A very simple procedure to extract pseudopotentials from ab initio atomic calculations is presented. The pseudopotentials yield exact eigenvalues and nodeless eigenfunctions which agree with atoxnic wave functions beyond a chosen radius x,. Moreover, logarithmic derivatives of real and pseudo wave functions and their first energy derivatives agree for r &r, guaranteeing excellent transferability of the pseudopotentials. Pseudopotentials were originally introduced to simplify electronic structure calculations by eliminating the need to include atomic core states and the strong potentials responsible for binding them. ' Two roughly distinct lines of recent development are discernable: In one, ion pseudopotentials of enforced smoothness were empirically fitted to reproduce experimental energy bands. ' Consequently, wave functions were only approximately described. In the other, the orthogonalized-plane-wave (OPW) concept underlying the pseudopotential method was used to derive "first principles" pseudopotentials from atomic calculations. ' These latter potentials are generally "hard core" in character, that is, strongly repulsive at the origin. The resulting wave functions generally exhibit the correct shape outside the core region; however, they differ from the real wave functions by a normalization factor. 'It is the purpose of this Letter to demonstrate that the normalization and hard-core problems can be solved simultaneously, while also maximizing the range of systems in which a pseudopotential gives accurate results.The new family of energy-independent pseudopotentials introduced here have the following desirable properties:(1) Real and pseudo valence eigenvalues agree for a chosen "prototype" atomic configuration.(2) Real and pseudo atomic wave functions agree beyond a chosen "core radius" r,.(3) The integrals from 0 to r of the real and pseudo charge densities agree for r &r, for each valence state (norm conservation).(4) The logarithmic derivatives of the real and pseudo wave function and their first energy deriv atives agree for r &r,.Properties (3) and (4) are crucial for the pseudopotential to have optimum transferability among a variety of chemical environments in self-consistent calculations in which the pseudo charge density is treated as a real physical object. 'This approach starids in contrast to earlier OPW-like approaches" ' in which the normalized pseudo wave functions have to be orthogonalized to core states and renormalized in order to yield accurate charge densities outside the core region. ' Property (3) guarantees, through Gauss's theorem, that the electrostatic potential produced outside r, is identical for real and pseudo charge distributions. Property (4) guarantees that the scattering properties of the real ion cores are reproduced Mlitk rnininzum error as bonding or banding shifts eigenenergies away from the atomic levels. A central point of our approach is that these two aspects of transferability are related by a simple identity. The method permits the potentials to be intrinsically ...

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Algebraic Riccati equations are encountered in many applications of control and engineering problems, e.g., LQG problems and H ∞ control theory. In this work, we study the properties of one type of discrete-time algebraic Riccati equations. Our contribution is twofold. First, we present sufficient conditions for the existence of a unique positive definite solution. Second, we propose an accelerated algorithm to obtain the positive definite solution with the rate of convergence of any desired order. Numerical experiments strongly support that our approach performs extremely well even in the almost critical case. As a byproduct, we provide show that this method is capable of computing the unique negative definite solution, once it exists.

Nonlinear matrix equations play a crucial role in science and engineering problems. However, solutions of nonlinear matrix equations cannot, in general, be given analytically. One standard way of solving nonlinear matrix equations is to apply the fixed-point iteration with usually only the linear convergence rate. To advance the existing methods, we exploit in this work one type of semigroup property and use this property to propose a technique for solving the equations with the speed of convergence of any desired order. We realize our way by starting with examples of solving the scalar equations and, also, connect this method with some well-known equations including, but not limited to, the Stein matrix equation, the generalized eigenvalue problem, the generalized nonlinear matrix equation, the discrete-time algebraic Riccati equations to express the capacity of this method.

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This work is to provide a comprehensive treatment of the relationship between the theory of the generalized (palindromic) eigenvalue problem and the theory of the Sylvester-type equations. Under a regularity assumption for a specific matrix pencil, we show that the solution of the ⋆-Sylvester matrix equation is uniquely determined and can be obtained by considering its corresponding deflating subspace. We also propose an iterative method with quadratic convergence to compute the stabilizing solution of the ⋆-Sylvester matrix equation via the well-developed palindromic doubling algorithm. We believe that our discussion is the first which implements the tactic of the deflating subspace for solving Sylvester equations and could give rise to the possibility of developing an advanced and effective solver for different types of matrix equations.

This note is concerned with the linear matrix equation X = AX ⊤ B +C, where the operator (·) ⊤ denotes the transpose (⊤) of a matrix. The first part of this paper set forth the necessary and sufficient conditions for the unique solvability of the solution X. The second part of this paper aims to provide a comprehensive treatment of the relationship between the theory of the generalized eigenvalue problem and the theory of the linear matrix equation. In the finally part of this paper starts with a briefly review of numerical methods for solving the linear matrix equation. Related to the computed methods, knowledge of the residual is discussed. An expression related to the backward error of an approximate solution is obtained; it shows that a small backward error implies a small residual. Just like for the discussion of linear matrix equations, perturbation bounds for solving the linear matrix equation are also proposed in this work.