.3 1.2 0.30

11: [l01AGA 100 58 100 100 100 100 94 100 100 100 1.1 1 .. 1 0.16

CONSISTENCY: 6.321E-04

EFFICIENCY TABLE INPUT IDENTIFICATION GROUP :;:

EPS=~-3

NUMBER OF SELECTED INTEGRALS: 140

METHOD 1

1: ROMBERGINT

3: SIMPSONINT 97

4: TNT. TRAPEX 56

5: INTEGRAL 98

6: QADHAT 4'') "-

"7 n .' . I NTEGRr.IAL. '7 '

l b

B: [to:l. F1At':1 14

9: [101ABA 53

10: D01ACA 49

U. = DO:l.AGA 73


"1 'J

27.0

'36

99

4') ,-

77

14

53

49

73

4 ~:j

1.(114.0

o u ~.~ 0 .. 5

3.3

:::;6

:35 42

44 'i'B

6 14

46 ~;3

36 49

48 74

6

B.5

LO

4 .. 5

2.0

T7

11

4r) "-

:37

41

('y :LO ""

'" 8 1:1.

5.4 14.0 :I. • :,'$ ;5.7 16.::;

o ~1 . .;.. l'l2 0.1 0.4 0 .. 9

1.4 3.1 O. '7 1..5 4.7

0 .. 5 2.9 O. :.3 0.8 L?

0.2 1 • :3 0.1 0.4 0 .. 8

4.0 0.6 1.7 3 .. 9

13 0.1 0.6 0.6

47 11 2.6 5.6

4':' •• J 1.4 41 2.0

62 14 49 46

REL.EFF.NUMBERS MEAN EIG. LOG. ROW VEC. SCAD SUM

1<J.~~ 17.:\. 4.10

:1..1 LO 0.00

t:- 1::-oJ tt • .J 505 2.47

2 .. 3 2.0 1.02

1.2 L2 () .. 22

4.6 4 .. :1. 2 u O~j

1.0 1..:1. 0.10

7 t:' .J 7.7 2.94

2.5 2.6 1.3(1

1..4 1 • :;,~ 0.29

""

UJ LI'I

EFFICIENCY TABLE INPUT IDENTIFICATION GROUP 2

E:PS=e-.. t:> NUMBER OF SELECTED INTEGRALS: 140

REL.. EFT. NUMBEF(S MEJiN EIG. LOG. ROW VEC. SCAR

METHOD 1 3 4 ~) 6 7 8 <1 10 11 SUM

--------------------------------------------------------------------------------------1: FWMBERGINT LL<1 ,f") P) 7.0 30.4 12.5 19.9 :L7 (/.4 30 .. 6 2~j .. 6 24. :5 4.60 ~ • ., 4,...

3: SIMPSONINT f.1O o '7 . ..; 0.5 '"j -y &.. ... :) o • <y 1.7 0.3 1.3 -) '"'1

A- ....... ' 2ltl '1 '') ,,;;., .. A .... 1.11

4: TNT. TRAPEX 51 51 1.6 7.1 3.0 7.0 0 .. 8 3.2 6. <1 6.B 7.3 2.86

5: INTEGRAL 80 100 51 4.1 1.7 4.4 0.6 r) , ..:,."0 4.3 4. :3 4.3 2.0<1

6: QADRAT 80 100 51 100 0.4 1.2 () • 1 o '7 . , 1.0 1.1 1..0 0.05 w

'" '?~ INTEGf-\:AAL. 80 98 51 98 98 2.9 0.3 l.6 2.6 :::: h C) ...., ,::-

.-:.~ n ... J :1. .. 33

8: D01AAA 9 9 9 9 (1 9 0.1 0.9 1.0 1.0 1.0 0.05

<1: D01ABA 38 38 38 38 38 :58 8 5.6 7.5 8.9 9.3 3 .. 22

10: DOIACA :!~3 29 '') 1 ';'.,t. 29 29 29 9 21 1.6 1.5 1. '7 0.76

11.: D01AG~1 79 99 :'51 99 99 97 9 3D 29 :1,.0 1..0 0.00


...

EFFICIENCY TABLE INPUT IDENTIFICATION GFWUP 2

EPB:::f,::! .. ··? NUMBER OF SELECTED INTEGRALS: 138

1:~El. EFF "NUMBER~3 MEAN EIG. U)(3. ROW VEC. SCA.

METHOD , 3 4 J::* 6 "1 8 9 10 11 SUM .L ,.} I

--------------------------------------------------------------------------------------1: ROMBERGINT 6.3 ~L 1 1.4 39.4 12.4 58n8 2.2 24.1 42.1 39.9 32.9 5.04

3: SIMP~3()NINT 79 o • 7' 0.4 6" 1 2.8 4.0 1.2 4.8 '7" :2 6.0 6.2 ?63

4 = I NT . TI;:APEX 44 44 0.4 7.1 3.4 LLO 0.9 6" ;2 7.4 8.6 B.l 3.01

I::' I NT EG f-t: A I... 28 3"'" :~~6 15.5 7.6 10.0 3.7 11.9 14.7 :1.4.6 :1.6.7 4.06 -.. J :: .J

6: QADF~AT 79 91 44 35 0.5 On? 0.2 0.8 :l..2 LO LO o .0:.5 w -.J

"7 II INTEGRA(-1L. 51 54 42 30 ~54 2 .. 0 0.4 1.7 2., :~ ;? .. :~~ r~ r) 1.16 l • .... "4

8: DO:lAAA 7 7 I". 7 7 7 0.1 1.0 LO :I. • 1 1.1 O.OS -..}

9: DOl ABA 13 13 13 1. ~~ 13 12 :1. 4.3 ~:i .. 0 5.9 6.n 2.77

10: DOl.ACA 14 15 12 14 1:5 le:-. ..1 ...•

" '7 1.l 1.2 1."2 0.32

11: D01AGA 76 86 44 3~5 El6 54 /' :1.3 15 1.0 :1..0 0.00


EFFICIENCY TABLE INPUT IDENTIFICATION mwup 3

EPS""e"·3 NUMBER OF SELECTED INTEGRALS: 200

REL. EFF • NUMBEf-<S MEAN I::.IG. LOG. ROW VET. SCAN

METHOD 1 2 " .. ""J '*

r.:-' .•. } 6 "? 8 9 1.0 1.1 SUM

-------------------------------------------------------------------------------------------1.: ROMBEI:;:GINT 2.6 2"7 2.9 On :'i :3 • f:l 1.0 4. "7 1.4 5.0 3.8 4.4 4.6 ~? • 1 (}>

