A counterexample to a hadamard matrix pivot conjecture

In the study of the growth factor of completely pivoted Hadamard matrices, it becomes natural to study the possible pivots. Very little is known or provable about these pivots other than a few cases at the beginning and end. For example it is known that the first four pivots must be 1,2,2 and 4 and the last three pivots in backwards order must be n/2, and n/2. Based on computational experiments, it was conjectured by Day and Peterson, that the n—3rd pivot must always be n/4. This conjecture would have suggested a kind of symmetry with the first four pivots. In this note we demonstrate a matrix whose n-3rd pivot is n/2 showing that the conjecture is false.     

Gaussian elimination is stable for the inverse of a diagonally dominant matrix

Let be a row diagonally dominant matrix, i.e.,

where with We show that no pivoting is necessary when Gaussian elimination is applied to Moreover, the growth factor for does not exceed The same results are true with row diagonal dominance being replaced by column diagonal dominance.

     

A Hadamard matrix of order 428

Four Turyn type sequences of lengths 36, 36, 36, 35 are found by a computer search. These sequences give new base sequences of lengths 71, 71, 36, 36 and are used to generate a number of new T‐sequences. The first order of many new Hadamard matrices constructible using these new T‐sequences is 428. © 2004 Wiley Periodicals, Inc.     

Values of Minors of (1, −1) Incidence Matrices of SBIBDs and Their Application to the Growth Problem

We obtain explicit formulae for the values of the v-j minors, j=0,1,2 of (1, –1) incidence matrices of SBIBD(v,k,). This allows us to obtain explicit information on the growth problem for families of matrices with moderate growth. An open problem remains to establish whether the (1, –1) CP incidence matrices of SBIBD(v,k,), can have growth greater than v for families other than Hadamard families.     

On the pivot structure for the weighing matrix W(12,11)

In the present article we concentrate our study on the growth problem for the weighing matrix W(12,11) and show that the unique W(12,11) has three pivot structures. An improved algorithm for extending a k?×?k (0,+,-) matrix to a W(n,n-1), if possible, has been developed to simplify the proof. For the implementation of the algorithm special emphasis is given to the notions of data structures and parallel processing.     

Enumeration of generalized Hadamard matrices of order 16 and related designs

We investigate signings of symmetric GDD( , 16, )s over for . Beginning with , at each stage of this process a signing of a GDD( , 16, ) produces a GDD( , 16, ). The initial GDDs ( ) correspond to Hadamard matrices of order 16. For , the GDDs are semibiplanes of order 16, and for the GDDs are semiplanes of order 16 which can be extended to projective planes of order 16. In this article, we completely enumerate such signings which include all generalized Hadamard matrices of order 16. We discuss the generation techniques and properties of the designs obtained during the search. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 119–135, 2009     

On the growth factor in Gaussian elimination for generalized Higham matrices

A Higham matrix is a complex symmetric matrix A=B+iC, where both B and C are real, symmetric and positive definite. We prove that, for such A, the growth factor in Gaussian elimination is less than 3. Moreover, a slightly larger bound holds true for a broader class of complex matrices A=B+iC, where B and C are Hermitian and positive definite. Copyright © 2002 John Wiley & Sons, Ltd.     

Bounding the growth factor in Gaussian elimination for Buckley's class of complex symmetric matrices

A Buckley matrix is an n × n complex symmetric matrix A = I _n + iC, where C is real symmetric positive definite. We prove that, for such A the growth factor in Gaussian elimination is not greater than $${1 + \sqrt{17} \over 4} \simeq 1.28078\ldots$$ Copyright © 2000 John Wiley & Sons, Ltd.     

On the classification of Hadamard matrices of order 32

All equivalence classes of Hadamard matrices of order at most 28 have been found by 1994. Order 32 is where a combinatorial explosion occurs on the number of Hadamard matrices. We find all equivalence classes of Hadamard matrices of order 32 which are of certain types. It turns out that there are exactly 13, 680, 757 Hadamard matrices of one type and 26, 369 such matrices of another type. Based on experience with the classification of Hadamard matrices of smaller order, it is expected that the number of the remaining two types of these matrices, relative to the total number of Hadamard matrices of order 32, to be insignificant. © 2009 Wiley Periodicals, Inc. J Combin Designs 18:328–336, 2010     

The D-optimal saturated designs of order 22

This paper attempts to prove the D-optimality of the saturated designs X${}^1$ and X${}^{11}$ of order 22, already existing in the current literature. The corresponding non-equivalent information matrices M${}^1$=(X${}^1$)${}^T$X${}^1$ and M${}^{11}$=(X${}^{11}$)${}^T$X${}^{11}$ have the maximum determinant. Within the application of a specific procedure, all symmetric and positive definite matrices M of order 22 with determinant the square of an integer and $\ge$det(M${}^1$) are constructed. This procedure has indicated that there are 26 such non-equivalent matrices M, for 24 of which the non-existence of designs X such that X${}^T$X =M is proved. The remaining two matrices M are the information matrices M${}^1$ and M${}^{11}$.     

