MAT-TRIAD 2005 Three Days Full of Matrices Program and Abstracts

MAT-TRIAD 2005 three days full of matrices

B¦dlewo, Poland March 35, 2005

Program and abstracts

Organizers

• Stefan Banach International Mathematical Center, Warsaw • Faculty of Mathematics and Computer Science, Adam Mickiewicz Uni- versity, Pozna« • Institute of Socio-Economic Geography and Spatial Management, Faculty of Geography and Geology, Adam Mickiewicz University, Pozna« • Department of Mathematical and Statistical Methods, Agricultural Uni- versity, Pozna« edited by

A. Markiewicz Department of Mathematical and Statistical Methods, Agricultural University, Pozna«, Poland and W. Woªy«ski Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Pozna«, Poland Contents

Part I. Information

Part II. Program

Part III. Abstracts

Some results on magic matrices ...... 13 Marek Aleksiejczyk

On linear combinations of idempotent matrices ...... 14 Jerzy K. Baksalary and Oskar Maria Baksalary

Computing the stationary vector of Markov chains by Schwarz Iterations ...... 15 Rafael Bru, Francisco Pedroche, and Daniel B. Szyld

On the minimum rank for pattern matrices∗ ...... 16 Rafael Cantò and Beatriz Ricarte

Projection methods for the linear split feasibility problems ... 17 Andrzej Cegielski

Some subclasses of H-matrices and how to use them for eigenvalue localization ...... 18 Ljiljana Cvetkovi¢

Matrix comparisons via eigenvector majorization: some results ...... 19 Pierre Druilhet

Some observations on periodic matrices ...... 20 Jürgen Groÿ

Weighted symmetry and hermitancy of matrices ...... 21 Jan Hauke

"When Leading Imply All" and other recent results ...... 22 Charles R. Johnson

Structure preserving algorithms ...... 23 Beata Laszkiewicz IV

A regularization of large linear least squares problems via rank revealing Hauseholder postmultiplications ...... 24 Andrzej Ma¢kiewicz

A robust approximate inverse preconditioner based on the Sherman-Morrison formula∗ ...... 26 J. Cerdàn, J. Marìn, and J. Mas

Norm estimates for eigenvalues of Riesz operators and matrices ...... 28 Mieczysªaw Mastyªo

Nonlinear polynomial eigenvalue problems with palindromic and symplectic structure ...... 29 Volker Mehrmann

G-majorization and matrix inequalities ...... 30 Marek Niezgoda

Mixed prudential choice in matrix decision problem under uncertainty ...... 31 Tadeusz Ostrowski

The conjugate gradients methods with indenite preconditioning ...... 32 Miroslav Rozloºník

Inverse eigenvalue problem for nonnegative matrices ...... 33 Helena migoc

Structured matrices and ecient algorithms of polynomial interpolation ...... 34 Alicja Smoktunowicz, Iwona Wróbel, and Przemysªaw Kosowski

Issai Schur (18751941) and the early development of the Schur complement: photographs, documents and biographical remarks ...... 35 George P. H. Styan

Characterizing the set of doubly stochastic matrices having unitary preimages ...... 36 Wojciech Tadej

MINRES residual norms of diagonally translated linear systems 37 Jurjen Duintjer Tebbens

Corollary 6 Route 66 to the structure of square matrices .. 39 Götz Trenkler V

Augmented systems in the potential uid ow problem ...... 40 Miroslav T·ma

On some uses of Sylvester-type equations ...... 42 Kresimir Veselic

Some properties of equiradial and equimodular sets ...... 43 Dominika Wojtera-Tyrakowska

Part IV. List of Participants

Index ...... 53

Part I

Information

The workshop will be held on 3-5 March 2005 at B¦dlewo (about 30 km from Pozna«). B¦dlewo is the Mathematical Research and Conference Center of the Polish Academy of Sciences - the setting is similar to Oberwolfach, with accommodation on site (for further info about this place please visit www.impan.gov.pl/Bedlewo). The aim of the workshop is to bring together researchers sharing an interest in a variety of aspects of matrix analysis and its applications and oer them a possibility to discuss current developments in these subjects. The format of this meeting will involve plenary talks and discussion sessions.

Organizers

Organizing committee:

• Jan Hauke, Institute of Socio-Economic Geography and Spatial Manage- ment, Faculty of Geography and Geology, Adam Mickiewicz University, Pozna«, • Augustyn Markiewicz (chair), Department of Mathematical and Statisti- cal Methods, Agricultural University, Pozna«, • Tomasz Szulc, Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Pozna«, • Waldemar Woªy«ski, Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Pozna«.

Part II

Program

Thursday, March 3, 2005

8:009:00 Breakfast Session I Chair C.R. Johnson 9:009:45 G.P.H. Styan: Issai Schur (18751941) and the early development of the Schur complement: photographs, documents and biographical remarks 9:4510:30 G. Trenkler: Corollary 6 Route 66 to the structure of square matrices 10:3011:00 Coee break Session II Chair T. Szulc 11:0011:45 M. Mastyªo: Norm estimates for eigenvalues of Riesz operators and matrices 11:4512:30 R. Cantò: On the minimum rank for pattern matrices 13:0015:00 Lunch Session III Chair G. Trenkler 15:0015:45 O.M. Baksalary: On linear combinations of idempotent matrices 15:4516:30 P. Druilhet: Matrix comparisons via eigenvector majorization: some results 16:3017:00 Coee break Session IV Chair M. Mastyªo 17:0017:30 J. Mas: A robust approximate inverse preconditioner based on the Sherman-Morrison formula 17:3018:00 J. Hauke: Weighted symmetry and hermitancy of matrices 18:0018:30 T. Ostrowski: Mixed prudential choice in matrix decision problem under uncertainty 19:00 Dinner 8

Friday, March 4, 2005

8:009:00 Breakfast Session V Chair K. Veselic 9:009:45 C.R. Johnson: "When Leading Imply All" and other recent results 9:4510:30 H. Smigoc: Inverse eigenvalue problem for nonnegative matrices 10:3011:00 Coee break Session VI Chair M. Tuma 11:0011:45 L. Cvetkovi¢: Some subclasses of H-matrices and how to use them for eigenvalue localization 11:4512:30 R. Bru: Computing the stationary vector of Markov chains by Schwarz Iterations 13:0015:00 Lunch Session VII Chair L. Cvetkovi¢ 15:0015:45 M. T·ma: Augmented systems in the potential uid ow problem 15:4516:30 J.D. Tebbens: MINRES residual norms of diagonally translated linear systems 16:3017:00 Coee break Session VIII Chair K. Zi¦tak 17:0017:30 A. Smoktunowicz and I. Wróbel: Structured matrices and ecient algorithms of polynomial interpolation 17:3018:00 B. Laszkiewicz: Structure preserving algorithms 18:0018:30 D. Wojtera-Tyrakowska: Some properties of equiradial and equimodular sets 19:00 Dinner 9