2: BULIF<SCHHH "-Z L....J 0.6 1.0 0.1 1.0 0.4 Ll 0.5 1.1 0.9 1.2 1.3 0.41

3: SIMF'BONINT 100 23 1.3 o ') . ~" L6 0.4 1.8 o 0:.:-• ..J 1.9 1.4 1.8 1.8 0.88

4: INT. TRAF'EX 72 19 72 0.1 1..2 0.3 1 • :~ o "'" • ..J 1..3 1.l. 1.4 1.4 0.50

5: INTEGRAL 100 23 100 ., ... , I r::. "7.9 1.8 9.5 2.7 9 r.:-...J "7.2 9.5 9 .. ~:! .. 'ill:"

V .. A,' •• """ W ():)

6: WHiRAT 74 21 74 5B ""1 , 0.3 1.0 0.4 1..1 0.9 1.1 i 'i 0.23 I

I "+ .1. q 14M

7: I NTEGf-<AAL 100 2] 100 72 100 74 ~5 .. 0 1.5 5112 3.9 4.6 4.7 2.24

8: D01AAA 98 -'"Z 98 72 98 71 98 0.3 1.0 0.8 1.0 1.0 0.04 ........ ..1

9: D01ABA ('19 23 99 71 99 72 99 97 3.5 2.6 3.:1. 3.2 1.68

10: D01ACA 100 ... ) .. " ~ ... } 100 ?:~ 100 74 100 9B 99 0 .. 8 :1. .. 0 1..0 0 .. 00

11: DO:l.AGA 99 23 ("19 71 99 74 99 9"7 SiB 99 L~~ 1.3 0.36



EPS=(!-6 NUMBER OF SELECTED INTEGRALS: 200

REL.EFF.NUMBERS MEAN EIG. LOG. ROW VEC. SCAM

METHO[I 1 3 4 5 6 7 8 9 10 11 SUM ----------------------------------------------~---------------------------------------

- 1: ROMBERGINT 21.0 3.4 6 .. 5 16.2 7 .. 3 43.5 1.6 44 .. 0 24.2 46.1 43_0 5.43

3: SIMPSONINT 95 0.4 0 .. 3 0.8 0.3 2 .. 4 0.2 2.0 1.2 2.4 2.4 1 .. 29

4: INT.TRAPEX 55 55 1.0 2.5 1.1 8.1 0.5 7.1 4 .. 2 7.8 8.0 3 .. 01

5:: INTEGRAL 85 86 53 2.3 1.0 7.2 0 .. 6 6.2 3.6 7.1 7.3 2.87 w

6: QADRAT 94 100 55 85 0.4 3.0 0.2 2.6 1..5 3.0 3.0 1.61 1.0

7: INTEGRAAL 94 99 54 85 99 6.9 0.5 5.8 3.4 6.8 7.0 2.80

8: It01AAA 85 86 55 77 85 85 0.1 0.9 0.5 1.0 1.0 0.00

9: [l01ABA 52 52 46 51 51 51 51 12.3 7.5 13.9 15.2 3.93

10:: [101ACA 95 100 55 86 100 99 86 52 0.6 1.1 1.1 0.19

11: D01AGA 92 97 55 83 97 96 84 51 97 2.0 2.0 0.97



EPS=e-9 NUMBER OF SELECTED INTEGRALS: 200

REL.EFF.NUMBERS MEAN E1G. LOG. ROlJ VEC. SCAM

METHOD 1 3 4 5 6 7 8 9 10 11 SUM --------------------------------------------------------------------------------------

1: ROMBERGINT 29.4 3.4 9.5 67.4 34.4120.3 1.7131.6 70.4 118.1 97.9 6.61

3: SIMPSONINT 52 0.7 0.5 1.8 1.2 4.3 0.7 3.6 2.2 4.0 4.8 ~.26

4: INT .. TRAPEX 12 12 0.9 4.8 2.0 8.5 0.5 9.5 4 .. 1 8.3 9.9 3.31

5: INTEGRAL 29 29 11 4.5 2.2 9.0 0.9 10.1 4.8 9.0 9.4 3.24 .c::--

6: QADRAT 52 95 3.2 29 0.6 1.9 0.1 1.8 1.2 1.9 2.0 0.98 0

7: INTEGRAAl 51 68 11 28 68 3.7 0.4 3.2 1.9 3 .. 5 3.8 1.91

8; D01AAA 48 48 12 29 48 47 0.1 1.1 0.6 1.0 1.0 0.00

9: D01ABA 9 9 8 8 9 9 9 12.1 5.5 11.6 16.1 4.01

10: D01ACA 52 76 12 29 76 68 48 9 0 .. 6 1.0 1.0 0.02

11: D01AGA 51 93 12 28 98 67 47 9 75 1.7 1.9 0.90



EF's=e-3 NUMBER OF SELECTED INTEGRALS: 200

REL.EFF.NUMBERS MEAN EIG. LOG. ROY VEC. SCA.

METHOD 1 2 3 4 5 6 7 8 9 10 11 SUM

-------------------------------------------------------------------------------------------1: ROMBERGINT 1.9 3.5 1.4 0.8 2.7 0.9 3.4 1.4 1.4 2 .. 2 3N~ 3.5 1.79

* 2: BULIRSCHINT 21 0.9 0.8 0.3 0.9 0.3 1.2 0.7 0.4 0.7 1.3 1.3 0.41

3: SIMPSONINT 96 21 0.4 0.2 1..0 0.3 1.5 0.4 0.4 0.7 1.2 1.2 0.22

4: I NT • TRAF'EX 71 19 73 0.5 1.6 0.6 2.8 1.0 0.9 1.4 2.3 2.3 1.22 ~

5: INTEGRAL 97 21 100 73 3.7 1.1 4.8 1.7 1.8 2.9 4.7 4.6 2.20 -6: QADRAT 57 17 59 43 60 0.3 1.1 0.5 0.5 0.8 1.3 1.3 0.37

7: INTEGRAAL 86 21 89 65 89 56 4.1 1.5 1.6 2 .. 6 4.1 4.0 2.02

8: D01AAA 33 11 34 26 34 25 31 0.4 0.5 0.7 1.0 1.0 0.00

9: I101ABA 93 21 93 71 93 56 84 32 1.0 1.6 2.5 2.5 1.31

10; [I01ACA 80 20 82 60 83 52 74 28 79 1.6 2.6 2.6 1.36

11: D01AGA 70 17 73 55 73 47 66 28 68 60. 1.6 1.6 0.68


EFFICIENCY TABLE INPUT IDENTIFICATION :, GROUP 4

EPS=e-6 NUMBER OF SELECTED INTEGRALS: 200

REL.EFF.NUMBERS MEAN EIG. LOG. ROW VEC. SCA.

