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Computer Mathematics Languages — An Overview
Published in Dingyü Xue, YangQuan Chen, Scientific Computing with MATLAB®, 2018
It is good news that the mathematical software packages are continuously developing, implementing the leading-edge numerical algorithms, providing more efficient, more reliable, faster and more stable packages. For instance, in the area of numerical algebra, a new LaPACK is becoming the leading package. Unlike the original purposes of EISPACK or LINPACK, the objectives of LaPACK have been changed. LaPACK is no longer aiming at providing libraries or facilities for direct user applications. Instead, LaPACK provides support to mathematical software and languages. For example, MATLAB and a freeware Scilab have abandoned the packages of LINPACK and EISPACK, and adopted LaPACK as their low-level library support.
Stability
Published in Richard L. Shell, Ernest L. Hall, Handbook of Industrial Automation, 2000
Allen R. Stubberud, Stephen C. Stubberud
the stability is determined by the location of the eigenvalues of the matrix A. For continuous time systems, the eigenvalues must be in the left half plane. Similarly, for discrete-time systems, the magnitude of the eigenvalues must be less than one. The question becomes how do we find the eigenvalues. There are many techniques to compute the eigenvalues of a matrix. Several can be found in Wilkinson [6] and Golub and Van Loan [7]. New techniques are probably being developed as you read this. A computer implementation can be found in any numerical linear algebra package such as EISPACK. In this section we outline one technique, the real Schur decomposition.
Numerical and Computational Issues in Linear Control and System Theory
Published in William S. Levine, Control System Fundamentals, 2019
A.J. Laub, R.V. Patel, P.M. Van Dooren
Quality mathematical software for eigenvalues and eigenvectors is available; the EISPACK [5], [11] collection of subroutines represents a pivotal point in the history of mathematical software. The successor to EISPACK (and UNPACK) is the recently released LAPACK [1] in which the algorithms and software have been restructured to provide high efficiency on vector processors, high performance workstations, and shared memory multiprocessors.
Quantum dynamics and spectra of the iodine atom in a strong laser field as calculated with the URIMIR package
Published in Molecular Physics, 2019
R. Marquardt, M. Quack, J. Stohner, I. Thanopulos
It is inherent to the Floquet approximation that the calculation of is performed by matrix-vector multiplications up to times T in steps of one period τ of the Hamiltonian. In our case, τ is approximately 30 fs and T is chosen either 300 ns or 10 µs, which implies or matrix-vector multiplications, respectively. Although high accuracy is almost always achieved for the numerical integration, after the diagonalisation of the complex evolution matrix , severe numerical errors can be introduced in the subsequent time propagation of the initial state; in particular, in our case here, where the diagonalised matrix is used for or sequential matrix-vector operations. For the numerical integration part of the computation, we used the Adams method for the solution of the numerical initial value problem [67] and EISPACK routines [68] for the matrix operations.
Personal reflections on 50 years of scientific computing: 1967–2017
Published in International Journal of Parallel, Emergent and Distributed Systems, 2020
When EISPACK (a precursor of LAPACK) was under development by a consortium of universities and national laboratories, Jim Wilkinson would visit Michigan for a summer school on numerical linear algebra, hosted by Cleve Moler. As a graduate student and aspiring numerical analyst, I attended all of Wilkinson's lectures. At the conclusion of the school, Moler hosted a party for all the attendees at his house. My wife and I, on the way to the party, encountered Wilkinson on the Michigan diag (campus landmark), looking a bit lost. We offered him a ride to the party, and in thanks he gave me an autographed copy of Handbook for Automatic Computation that he was using for his lectures. That is one of my most treasured books.