Saturday, March 5, 2005

8:009:00 Breakfast Session IX Chair R. Bru 9:009:45 V. Mehrmann: Nonlinear polynomial eigenvalue problems with palindromic and symplectic structure 9:4510:30 M. Rozloºník: The conjugate gradients methods with indenite preconditioning 10:3011:00 Coee break Session X Chair V. Mehrmann 11:0011:45 K. Veselic: On some uses of Sylvester-type equations 11:4512:30 A. Ma¢kiewicz: A regularization of large linear least squares problems via rank revealing Hauseholder postmultiplications 13:0015:00 Lunch Session XI Chair G.P.H. Styan 15:0015:45 J. Groÿ: Some observations on periodic matrices 15:4516:30 M. Niezgoda: G-majorization and matrix inequalities 16:3017:00 Coee break Session XII Chair J. Mas 17:0017:30 M. Aleksiejczyk: Some results on magic matrices 17:3018:00 W. Tadej: Characterizing the set of doubly stochastic matrices having unitary preimages 18:0018:30 A. Cegielski: Projection methods for the linear split feasibility problems 19:00 Dinner

Part III

Abstracts

Marek Aleksiejczyk 13 Some results on magic matrices

Marek Aleksiejczyk

University of Warmia and Mazury in Olsztyn, Poland

Abstract

The aim of this work is to investigate certain properties of magic matrices, i.e. square matrices of order n satisfying the following conditions: (m1) the entries of A belong to the set {1, 2, . . . , n2} (m2) if (i, j) 6= (k, l) then aij 6= akl (m3) the sums of each row, each column, main diagonal and antidiagonal are equal. We focus our attention to their squares, higher powers and numerical ranges of magic matrices and their powers (we present an estimation in general case and an exact formulas in special cases). Some open problems are also proposed.

Keywords

Magic matrix, Numerical range.

References

M. Aleksiejczyk (2002) Properties of numerical ranges of matrices, doctoral disser- tation, Institute of Mathematics of the Polish Academy of Sciences, Warsaw (in Polish) S. J. Kirkland, M. Neumann (1995). Group inverses of Mmatrices associated with nonnegative matrices having few eigenvalues. Linear Algebra Appl. 220, 181 213. A. ZalewskaMitura, J. Zemánek (1997). The Gerschgorin discs under unitary simi- larity, Banach Center Publications, Vol. 38, 427441. Institute of Mathematics, Polish Academy of Sciences. 14 Jerzy K. Baksalary and Oskar Maria Baksalary On linear combinations of idempotent matrices

Jerzy K. Baksalary1 and Oskar Maria Baksalary2

1 Zielona Góra University, Zielona Góra, Poland 2 Adam Mickiewicz University, Pozna«, Poland

Abstract

Two problems considered recently in the literature, both dealing with linear combinations of idempotent matrices, are revisited. The rst of them was posed by Coll and Thome (2003) and concerns the question of when a linear combination T = c1P1 + c2P2 of nonzero dierent complex idempotent matrices P1, P2, with nonzero complex numbers c1, c2, is the group involutory matrix. According to the solution provided by Coll and Thome (2003) as Theorem 1, it is possible in a nite number of cases, each characterized by denite values of scalars c1 and c2. In the present paper it is shown that this theorem is not correct, for the actual number of cases in which T is the group involutory matrix is innite. The second problem was posed by Özdemir and Özban (2004) who considered the question of when a linear combination P = c1P1 + c2P2 + c3P3 of nonzero dierent, mutually commuting, complex idempotent matrices P1, 2 P2, P3, with nonzero complex numbers c1, c2, c3, satises P = P. The answer to this question given in the above mentioned paper as Theorem 3.2 does not provide the complete solution to the problem, for it characterizes particular sets of sucient conditions only. This lacuna is covered by the present paper, in which the list of necessary and sucient conditions ensur- ing that P is an idempotent matrix is obtained. Parenthetically notice that this result generalizes part (a) of Theorem 1 in Baksalary (2004).

Keywords

Commutativity, Group inverse, Group involutory matrix, Idempotency, Obli- que projector, Projector, Tripotent matrix.

References

Baksalary, O.M. (2004). Idempotency of linear combinations of three idempotent matrices, two of which are disjoint. Linear Algebra Appl. 388, 6778. Coll, C. and N. Thome (2003). Oblique projectors and group involutory matrices. Appl. Math. Comput. 140, 517-522 . Özdemir, H. and A.Y. Özban (2004). On idempotency of linear combinations of idempotent matrices. Appl. Math. Comput. 159, 439-448 . Rafael Bru, Francisco Pedroche, and Daniel B. Szyld 15 Computing the stationary vector of Markov chains by Schwarz Iterations

Rafael Bru1, Francisco Pedroche1, and Daniel B. Szyld2

1 University Politecnica de Valencia, Spain 2 Temple University, Philadelphia, USA

Abstract

A convergence analysis is presented for additive Schwarz iterations when applied to consistent singular systems of equations Ax = b. The theory applies to singular M-matrices with one-dimensional null space, and is applicable in particular to systems representing ergodic Markov chains. The results are based on an algebraic formulation of Schwarz methods, in particular the convergence theorem for additive Schwarz iterations and the existence of a splitting of the matrix A with the same iteration matrix as the additive Schwarz scheme. This work complements the results of [Marek and Szyld, LAA, in press], where multiplicative Schwarz iterations are shown to converge for singular systems.

Keywords

Schwarz method, Markov chains, Iterative methods.

References

Marek, I. and B.D. Szyld (2004). Algebraic Schwarz Methods for the Numerical Solution of Markov Chains. Linear Algebra Appl. 386, 67-81. 16 Rafael Cantò and Beatriz Ricarte On the minimum rank for pattern matrices∗

Rafael Cantò and Beatriz Ricarte

Universitat Politécnica de Valéncia, Spain

Abstract

By a partial matrix we mean an m × n matrix P some of whose entries are specied and the remainder entries are free variables of some indicated set. A completion matrix of P is an specication of the free variables obtaining a conventional matrix. Matrix completion problems try to obtain conditions for the existence of a completion for a given partial matrix in a class of interest. The historical motivation for the study of these problems appears in subjects as mathematical economics, biology, social sciences, etc. in which models some matrix elements give qualitative instead of quantitative information. A pattern matrix P is a partial matrix with the specied entries equal to zero and the remainder entries are nonzero free variables over a eld IF. We say pattern(A) = P if A is a completion matrix of P. An interesting matrix completion problem asks for those completions for a given pattern matrix P with the lowest possible rank. By mr(A) we denote the minimum rank of P that is mr(P) = min{rank(A)|pattern(A) = P}. In this work we consider a graph theoretic approach to study the minimum rank completion problem and obtain dierent lower and upper bounds for solving the question.

Keywords

Pattern matrix, Minimum rank, Bipartite graph.