METHOD 1 3 4 5 6 7 9 10 11 SUM ---------------------------------------------------------------------------------

1: ROHBERGINT 26.4 1.5 6.8 17.1 5.3 1 .. 8 1.6 18.1 22.1 18.4 4.20

3: SIHf'SONINT 94 0.1 0.3 0.6 0.2 0 .. 2 0.3 0.8 1.0 1.0 0.00

4: INT.TRAF'EX 43 43 2.5 4.4 1.5 1.2 1.1 5.6 7.0 7.4 2.89

5: INTEGRAL 80 84 43 1.8 0.6 0.5 0.8 2.2 2.9 2.9 1.56 .j::-

6: QADRAT 91 96 42 81 0.3 0.3 0.5 1.2 1.6 1.6 0.67 N

7: INTEGRAAL 87 88 38 75 86 0.8 1.1 3.9 5.0 4.8 2.27

9: D01ABA 54 54 37 54 52 49 1.2 4.6 5.8 6.3 2.65

10: D01ACA 22 22 16 22 22 19 22 3.1 3.9 5.1 2.34

11: D01AGA 65 68 31 59 66 61 40 16 1.3 1.3 0.34



EPS=(!-9 NUMBER OF SELECTED INTEGRALS: 194

REL .. EFF .. NUMBERS MEAN EIG. LOG .. ROW VEC. seA.

METHOD 1 3 4 5 6 7 9 11 SUM ----------------------------------------------------------------------------

1: ROMBERGINT 33.3 2.0 15.1 42.0 17.9 1.8 46.5 48.5 39 .. 4 5.30

3: SIMPSON!NT 51 0.2 0.5 1 .. 0 0.5 0.4 1.6 1.6 1.7 0.78

4: INT.TRAPEX 14 14 2.6 7.5 2.9 1.0 8.5 8.5 9.4 3.23

5: INTEGRAL 35 46 13 2.4 1.0 0.6 2.9 3.1 3.1 1.63 .j:-.

6: QADRAT 51 98 14 46 0.4 0.2 1.5 1 .. 5 1.3 0.40 w

7: INTEGRAAL 43 60 13 40 60 0 .. 4 2 .. 9 3.0 2 .. 9 1.52

9: D01ABA 12 12 9 11 12 11 5 .. 7 6 .. 0 8 .. 1 3 .. 01

11: D01AGA 39 70 11 35 70 45 10 1 .. 0 1.0 0.00



EPS=~-3

NUMBER OF SELECTED INTEGRALS: 200

METHOD 1 2 '7 ' • .1 4 !=) 6 7 8 9 :1.0 1.1

t: RDMJ.3ERGINT 2.9 6.5 2.2 2.0 4.0 3.6 6.8 1.2 7.6 3.0

2: BULIRSCHINT 10 0.4 0.9 0.2 0.6 0.4 1.0 0.4 0.6 0.3

3: SIMF'SDNINT 81 10

4 = INT. TRAPEX 67 9 ~S6

5: INTEGRAL 92 10 84

6: OADRAT 57 9 46

7: INTEGRAAL 85 9 76

8: D01AAA 4') 8 ~n

9: DOtABA 73 9 59

10: D01ACA 94 10 f:\4

11: D01AGA '77 9 67

CONSISTENCY: 3.481[-02

0.6 0.3 1.4 0.5 2.4 0.3 1.2 0.7

0.6 1.6 1.2 2.9 0.5 2.0 1.2

66 3.9 1.7 6.8 1.0 3.8 2.1

49

60

37

61

6tl

63

c·.!:' ;;;J "I

87

4:1

71

96

77

0.5 1.7 0.3 1.0 0.5

51 3.9 0.5 2.2 1.1

35

i~'r ... ,.J

57

~:;4

40

64

89

72

0.2 0.6 0.3

41 3.9 2.0

42 73 0.5

41 67 79

REL..EFF.NUMBERS MEAN EIG. LOG. ROW VEC. SCAR SliM

8,,3

" '") .I, .....

2 n ~3

:;.~., 9

6.4

1 .7

3 .. 5

1.0

5,,6

1.7

3,,4

9·,0 3.16

1" 5 0" ~;f.l

2 .. 1 1.10

3.2 L6'7

6.1 2.60

1.7 0 .. 79

3 .. 4 1.15

LO 0 .. 00

5"9 2 .. ~:j5

1.6 0.71

3.2 1. bB

~ ~

[


EPS"::~-6

NUMBER OF SELECTED INTEGRALS: 2()0 REL.FFF.NUMBERS MEPIN EIG. LOG. ROW VFC. SCAD

METHOD 1 2 3 4 5 6 7 8 9 to 11 SUM -------------------------------------------------------------------------------------------1: ROMBERGINT 4.7 3.1 2.9 L 1 10 .. 9 6.2 10.5 1«1 13.3 6.5 14.0 13 .. 8 3.79

2: BULIRSCHINT 29 0.3 0.7 0.1 :1..0 0.6 1.7 0 .. 2 1.4 0» el t ,.8 :1..8 0.88

3: SIMPSONINT 79 28 1 .., . , 0.4 3.2 2.0 6.2 0.9 4.1 '") '") ..:.." ,-;,.. 5" <y 5.7 '") '::-1

4· .... .... J J,

4: INT.TRAPEX 64 27 ~j'7 0.2 .., " .... "4 1.3 3.7 0 .. 4 2.9 1.3 :3 It 6 3.5 1-82 ~ V1

5: INTEGRAL 85 29 '78 64 9.'7 5.6 :1.4.9 2.0 :1.2.2 5.8 l.~j", 4 14,,8 3.89

6: QADRAT 87 29 93 64 85 0.6 1.6 0.2 1.2 0.6 1.6 :1..6 0.67

7: INTEGRAAL 87 29 84 64 85 91 2.9 0.4 2.2 1.1 2 .. 9 2 .. 8 1.48

8: DOl.AAA 63 29 58 59 63 64 63 0.1 0.7 0.4 LO :l .0 0 .. 00

9: n01ABA ei4 ~~8 47 t::'" ,.J4 54 ~.:i4 54 ~~;2 5.8 2~5 7.5 7 .. 7 2.94

10: DOlf:ICA 86 29 86 64 85 94 88 64 54 0.5 1.3 :1..3 0.34

11.: DOlAGA 8'7 29 82 64 P"" .Id 9() 138 1.;3 54 B8 " . ., ~_ II .r 2,,6 1.:W

CONSISTENCY: i.115E-02

EFFICIENCY TABLE INPUT IDENTIFICATION mmup 5

EPs::::e-9 NUMBER OF SELECTED INTEGRALS: 195

RI:.L. FFF .. NUMBf·.RS ME?IN ETG. LOG. RDW VEe. SCAn

METHOD '1 2 3 4 5 6 7 8 9 10 11 SUM ... -------------------------------------------------------------------------------------------