超球级数Hadamard乘积的增长性质

研究了超球级数的Hadamard积的性质,得出了超球级数Hadamard积成正规增长的条件.     

Stability of a Pivoting Strategy for Parallel Gaussian Elimination

Gaussian elimination with partial pivoting achieved by adding the pivot row to the kth row at step k, was introduced by Onaga and Takechi in 1986 as means for reducing communications in parallel implementations. In this paper it is shown that the growth factor of this partial pivoting algorithm is bounded above by _n <#60; 3 ^n–1, as compared to 2 ^n–1 for the standard partial pivoting. This bound _n, close to 3 ^n–2, is attainable for class of near-singular matrices. Moreover, for the same matrices the growth factor is small under partial pivoting.This revised version was published online in October 2005 with corrections to the Cover Date.     

Accurate computations with collocation matrices of a general class of bases

In this paper, we provide algorithms for computing the bidiagonal decomposition of the collocation matrices of a very general class of bases of interest in computer‐aided geometric design and approximation theory. It is also shown that these algorithms can be used to perform accurately some algebraic computations with these matrices, such as the calculation of their inverses, their eigenvalues, or their singular values. Numerical experiments illustrate the results.     

Computing a logarithm of a unitary matrix with general spectrum

We analyze an algorithm for computing a skew‐Hermitian logarithm of a unitary matrix and also skew‐Hermitian approximate logarithms for nearly unitary matrices. This algorithm is very easy to implement using standard software, and it works well even for unitary matrices with no spectral conditions assumed. Certain examples, with many eigenvalues near ? 1, lead to very non‐Hermitian output for other basic methods of calculating matrix logarithms. Altering the output of these algorithms to force skew‐Hermitian output creates accuracy issues, which are avoided by the considered algorithm. A modification is introduced to deal properly with the J‐skew‐symmetric unitary matrices. Applications to numerical studies of topological insulators in two symmetry classes are discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

On the Properties of Accretive-Dissipative Matrices

Let A be a complex n × n matrix, and let A = B + iC, B = B*, C = C* be its Toeplitz decomposition. Then A is said to be (strictly) accretive if B > 0 and (strictly) dissipative if C > 0. We study the properties of matrices that satisfy both these conditions, in other words, of accretive-dissipative matrices. In many respects, these matrices behave as numbers in the first quadrant of the complex plane. Some other properties are natural extensions of the corresponding properties of Hermitian positive-definite matrices.__________Translated from Matematicheskie Zametki, vol. 77, no. 6, 2005, pp. 832–843.Original Russian Text Copyright ©2005 by A. George, Kh. D. Ikramov.     

Simultaneous backward stability of Gauss and Gauss–Jordan elimination

It is well known that some pivoting strategies are backward stable for Gauss elimination but are not backward stable for Gauss–Jordan elimination (GJE) when these procedures are used to solve a linear system Ax=b. We analyse the simultaneous backward stability for Gauss and GJE of several pivoting strategies, including a pivoting strategy which we call double partial pivoting. Copyright © 2002 John Wiley & Sons, Ltd.     

A Note on a Paper by P. Amodio and F. Mazzia

In 1999 Amodio and Mazzia presented a new backward error analysis for LU factorization and introduced a new growth factor _n . Their very interesting approach allowed them to obtain sharp error bounds. In particular, they derive nice results assuming that partial pivoting is used. However, the forward error bound for the solution of a linear system whose coefficient matrix A is an M-atrix given in Theorem 4.1 of that paper is not correct. They first obtain a bound for the condition number (U) assuming that one has the LU factorization of an M-matrix and then they apply the bounds obtained when partial pivoting is used. But if P is the permutation associated with partial pivoting then PA = LU can fail to be an M-atrix and the bound for (U) can be false, as shown in our Example 1.1. We also prove that, for a pivoting strategy presented in the paper, the growth factor of an M-matrix A is _n(A) = 1 and (U) (A), where U is the upper triangular matrix obtained after applying such a pivoting strategy.This revised version was published online in October 2005 with corrections to the Cover Date.     

A New Approach to Backward Error Analysis of Lu Factorization

A new backward error analysis of LU factorization is presented. It allows to obtain a sharper upper bound for the forward error and a new definition of the growth factor that we compare with the well known Wilkinson growth factor for some classes of matrices. Numerical experiments show that the new growth factor is often of order approximately log₂ n whereas Wilkinson's growth factor is of order n or .