Encyclopaedia of Mathematics: Supplement Volume I

594 Pages·1997·25.483 MB·English

by  M. Hazewinkel (auth.), M. Hazewinkel (eds.)

ENCYCLOPAEDIA OF MA THEMA TICS
Supplement Volume I
ENCYCLOP AEDIA
OF
MATHEMATICS
Supplement Volume I
..
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ENCYCLOPAEDIA OF MATHEMATICS
Managing Editor
M. Hazewinkel
List of Authors
M. Ablowitz, L. Accardi, P. N. Agrawal, R. Aharoni, F. Altomare, S. S. Antman, P. L. Antonelli,
F. Argoul, A. Arneodo, H. Attouch, T. Aubin, W. Auzinger, D. Aze, G. Bachman, E. Bach,
N. Balakrishnan, A. Banyaga, A. Baragar, V. Barbu, O. E. Barndorff-Nielsen, L. M. Barreira,
Ph. Barry, N. L. Bassily, K. J. Bathe, L. M. Batten, W. Beckner, A. Bejancu, D. Benson,
W. Benz, E. E. M. van Berkum, A. J. Berrick, F. Beukers, A. Bialostocki, H. Bieri, K. Binder,
N. H. Bingham, Ch. Birkenhake, P. Bhesild, T. S. Blyth, I. M. Bomze, C. Bonotto, C. de Boor,
P. Borwein, A. Bottcher, F. Bouchut, N. Bouleau, O. 1. Boxma, F. Brackx, P. Brandi, A. Brandstadt,
M. Braverman, E. Briem, R. W. Bruggeman, N. G. de Bruijn, B. Buchberger, F. Buekenhout, P. Bullen,
A. Bundy, G. Buskes, M. Campanino, L. M. B. C. Campos, C. Cannings, J. Carlson, J. M. F. Castillo,
C. Y. Chan, G. Chaudhuri, Qingming Cheng, A. Childs, K. Ciesielski, C. J. S. Clarke, A. M. Cohen,
M. Coornaert, C. Corduneanu, G. Crombez, M. Cwikel, A. Davydov, A. S. Deif, W. J. M. Dekkers,
F. M. Dekking, M. Denker, J. Desarmenien, H. Diamond, Y. Diers, K. Doets, H. Doss, R. M. Dudley,
R. Duduchava, R. Eddy, J. H. J. Einmahl, E. F. Eisele, P. C. Eklof, U. Elias, R. L. Ellis, H. Endo,
T. Erdelyi, F. H. L. Essler, B. Fine, P. C. Fishburn, R. W. Fitzgerald, Ph. Flajolet, G. B. Folland,
A. T. Fomenko, E. Formanek, J. E. Fornress, R. Frank, A. E. Frazho, R. Fritsch, L. Fuchs, M. Fujii,
J. Galambos, A. Garcia-Olivares, M. Gasca, M. O. Gebuhrer, A. Geroldinger, A. A. Giannopoulos,
A. P. Godbole, I. Gohberg, R. Goldblatt, S. W. Golomb, R. Gompf, D. H. Gottlieb, J. Grandell,
B. Green, B. Grigelionis, L. Gross, K. Gustafson, H. Gzyl, W. J. Haboush, Y. O. Harnidoune,
M. E. Harris, H. Hauser, M. Hazewinkel, H. J. A. M. Heijmans, L. Heinrich, J. L. van Hemmen,
B. M. Herbst, W. A. Hereman, C. Herrmann, A. Herzer, K. Hess, C. C. Heyde, T. Hida, A. Hildebrand,
T. Hill, J. Hinz, J. W. P. Hirschfeld, W. van der Hoek, K. H. Hofmann, H. Holden, C. S. Hoo,
C. B. Huijsmans, W. W. J. Hulsbergen, J. Hurrelbrink, D. Iagolnitzer, N. H. Ibragimov, M. Ikle,
A. Hehman, J. R. Isbell, A. N. Iusem, A. O. Ivanov, K. R. Jackson, B. Jansen, G. W. Johnson,
N. J. Johnson, D. Jungnickel, P. E. Jupp, D. V. Juriev (D. V. Yur'ev), M. A. Kaashoek, J.-P. Kahane,
V. A. Kaimanovich, V. Kalashnikov, W. C. M. Kallenberg, PI. Kannappan, H. G. Kaper, H. Kargupta,