1: RClMBERGINT 6 .. 0 1.2 3.9 0.3 :1.2.1 "7.6 16.7 1.4 22.6 10 .. 0 23·.7 ,.,,o) t::" .. ·_A .. ft ...J 4.49

2: BlILIRSCHINT 33 0.1 0.7 0.1 0.8 0.5 2.3 0.2 1.6 0.7 2.3 :~ 11 4 1.26

3= SIMPSONINT /9 ~~2 5.3 0.4 9.9 6.1 25.3 2 .. 4 18.0 8.1 25n5 23.7 4. ~.;i6

4: INT.TRAPFX :::;4 30 :;3 0.1 1.8 1.1 4.4 0.4 3.3 1.4 4.4 4 .. 3 2 .. 11. .,.. 0\

5: INTEGRAL 28 1.8 ~:.~8 27 19.5 11.4 46.9 4.1 3~~.7 1,6 .. 2 48·.3 49.5 5.6:.3

6: QADRAT 81 33 81 54 28 0.6 2.6 0.3 1.8 0.8 2.6 ''l> "" 110', .... J 1 .. 30

7: INTEGRAAL 79 ~52 79 S2 27 81 4.4 0.4 :5.0 1.;3 4 • :3 4.1 ;.~. 02

8: [l01AAA 50 33 50 47 2B 50 50 0.1 0.7 0.3 1.0 LO 0.00

9: nOl.ABA 37 29 37 :36 24 ~57 ~57 37 7.4 3.3 10 .. 6 1.0.9 :3.45

10: DOl.ACA 76 :53 '76 :'51 2B 76 '7/.) 50 37 0.5 1.4 1.3 0.43

11: D01AGA 80 32 81 !:;3 27 94 ~:W !':jO ~f,7 '76 3 .. ':;~ 3,,0 1 n :57


- 47 -

6. Relevance of accuracy indicators

In this section we discuss the accuracy indicators we have used in

the test. Some of them, like IFAIL, are described as such in the docu

mentation, other One?, like }1 in QADRAT or ERROR in INTEGRAAL, have been

adopted as such by us.

In order to investigate the relevance of a certain parameter as an

accuracy indicator we have established from the reliability tables

(cf. pp. 23-27) for every group and tolerance the following numbers:

The percentage of cases in which the parameter has given the wrong

indication, i.e. column (1) + column (3);

The percentage of cases in which the indication "accuracy not obtained"

is wrong, i.e. column (I)/(column (1) + column (2).

These numbers are given in two separate tables, for every considered

indicator.

The parameter M of QADRAT

The procedure QADRAT uses an internally defined minimal step length

HMIN. It is possible that during the computation of the integral a

subinterval is encountered with length less than HMIN, for which the

computed approximation of the integral does not satisfy the accuracy

condition used. If so, this approximation is nevertheless accepted.

The number of such subintervals is recorded in the output parameter M.

We have regarded M being zero or not an indication as to whether the

procedure has succeeded in computing the integral with the required

accuracy.

group

e: = 10-3

10-6

1:)-9

- 48 -

o

o

o

2

30

o

4

3

33

56

44

4

51

55

46

5

38

15

20

table 1: M gives the wrong indication

Over all five problem groups and all three tolerances together the

percentage of cases in which M gives the wrong indication turns out

to be 27%.

group

E: = o

o

o

2

29

o

44

3

100

100

100

4

92

96

94

5

47

100

100

table 2: M ¥ 0 is the wrong indication

Over all five groups and all three tolerances together the percentage

of problems with M ¥ 0 in which the error in the computed integral

is less than the tolerance turns out to be 81%.

From these results we conclude that the parameter M of QADRAT is

irrelevant as an accuracy indicator.

Remark. The parameter M of INTEGRAL with the same meaning is some-

what more relevant, but in our opinion equally not usable. Particularly,

the performance in the difficult problem groups 3 and 4 is rather bad.

- 49 -

The parameter ERROR of INTEGRAAL

Upon exit the parameter ERROR of INTEGRivu" contains, according to the

documentation, a realistic approximation (see footnote p. 10) of the

error in the computed integral. This value being less than the toler

ance might be used as a criterion of acceptance of the computed in-

tegral.

group

£ 10-3

10-6

10-9

o

6

5

2

34

20

11

3

o

30

56

4

II

48

41

.5

10

6

6

table 3: ERROR induces the wrong indication

The overall percentage of cases In which the actual error and the

value of ERROR lie on different sides of the tolerance turns out to

be 18%.

group 2 3 q. 5

'" =10-3 0 H'O 0 0 0

0 91 100 88 40

10-9 0 J6 64 50 22

table if! ection 011 account of ERROR ~s wrong

The overall percentage of problems with ERROR> € in which the actual

error is less than € turns out to be 60%.

From these results we conclude that the value of ERROR being less than

E: or not is not usable as an accuracy indicator.

- 50 -

In order to check how realistic the value of ERROR is, we have established

the percentages of the cases in which the error in the computed value is

less than twice the value of ERROR (table 5) and the percentages of the

cases in which the ratio of the two lies between! and 2 (table 6).

~.I.0Up 2 3 4

e: = 10-3 60 60 99 63

10-6 54 72 99 83

10-9 66 86 100 96

table 5: actual error < 2 value of ERROR

group 1 2 3 4

e: = 10-3 19 10 I 21

10-6 35 26 I 8

10-9 40 21 6 16

table 6: ! < actual error < 2 value of ERROR

5

74

97

99

5

20

26

13

From these last two tables we see that the error estimate supplied by the

procedure is conservative rather than realistic.

The parameters IFAIL and ACC of DOIACA

Upon exit the procedure DOIACA indicates via the parameter IFAIL whether

the computed integral is acceptable. The tables 7 and 8 reveal the

significance of this indication.

group 2 3 4 5

e: = 10-3 0 21 0 14 II

10-6 0 17 0 9 23

10-9 0 8 5 0 15

table 7: IFAIL gives the wrong indication

The overall percentage of cases in which IFAIL gives the wrong indication is 8%.

