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Linearized methods

We now discuss the most important theoretical methods developed thus far the augmented plane wave (APW) and the Korringa-Kolm-Rostoker (KKR) methods, as well as the linear methods (linear APW (LAPW), the linear miiflfm-tin orbital [LMTO] and the projector-augmented wave [PAW]) methods. [Pg.2210]

Andersen O K 1975 Linear methods in band theory Phys. Rev. B 12 3060... [Pg.2231]

Goresky CA. A linear method for determining liver sinusoidal and extravascular volumes. Am J Physiol 1963 204 626-40. [Pg.526]

In order to reduce the complexity of the problem, several approximation schemes have been developed. In the BGK model, the collision integral is replaced by a simple local term ensuring that the well-known Maxwell distribution is reached at thermal equilibrium [16]. The linearization method assumes that the phase space distribution is given by a small perturbation h on top of a (local) Maxwell distribu-tion/o (see, e.g., [17, 18]) ... [Pg.132]

Additionally, Breiman et al. [23] developed a methodology known as classification and regression trees (CART), in which the data set is split repeatedly and a binary tree is grown. The way the tree is built, leads to the selection of boundaries parallel to certain variable axes. With highly correlated data, this is not necessarily the best solution and non-linear methods or methods based on latent variables have been proposed to perform the splitting. A combination between PLS (as a feature reduction method — see Sections 33.2.8 and 33.3) and CART was described by... [Pg.227]

As an extension of perceptron-like networks MLF networks can be used for non-linear classification tasks. They can however also be used to model complex non-linear relationships between two related series of data, descriptor or independent variables (X matrix) and their associated predictor or dependent variables (Y matrix). Used as such they are an alternative for other numerical non-linear methods. Each row of the X-data table corresponds to an input or descriptor pattern. The corresponding row in the Y matrix is the associated desired output or solution pattern. A detailed description can be found in Refs. [9,10,12-18]. [Pg.662]

A significant development in pipeline network computation in the last few years was the introduction of the so-called linearization method first by Wood and Charles (Wll) and later, independently, by Bending and Hutchi-... [Pg.155]

For the special case for which n = 2, it can be shown that the linearization method defined above becomes identical to the Newton-Raphson method. The result may be generalized to apply to any homogeneous function of degree n. [Pg.156]

While it is technically erroneous to claim that the linearization method does not require any initialization (J2), it is true that the initialization procedure used appear to be quite effective. A more comprehensive discussion of initialization procedure will be given in Section III,A,5. With this initialization procedure, the linearization method appears to converge very rapidly, usually in less than 10 iterations for formulations A and B. Since the evaluation of f(x) and its partial derivatives is not required, the method is also simpler and easier to implement than the Newton-Raphson method. [Pg.156]

On the debit side, the linearization method is quite sensitive to the form of the network element model. Jeppson and Tavallaee (J2) reported that convergence rate was slow when the usual pump and reservoir models were incorporated, but they obtained significant improvements after the models had been suitably transformed. Although the number of iterations required is small using formulations A and B, the dimension of the matrix equation is substantial. Hence, it becomes essential to use sparse computation techniques if the method is to retain its competitive edge in larger problems. [Pg.156]

For formulations A and B, one general procedure is to solve the laminar flow equations which are linear and use the solution as the initial guesses for the nonlinear equations. Variations of this procedure have been used by Bending and Hutchison (B5), Wood and Charles (Wll), and Jeppson and Tavallaee (J2) in conjunction with the linearization method. [Pg.157]

The methodology takes the form of an MINLP, which must be linearised to find a solution. The linearization method used was the relaxation-linearization technique proposed by Quesada and Grossman (1995). During the application of the formulation to the illustrative examples it was found that only one term required linearization for a solution to be found. [Pg.171]

Unfortunately, even the planning of green biotechnology has now evolved into a wicked problem with complex structures and no obvious causal chains. This applies also to the PA. These problems cannot be determined completely in a quantitative and scientific manner, and there are no existing solutions in the sense of definitive and objective answers alone. Wicked problems have been addressed mainly through formalized (linear) methods that are suitable only for the solution of tame problems. [Pg.294]

A final consideration about PCA is concerned with its use as a preprocessor of non-linear methods such as neural networks [22], The assumption of a normal distribution of the data requires all following analysis steps to adhere to this hypothesis. If positive results are sometimes achieved they have to be considered as serendipitous events. [Pg.157]


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Algebraic equations linear, matrix method solution