∗Supported by the Spanish DGI grant number AGL2004-03263/AGR and the OCYT grant GRUPOS 03/062. Andrzej Cegielski 17 Projection methods for the linear split feasibility problems

Andrzej Cegielski

University of Zielona Góra, Poland

Abstract

Let C ⊂ IRn be a closed convex subset, A - an n×m real matrix and b ∈ IRm. Consider the following linear split feasibility problem (LSFP) ﬁnd x ∈ C such that AT x ≤ b, if such x exist. The problem has lot of applications, e.g., the problem of computed tomography or the problem of intensity modulated radiation therapy can be modelled as a large scale LSFP. We study a projection methods for the LSFP which generate a sequence (xk) by the following iterative scheme

xk+1 = T (xk), (1) where x1 ∈ C is arbitrary, the operator T : C −→ C is dened by the equality

T (x) = PC (x − µγ(x)Gw(x)), (2)

µ ∈ [0, 2], is called a relaxation parameter, γ : C −→ IR+ is called a step size function and IRm is called a weight operator. We call the method w : C −→ + (1) with T dened by (2) the projected surrogate constraints method (PSC- method). We study in the paper the Fejér monotonicity and the convergence of the PSC-method in dependence of step size γ and weights w. We show also that the convergence of the surrogate constraints method of Yang-Murty and of the CQ-method of Byrne applied to the LSFP follows from our main result.

Keywords

Linear split feasibility problem, Projection methods.

References

Byrne, C. (2002). Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Problems 18, 441-453. Cegielski, A. Convergence of the projected surrogate constraints method for the linear split feasibility problems. (submitted). Yang, K. and K.G. Murty (1992). New iterative methods for linear inequalities. JOTA 72, 163-185. 18 Ljiljana Cvetkovi¢ Some subclasses of H-matrices and how to use them for eigenvalue localization

Ljiljana Cvetkovi¢

University of Novi Sad, Serbia and Montenegro

Abstract

Well known characterization of H-matrices is given by the fact that the matrix is an H-matrix i it can be scaled to a strictly diagonally dominant (SDD) matrix by nonsingular diagonal matrix (from the right side, of course). How to nd such a scaling matrix is still open question. Because of that, some subclasses of H-matrices are very useful, in particular if they are described by "checkable" conditions, meaning simple functions of matrix elements only. Several such subclasses will be presented. Some of them will be used for obtaining various Ger²gorin-type localization theorems. On the other hand, some of them will help us to obtain more information about the eigenvalues on the boundary of such a Ger²gorin-type localization area.

Keywords

H-matrices, Eigenvalue localization.

References

Cvetkovi¢, Lj. (1998). Convergence theory for relaxation methods to solve systems of equations. Monographical Booklet MB-5 PAMM, Technical University of Bu- dapest. Cvetkovi¢, Lj., V. Kosti¢, and R.S. Varga (2004). A new Ger²gorin-type eigenvalue inclusion area. ETNA 18, 73-80. Cvetkovi¢, Lj. and V. Kosti¢ (2005). New criteria for identifying H-matrices. Journal of Computational and Applied Mathematics, in print. Varga, R.S. (2004). Ger²gorin and his circles. Springer Series in Computational Mathematics 36. Pierre Druilhet 19 Matrix comparisons via eigenvector majorization: some results

Pierre Druilhet

CREST-ENSAI, Bruz, France

Abstract

We discuss some aspect of matrix comparison w.r.t some class of criteria. Consider a set M of (t,t) symmetric matrices, we dene a criteria to be a map from M onto (−∞, +∞). A matrix M is minimal w.r.t. φ if it minimizes φ(M) over M. Such kind of problem occurs typically in experimental designs, where M is a set of information matrices and φ is usually a convex, non- increasing function. We are interesting particulary in the Φp criteria dened by: 1/p 1 −p for Φp(M) = = t−1 tr(M ) , p 6= 0 −1/t Φ0(M) = lim Φp(M) = det(M) , p→0 We present some sucient condition for a matrix to minimize simultaneously all the Φp criteria for p greater than some po. For example, we have:  Φ (M ) = Φ (M )  po d1 po d2 λ(M −po ) ≺ λ(M −po ) =⇒ Φ (M ) ≤ Φ (M ) for p > p , d1 d2 p d1 p d2 o  for Φp(Md1 ) ≥ Φp(Md2 ) p < po where ≺ is the majorization ordering (see Marshall & Olkin, 1975). Note that the converse part does not hold. We also present similar results, some of them using weak majorization and we give some applications

Keywords

Majorization, Schur convexity, Matrix comparison.

References

Druilhet, P. (2004). Optimality criteria in experimental designs. Linear Algebra Appl. 388, 147-157. Marshall, A.W. and I. Olkin (1979). Inequalities: Theory of Majorization and its Applications. New York: Academic Press. Hedayat, A. (1981). Study of optimality criteria in design of experiment. In: M. Csörgö, D.A. Dawson, J.N.K. Rao, and A.K.Md.E. Sahel (Eds.), Statistics and Related Topics, (pp. 39-56). North-Holland Publishing Company. 20 Jürgen Groÿ Some observations on periodic matrices

Jürgen Groÿ

University of Dortmund, Germany

Abstract

We recall the concept of a periodic square matrix, see Mirsky (1955), ad- mitting a positive integer power greater than one which is identical to the matrix itself. Some basic observations are given and a discussion of recent results in the literature is carried out, showing the possibility of strengthening and generalizing some of them.

Keywords

Matrix power, Diagonalizable matrix, EP matrix, Normal matrix, Group inverse, Moore-Penrose inverse.

References

Mirsky, L.M. (1955). An Introduction to Linear Algebra. Oxford: Clarendon Press. [Reprinted 1990, New York: Dover Publications.] Jan Hauke 21 Weighted symmetry and hermitancy of matrices

Jan Hauke

Adam Mickiewicz University, Pozna«, Poland

Abstract

Symmetry properties of matrices (operators) play important role in presentation of many charactersistcs of matrices (operators). There is used non homogenous terminology for some of properties {compare Horn and Johnson

[1] and Sadun [2]}. We propose to use a notation of weighted symmetry (WL- WR-symmetry) and weighted hermitancy (WL − WR-hermitancy). For some matrices WL, WR ∈ Cn,n we say that

A ∈ Cn,n is WL-WR-symmetric i WLAWR = A and we say that

∗ A ∈ Cn,n is WL-WR-hermitian i WLAWR = A . In similar way we dene other symmetry properties investigated by Presm- man [3] and Trench [4,5]. In the presentation we generalize results presented in papers [3], [4], and [5]

Keywords

Weighted symmetry of matrices, Weighted hermitancy matrices.