- 51 -

group 2 3 4 5

e: = 10-3 0 29 0 20 100

10-6 0 19 0 8 81

10-9 0 9 17 0 37

table 8: IFAIL = I is the wrong indication

The overall percentage of cases with IFAIL = ] (accuracy not obtained)

in which the error is less than the tolerance turns out to be 13%.

From these two tables we conclude that IFAIL is a significant indi-

cator, with a possible exception of group 5.

T!he parameter ACC of DOIACA has the same meaning as ERROR of INTEGRAAL.

For this parameter the tables corresponding to the tables 5 and 6 are:

group 2 3 4 5

e: = 10-3 100 86 100 79 90

10-6 89 86 100 94 79

10-9 6 86 91 95 70

table 9: actual error < 2

value of ACe

group 2 3 4 5

I:: = 10-3 0 10 31 ]6

10-6 6 9 5 19 22

10-9 22 II 52 20 27

table ]0: ~ < actual error < 2 value of ACC

- 52 -

From these two tables we see that the value of ACC contains in most

cases (convergent or not) an upper bound for the actual error, but not

a good estimate.

The parameter IFAIL and ERROR of DOIAGA

The parameter IFAIL of DOIAGA has the same meaning as the same parameter

of DOIACA. The table corresponding to table 7 is

group 2 3 4 5

£ == 10-3 0 16 26 5

10-6 a 3 30 0

10-9 0 16 2 28 5

table 11: IFAIL gives the wrong indication

From this table and the fact that IFAIL = I and yet the integral is

-6 -9 computed correctly, occurs only twice (group 4, £ = 10 ,10 ) we

conclude that IFAIL ~s a significant indicator.

Upon exit the parameter ERROR contains, according to the documentation,

an estimate of the absolute error in the computed integral. For this

parameter the tables corresponding to the tables 5 and 6 are:

group 2 3 4 5

c: 10-3 lOa 73 90 48 79

10-6 100 99 97 47 90

10-9 100 100 99 58 96

table 12: actual error value of ERROR < 2

group

IS = 10-3 5

10-6

10-9 2

table 13: ! <

- 53 -

2 3 4

7 17 15

10 17 14

1 1 12

actual error value of ERROR

5

2

3

< 2

From these two tables we see that ERROR contains nearly always an upper

bound for the actual error, as the documentation states.

The results with respect to group 4 for both IFAIL and ERROR are re-

markably bad (cf. Remarks on group 4, page 30).

7. The Robinson test set

In [11J Robinson gives a test set of integrals divided in groups that

are more or less comparable with the groups on page 13, namely:

A - Well-behaved functions;

B - Functions with one or more integrable singularities within the

integration interval [a,bJ;

C - Functions with one or more sharp peaks 1n [a,bJ, or with a singu-

larity "near" [a,bJ;

D - Rapidly oscillating functions;

E - Step functions; functions with a number of finite discontinuities

in the function or its first derivative within [a,bJ.

A test performed with Robinson's set may give an idea of the validity

of the results and the ensuing conclusions in section 5. In this section

- S4 -

we will discuss the confrontation of our results with the results ob

tained by test runs with the types A,B,C and D of Robinson's set.

Remark. In contrast to our test set the functions 1n Robinson's set

are not made positive, but the integrals as given in [IIJ are computed.

Type A

This set of integrals agrees with group 1. The reliability score·

agrees practically with that of group 1, though most of the pro

cedures do not have, 100% correctly computed integrals.

The differences in the efficiency are small. The corresponding entries

of the log. scale columns of the efficiency tables differ about 0.2 - 0.3,

except INTEGRAL and DOIAGA where the difference is about 0.6. If the

function A2I: (x - 1)(x - 2) ••• (x - n)/n:, which might be considered

oscillatory rather than smooth particularly for the larger values of n,

is omitted, then the difference for INTEGRAL and DOIAGA reduces to the

level of the other procedures.

Type B

The integrals of type B may be divided into the following three groups.

i) Unbounded functions: BI, B2, B8, BII, B17-22. These functions be

long to none of the five groups.

All procedures, except SIMPSONINT sometimes, fail in computing the

correct value of these integrals, in agreement with our earlier

experience, see p. 13.

- 55 -

ii) Functions with a singularity in one of the endpoints: B3, B4, B7,

B9, BI2-16. These functions belong to group 3.

The high reliability of SIMPSONINT and DOIAGA for all tolerances

is affirmed as well as the bad behaviour of QADRAT for 8 = 10-3

,

DOIACA is, just as before, only for E = 10-9 less reliable, but

very faithful yet, viz. 22% correctly computed integrals and 98%

correctly indicated results by IFAIL. For the rest the reliability

is nearly always lower than that of group 3,

For the procedures with an acceptable reliability the agreement

of the efficiency scores is satisf~ctory, as may be seen from the

following table where the log. scale scores of the Robinson inte

grals and the integrals of group 3 are given

€ = 10-3 £ = 10-6

£ = 10-9

SIMPSONINT 0.98 0.88 0.82 1.29 I. 76 2.26

INTEGRAL 3.56 3.25 2.44 2.87 2.90 3.24

QADRAT 0.63 0.23 1 .56 1 .61 1.03 0.98

INTEGRAAL 3.25 2.24 2.47 2.80 1.56 1.91

DOlAAA 0.08 0.04 0.00 0.00 0.00 0.00

DOIACA 0.00 0.00 0.17 0.19 0.29 0.02

DOIAGA I • I I 0.36 0.96 0.97 0.72 0.90

iii) Functions with a singularity within the open interval: B5, B6, BIO.

These functions belong to group 4.

- 56 -

This set is too small to make a detailed comparison. So we confine

ourselves to the following remarks.

SIHPSONINT is the only procedure with high reliability for all

-3 tolerances. The rptner bad reliability of QADRAT for e: ::: 10 and

of DOIAGA for all tolerances is affirmed. The efficiency scores

of SIMPSONINT,QADRAT and DOIAGA are roughly the same as in group 4.

Type·C

This set of integrals should have to agree with group 2. But there is a

difference. Nearly all functions of group 2 have a peak within the open

interval, whereas the set of type C contains only one such function

(C7). This turns out to be in favour of DOIAC~ which does not use the

endpoints of the interval.