G. Karpilovsky, K. Keimel, R. Kerman, M. Kibler, T. Kimura, A. U. Klimyk, H.-B. Knoop,
S. O. Kochman, H. T. Koelink, J. N. Kok, Yu. S. Kolesov, V. Kolmanovskil, M. Kolster,
T. H. Koornwinder, V. M. Kopytov, V. E. Korepin, V. S. Korolyuk, Y. Kosmann-Schwarzbach,
A. M. Krall, A. M. Krasnosel'skil, M. A. Krasnosel'skil, A. Krebsz, E Kreimer, E-V. Kuhlmann,
S. Kuhlmann, Yu. A. Kuznetsov, I. S. Labouriau, J. C. Lagarias, E. R. Lamken, E. Lanckau,
E. Landvogt, S. Lang, M. A. A. van Leeuwen, K. S. Lim, J. H. van Lint, G. L. Litvinov, E. K. Lloyd,
G. Longo, J. D. Louck, E. Lutwak, W. Luxemburg, S. N. MacEachern, J. Madden, J. Magnen,
M. Mahowald, S. Majid, E. Malkowsky, M. Masumoto, A. Mateescu, S. V. Matveev, G. A. Maugin,
T. D. Mavridou, J. C. van der Meer, G. Melan~on, A. Micali, Ch. Micchelli, H. Mikawa, J. S. Milne,
M. Mitrea, I. M. Mladenov, P. van Moerbeke, J. M. M~ller, O. Moreno, A. Muravitsky, O. R. Musin,
C. Nash, J. van Neerven, V. Nesterov, R. Nest, I. Netuka, Siu-Ah Ng, E Nielson, H. R. Nielson,
R. Nishii, R. Norberg, E. Novak, K. Nowak, D. Nualart, S. Ochanine, G. J. Olsder, P. Orlik, L. Ornea,
J. O'Rourke, T. W. Palmer, A. Papadopoulos, P. M. Pardalos, R. G. Parker, K. R. Parthasarathy,
A. Pasini, V. D. Pathak, I. R. Petersen, R. Phelps, A. Pietsch, A. Pillay, I. Pinelis, Z. Piotrowski,
V. Pless, A. V. Pokrovskil, L. Polkowski, A. J. van der Poorten, A. Pott, B. L. S. Prakasa
Rao, E. Previato, G. Priest, K. Przeslawski, D. Przeworska-Rolewicz, Gh. Paun, J. M. Rassias,
B. D. Reddy, R. A. Renaut, E. Renshaw, J. W. Rice, P. Richetti, M. de Rijke, G. R. Robinson,
A. C. M. van Rooy, M. R~rdam, I. Ro~ca, W. Roth, J. Rovnyak, R. Roychoudhury, A. L. Rukhin,
A. M. Rushdi, E Ruskey, D. Saari, A. Salomaa, N. W. Sauer, M. Scafati Tallini, A. R. Schep,
Ch. Schlindwein, C. P. Schnorr, B. S. W. Schroder, W. Schwarz, B. Schweizer, S.-Y. Shaw,
L. A. Shepp, R. T. Shield, B. K. Shivamoggi, T. A. K. Sinha, A. Sklar, N. J. A. Sloane, J. Slominska,
P. Sole, B. Solomon, A. J. Sommese, G. Starke, W.-H. Steeb, F. W. Steutel, H. Stichtenoth, M. Stoll,
P. Stovicek, E. Straume, S. van Strien, S. J. Summers, L. Takacs, D. Talay, T. Tanino, R. Tolimieri,
A. A. Tuzhilin, C. Udri~te, R. G. Underwood, A. S. Ustiinel, M. van de Vel, A. Venkov, J. Verwer,
R. Viertl, P. Vopenka, W. R. Wade, M. Waldschmidt, W. D. Wallis, E. Wattel, G. Weaver, H. Wehlan,
R. Weiner, D. West, J. Wiegerinck, S. D. R. Wilson, W. S. Wilson, J. A. Winn, H. J. Woerdeman,
M. Yor, A. E. Zalesskil, W. Zelazko, R. Zivaljevic
PREFACE TO THE FIRST SUPPLEMENT VOLUME
The present volume of the ENCYCLOPAEDIA OF MATHEMATICS is the first of several (planned are
three) supplementary volumes.