References

Horn, R.A and C.R. Johnson (1990). Matrix Analysis. Cambridge: Cambridge Uni- versity Press. Sadun, L. (2001). Applied Linear Algebra - The Decupling Principle. New Jersey: Prentice Hall. Pressman, I. S. (1998). Matrices with multiple symmetry properties: applications of centrohermitian and perhermitian matrices. Linear Algebra Appl. 377, 207 218. Trench, W. F. (2004). Characterization and properties of matrices with generalized symmetry or skew-symmetry. Linear Algebra Appl. 377, 207218. Trench, W. F. (2004). Hermitian, hermitian R-symmetric and hermitian R-skew symmetric Procrustes problem. Linear Algebra Appl. 387, 8398. 22 Charles R. Johnson "When Leading Imply All" and other recent results

Charles R. Johnson

College of William and Mary, USA

Abstract

Under what circumstances are all principal minors of a given matrix positive when the leading ones are? Two circumstances are known: symmetric matrices (implication positive deniteness) and Z-matrices (o-diagonal entries negative; implication M-matrices). We give a recent characterization of all such circumstances, which identies a new and interesting class of matrices that unies symmetric and Z-matrices. Time permitting, other recent results about LU factorization, etc will be given. Beata Laszkiewicz 23 Structure preserving algorithms

Beata Laszkiewicz

Wrocªaw University of Technology, Poland

Abstract

In recent years interests in the structure preserving algorithms and structured tools for structured matrices have increased (see for example [1]-[5]). In the talk we review some properties of the structure preserving functions in automorphism groups associated with bilinear or sesquilinear forms. For example, the principal square root is a function that preserves every automorphism group. We present numerical experiments with algorithms for computing the polar decomposition and the matrix sign decomposition in matrix groups.

Keywords

Structured matrix, Structure preserving function, Polar decomposition, Ma- trix sign, Matrix square root.

References

Benner, P., V. Mehrmann, and H. Xu (1998). A numerically stable, structure preserving method for computing the eigenvalues of real Hamiltonian or sympletic pencils. Numerische Math. 78, 329-358. Benner, P., D. Kressner, and V. Mehrmann (2003). Structure preservation: a chal- lenge in computational control. Future Generation Computer Systems 19, 1243- 1252. Higham, N.J., D.S. Mackey, N. Mackey, and F. Tisseur (2004). Computing the polar decomposition and the matrix sign decomposition in matrix groups. SIAM J. Matrix Anal. Appl., SIAM J. Matrix Anal. Appl. 25, 1178-1192. Higham, N.J., D.S. Mackey, N. Mackey, and F. Tisseur (2004). Functions preserving matrix groups and iterations for the matrix square root. SIAM J. Matrix Anal. Appl., to appear. Mackey, D.S., N. Mackey, and F. Tisseur (2003). Structured tools for structured matrices. Electr. J. Linear Algebra 10, 106-145. 24 Andrzej Ma¢kiewicz A regularization of large linear least squares problems via rank revealing Hauseholder postmultiplications

Andrzej Ma¢kiewicz

Technical University of Pozna«, Poland

Abstract

In this paper we discuss a new iterative method that is suited for regularization of the series of large linear least squares problems. In each problem in the series the same rank-decient coecient matrix A is used and weighted in a specic manner. The main feature of these problems is that the matrix A (not necessarily sparse) is having a cluster of small singular values, and there is a well-determined gap between its large and small singular values. The new algorithm uses (only once) properly chosen Hauseholder postmultiplications. These transformations provide an elegant way to extract a well-conditioned core subproblems of minimum dimension both for the linear least squares and the total least squares problem. Next, a modied version of the LSQR algorithm of Paige and Saunders is used to solve the particular weighted problems in a row. A partial reorthogonalization for maintaining semi-orthogonality among the Lanczos vectors is used. Examples showing promising results from numerical experiments are presented. As a by-product a new and eective spectral matrix norm estimator is given. Possible applications to signal processing, image processing, multiple linear regression and geographically weighted regression (GWR) are mentioned.

References

Björck, Å. (1996). Numerical Methods for Least Squares Problems. SIAM, Philadel- phia. Björck, Å., T. Elfving, and Z. Strako¢s(1998). Stability of conjugate gradient and Lanczos methods for linear least squares problems SIAM J. Matrix Anal. Appl. 19, 720-736. Fotheringham, S., Ch. Brunsdon, and M. Charlton (2002). Geographically Weighted Regression. J. Wiley. Golub, G. H. and W. Kahan (1965). Calculating the singular values and pseudo- inverse of a matrix. J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2, 205-224. Golub, G. H. and C. F. Van Loan (1989). Matrix Computations 2 Ed. Johns Hopkins University Press, Baltimore. Andrzej Ma¢kiewicz 25

Golub, G. H. and U. von Matt (1997). Tikhonov Regularization for Large Scale Problems. In: G. H. Golub, S. H. Lui, F. Luk. and R. Plemmons (Eds.), Work- shop on Scientic Computing. Springer, New York. Hansen, P. C. (1998). Rank-Decient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion. SIAM, Philadelphia. Lanczos, C. (1950). An Iteration methods Method for the Solution of the Eigenvalue Problem of Linear Diferential and Integral Operators. J. of Res. Nat. Bur. Stand. 45, 255-282. Larsen, R. M. (1997). Iterative algorithms for two-dimensional helioseismic inversion. In: B. H Jacobsen (Ed.), Proc. of the Interdisciplinary Inversion Workshop 5 (pp.123-138). Dept. Earth Sciences, Aarhus University. Larsen, R. M. and P. C. Hansen (1997). Ecient Implementations of the SOLA Mollier Method. Astron. Astrophys. Suppl. 121, 587-598. Ma¢kiewicz, A. Tikhonov Regularization via Lanczos Bidiagonalization Algorithm with Selective Reorthogonalization, in preparation. Ma¢kiewicz, A. and R. Rhalia (1996). An ecient orthogonalization method for accurate computation of singular values. In: M. Doblare, J.M. Correas (Eds.), Metodos Numericos en Ingenieria, Vol.2 (pp.1371-1382). SEMNI Barcelona, Spain. Paige, C. C. and M. A. Saunders (1982). LSQR: an algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Software 8, 43-71. Saad, Y. (1980). On the rates of convergence of the Lanczos and the block-Lanczos Methods. SIAM J. Numer. Anal. 17, 687-706. Simon, H. D. (1984). Analysis of the symmetric Lanczos algorithm with reorthogonalization methods. Linear Algebra Appl. 61, 101-131. 26 J. Cerdàn, J. Marìn, and J. Mas A robust approximate inverse preconditioner based on the Sherman-Morrison formula∗

J. Cerdàn, J. Marìn, and J. Mas

Universitat Politécnica de Valéncia, Spain

Abstract

To solve a large, and sparse linear system

Ax = b; (1) an approximate solution of (1) is usually obtained by using a preconditioned iterative Krylov subspace method [6]. In this work we focus on factorized approximate inverse preconditioners, in this class of preconditioners two matrices such that its product is an approximation of the inverse of A are computed and stored explicitely. Therefore, the application of the preconditioner reduces to a matrix-vector product which can be easily implemented on par- allel computers. Recently a new framework for computing sparse approximate inverse preconditioners for nonsymmetric matrices has been presented in [2]. This algorithm, referred to as AISM, computes an approximate inverse of A using the Sherman-Morrison formula. The authors show that its computation is stable for M-matrices and H-matrices [3]. For symmetric positive denite matrices a question which remained open is how to exploit the symmetry in order to avoid the computation of two factors. In this work we answer this question and we present a modication of AISM which is well dened for this class of matrices. In addition, the new formulation seems to be more robust in the nonsymmetric case. The results of numerical experiments performed with the modied approximate inverse preconditioner based on the Sherman- Morrison will be presented. We will refer to it as AISMr preconditioner. The matrices used in the test come from the Harwell Boeing collection [5] and Tim Davis' collection [4].