As for the reliability, the percentage of correctly computed integrals

of DOlAAA and DOIACA for all tolerances, and of INTEGRAL for e: = 10-9

are much higher than in group 2. For the rest the agreement is satis-

factory; the reliable procedures being ROMBERGINT, SIMPSONINT, INTEGRAL,

QADRAT except for e: ::: 10-3, INTEGRAAL and DOIAGA.

The efficiency of DOIACA is also better than in group 2. Its score is

about that of DOIAGA which remains one of the best. The log. scale

values of ROMBERGINT and DOIABA are much higher (0.8-1.5) than the cor-

responding values in group 2. For the rest there is a good agreement,

with differences of 0.4 or less (where the values of the Robinson set

are mostly higher than those of group 2).

- 57 -

Type D

This set of integrals agrees with group 5. Bllt most of the functions in the

Robinson set oscillate more rapidly than the functions in group 5.

This may be responsible for the differences between the two sets.

With respect to the reliability SIMPSONINT gives the most remarkable

difference: 28% correctly computed integrals for E = 10-3 and 74% for

-6 E = 10 • Other different percentages of correctly computed integrals

are: INT. TRAPEX 88% for E = 10-3 and 10-6, DOIABA 16% and DOIACA 47%

for E = 10-9•

For the rest the scores are comparable,with mostly a difference less

than 10%.

For the procedures with an acceptable reliability the log. scale scores

of the efficiency tables of the Robinson set and group 5 are given in

the following table.

E = 10-3 E 10-6

£ = 10-9

ROMBERGINT 2.08 3.16 3. J 5 3.79 3.80 4.49

SIMPSONINT 1.57 I • 10 3.19 2.51 4.89 4.56

INT. TRAPEX 1.44 1.67 1.77 1.82 2.09 2. 1 I

INTEGRAL 3.02 2.60 4.58 3.89 5.98 5.63

QADRAT 1.01 0.79 1.47 0.67 1.68 1.30

INTEGRAAL 1.60 I. 75 1.94 1.48 2.35 2.02

DOIACA 0.55 0.71 0.18 0.34 0.35 0.43

DOIAGA 2.67 1.68 2.45 1.39 2.45 1.57

We see that the agreement is satisfactory, except for DOIAGA where the

high score for the Robinson set is remarkable.

- 58 -

Conclusion

We conclude that the test with the Robinson set endorses greatly the

results of section 5. The main difference occurs in the group of

oscillating function~> where enlargement of the oscillations affects

adversily the perfo~mance of SIMPSONINT and DOIAGA particularly.

8. Concluding remarks

The information provided in this report, particularly the pages 23-27,

may help a customer of the Eindhoven University of Technology to choose

a procedure appropriate to his quadrature problem. However, the follow

ing remarks by Robinson [llJ are worth quoting: " ••• it is not possible

to make conclusive absolute statements about the performance of any

method based on an empirical study such as this one (or any other).

The results of these tests must be viewed only as a guide to the true

capacities of each method." and II ••• if the user has many integrals to

evaluate, some preliminary testing may be worthwhile to reveal the

method which is best for his particular problem. Indeed, where possible,

it is only sensible that simple strategies directed at a very special

type of integral should be used when evaluating several such integrals

in preference to any of these more general approaches."

The results of section 5 show that no procedure is the best for all

types of integrals with respect to both reliability and efficiency.

The overall most reliable procedure is SIMPSONINT, followed by QADRAT

except for low accuracy and DOIAGA except for functions with a singu

larity within the open integration interval.

- 59 -

The overall most efficient procedure is DOlAAA (but remind the remark

in its description, page 10), followed by QADRAT, DOIACA and DOIAGA.

SIMPSONINT, though very reliable, is rather expensive.

The parameter M of IN:£GRAL and QADRAT is not significant with respect

to the accuracy of~he computed value. For convenience of the user it

should be better removed.

On the other hand the parameter IFAIL turns out to be a reliable in

dicator with respect to the question whether the procedure has computed

the integral with the required accuracy. The parameter ERR (OR) of

several procedures, pretending to supply an estimate of the error

the computed value, is of little use to the user, since it substantial

ly overestimates the error mostly.

Within the scope of the test we have considered the procedures to be

black boxes, i.e., we have not looked at the implementations. However,

we have the impression that something is wrong with the implementation

of BULIRSCHINT, and that it is possible and worthwhile as well to im

prove the reliability of QADRAT and DOIAGA for the cases above men

tioned, in a rather simple way.

- 60 -

Appendix A. The test integrals

Group I: Smooth or well-behaved functions.

nr.

101 exp(-x)

102 cos (x)

2 103 exp(-x )

f(x)

104 1/(1 + cos(x»

105 1/(1. + II cos(x», A = 5, A == 4

106 1/(1 + x)

2 2 107 1/(1 + A x ), A == 1

108 idem, A == 5

110 1/(x2 + x + 1)

111 exp(" cos(x»cos(" sin(x», A =

112 If , A == 3

114 " A == 4

115 x sin(2x)/(1 +A cos 2(x», A = 0.5

116 " , 1.==5

a

-1

0

0

0

0

0

0

" -1

0

0

II

0

"

0

"

Ii> I

e - l/e

1f 2"

8 ;; """2 erf(b)

1f "2

1f / /,.,2 2 1T - 11

In(2)

arctan(A)/A

" idem

I IC+a+A\ 2aA n 1 - a + AI

+ _1_ eA

[arctan(2;a) + arctan (2;a)] •

a:= I2A - I, 8:= 12A + I.

1f/3/3

1f 1f

II idem

1T/4A

" idem

1f 11n( (1 + 1l'+I)/2) "2

" idem

nr. f(x)

117 3/(x4 + x3

+ x + 1)

118 idem

119 2 exp(x )(1

2 + 2x )

120 A 1/(x(1 + x»), A = 2

121 It A :: 5

122 In(l + x) / (l 2

+ x )

- 61 -

a b I

b o 10 b+1 + In( b + I ) +

1r,2- b + 1

__ 1 (arctan{2b - 1\) + arctan ( __ I \)) 13 \ \ 13 \13 :

o 5 idem

0 e

2 .!. In(2A+1/(1 A

+ 21.»