Analytical method validation linearity

Analytical methods linearity

Analytical methods multiple linear regression

Another method to obtain canonical models with linear terms

Approximate Methods and Linear Algebra

Calibration linear versus nonlinear methods

Conduction, heat linear methods

Configuration interaction linear variations method

Conjugate Gradient method linear algebraic systems

Correlation Methods for Kinetic Data Linear Free Energy Relations

Coulomb linear scaling methods

Deconvolution methods linear

Density functional full-potential linearized augmented plane wave method

Density linear-scaling method

Electrochemical methods linear sweep voltammetry

Electronic structure methods linear scaling

Elimination methods for solving linear systems

Energy-linearized methods

Exponential matrix method, linear

Finite element method linear interpolation

Free energy calculations linear response method

Full-potential linear augmented plane wave method

Full-potential linearized augmented plane wave method

Iterative linear solvers Conjugate Gradient method

Iterative methods to solve the linear system

Kohn-Sham method linear-scaling methods

Kohn-Sham potential linear-scaling methods

Large linear system solution, with iterative methods

Least squares method linear fits

Linear Attic method

Linear Augmented Plane Wave method

Linear CG method

Linear Methods for Diagnosis

Linear Multistep Methods for DAEs

Linear R12 methods

Linear Recursive Methods - Kekule Structure Counting

Linear Scaling Electrostatic and Generalized Solvent Boundary Methods

Linear Synchronous Transit optimization method

Linear System Solution with Iterative Methods

Linear algebra methods

Linear algebra numerical methods

Linear algebraic methods

Linear calibration methods

Linear coefficients method

Linear combination atomic orbital method

Linear combination of atomic orbitals LCAO) method

Linear combination of atomic orbitals method

Linear congruential method

Linear convolution method

Linear differential correction method

Linear extrapolation method

Linear free energy relationship method

Linear free energy relationship method solvents, effect

Linear free-energy-related estimation methods

Linear frequency response, methods

Linear heat flux methods

Linear interaction energy method

Linear interpolation method

Linear inversion methods

Linear least-squares fitting methods

Linear methods

Linear methods

Linear muffin-tin orbital method

Linear muffin-tin orbital method LMTO)

Linear multistep methods

Linear operator annihilation method

Linear operator equations and their solution by iterative methods

Linear polarization method

Linear preconditioning conjugate method

Linear programming simplex method

Linear projection methods

Linear regression, forecasting method

Linear response method

Linear scaling method, Hartree-Fock methods

Linear scaling methods

Linear scaling methods applications

Linear semi-integral method

Linear simplex method

Linear solution methods

Linear solvation energy method

Linear solvation energy relationship methods

Linear statistical methods

Linear synchronous transit method

Linear systems approach methods

Linear traverse method

Linear variation method

Linear variation method formulation

Linear variation method hamiltonian

Linear variation method matrix

Linear, generally method

Linear-equation method

Linear-graph method

Linear-least-squares method

Linear-programming method

Linear-response free energy method

Linear-scaling DFT LCAO Methods for Solids

Linear-scaling methods fast multipole method

Linear-scaling methods theory

Linearity dilution method

Linearity method development

Linearity of an analytical method

Linearization method analytical

Linearization methods

Linearized augmented plane wave method

Linearized least squares method

Linearized muffin tin orbital method

Localized quasi-linear inversion based on the Bleistein method

Matrix Formulation of the Linear Variation Method

Method Characteristic Parameters of a Linear Calibration Function

Method linearity

Method validation high-level linearity

Method validation linearity

Molecular orbital theory LCAO method (linear combination

Molecular orbitals LCAO method (linear combination

Molecular volume difference method linear

Multiple scattering theory linearized methods

Non-linear least squares method

Non-linear least-squares fit method

Non-linear methods

Non-linear optical methods (

Numerical integration, linear-scaling methods

Numerical methods linear equations

Numerical methods linear regression

Orbitals LCAO method (linear combination

Other linear methods

Overview of Linear Methods

Perturbation methods linear response

Poisson-Boltzmann linearized, method

Quantitative structure-activity relationship linear regression methods

Reactivity linear free energy relationship method

Recursion method linear limit

Reduced linear equations method

Regression analysis linear least squares method

Regression methods, assumptions linear

Regression, linear method

Solution Methods for Linear Algebraic Systems

Solution Methods for Linear Finite Difference Equations

Solving linear equations (Newtons method)

Standard Test Method for Determination of Phenolic Antioxidants and Erucamide Slip Additives in Linear Low-Density Polyethylene Using Liquid Chromatography

Standard Test Method for Linear Thermal Expansion of Solid Materials with a Vitreous Silica Dilatometer, (Withdrawn)

Stern-Geary linear polarization method

Symmetry constraints linear variation method

System of implicit non-linear equations the Newton-Raphson method

The Incremental Methods of Linearity Measurement

The Linear Interaction Energy (LIE) Method

The Linear Method

The Logarithmic Dilution Method of Linearity Measurement

The Method of Least Squares and Simple Linear Regression

The regularization method in a linear inverse problem solution

Transform Methods for Linear PDEs

Triangle method linear isotherm

Variational theory of linearized methods

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