∗ Supported by Spanish DGI Grant MTM2004-02998.

References

Benzi, M., C. D. Meyer, and M. T·ma (1996). A sparse approximate inverse preconditioner for the conjugate gradient method. SIAM J. Sci. Comput. 17, 1135- 1149. J. Cerdàn, J. Marìn, and J. Mas 27

Cerdàn, J., J. Marìn, and J. Mas (2003). Preconditioning sparse nonsymmetric linear systems with the Sherman-Morrison formula. SIAM J. Sci. Comput. 25(2), 701-715. Cerdàn, J., T. Faraj, J. Marín, and J. Mas. A block approximate inverse preconditioner for sparse nonsymmetric linear systems. Submitted. Davis, T. University of Florida sparse matrix collection. NA Digest, 92(42), October 16, 1994. http://www.cise.u.edu/research/sparse/matrices. Retrieved April, 2002. Du, I. S., R. G. Grimes, and J. G. Lewis (1992). User's guide for the Harwell- Boe- ing sparse matrix collection. Technical Report RAL-92-886, Rutherford Apple- ton Laboratory, Chilton, England. Saad, Y. (1996). Iterative Methods for Sparse Linear Systems. PWS Publishing Company, Boston. 28 Mieczysªaw Mastyªo Norm estimates for eigenvalues of Riesz operators and matrices

Mieczysªaw Mastyªo

Adam Mickiewicz University, Pozna«, Poland

Abstract

We will give a brief survey concerning recent joint work with A. Defant and C. Michels. Using abstract interpolation theory, we study eigenvalue distri- bution problems for Riesz operators on complex symmetric Banach sequence spaces. We prove eigenvalue estimates for matrices. Combining the results with some geometrical estimates for Kronecker's matrices, we show appli- p cations to Orlicz sequence spaces `ϕ. For the power function ϕ(t) = t with 2 ≤ p < ∞, we obtain a celebrated result of W.B. Johnson, H. König, B. Mau- rey and J.R. Retherford from 1979 which says that each complex n×n matrix T = [τij] satises the following eigenvalue estimate:

n 1/p n n p/q 1/p X p X X q |λi(T )| ≤ |τij| , i=1 j=1 i=1 where n is a sequence of eigenvalues of and , {λi(T )}i=1 T 2 ≤ p < ∞ 1/p + 1/q = 1. Volker Mehrmann 29 Nonlinear polynomial eigenvalue problems with palindromic and symplectic structure

Volker Mehrmann

Technische Universität Berlin, Germany

Abstract

We will present several practical examples for nonlinear palindromic and symplectic eigenvalue problems and discuss their properties. Allthough these problems on rst sight look quite dierent we show how they are related via a non-unimodular transformation that preserves the important part of the spectrum. We discuss structure preserving linearization and canonical forms for such matrix polynomials and we introduce numerical methods for the solution of the eigenvalue problem. 30 Marek Niezgoda G-majorization and matrix inequalities

Marek Niezgoda

Agricultural University of Lublin, Poland

Abstract

A unied approach is presented to the vector inequalities of the form Lx Kx↓, where L and K are linear operators, is a group induced cone ordering and (·)↓ is the normal map associated with the ordering. In particular, a G- majorization inequality involving two orthoprojectors is given. The inequalities generalize a variety of majorization results on eigenvalues and singular values of matrices. The results are interpreted for various classes of matrices.

Keywords

G-majorization, Group induced cone ordering, Singular value, Eigenvalue, Doubly graded matrix, Tournament matrix, Absolutely doubly substochastic matrix, Hadamard product.

References

Ando, T. (1994). Majorization and inequalities in matrix theory. Linear Algebra Appl. 199, 1767. Ando, T., R. A. Horn and C. R. Johnson (1987). The singular values of a Hadamard product: A basic inequality. Linear and Multilinear Algebra 21, 345365. Bapat, R. B. (1991). Majorization and singular values III. Linear Algebra Appl. 145, 5970. Dahl, G. (2003). The doubly graded matrix cone and Ferrers matrices. Linear Al- gebra Appl. 368, 171190. Gregory, D. A. and S. J. Kirkland (1999). Singular values of tournament matrices. Electron. J. Linear Algebra 5, 3952. Horn, R. A. and F. Kittaneh (1998). Two applications of a bound on the Hadamard product with a Cauchy matrix. Electron. J. Linear Algebra 3, 412. Lewis, A. S. (1996). Group invariance and convex matrix analysis. SIAM J. Matrix Anal. Appl. 17, 927949. Marshall, A.W. and I. Olkin (1979). Inequalities: Theory of Majorization and its Applications. New York: Academic Press. Niezgoda, M. (1998). Group majorization and Schur type inequalities. Linear Al- gebra Appl. 268, 930. Niezgoda, M. (1999). Gmajorization inequalities for linear maps. Linear Algebra Appl. 292, 207231. Tadeusz Ostrowski 31 Mixed prudential choice in matrix decision problem under uncertainty

Tadeusz Ostrowski

The State Vocational University, Gorzów Wlkp., Poland.

Abstract

Decision problem under uncertainty is considered. The problem is given as square outcome matrix. We show how in that kind of matrix decision problem the mixed prudential choice (MPC for short) can be found with the determinant and cofactors of the given matrix. We also give a formula for the value of that choice. We use an example to analyse the MPC in comparison to all very well known criteria like: Criterion of Optimism (called often Maximax Rule), Wald Criterion (known also as Criterion of Pessimism or Maximin Rule), Hurwicz Criterion, Laplace Criterion (The Principle of Insucient Reason), and Savage Criterion (called sometimes Regret Criterion or Minimax Rule). At last we show the MPC paradox connected with Alos-Ferrer's payo matrix, where some decision is excluded no matter how large, or just opposite, how small is the value of a parameter and consequently how much you can win or lose.

Keywords

Decision under uncertainty, Mixed prudential choice.

References

Alos-Ferrer, C. (2000). Finite Population Dynamics and Mixed Equilibria. Univer- sity of Vienna, Department of Economics, Working Papers, No: 0008 32 Miroslav Rozloºník The conjugate gradients methods with indenite preconditioning

Miroslav Rozloºník

Institute of Computer Science Academy of Sciences of the Czech Republic, Prague

Abstract

In this contribution we consider the solution of symmetric indenite linear systems of the form AB x f = , BT 0 y g where A is a symmetric positive denite matrix and B has full column rank. Saddlepoint problems of this type arise in many application areas such as computational uid dynamics, electromagnetism, optimization and nonlinear programming. Particular attention has been paid to the iterative solution of these systems and to preconditioning techniques. Several structure dependent schemes have been proposed and analyzed. Indeed, the block pattern of saddle-point systems enables to take into account not only simple preconditioning strategies and scalings, but also preconditioners with a particular block structure. Here we analyze the null-space projection (or con- straint) indenite preconditioner. Since it was shown that the behavior of most of nonsymmetric Krylov subspace methods can be in this case related to the convergence of preconditioned conjugate gradient method (PCG) we study in detail its theoretical properties and propose simple procedures for correcting its possible misconvergence. The numerical behavior of the scheme is discussed and the maximum attainable accuracy of the approximate solution computed in nite precision arithmetic is estimated. This contribution is a joint work with V. Simoncini and it is partially supported by the project 1ET400300415 within the National Program of Research "Information Society".

