Existing routines, such as xGELSY or xGELSD in LAPACK, for solving rank-deficient least squares problems require O(mn) operations to solve min ‖b−Ax‖ where A is an m by n matrix. We present a modification of the LAPACK routine xGELSY that requires O(mnk) operations where k is the effective numerical rank of the matrix A. For low rank matrices the modification is an order of magnitude faster tha...

A C++ library, named ZKCM, has been developed for the purpose of multiprecision matrix calculations, which is based on the GNU MP and MPFR libraries. It is especially convenient for writing programs involving tensorproduct operations, tracing-out operations, and singular-value decompositions. Its extension library, ZKCM QC, for simulating quantum computing has been developed using the time-depe...

In this partly expository paper we discuss and describe some of our old and recent results on partial orders on the set G n of graphs with n vertices and m edges and some operations on graphs within G n that are monotone with respect to these partial orders. The partial orders under consideration include those related with some Laplacian characteristics of graphs as well as with some probabilis...

A complex square matrix A is called an orthogonal projector if A2 = A = A∗, where A∗ denotes the conjugate transpose of A. In this paper, we give a comprehensive investigation to matrix expressions consisting of orthogonal projectors and their properties through ranks of matrices. We first collect some well-known rank formulas for orthogonal projectors and their operations, and then establish v...

We describe a datatype for (dense) matrices whose primitive operations are decomposition and composition (of submatrices), as opposed to indexed element access which is the primitive operation on conventional arrays. Using the composition and decomposition operations it is for example possible to express both recursive and traditional block matrix algorithms (e.g., Cholesky factorization, QR-fa...

Ahstruct-A “coordinate recurrence” method for solving sparse systems of linear equations over finite fields is described. The algorithms discussed all require O( n,( w + nl) logkn,) field operations, where nI is the maximum dimension of the coefficient matrix, w is approximately the number of field operations required to apply the matrix to a test vector, and the value of k depends on the algor...

This paper proposes a new, large diffusion layer for the AES block cipher. This new layer replaces the ShiftRows and MixColumns operations by a new involutory matrix in every round. The objective is to provide complete diffusion in a single round, thus sharply improving the overall cipher security. Moreover, the new matrix elements have low Hamming-weight in order to provide equally good perfor...

Let K be a Kleene algebra with tests B. As argued in Lecture ??, the structure (Mat(n,K), ∆(n,B), +, ·, ∗, , 0n, In) again forms a Kleene algebra with tests, where Mat(n,K) denotes the family of n × n matrices over K, the operations + and · are the usual operations of matrix addition and multiplication, respectively, 0n is the n × n zero matrix, and In the n × n identity matrix. The operation ∗...

Evaluating the intersection of two rational parameterized algebraic surfaces is an important problem in solid modeling. In this paper, we make use of some generalized matrix based representations of parameterized surfaces in order to represent the intersection curve of two such surfaces as the zero set of a matrix determinant. As a consequence, we extend to a dramatically larger class of ration...

One may consider that the algebraic complexity of basic linear algebra over an abstract field K is well known. Indeed, if ω is the exponent of matrix multiplication over K, then for instance computing the determinant, the matrix inverse, the rank or the characteristic polynomial of an n×n matrix over K can be done in O (̃n) operations in K. Here the soft “O” notation indicates some missing logar...