Solving Sets of Simultaneous Linear Equations

Sometimes a chemical system can be represented by a set of n linear equations in n imknowns, i.e., 

A familiar example is the spectrophotometric determination of the concentrations of a mixture of n components by absorbance measurements at n different wavelengths. The coefficients a j are the e, the molar absorptivities of the components at different wavelengths (for simplicity, the cell path length, usually 1.00 cm, has been omitted from these equations). For example, for a mixture of three species P, Q and R, where absorbance measurements are made at X, A,2 and, 3 the equations are  

Thus nine coefficients are required for the determination of three unknown concentrations. 

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A determinant is simply a square matrix. There is a procedure for the numerical evaluation of a determinant, so that anN xN matrix can be reduced to a single numerical value. The value of the determinant has properties that make it useful in certain tests and equations. (See, for example, "Solving Sets of Simultaneous Linear Equations" in Chapter 10.)... 

SimultEqns.xls illustrates ways to solve sets of simultaneous linear equations by using matrices. 

We begin our discussion of linear systems by introducing the determinant as a tool for solving sets of simultaneous linear equations in which the indices of the unknown variables are all unity. Consider the pair of equations ... 

The application of matrix algebra for solving sets of simultaneous linear equations - homogeneous and inhomogeneous equations. 

A set of simultaneous linear equations can also be solved by using matrices, as shown in Chapter 9. The solution matrix is obtained by multiplying the matrix of constants by the inverse of the matrix of coefficients. Applying this simple solution to the spectrophotometric data used above, the inverted matrix is obtained by selecting a 3R x 3C array of cells, entering the array formula... 

For each of the three interference functions /x( )> h( )> and b(q) determined as mentioned above, Equation (4.19) is applicable, where the iap (q)s [i.e., cc( ) ch( )> and i hh( )] are common among the three cases and are unknown and yet to be determined. The weighting factors wa, on the other hand, depend, as seen from (4.16), on the scattering lengths ba and assume different values in the three measurements. For a given q value, Equation (4.19) therefore constitutes a set of simultaneous linear equations with three unknowns iap(q), whose values can be determined by solving the simultaneous equations. The partial pair distribution functions gap(r) are then obtained from them by Fourier inversion as implied by (4.18). 

Equation (15) is the key equation of the Kohn variational principle for the -matrix (21). For small problems, when the spectral representation of ft can be obtained, both methods are essentially equivalent. If the linear equations are to be solved iteratively, the present method, Eq. (14), effectively requires to solve half the number of sets of simultaneous linear equations as the basis and xT can chosen real making Eq. (14) real while (15) remains complex. 

The eigenvalues and eigenvectors can be found by solving the set of simultaneous linear equations represented by Eq. (9.63) ... 

A set of simultaneous linear equations can be solved craiveniently by Cramer s rule, which involves finding the quotient of two determinants. Given the equations... 

This set of simultaneous linear equations is solved by any of a number of numerical methods. [James and Coolidge, 1933] utilizes a method employing Lagrangian multipliers. 

The Newton-Raphson method consists in solving simultaneously the conservation and mass action equations. Because of its simplicity and rather fast convergence, it is well-fitted to sets of non-linear equations in several unknowns, as described in Chapter 3. 

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

SC) CAS-SDCI may be viewed as a set of d equations to be solved simultaneously (actually, the implementation of the method is a recursive diagonalization process). In contrast, ec-CCSD implies a unique diagonalization and a further resolution of a rather small set of non-linear equations. 

This is just a pair of simultaneous linear equations. To solve for E we set the determinant (product of diagonal terms minus product of antidiagonal terms) to zero ... 

In this section we will briefly review the most salient aspects of matrix algebra, insofar as these are used in solving sets of simultaneous equations with linear coefficients. We already encountered the power and convenience of this method in section 6.2, and we will use matrices again in section 10.7, where we will see how they form the backbone of least squares analysis. Here we merely provide a short review. If you are not already somewhat familiar with matrices, the discussion to follow is most likely too short, and you may have to consult a mathematics book for a more detailed explanation. For the sake of simplicity, we will restrict ourselves here to two-dimensional matrices. 

This set of simultaneous linear algebraic equations can be solved by inverting the ABC matrix. This can... 

In summary, to do a Cl calculation, we choose a one-electron basis set Xh iteratively solve the Hartree-Fock equations (11.12) to determine one-electron atomic (or molecular) orbitals as linear combinations of the basis set, form many-electron configuration functions d>i using the orbitals cf>i, express the wave function as a linear combination of these configuration functions, solve (11.18) for the energy, and solve the associated simultaneous linear equations for the coefficients c, in (11.17). [In practice, (11.18) and its associated simultaneous equations are solved by matrix methods see Section 8.6.]... 

Results have been obtained for an OTB of pilot plant size (42 rows). WATER (10.25 t/h Tin=44°C Tout=500 C) FUMES (72.5 t/h Tin=592°C Tout=197°C). In VALI-Belsim software in which the simulation model has been implemented, the simulation of the OTB needs 42 modules, one for each row of tubes. Since VALI implements an numerical procedures to solve large sets of non-linear equations, all model equations are solved simultaneously. The graphical user interface allows easy modification of the tube connections and the modelling of multiple pass bundles. 

In Chapter 7 we developed a method for performing linear variational calculations. The method requires solving a determinantal equation for its roots, and then solving a set of simultaneous homogeneous equations for coefficients. This procedure is not the most efficient for programmed solution by computer. In this chapter we describe the matrix formulation for the linear variation procedure. Not only is this the basis for many quantum-chemical computer programs, but it also provides a convenient framework for formulating the various quantum-chemical methods we shall encounter in future chapters. 

Solving this set of six linear equations in six unknowns by any of the standard simultaneous linear equation methods, we find the values shown in Table 3.A. ... 

For the numerical solution of the mass transfer model equations using the finite element method, the system must first be clearly defined. The finite element method is based on the numerical approximation of the dependent variables at a specific nodal location, where a set of simultaneous linear algebraic equations is produced that can be solved either directly or iteratively. 

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