In the prefaces to the original first ten volumes I wrote:
'Ideally, an encyclopaedia should be complete up to a certain more-or-Iess well defined
level of detail. In the present case I would like to aim at a completeness level whereby
every theorem, concept, definition, lemma, construction, which has a more-or-Iess
constant and accepted name by which it is referred to by a recognizable group of
mathematicians occurs somewhere and can be found via the index.'
With these three supplementary volumes we go some steps further in this direction. I will try to say
a few words about how much further.
The first source of (titles of) articles was the collective of users of the original 10 volume ENCY
CLOPAEDIA OF MATHEMATICS. Many users transmitted suggestions for additional material to be covered.
These suggestions were taken seriously and checked against the 3.5M keyword list of the FIZISTN
database MATH in Karlsruhe. If the hit rate was 10 or better, the suggestion was usually accepted.
For the second source I checked the index of volumes 1-9 against that same key phrase list (normal
ized). Everything with a hit frequency in the normalized list of 40 or better was checked and, if not really
present-a casual mention did not suffice-resulted in an invitation to an expert to contribute something
on it.
This 'top 40' supplementary list already involves more articles than would fit in a single volume
alone and the simple expedient was followed of processing first what came in first (while being carefull
about groups of articles that refer heavily to each other and other matters such as timelyness). However,
the three supplementary volumes together will surely cover the whole 'top 40' and actually go one step
deeper, roughly to the level of the 'top 20' .
For the final (as far as I can see at the moment only electronic) version of the ENCYCLOPAEDIA OF
MATHEMATICS (WEB and CDROM both) I hope and expect to go as far as the 'top 6'. This means
an estimated 32000 articles and an 120K standard key phrase list, a four-fold increase over the printed
13-volume version. It should be noted that if one actually checks one of these 'top 6' standard key
phrases in the database MATH, the number of hits is likely to be quite a bit higher; such a search will
also pick mentions in title and abstract (and not only those in the key-phrase field).
The present volume has its own index. This index is structured exactly like Volume 10, the index to
Volumes 1-9. For details I refer to the Introduction to that index volume.
The number of authors involved in this volume is substantial and in a sense this ENCYCLOPAEDIA
is more and more a community effort of the whole mathematical world. These authors are listed
collectively on one of the preliminary pages, and individually below their contributions in the main body
vii
PRERFACE TO THE SUPPLEMENT VOLUME
of this volume. I thank all of them most cordially for their considerable efforts. The final responsability
for what to include and what not, etc., however, is mine.
The steps from electronic file to printed copy can involve more work than is sometimes realized. For
one important step in this process we used to rely on Teun de Graaf of KAP. Regretfully he died a few
months ago, just before he was to retire. I thank him gratefully for his work.
As is clear from the above, I have made heavy use of that invaluable resource the FIZiSTN MATH
database in Karlsruhe. I thank that institution, in particular Dr. Olaf Ninnemann and the 'MATH group',
for their assistance and the facilities put at my disposal. As in the case of the original 10 volumes, this
one would not have existed without the very considerable efforts of Rob Hoksbergen, who took care of
all coordination and administration, and an awful lot of other detail work besides.
