Improved cost estimates are given for the problem of computing the inverse of an n×n matrix of univariate polynomials over a field. A deterministic algorithm is demonstrated that has worst case complexity (n3s) field operations, where s ≥ 1 is an upper bound for the average column degree of the input matrix. Here, the “+o(1)” in the exponent indicates a missing factor c1(log ns)2 for positive r...

The SR algorithm is a structure-preserving algorithm for computing the spectrum of symplectic matrices. Any symplectic matrix can be reduced to symplectic butterfly form. A symplectic matrix B in butterfly form is uniquely determined by 4n− 1 parameters. Using these 4n− 1 parameters, we show how one step of the symplectic SR algorithm for B can be carried out in O(n) arithmetic operations compa...

In this paper we describe block algorithms for the reduction of a real symmetric matrix to tridiagonal form and for the reduction of a general real matrix to either bidiagonal or Hessenberg form using Householder transformations. The approach is to aggregate the transformations and to apply them in a blocked fashion, thus achieving algorithms that are rich in matrix-matrix operations. These red...

In this paper we describe block algorithms for the reduction of a real symmetric matrix to tridiagonal form and for the reduction of a general real matrix to either bidiagonal or Hessenberg form using Householder transformations. The approach is to aggregate the transformations and to apply them in a blocked fashion, thus achieving algorithms that are rich in matrix-matrix operations. These red...

In this paper we describe block algorithms for the reduction of a real symmetric matrix to tridiagonal form and for the reduction of a general real matrix to either bidiagonal or Hessenberg form using Householder transformations. The approach is to aggregate the transformations and to apply them in a blocked fashion, thus achieving algorithms that are rich in matrix-matrix operations. These red...

A process-planning model (PP model) is proposed to convert the geometric features into manufacture machining operations and sequence the machining operations of the part in a feasible and effective order. The process-planning model (PP model) construct a feature framework that makes a mapping from geometric features into machining operations. A semantic net named the Precedence-Relations-Net is...

An efficient numerical method is developed for evaluating φ(A), where A is a symmetric matrix and φ is the function defined by φ(x) = (ex − 1)/x = 1+ x/2 + x2/6+ .... This matrix function is useful in the so-called exponential integrators for differential equations. In particular, it is related to the exact solution of the ODE system dy/dt = Ay + b, where A and b are t-independent. Our method a...

A real symmetric matrix of order n has a full set of orthogonal eigenvectors. The most used approach to compute the spectrum of such matrices reduces first the dense symmetric matrix into a symmetric structured one, i.e., tridiagonal matrices or semiseparable matrices. This step is accomplished in O(n 3) operations. Once the latter symmetric structured matrix is available, its spectrum is compu...

Karatsuba and Toom-Cook are well-known methods used to multiply efficiently two long integers. There have been different proposal about the interpolating values used to determine the matrix to be inverted and the sequence of operations to invert it. A definitive word about which is the optimal matrix (values) and the (number of) basic operations to invert it seems still not to have been said. I...