II " idem

0 1 'IT '8 In(2)

- 62 -

Group 2: Functions with one or more sharp peaks within the interval.

nr. f(x) a b I

201 ? 2 13, 10-2 2 ( 2 - A ) ( 1 - A ) 1/ «x - A) - + 11 ), A A = arctan\--l1- - arctan --ll-

202 II A = 12, II = 10-2 " " idem

203 " A = 13, 11 = 10-3 " " idem

204 " A 12, 11 = 10-3 " " idem

205 " A /3, 11 = 10-4 " " idem

206 " A 12, II = 10-4 " " idem

4 2 12, 10-5 2 ~(2-A,lJ) -<P(I-A,ll) *) 207 111fil / « x-A) + \l ), A \l =

208 " A = 13, 10-5 " " idem ]J

209 " A 12, ]J 10-3 " II idem

210 " A /3, ]J 10-3 " II idem

211 II A 12, Jl = 10-4 II II idem

212 II A 13, = 10-4 " II idem , ]J

213 sin(ln(x» -7f 7f sinh(7f) 2 e

I

1214 l/(x+ 10-4) 0 In(I+104)

*)Iil(x,y):= ! ( ! In(X~ +ax+y ) + arctan ( ax;¥. ) + arctan ( axy-y

)), a:= /2il. x - ax + y

- 63 -

Group 3: Functions with a singularity in (one of) the endpoints

nr. f(x) a b I

301 x In( 1 Ix) 0 1 0.25

302 In(sin(x) ) sin(2x) 0

1T 1 In(].1/A) -(A sin2(x) + \l cos 2(x» 2

2 2].1(A -)..I)

A ::: 2.3, j,l :: 1.8

303 idem A ::: 1. 7, j,l :::: 1.3 II " idem

304 A A == 0.5 a 1 1 I (I + A) x ,

305 idem " ::: 1 .5 " " idem

A-I ll-I *) 306

x +x A = 1 • I , 3.8 a 1 BCA, j,l) (l + x) A+ll

, ].1 ::::

307 idem A == 2.1, 5.7 tt " idem , ].l =

308 xA- 1 (1 - x) ].1-1

0 1 (a + y) -A(S + y) -)..IB (A. , ].1) A+].1

(ax + S( I - x) + y)

A = 1. 7, ].1 ::: 5.4

a ::: 0.5 , 13 ::: 0.3, y :::: 0.6

309 idem A ::: 2.3, == 3.9 " tt idem , ].1

a == 0.3, 13 :: 0.6, y ::: 0.5

310 x,,-l (I - x) ].1-1 • A = ].1 =:; 1.5 a 1 B(A,ll)

311 ide"'1 A 3 5 " II idem • =-Z , )l = -

3

312 idem A I 1 9 " " idem • :::- , 11 5 4

313 iJ - x2 0 I 1T/4

314 /1 - x2 -1 ) if/2

- 64 -

nr. f (x) a b I

315 A x InU/x), A == 0 I/O + I) 2

316 idem A 0.71 " " idem

317 idem A == 1. 21 " " idem

318 idem A '" 2.7 " II idem ,

319 In(l/x)arcsin(x) 0 2L+ 2

In(2) - 2

320 arccos (x) -) 1T

- 65 -

Group 4: Functions with a singularity within the open interval

nr. f(x)

401 1T if x < "2 then I else sin(x)

402

a b

o

o

'IT 1T - +

2

403 x < A then 0 else (x - A), A'" -.;,... 0 12

404 idem A = O. 5E

405 if x < A then 0 else !(x - 1,)2

406 idem

I

rz A = 0.31

fI " idem

o

" " idem

I

407 ln(l + Ixi) -a b (1 +a)ln(l +a) + (1 +b)ln(1 +b)

. x 2 I-x 2 40S 1.£ x < A then (I) else (I-A) , A == 0.9

409 [x] + sin(I(x - [x])

410 idem

411 1 + Ix - A Ill, A 0.85, jJ

412 idem , A == 0.64, 11 O.SI

413 1 - exp(-Alxl), A 5

414 idem 10

416 idem

417 It - x2 + 4[x](x - 1)

41S idem

419 It - (x - [x])2

420

a==0.37,13=0.15

-a - b, a == 0.85, b == 0.64

a 1 '3

o n ~n(n-l) +2n/1T, n = 3

" " idem , n == 5

o

" " idem

-1 2(1 - (1 - exp(-A»!A)

" " idem

o b

" I! idem

1T a a+2 "2

" " • a -0.31

o 'IT 2 "2

o 2

9 b == IS9 64

9 , b '" 8

Group 5: Rapidly oscillating functions

nr. f(x)

501 2 x cos (nx), n = 10

502 idem , n = 20

1 503 - cos(z sin(x», z = 10 1T

504 (I + sin(l/x»/x

505 idem

506 cos (l/x)

507 idem

508 sin(ln(x»

509 idem

510 idem

511 sin(l/x) + 1 + x eos(l/x)

x

512 idem

513 InCl - 2 A cos(x) + A2)sin(nx)sin(x)

, A ... 0.5, n ... 10

514 idem , A ... 0.71, n ... 20

515 n cos (x)cos(nx) , n ;: 20

516 idem , n =40

- 66 -

a b I

o 1T

tI " idem

o 1T JO(z)

a .! SiCa) - Si(1T) + In(!!.), a = 201T. 1T 1T

" II idem , a = 40n

a 1 I 1 :rr -(;r+-a cos(a)+Sl(a)-Si(1T»

, a ... 20n

II " idem , a ... 401T

-n1T n 1 e - H 1 - (-\) exp ( -n1T ) ,n = 1 0

II II idem , n ... 20

" " idem , n ... 30

a 1 - 1 - 1 COS (a) + In(!!.) , a ... 251T 1T 1T a 1T

tI " idem , a = 501T

o

" It idem

o

It " idem

517 sin2n(x)cos(z cos(x», n= L z=IOr It

518 idem , n =1-, z =1 Or " " idem

519 exp(A cos(x»COS(A sin(x», ).=10 o 1T 1T

520 exp(-x)sin(n1T), n = 50 n --2 (1 - exp(-21T»

1 + n

- 67 -

Group 6: Pest functions

nr. f(x) al b I

601 2 'IT (cos()\x) +cos(A/x»/(1 +x ),A = 0 ";) exp(-A)

....

602 idem A 3 II " idem ,

603 cos(A(x+ l/x»cos(A(x-l/x» 'IT

1 + x 2 0 - exp(-ZA) 4

A =

604 idem A = 2 " II idem

sin(A jl-I 0 7T 7T tan(x) + flX) cos (x) -

605 2 2 sin(x)

A = I , fl 2

606 idem A 5, fl 2 II " idem

607 casCn A 'IT -A-I exp(-n) +2:. tan(x) + Ax) cos (x) 0 - 2 'IT

2 2

A I , n = 5

608 idem A "" 2, n = 5 II It idem

609 exp(n cos(x»cos(n sin(x», n = 20 0 'IT 'IT

610 sin(ln(x» -2'IT 27T sinh(rr) e e

611 ifx < )" o else I , A 0.71 0 I - A

612 idem )" 0.31 II It idem ,

613 I a 2 - A2/6 [_1_] + I I-x

614 • (IT S1n 2 (x - [x]» 0 n 2n/'IT, n = 3

615 idem II " idem, n 5

- 68 -

Appendix B. Reciprocal matrices

The following results are taken from Saaty [12].