Bussum, May 1997
PROF. DR. MICHIEL HAZEWINKEL
email: [email protected]
CWI
P.O.Box 94079
1090GB Amsterdam
The Netherlands
Telephone: +31 - 20 - 5924204
Fax: +31 - 20 - 5924199
viii
________ A _______ _
A PRIORI AND A POSTERIORI BOUNDS IN MA- which implies that under the conditions IIA-18AII < 1,
TRIX COMPUTATIONS - Three sources of errors are
behind the lack of accuracy in numerical computations: 118xll < IIA-18AII + IIAI~:lfbll
(4)
data errors associated with the physical model, trun Ilxll - 1 - IIA-18AII
cation errors when series are truncated to a number
of terms, and rounding errors resulting from finite ma- Setting k(A) IIA -lIIIIAII, from Ilbll :::; IIAllllxl1 one
chine precision. Therefore, for the equation f (a, x) = 0 finds
in which a is a parameter, if an error occurs in a, the al
gorithm returns x and not x. How much the computed W8x :::; 1 - kk((AA)) 11.lAII (1l1f8AAIlII + l1I18bblll)I ' (5)
solution x differs from x is an indication of the com IIAII
putational accuracy of the result. Since x is unknown,
which measures the relative error in the solution x re
xl,
the norm Ix - taken as an indication of the accu
sulting from the errors 8A and 8b in the data. Thus,
racy, cannot be calculated, and numerical analysts have
the factor k(A), called the condition number of A is the
devised bounds for the latter in terms of the variation
main factor affecting computational accuracy, since by
8a in a. These bounds are of two types: a priori bounds
setting 118AII :::; EIIAII, 118bll :::; Ellbll, where E is the rela
and a posteriori bounds. The first are applied prior to
tive error in the data, it follows that
computation and are usually poor in nature, whereas
the second use information extracted from a computa k(A) = lim ~ 118xll (6)
tion, and are more indicative. In this article, attention €-tO E Ilxll '
is focussed on the most widely available bounds which
i.e., representing the sensitivity of the solution to rel
can be implemented to estimate the accuracy of results
ative changes E in either A or b, that is, 118xll Illxll ~
and to check the stability of numerical algorithms (cf.
E' k(A). So, if k(A) = lOP and E = 5· lO-t, where t is
also Stability of a computational process).
the machine mantissa, the number of significant digits
Linear matrix equations Ax = b, A E Rnxn, x
x in becomes t - p.
det(A) =I- O. Assuming that the computed solution
is equal to x + 8x, where x = A-I band 8x is the error The above analysis lacks accuracy, for the error in
IIAII is not equal to EIIAII, since other factors intervene,
in the solution resulting from a perturbation 8A in A
one of which is the growth factor in the elements as
and 8b in b, the perturbed problem
sociated with the interchange strategy as well as the
size of IIAII, yet it provides an a priori estimate for the
(A + 8A)(x + 8x) = b + 8b (1)
expected accuracy. Before the late J. Wilkinson it was
thought that rounding errors can ruin the solution com
implies, by cancelling equal terms,
pletely; he showed than an upper bound exists for 118AII
8x = A-1( -8Ax - 8A8x + 8b). (2) and that fear of obtaining poor results is unwarranted
[14].
On the contrary, one can also reach total machine
It follows that for a consistent matrix-vector norm [4]
accuracy. Consider, for example, the matrix
118xll :::; IIA-18Allllxll + IIA-18A11118xll + IIA-18bll ' (10 0)
A = 05
(3)
10-5 '
A PRIORI AND A POSTERIORI BOUNDS IN MATRIX COMPUTATIONS
IIAII = 105, IIA-lil = 105, and k(A) = 1010, so one c) xnew = X - 8x, re-do from a).
expects to loose all significant digits on a lO-digit ma x
The algorithm converges very rapidly and -+ x.
chine, contrary to the fact that for any right-hand side x
Finally, accuracy of is not the only issue which mat
b, full accuracy in this situation is reached. This led R.
ters to users. There is also the question of stability of the
Skeel [10] to consider componentwise estimates of 8x.
algorithms. By the stability of x one means that it sat
For 18AI ~ EIAI, 18bl ~ Elbl, where 1·1 stands for the
isfies the perturbed problem
modulus of the vector taken componentwise, one obtains
from (2): (A+8A)x=b+8b, (13)
18xl ~ IA-18Allxl + IA-18A118xl + IA-18bl ' (7)
in which 8A and 8b are of the order of the machine al
giving for p(IA-18AI) < 1, where p is the spectral ra lowed uncertainty. An algorithm which returns x such
dius, that that its backward errors 8A and 8b are larger than what
18xl ~ (I -IA-18AI)-1 (IA-18Allxl + IA-18bl)· (8) is anticipated in terms of the machine eps, is unstable.
