Linear simultaneous equations

Equation (2.45) represents the weighted residual statement of the original differential equation. Theoretically, this equation provides a system of m simultaneous linear equations, with coefficients Q , i = 1,... m, as unknowns, that can be solved to obtain the unknown coefficients in Equation (2.41). Therefore, the required approximation (i.e. the discrete solution) of the field variable becomes detemfined. 

Because of the work involved in solving large systems of simultaneous linear equations it is desirable that only a small number of us be computed. Thus the gaussian integration formulas are useful because of the economy they offer. See references on numerical solutions of integral equations. 

Solution of Sets of Simultaneous Linear Equations 71. Least Squares Curve Fitting 76. Numerical Integration 78. Numerical Solution of Differential Equations 83. 

Because of round off errors, the Regula Falsa method should include a check for excessive iterations. A modified Regula Falsa method is based on the use of a relaxation factor, i.e., a number used to alter the results of one iteration before inserting into the next. (See the section on relaxation methods and Solution of Sets of Simultaneous Linear Equations. )... 

Sets of simultaneous linear equations are frequently defined as [12] ... 

The bold-face characters employed in Eq. (33) imply that each symbol represents a matrix. The problem of the resolution of simultaneous linear equations will be discussed in Section 7.8, as certain properties of matrices must first be explained. 

Note the symmetry of the summation terms in xtj and that numbering of x s in the summations corresponds to matrix indices (rows, columns). This set of p equations in p unknowns can be solved on a computer using one of the many readily available routines for solving simultaneous linear equations. 

The matrix solution for the parameters of the simultaneous linear equations is stated here without proof ... 

The matrix approach to the solution of a set of simultaneous linear equations is entirely general. Requirements for a solution are that there be a number of equations exactly equal to the number of parameters to be calculated and that the determinant... 

Suppose these solutions had been attempted using simultaneous linear equations. One reasonable set of linear equations might appear to be... 

Thus, the intercept bQ= and the slope b = 213 are identical to those values obtained using the simultaneous linear equations approach (see Equation 5.19 and Figure 5.1). Because there is an exact fit, the residuals are equal to zero. 

Example 1.3.3 Solution of simultaneous linear equations by LU decomposition... 

Constraints may be imposed on a set of simultaneous linear equations by the method of Lagrangian multipliers. Let the Lagrangian multipliers be — - Therefore, add to equation (A.28) the quantity... 

Evaluation of these integrals and expansion of the above determinant leads to a polynomial of degree N in , the roots of which give the energies of the molecular orbitals constructed in the above manner. The coefficients Cfj may be determined by substitution of the particular E into the simultaneous linear equations covered by the above determinant together with the normalization equation... 

The equations just derived are inhomogeneous simultaneous linear equations in 3N unknowns, the three cartesian components of each relative velocity V — ft . As such, they have the solution... 

Application of the Lagrange method, used above, leads to the following set of homogeneous simultaneous linear equations as the condition for minimum energy ... 

Solving Simultaneous Linear Equations with Excel Matrix Operations ... 

Figure 19-5 Solving simultaneous linear equations with Excel.

Similarity transformation, 87, 92,101,412 Similar matrices, 87,101-102 Simultaneous linear equations, 15-16, 87 Singlet, 58,310-311 Singly excited function, 311 Singularity, 17 SI units, 23... 

Simultaneous linear equations. A set of n simultaneous linear equations in the n unknowns xlyx2,...,xn has the form... 

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