An n x n matrix A is called posi ti ve if it satisfies the condition

a. A .. > 0, I .'" i,j :':: n. 1.J

From the Perron-Frobenius theorem we know that if A is positive, then

the spectral radius peA) is a simple eigenvalue and the corresponding

eigenvector, w say, is positive.

A positive n x n matrix A is called reciprocal, if its elements satis-

fy the condition

, D. A ••

1.J -I

A .. , I s i,j :':: n; in particular A .. J1. 1.1.

I for all i.

Lemma I. The spectral radius peA) of a reciprocal n x n matrix A

satisfies the inequality

(I) p (A) ~ n.

Proof. From Aw p(A)w it follows that for all 1.

n

.I J=l

peA) A .. w./w., 1.J J 1.

from which we derive by summation over i

n n np(A) = I .I

i=l J=l A •• w./w.

1.J J 1.

n n

I .I i=l J=l

(A .. w./w. + A .. w./w.) 1.J J 1. J 1. 1. J

= ! 2 ~ n

since A .. w./w. J1. 1. J

-I -1 (A .. w./w.) and y + y ;? 2 for any y > 0.

1.J J 1. o

- 69 -

A reciprocal n x n matrix'is called consistent if its elements satisfy

the condition

c.

Lemma 2. The followin~ properties are equivalent for a reciprocal

n x n matrix A.

i. A is consistent,

ii. rank(A) I,

iii. p (A) = n,

iv. There exists positive numbers vk ' 1 ~ k ~ n,such that A .. ~J

v./v .• ~ ]

Proof. i + ii, since the i-th row is A .. times the j-th row for all ~J

~ and j.

ii .+ iii. since p (A) is the only non-zero eigenvalue of A

and consequently peA) = trace(A) = n.

iii + iv, since it follows from (2) that peA) = n implies

A .. w./w. = 1 for all i and j. So vk = cwk • 1 ~ k ~ n, with c ~J J ~

an arbitrary positive constant.

iv + i. is trivial.

Lemma 3: Let A be a reciprocal matrix with spectral radius peA) and

corresponding eigenvector w > O. Let 0 .. be the "local deviation from ~J

consistency", defined by

(3) A .• ~J

w. ~

w· J

(l + 0 .• ). ~J

Let -~ be the average of the remaining n - I eigenvalues

n ~ = - ( I

i=l A.)/(n - 1) = (p(A) - n)/(n - 1). ~

o

- 70 -

Then

2 o .. 1J 1

f.I = --:;--":'7" n(n - I)

(4) + 0 •• ' 1J

Proof. Since A .. w./w. = 1 + 6 .. it follows from (2) that 1J J 1 1J

n n np(A) = n

2 + ! I I (6 .. + 0 .. )

1J J1 i=l j=l

2 I (0 •• n + + 1J l~i<j~n

2 2

6 .•

I ( 1 1J n + + 0 ..

I~i<jsn 1J

from which (4) immediately follows.

Remark. 2 Jl is the variance of the 0 ..• 1J

+ 6 .• 1J

)

- 1 )

It is easy to verify that the matrix E of section 4 is a reciprocal

matrix, which is consistent if one and the same problem set J is used

for all procedures.

o

- 71 -

References

I. Boor, C. de, On Writing an Automatic Integra~ion Algorithm; in

Mathematical Software (ed. J.R. Rice) Academic Press, New York

and London (1971), p. 201-209.

2. Boor, C. de, Chdre: an Algorithm for Numerical Quadrature; 1n Mathe

matical Software (ed. J,R. Rice), Academic Press, New York and

London (1971), pp. 417-449.

3. Cody, W.J., The Evaluation of Mathematical Software; in Program

Test Methods (ed. W.C. Hetzel) Prentice Hall, Inc., Englewood

Cliffs, New Jersey, J973.

4. Einarsson, B., Testing and Evaluation of Some Subroutines for

Numerical Quadrature; in Software for Numerical Mathematics

(ed. D.J. Evans). Academic Press, London and New York (1974),

p. 149-157.

5. Kahaner, D.K., Comparison of Numerical Quadrature Formulas; in

Mathematical Software (ed. J.R. Rice). Academic Press, New York

and London, 197J, p. 229-259.

6. Lyness, J.N. and J.J, Kaganove , Comments on the Nature of Auto-

matic Quadrature Routines. TOMS ~, 65-81 (1976).

7. Lyness, J.N. and J.J. Kaganove , A technique for comparing auto-

matic quadrature routines. The Computer Journal 20, 170-177 (1977) .

8. Lyness, J.N., Performance Profiles and Software EvaluatioD; in

Performance Evaluation of Numerical Software (ed. L.D. Fosdick),

North-Holland Publishing Company, Amsterdam-New York-Oxford J979,

p. 51-58.

72-

9. O'Hara, H. and F.J. Smith, The Evaluation of Definite Integrals

by Interval Subdivision. The Computer Journal 11, 179-182 (1969).

10. J.R. Rice., Software for Numerical Computation, Purdue University,

Technical Report CSD-TR 214 (1977).

II. Robinson, I., A comparison of numerical integration programs. Journal

of Computational and Applied Mathematics 5, 207-223 (1979).

12. Saaty, T.L., A Scaling Method for Priorities in Hierarchical Structures.

J. of Math. Psych. ~, 234-281 (1977).

13. Vorst, H.A. van der , Vergelijking van Programmatuur in Algol en

Fortran voor Automatische Integratie (Dutch). ACCU-Reeks 12 (1974).

Academisch Computer Centrum Utrecht.

14. BURROUGHS LARGE SYSTEMS NUMERALS, Numerical Analysis Program Library.

Burroughs Corporation, Detroit, Michigan 48232 (1974).

IS. NAG ALGOL 60 Library Manual - Mark 7. Numerical Algorithm Group Ltd.

Oxford, june 1980.

16. RG-INFORMATIE, PP-3.1.1. (maart 1977) (Dutch).

Technische Hogeschool Eindhoven

Rekencentrum.

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