This is checked using the Oettli-Prager criterion [9]
For relative perturbations E in A and b, this implies that
_I8 Ix _1I < _2_EIc...,:II,--,-A_-l..:.,.;1I c....,.A-,--,:I1 ~I lAx - bl ~ ~A Ixl + ~b, (14)
(9)
Ilxll - 1 - Ell lA-II lAlli'
which has been derived for interval equations [A ±
giving
~A]x = [b ± ~b] but is also suitable for systems subject
k(A) = IliA-II lAlli, (10) to uncertainties of the form ~A = EIAI and ~b = Elbl.
which is a better sensitivity criterion. For the above Thus follows the stability criterion Irl ~ E(IAllxl +Ibl); a
(2 x 2)-matrix one has k(A) = 1, as expected. result implemented in most available packages by what
However, a practical bound for the error in x re is called a performance index. (For instance, in the IMSL
lies on information extracted from computation, since it is given by
it shows the true state of affairs. If r = Ax - b is
trh =e rAes(ixd u+a l8 xe)r r-or bo =f t hAe8 xtr uoen es ofilnudtiso nIS,x th=e nA -blyr ,w hrietnincge p = l:mS,a:Sxn BN + A N· "L,..nJj =l IX A j I' (15)
the a posteriori bound
where BN = maxl:Si:Sn Ibil, AN = maxl:Si,j:Sn laijl.)
Illf~:1 ~ -111 ~ ::~:: If p < machine eps, the solution is stable [2]. The back
IIA :11:1111 k(A) . (11)
ward error can be given by 8A = -H ·IAI·diag(sgn(xi)),
This shows that k(A) intervenes also here and that Ilrll 8b = H . Ibl, where H is a diagonal matrix and hii =
x
can be found very small, yet is totally inaccurate as a r;j(IAllxl + Ibl)i, with Ihiil < machine eps.
result of the large k( A), [2]. The above bound can still Linear equations Ax = b, A E Rmxn. Independent of
be improved by noticing that since A-I is unknown and whether the equations have a unique solution (as in the
only an approximate inverse matrix B of A is known, it previous section), more than one solution or no solution
can be written in the practical form (only a least-squares solution), the general solution x is
118xll = IIA-l B-1 Brll = (12) given by
= II(I - (I - BA))-lBrll < (16)
< IIBrl1
where c is an arbitrary vector and A + is the Penrose
- 1 - III - BAil'
Moore inverse of A, satisfying AA+ A = A, A+ AA+ =
provided that III - BAil < 1.
A +, (A + A) T = A + A and (AA +) T = AA+ . For a consis
Although scientists have elaborated on various
tent set of equations the residual Ax - b = (AA + - I)b =
bounds for estimating the accuracy of their results, the
0, is a special case. The minimal least-squares solution
situation is not so bleak, for the above bounds are not
x becomes, therefore, x = A + b. Unlike in the previous
the end of the story. After all, can be ameliorated by
section, if A undergoes a perturbation EAll in general
a technique called the method of iterative refinement,
(A + EAd+ cannot be expanded into a Taylor series in
very similar to Newton's method for solving non-linear
E; it does have a Laurent expansion [2]. For acute per
equations (cf. Newton method). Today it is an avail
turbations (rank(A) = rank(A + EAl)), from [13] one
able facility in most packages (Linpack, Lapack, etc.)
finds that
and runs as follows:
a) compute r = Ax - b;
(17)
b) solve A8x = r;
2
A PRIORI AND A POSTERIORI BOUNDS IN MATRIX COMPUTATIONS
if c) A;t"ew = B - 8A+.
II (A + 8A)+ 112 = ar(A ~ 8A)' The algorithm converges and B --+ (AT A)-lAT.