References

Rozloºník, M. and V. Simoncini (2002). Krylov subspace methods for saddle point problems with indenite preconditioning. SIAM J. Matrix. Anal. Appl. 24, 368391. Helena migoc 33 Inverse eigenvalue problem for nonnegative matrices

Helena migoc

University of Ljubljana, Slovenia

Abstract

The nonnegative inverse eigenvalue problem is the problem of nding necessary and sucient conditions for a list of n complex numbers σ to be the spectrum of a nonnegative matrix. The problem has been completely solved only for n ≤ 4. Boyle and Handelman have shown that if σ = (λ1, λ2, ..., λn) is such list of nonzero complex numbers that Pn k for all positive sk = i=1 λi > 0 integers k; then there exists a nonnegative matrix A with nonzero spectrum

(λ1, λ2, ..., λn). However, their proof is not constructive. The aim of the talk is to show how to construct such matrices for some lists of complex numbers σ. The talk is based on a joint work with Thomas J. Laey. 34 Alicja Smoktunowicz, Iwona Wróbel, and Przemysªaw Kosowski Structured matrices and ecient algorithms of polynomial interpolation

Alicja Smoktunowicz, Iwona Wróbel, and Przemysªaw Kosowski

Warsaw University of Technology, Poland

Abstract

The problem of constructing ecient algorithms of polynomial interpolation involves matrices with special structure. Numerical algorithms should exploit the structure of such matrices. Horner's, Goertzel's, Clenshaw's methods and the scheme and the scheme of divided dierences are frequently used in the interpolation and approximation problems and in signal processing. In order to improve accuracy we propose modications of these algorithms. A comparison of various techniques with respect to eciency, numerical stability and accuracy is given. The numerical tests in Matlab demonstrate the computational advantages of the proposed modications.

Keywords

Structured matrices, Polynomial evaluation, Interpolation, Clenshaw's method, Goertzel's algorithm, FFT, Condition number, Numerical stability, Iterative renement.

References

Björck, Å. and V. Pereyra (1970). Solution of Vandermonde systems of equations. Math. Comput. 24 (112), 893-903. Higham, N. J. (1996). Accuracy and Stability of Numerical Algorithms. Philadel- phia: SIAM. Smoktunowicz, A. (2002). Backward stability of Clenshaw's algorithm. BIT 42 (3), 600-610. Smoktunowicz, A. and I. Wróbel. On improving the accuracy of Horner's and Go- ertzel's algorithms. To appear. Smoktunowicz, A., P. Kosowski, and I. Wróbel (2004). How to overcome the numerical instability of the scheme of divided dierences? arXiv.math.NA/0407195. George P. H. Styan 35 Issai Schur (18751941) and the early development of the Schur complement: photographs, documents and biographical remarks

George P. H. Styan

McGill University, Montréal, Canada

Abstract

We comment on the development of the Schur complement from 1812 through 1968 when it was so named and given a notation by Emilie Virginia Haynsworth (19161985). The adjectival noun Schur was chosen because of the Hilfssatz in the paper by Issai Schur (18751941) published in 1917 in the Journal für die reine und angewandte Mathematik (vol. 147, pp. 205232), in which the Schur determinant formula was introduced. Following some biographical remarks about Issai Schur, we present the Banachiewicz inversion formula for the inverse of a nonsingular partitioned block matrix which was introduced in 1937 by the astronomer Tadeusz Banachiewicz (18821954). We note, however, that closely related results were obtained earlier in 1923 by the geodesist Hans Boltz (18831947) and in 1933 by Ralf Lohan. We illustrate our ndings with several photographs and copies of original documents. [Joint research with Simo Puntanen (University of Tampere) supplement- ing Chapter 0 in The Schur Complement and Its Applications, Series: Nu- merical Methods and Algorithms 4 (Fuzhen Zhang, ed.), Springer 2005; see also Issai Schur (18751941) and the Early Development of the Schur Com- plement, with Applications to Statistics and Probability by Simo Puntanen & George P. H. Styan, Report A 346, Dept. of Mathematics, Statistics & Philosophy, University of Tampere, July 2004.] 36 Wojciech Tadej Characterizing the set of doubly stochastic matrices having unitary preimages

Wojciech Tadej

Cardinal Stefan Wyszy«ski University, Warszawa, Poland

Abstract

The problem of characterizing doubly stochastic n x n matrices that can be obtained by squaring the moduli of the entries of some unitary matrix will be presented. It is associated with the experimental physics problem of determining a discrete time unitary transformation of the state space of a nite dimensional quantum system undergoing a quantum process, corresponding to a doubly stochastic matrix of transition probabilities between basis states, measured in this process. It is also related to the problem of nding all real and complex Hadamard matrices, having various applications in mathematics and theoretical physics. Results concerning the geometry and spectra of the set of doubly stochastic matrices and those originating from unitary matrices will be presented, for n=3 and n=4.

References

Bengtsson, I., A. Ericsson, M. Kus, W. Tadej, K. Zyczkowski. Birkho's polytope and unistochastic matrices, N=4 and N=4. www.arxiv.org/abs/math.CO/0402325 Jurjen Duintjer Tebbens 37 MINRES residual norms of diagonally translated linear systems

Jurjen Duintjer Tebbens

Institute of Computer Science Academy of Sciences of the Czech Republic, Prague

Abstract

Symmetric indenite linear systems are currently solved by either preconditioned direct [2, 3, 4] or preconditioned iterative [1, 6] methods. Among the methods that are being preconditioned, the MINRES method [10] belongs to the most popular ones. As a well-known matter of fact, convergence behavior of MINRES can be related to the eigenvalues of the involved system matrix. Hence a diagonal translation, shifting the spectrum, can have an important inuence on the convergence speed of MINRES processes. In this talk we compare the MINRES residual norms obtained from two linear systems whose matrices dier by a diagonal translation. The equation we derive does not refer to spectral properties of the matrix, but it is based on an exact expression for the residual norm of residual minimizing methods such as MINRES [7]. The comparison is a generalization of the relation between the convergence for systems with translated matrices that has been formulated in [8, 9] for the special case of tridiagonal Toeplitz matrices. Surprisingly, the residual norms appear to be mutually related in a rather complicated way [5]. The result was obtained in joint work with Zden¥k Strako².

Keywords

Submission of abstract, Reference style, Submission date, How to submit.

