Simultaneous algebraic equations linear

This approximation resolves the computational difficulty encountered in the direct exact formulation that requires repeated computations of the solution of linear simultaneous algebraic equations and determinants of the matrices with huge dimensions. The efficiency in the approximated expansion is gained by the appreciation that the conditioning information can be truncated within one period of the system only. For linear systems, the expressions for the reduced-order likelihood function p(yi, yj, - - -, yNp W, C) and the conditional PDFs p(.yn 0, yn-Np, yn-Np+1, , y -i, C) are available since they are Gaussian and the correlation functions are known in closed forms regardless of the stationarity of the response. For stationary response, the method is very efficient in the sense that evaluation of all the conditional PDFs p(ynW, yn-Np,yn-Np+i,, y -i, C) requires the inverse and determinant of two relatively small matrices only. 

Fock modified Hartree s SCF method to include antisymmetrization. Roothaan further modified the Hartree-Fock method by representing the orbitals by linear combinations of basis functions similar to Eq. (16.3-34) instead of by tables of numerical values. In Roothaan s method the integrodifferential equations are replaced by simultaneous algebraic equations for the expansion coefficients. There are many integrals in these equations, but the integrands contain only the basis functions, so the integrals can be calculated numerically. The calculations are evaluated numerically. This work is very tedious and it is not practical to do it without a computer. 

Implicit equations cannot be solved individually but must be set up as sets of simultaneous algebraic equations. When these sets are linear, the problem can be solved by the application of the Gauss eiimination methods developed in Chap. 2. If the set consists of nonlinear equations, the problem is much more difficult and must be solved u.sing Newton s method for simultaneous nonlinear algebraic equations developed in Chap. I,... 

GAUSS Subroutine GAUSS solves a system of simultaneous linear algebraic equations by Gaussian elimination and back substitution. The number of equations (equal to the number of unknowns) is NROW. The coefficients are in array SLEQ(NR0W,NR0W+1), where the last column is the constants. 

The steady-state problem yields a system of simultaneous linear algebraic equations that can be solved by Gaussian elimination and back substitution. I shall turn now to calculating the time evolution of this system, starting from a phosphate distribution that is not in steady state. In this calculation, assume that the phosphate concentration is initially the same in all reservoirs and equal to the value in river water, 10 I 3 mole P/m3. How do the concentrations evolve from this starting value to the steady-state values just calculated ... 

The equations can now be rearranged to give a system of simultaneous linear algebraic equations for the unknown values dely in terms of the known values of y(x) and the ex coefficients ... 

Then this system of simultaneous linear algebraic equations can be solved using the subroutine GAUSS developed in Section 3.3. Because I have dropped the nonlinear term, I must always use a delx sufficiently small to ensure that all the dely values are indeed much smaller than the y values. 

GAUSSO Subroutine GAUSSO solves a system of simultaneous linear algebraic equations by Gaussian elimination and back substitution. 

In Chapter 7 I showed how much computational effort could be avoided in a system consisting of a chain of identical equations each coupled just to its neighboring equations. Such systems arise in linear diffusion and heat conduction problems. It is possible to save computational effort because the sleq array that describes the system of simultaneous linear algebraic equations that must be solved has elements different from zero on and immediately adjacent to the diagonal only. 

POLYMATH. AIChE Cache Corp, P O Box 7939, Austin TX 78713-7939. Polynomial and cubic spline curvefitting, multiple linear regression, simultaneous ODEs, simultaneous linear and nonlinear algebraic equations, matrix manipulations, integration and differentiation of tabular data by way of curve fit of the data. 

It solves sets of simultaneous (non-linear) algebraic equations by a modified... 

Specific type of free stream condition can be represented by finding the appropriate functions, B a, u ) and D a, cu) defining the tangential and normal velocity components. The additional subscript (oo) refers to the conditions being evaluated at the free stream y = Y). Now one can solve for the constants C to C4 by simultaneously solving (2.6.94) and (2.6.96) - (2.6.98). All these can also be written as the following linear algebraic equation,... 

The calculation of temperatures and equilibrium compositions of gas mixtures involves simultaneous solution of linear (material balance) and nonlinear (equilibrium) algebraic equations. Therefore, it is necessary to resort to various approximate procedures classified by Carter and Altman (Cl) as (1) trial and error methods (2) iterative methods (3) graphical methods and use of published tables and (4) punched-card or machine methods. Numerical solutions involve a four-step sequence described by Penner (P4). 

Equations (1) through (5) form a linear system of five algebraic equations In five unknowns. Solving them simultaneously using an equation solver gives. ... 

The one-electron, Coulomb and exchange integrals are analogous to Eqs (9.5)-(9.7), but in terms of MO s rather than AO s. (The 4 must now contain contributions from all the nuclei in the molecule.) The optimized wavefunction of the form (11.43) involves, in principle, the solution of N simultaneous integrodifferential Hartree-Fock equations. It is much more computationally efficient to transform these into a set of N linear algebraic equations. To do this, each of the MO s is expressed in terms of a set of n basis functions ... 

There are two basic families of solution techniques for linear algebraic equations Direct- and iterative methods. A well known example of direct methods is Gaussian elimination. The simultaneous storage of all coefficients of the set of equations in core memory is required. Iterative methods are based on the repeated application of a relatively simple algorithm leading to eventual convergence after a number of repetitions (iterations). Well known examples are the Jacobi and Gauss-Seidel point-by-point iteration methods. 

Equations (9), (20), and (21), and the boundary conditions define a nonlinear and coupled system of partial differential equations, solved by an FVM. The equations were linearized around a guessed value. The guessed values were updated iteratively to convergence before executing the next time step. Since the electroneutrality constraint tightly couples the potential and concentration fields, the discretized sets of algebraic equations at each node point were solved simultaneously. Attempts were made to employ a sequential solver in which the electrical field was assumed for determination of the concentration of each species. In this way, the concentration fields appear decoupled and could be determined easily with a commercial, convection-diffusion solver. A robust method for converging upon the correct electrical field was, however, not found. 

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