=
The eigenvalue problem Ax AX, A E Rnxn. Un
ar~A)' like in the previous sections, a perturbation 8A in A af
IIA+112 =
fects both the eigenvalue A and its corresponding eigen
ai(A) - al(8A) ::::: ai(A + 8A) ::::: ai(A) + ai(8A), vector x. One of the earliest a priori bounds for I). - AI,
i = 1, ... ,r, where)' is the eigenvalue of A + 8A, is the Bauer-Fike
theorem [1]. It states that the eigenvalues of A + 8A lie
with al ~ ... ~ ar the singular values of A. P. Wedin
in the union of the discs
also showed that for any A and 8A,
Mi = {z: Iz - Ail::::: IIT-lIIIITIII18AII}, (23)
(18)
i = 1, ... ,n,
where J-l has tabulated values for various cases of where T is the modal matrix of A (T-l AT = A) as
rank(A) in relation to m and n. It therefore follows that suming A to be non-defective. The bound follows upon
the bound for the relative error in A + is given by noticing that since
II(A + 8A)+ - A+II < k(A) IIIIMAII1122 ).1 - A - 8A = ().1 - A) [I - ().1 - A)-18A]
(19)
IIA+112 -J-l1_k(A)~'
IIAI12 is singular, one has 11()'1 - A)-18AII > 1. Writing
where k(A) = IIA11211A+112 is the spectral condition num ().1 - A)-l = T()'1 - A)-IT-I,
ber, which measures the sensitivity of A+ to acute per
one finds that
turbations in A. To obtain a bound for 118xll/llxli one
starts with x + 8x = (A + 8A)+(b + 8b) to find that 1 ::::: IIT()'1 - A)-lT-18AII :::::
8x= [(A+8A)+-A+]b+(A+8A)+8b. (20)
~
::::: IITII liT-III 118AII . ,
Using a decomposition theorem [13] for (A+8A)+ -A+, mIni IA - Ail
one finds that [8] which implies (23). The condition number to the prob
lem is therefore of the order of k(T) = IITIIIIT-lil. In
:::::
l:f:rI~2 k [(2 + 17k)a + (J,] , (21) deed, if the above discs are isolated, then mini I). - Ai I =
18Ail and 18Ail ::::: k(T)118AII. However, unless A is a nor
where
mal matrix (11T112 = IIT-1112 = 1), the bound in (23) is
known to yield pessimistic results. A more realistic cri
terion can be obtained. As 1-().1 - A)-18A is singular,
then for a vector z =I- 0,
z::::: I( ).1 - A)-11IT-1118AIITllzl ,
Note that the error is dominated by k2(A), unlike the
bound in (19), which is dominated by k(A) only. For implying that [3]
the special case rank(A) = n < m the above expression min I). - Ail::::: p(IT-1118AIITI), (24)
becomes k[(l + 17k)a + (J,]. For rank(A) = m < n it
becomes k(2a + (J). Finally, for rank(A) = m = n it which is tighter than (23) for non-normal matrices.
reduces to k(a + (J), which is the situation of the previ Moreover, (24) is scale-invariant under the transforma
ous section. It should be noted that the above bounds tion A = DBD-l , where D is diagonal.
are valid for acute perturbations and that the situation Both the Bauer-Fike bound and the above are valid
worsens if 8A increases the rank of A, in which case the for the whole spectrum; thus, a group of ill-conditioned
accuracy of A + deteriorates further as eigenvalues affects the bound for the well-conditioned
II
ones. This led Wilkinson to introduce his Si factors,
measuring the sensitivity of the ith eigenvalue alone.
From first-order perturbation theory one finds, [4],
Again one can follow an iterative scheme of refinement
similar to the one mentioned in the previous section to
(25)
improve on A +. It runs as follows for the full rank situ
ation: where xi and yi' are the eigenvector and eigenrow asso
a) R = AT - AT AB; ciated with a simple Ai. So, normalizing both of them,
b) 8A+ = -BBT R; 1/1yi' xii becomes the Wilkinson factor and is equal
3