/pdf/modern-methods-and-algorithms-of-quantum-chemistry-1l333qh8zp.pdf

Modern methods and algorithms of quantum chemistry

The authors' survey paper is devoted to the present state of computational methods in linear algebra. Questions discussed are the means and methods of estimating the quality of numerical solution of computational problems, the generalized inverse of a matrix, the solution of systems with rectangular and poorly conditioned matrices, the inverse eigenvalue problem, and more traditional questions such as algebraic eigenvalue problems and the solution of systems with a square matrix (by direct and iterative methods).

Computational methods of linear algebra

Spline smoothing can be reduced to the minimization of a certain quadratic form with positive semidefinite matrix. For polynomial splines this matrix is closely related to an oscillation matrix and its eigenvectors show the typical sign distribution. This fact is the basis for a variant of spline smoothing.

Oscillation matrices with spline smoothing

The authors describe a method of calculating highly excited states of hydrogen in uniform magnetic fields of moderate strength ( approximately 5T). Results for Balmer emission lines and an approximate photoabsorption spectrum for Cs are presented. The authors comment on the use of adiabatic approximations and the role of the spin-orbit interaction.

http://iopscience.iop.org/article/10.1088/0022-3700/15/8/011/pdf

The quadratic zeeman effect in hydrogen rydberg series: application of sturmian functions

LAPACK is a transportable library of Fortran 77 subroutines for solving the most common problems in numerical linear algebra: systems of linear equations, linear least squares problems, eigenvalue problems and singular value problems. LAPACK is designed to supersede LINPACK and EISPACK, principally by restructuring the software to achieve much greater efficiency on vector processors, high-performance ``superscalar`` workstations, and shared-memory multi-processors. LAPACK also adds extra functionality, uses some new or improved algorithms, and integrates the two sets of algorithms into a unified package. The LAPACK Users` Guide gives an informal introduction to the design of the algorithms and software, summarizes the contents of the package, describes conventions used in the software and documentation, and includes complete specifications for calling the routines. This edition of the Users` guide describes Release 1.0 of LAPACK.

/pdf/lapack-users-guide-release-1-0-4slivdr39c.pdf

LAPACK users' guide: Release 1. 0

An algorithm is described for reducing the generalized eigenvalue problem Ax = lBx to an ordinary problem, in case A and B are symmetric band matrices with B positive definite. If n is the order of the matrix and m the bandwidth, the matrices A and B are partitioned into m-by-m blocks; and the algorithm is described in terms of these blocks. The algorithm reduces the generalized problem to an ordinary eigenvalue problem for a symmetric band matrix C whose bandwidth is the same as A and B. The algorithm is similar to those of Rutishauser and Schwartz for the reduction of symmetric matrices to band form. The calculation of C requires order n2m operation. The round-off error in the calculation of C is of the same order as the sum of the errors at each of the n/m steps of the algorithm, the latter errors being largely determined by the condition of B with respect to inversion.

C. R. Crawford

Papers

Reduction of a band-symmetric generalized eigenvalue problem