References

M. Benzi (2002). Preconditioning Symmetric Indenite Linear Systems. House- holder Symposium XV, June 17-21, Peebles, Scotland. I. Du (2004). MA57A Code for the Solution of Sparse Symmetric Denite and Indenite Systems. ACM Trans. Math. Software 30, 118144. Du I. and J. Reid (1982). MA27A Set of FORTRAN Subroutines for Solving Sparse Symmetric Sets of Linear Equations. Technical Report AERE R10533, London, UK. Du, I. and J. Reid, (1983). The Multifrontal Solution of Indenite Sparse Sym- metric Linear. ACM Trans. Math. Software 9, 302325. 38 Jurjen Duintjer Tebbens

Tebbens, J. D., A. Greenbaum, and Z. Strako². GMRES and Ideal GMRES for a Convection Diusion Model Problem, in preparation. Hagemann, M. and O. Schenk. Weighted Matchings for the Preconditioning of Sym- metric Indenite Linear Systems, Technical Report CS-2004-005, Department of Computer Science, University of Basel. Submitted. Ipsen, I. C. F. (2000). Expressions and Bounds for the GMRES Residual. BIT 40, 524535. Liesen, J. and Z. Strako² (2004). Convergence of GMRES for Tridiagonal Toeplitz Matrices. SIAM J. Matrix Anal. Appl., accepted for publication. Liesen, J. and Z. Strako² (2004). GMRES Convergence Analysis for a Convection- Diusion Model Problem. SIAM J. Sci. Comput. , accepted for publication. Paige, C. C. and M. Saunders (1975). Solution of Sparse Indenite Systems of Linear Equations. SIAM J. Numer. Anal.12, 617629.

This work is supported by the GA AS CR under the project 1ET400300415. Götz Trenkler 39 Corollary 6 Route 66 to the structure of square matrices

Götz Trenkler

University of Dortmund, Germany

Abstract

Starting from Corollary 6 in Hartwig and Spindelböck (1983), several matrix types including EP and normal matrices are discussed. Special attention is paid to oblique and orthogonal projectors and their relationship to the group inverse.

Keywords

EP matrix; Normal matrix.

References

Hartwig, R. E. and K. Spindelböck (1983). Some Closed Form Formulae for the In- tersection of Two Special Matrices under the Star Order. Linear & Multilinear Algebra. 13, 323331. 40 Miroslav T·ma Augmented systems in the potential uid ow problem

Miroslav T·ma

Institute of Computer Science Academy of Sciences of the Czech Republic, Prague

Abstract

Mixed-hybrid nite element discretization of the Darcy's law and the con- tinuity equation which describe the potential ow problem in porous media leads to a saddle-point problem with the symmetric and indenite system matrix of the following block structure [1].

 ABC BT  . CT Several approaches for solving these problems have been considered recently. They range from a pure iterative solution based on the preconditioned conjugate gradient or MINRES methods, block elimination based on the Schur complement to the dual variable approach and consecutive iterative solution of the resulting system projected on the null space of the matrix block (B,C)T . (see, e.g., [2], [3], [4] and [5]). In this contribution, our work on both theory and algorithms in this eld will be presented. In addition, we will describe the real-world application which provides these saddle-point problems. The results were obtained in joint work with Mario Arioli, Miroslav Rozloºník and Ji°í Mary²ka.

References

Mary²ka, J., M. Rozloºník, and M. T·ma (1995). Mixed-hybrid nite element approximation of the potential uid ow problem. J. Comput. Appl. Math. 63, 383-392. Mary²ka, J., M. Rozloºník, and M. T·ma (1996). The potential uid ow problem and the convergence rate of the minimal residual method. Num. Lin. Alg. with Appl. 3(6), 525-542. Mary²ka, J., M. Rozloºník, and M. T·ma (2000). Schur complement systems in the mixed-hybrid nite element approximation of the potential uid ow problem. SIAM J. Sci. Comput. 22, 704723. Mary²ka, J., M. Rozloºník, and M. T·ma (2000). Schur complement reduction in the mixed-hybrid approximation of Darcy's law: rounding error analysis. J. Comput. Appl. Math. 117, 159173. Miroslav T·ma 41

Arioli, M., J. Mary²ka, M. Rozloºník, and M. T·ma (2004). The dual variable approach for the mixed-hybrid approximation of the potential uid ow problem. Submitted to ETNA. 42 Kresimir Veselic On some uses of Sylvester-type equations

Kresimir Veselic

Fernuniversitaet Hagen, Germany

Abstract

The Sylvester, Lyapunov and other similar equations have been known to yield useful informations on many objects attached to a given matrix. We follow up by showing some recent/new results in this direction. Among others we will consider (i) bounds on the matrix exponential and stability properties, (ii) optimizing exponential decay and (iii) relative bounds for the eigenvectors of symmetric matrices/operators. Dominika Wojtera-Tyrakowska 43 Some properties of equiradial and equimodular sets

Dominika Wojtera-Tyrakowska

Adam Mickiewicz University, Pozna«, Poland

Abstract

Our goal is to describe some properties of matrices belonging either to the set of matrices equiradial with a matrix A, or to the set of matrices equimodular with A. The rst set is specied by the standard Gerschgorin-type information about A, i.e. by the vector

r ItG(A) = (|a11|,R1(A), |a22|,R2(A),..., |ann|,Rn(A)), where Ri(A) is the sum of moduli of the o-diagonal entries in the i'th row, and the second set is distinguished by the spreaded Gerschgorin-type information about A, i.e. by the vector

m ItG(A) = (|a11|, |a12|,..., |a1n|, |a21|, |a22|,..., |a2n|,..., |an1|, |an2|,..., |ann|).

References

Johnson, C. R., T. Szulc, and D. Wojtera-Tyrakowska. Optimal Gersgorin-Style Estimation of Extremal Singular Values. Accepted for publication in Linear Algebra Appl. Wojtera-Tyrakowska, D. (2002). Gersgorin-style Estimation of the Spectral Radius and of the Smallest, in the Modulus, Eigenvalue. Journal of Electrical Engi- neering 52(12/s), 1719.

Part IV

List of Participants

Participants

1. Marek Aleksiejczyk: University of Warmia and Mazury in Olsztyn Faculty of Mathematics and Computer Sciences Department of Physics and Computer Methods ul. oªnierska 14 10-561 Olsztyn, Poland [email protected] 2. Ewa Bakinowska: Departament of Statistical and Mathematical Methods Agricultural University of Pozna« Wojska Polskiego 28 60-637 Pozna«, Poland [email protected] 3. Jerzy K. Baksalary: Faculty of Mathematics, Computer Science and Econometrics University of Zielona Góra Szafrana 4a 65-516 Zielona Góra, Poland [email protected] 4. Oskar M. Baksalary: Faculty of Physics Adam Mickiewicz University Umultowska 85 61-614 Pozna«, Poland [email protected] 5. Rafael Bru: Departament de Matemàtica Aplicada Universitat Politècnica de València Apartat de Correus 22012 46071 València, Spain [email protected] 6. Rafael Cantó: Departament de Matemàtica Aplicada Universitat Politècnica de València Apartat de Correus 22012 46071 València, Spain [email protected] 48

7. Andrzej Cegielski: Faculty of Mathematics, Computer Science and Econometrics University of Zielona Góra Szafrana 4a 65-516 Zielona Góra, Poland [email protected] 8. Ljiljana Cvetkovi¢ Department of Mathematics and Informatics Faculty of Science University of Novi Sad Trg Dositeja Obradovi¢a 4 21000 Novi Sad, Serbia and Montenegro [email protected] 9. Pierre Druilhet: ENSAI Rue Blaise Pascal, Campus de Ker Lann 35 170 BRUZ, France [email protected] 10. Katarzyna Filipiak: Departament of Statistical and Mathematical Methods Agricultural University of Pozna« Wojska Polskiego 28 60-637 Pozna«, Poland kas[email protected] 11. Jürgen Groÿ: Department of Statistics University of Dortmund, Vogelpothsweg 87 44221 Dortmund, Germany; [email protected] 12. Jan Hauke: Institute of Socio-Economic Geography and Spatial Management Faculty of Geography and Geology Adam Mickiewicz University Dzi egielowa 27 61-680 Pozna«, Poland [email protected] 13. Charles R. Johnson: Department of Mathematics College of William and Mary Williamsburg, Virginia 23187, USA [email protected] 49

14. Barbara Koªodziejczak: Faculty of Mathematics and Computer Science Adam Mickiewicz University Umultowska 87 61-614 Pozna«, Poland [email protected] 15. Aneta Konieczna: Faculty of Mathematics and Computer Science Adam Mickiewicz University Umultowska 87 61-614 Pozna«, Poland [email protected] 16. Tomasz Kossowski: Institute of Socio-Economic Geography and Spatial Management Faculty of Geography and Geology Adam Mickiewicz University Dzi egielowa 27 61-680 Pozna«, Poland [email protected] 17. Beata Laszkiewicz: Institute of Mathematics Wrocªaw University of Technology Z. Janiszewskiego 14 50-372 Wrocªaw, Poland [email protected] 18. Andrzej Ma¢kiewicz: Institute of Mathematics Technical University of Pozna« Piotrowo 3a/716 60-965 Pozna«, Poland [email protected] 19. Jose Mas: Departament de Matemàtica Aplicada Universitat Politècnica de València 46022 València, Spain [email protected] 20. Augustyn Markiewicz: Departament of Statistical and Mathematical Methods Agricultural University of Pozna« Wojska Polskiego 28 60-637 Pozna«, Poland [email protected] 50

21. Mieczysªaw Mastyªo: Faculty of Mathematics and Computer Science Adam Mickiewicz University Umultowska 87 61-614 Pozna«, Poland [email protected] 22. Volker Mehrmann: Technische Universität Berlin Sekretariat MA 4-5, Straÿe des 17. Juni 136, D-10623 Berlin, Germany [email protected] 23. Marek Niezgoda: Department of Applied Mathematics Agricultural University of Lublin Akademicka 13 20-950 Lublin, Poland [email protected] 24. Tadeusz Ostrowski: The State Vocational University Chopina 52 66-400 Gorzów Wlkp., Poland [email protected] 25. Miroslav Rozloºník Institute of Computer Science Academy of Sciences of the Czech Republic Pod vodárenskou v¥ºí 2 CZ-182 07 Prague 8, Czech Republic [email protected] 26. Rafaª Ró»a«ski: The State Vocational University Chopina 52 66-400 Gorzów Wlkp., Poland [email protected] 27. Helena migoc Faculty of Mathematics and Physics University of Ljubljana Jadranska 19 SI-1000 Ljubljana, Slovenia [email protected] 28. Alicja Smoktunowicz Faculty of Mathematics and Information Science Warsaw University of Technology Pl. Politechniki 1 00-661 Warszawa, Poland [email protected] 51

29. George P. H. Styan: Department of Mathematics and Statistics McGill University, Burnside Hall Room 1005, 805 rue Sherbrooke Street West, Montreal (Quebec), Canada H3A 2K6; [email protected] 30. Anna Szczepa«ska: Departament of Statistical and Mathematical Methods Agricultural University of Pozna« Wojska Polskiego 28 60-637 Pozna«, Poland [email protected] 31. Tomasz Szulc: Faculty of Mathematics and Computer Science Adam Mickiewicz University Umultowska 87 61-614 Pozna«, Poland [email protected] 32. Wojciech Tadej: Faculty of Mathematics and Natural Sciences Cardinal Stefan Wyszynski University ul. Dewajtis 5 01-815 Warszawa, Poland e-mail: [email protected] 33. Jurjen Duintjer Tebbens Institute of Computer Science Academy of Sciences of the Czech Republic Pod vodárenskou v¥ºí 2 CZ-182 07 Prague 8, Czech Republic [email protected] 34. Goetz Trenkler: Department of Statistics University of Dortmund Vogelpothsweg 87 44221 Dortmund, Germany; [email protected] 35. Miroslav T·ma Institute of Computer Science Academy of Sciences of the Czech Republic Pod vodárenskou v¥ºí 2 CZ-182 07 Prague 8, Czech Republic [email protected] 52

36. Kresimir Veselic LG Mathematische Physik Fernuniversitaet Hagen P.O. Box 940 D-58084 Hagen, Germany [email protected] 37. Dominika Wojtera-Tyrakowska: Faculty of Mathematics and Computer Science Adam Mickiewicz University Umultowska 87 61-614 Pozna«, Poland [email protected] 38. Waldemar Woªy«ski: Faculty of Mathematics and Computer Science Adam Mickiewicz University Umultowska 87 61-614 Pozna«, Poland [email protected] 39. Iwona Wróbel: Faculty of Mathematics and Information Science Warsaw University of Technology Pl. Politechniki 1 00-661 Warszawa, Poland [email protected] 40. Krystyna Zi etak: Institute of Mathematics Wrocªaw University of Technology Z. Janiszewskiego 14 50-372 Wrocªaw, Poland [email protected] Index

Aleksiejczyk M., 13 Mastyªo M., 28 Mehrmann V., 29 Baksalary J.K., 14 Baksalary O.M., 14 Niezgoda M., 30 Bru R., 15 Ostrowski T., 31 Cantò R., 16 Cegielski A., 17 Pedroche F., 15 Cerdàn J., 26 Cvetkovi¢ L., 18 Ricarte B., 16 Rozloºník M., 32 Druilhet P., 19 Smigoc H., 33 GroÿJ., 20 Smoktunowicz A., 34 Styan G.P.H. , 35 Hauke J., 21 Szyld D.B., 15

Johnson C.R., 22 T·ma M., 40 Tadej W. , 36 Kosowski P., 34 Tebbens J.D. , 37 Trenkler G., 39 Laszkiewicz B., 23 Veselic K., 42 Ma¢kiewicz A., 24 Marìn J., 26 Wojtera-Tyrakowska D., 43 Mas J., 26 Wróbel I., 34