Equation simultaneous

Traditionally, least-squares methods have been used to refine protein crystal structures. In this method, a set of simultaneous equations is set up whose solutions correspond to a minimum of the R factor with respect to each of the atomic coordinates. Least-squares refinement requires an N x N matrix to be inverted, where N is the number of parameters. It is usually necessary to examine an evolving model visually every few cycles of the refinement to check that the structure looks reasonable. During visual examination it may be necessary to alter a model to give a better fit to the electron density and prevent the refinement falling into an incorrect local minimum. X-ray refinement is time consuming, requires substantial human involvement and is a skill which usually takes several years to acquire. 

Equation (2.106) gives rise to an implicit scheme except for 0 = 0. The application of implicit schemes for transient problems yields a set of simultaneous equations for the field unknown at the new time level n + 1. As can be seen from Equation (2.111) some of the terms in the coefficient matrix should also be evaluated at the new time level. Therefore application of the described scheme requires the use of iterative algorithms. Various techniques for enhancing the speed of convergence in these algorithms can be found in the literature (Pittman, 1989). 

Among the ordinary numbers, only 0 has no inverse. Many matriees have no inverse. The question of whether a matr ix A has or does not have a defined inverse is elosely related to the question of whether a set of simultaneous equations has or does not have a unique set of solutions. We shall eonsider this question more fully later, but for now reeall that if one equation in a pair of simultaneous equations is a multiple of the other. 

Gaussian Elimination, hi the most elementary use of Gaussian elimination, the first of a pair of simultaneous equations is multiplied by a constant so as to make one of its coefficients equal to the corresponding coefficient in the second equation. Subtraction eliminates one term in the second equation, permitting solution of the equation pair. 

Note that the matrix from Exercise 2-8 is the matrix of coefficients in this simultaneous equation set. Note also the similarity in method between finding the least equation and Gaussian elimination. 

Write a program in BASIC for solving linear nonhomogeneous simultaneous equations by... 

This means that onee A is known, it ean be multiplied into several b veetors to generate a solution set x = A b for each b vector. It is easier and faster to multiply a matrix into a vector than it is to solve a set of simultaneous equations over and over for the same coefficient matrix but different b vectors. 

Procedure. Write a program for solving simultaneous equations by the Gaussian elimination method and enter the absorptivity matiix above to solve Eqs. (2-51). Set up and solve the problem resulting from a new set of experimental observations on a new unknown solution leading to the nonhomogeneous veetor b = 0.327,0.810,0.673. ... 

The purpose of this projeet is to gain familiarity with the strengths and limitations of the Gauss-Seidel iterative method (program QGSEID) of solving simultaneous equations. 

Derivation of bond enthalpies from themioehemieal data involves a system of simultaneous equations in which the sum of unknown bond enthalpies, each multiplied by the number of times the bond appears in a given moleeule, is set equal to the enthalpy of atomization of that moleeule (Atkins, 1998). Taking a number of moleeules equal to the number of bond enthalpies to be determined, one ean generate an n x n set of equations in whieh the matrix of eoeffieients is populated by the (integral) number of bonds in the moleeule and the set of n atomization enthalpies in the b veetor. (Obviously, eaeh bond must appear at least onee in the set.)... 

Procedure. Run one or more simultaneous equation programs to determine the C—C and C—H bond energies and interpret the results. The error veetor is the veetor of ealeulated values minus the veetor of bond enthalpies taken as tme from an aeeepted source. Caleulate the enor veetor using a standard souree of bond enthalpies (e.g., Laidler and Meiser, 1999 or Atkins, 1994). Expand the method for 2-butene (2-butene) = —11 kJ mol ] and so obtain the C—H, C—C,... 

The b vector in this equation set has been converted from kilocalories per mole (Benson and Cohen, 1998) to kilojoules per mol. Solve these simultaneous equations to obtain the energetic conPibutions for P, S, T, and Q. 

This is a pair of simultaneous equations in Ai and A2 called the secular equations... 

What we fomierly called the nonhomogeneous vector (Chapter 2) is zero in the pair of simultaneous nomial equations Eq. set (6-38). When this vector vanishes, the pair is homogeneous. Let us try to construct a simple set of linearly independent homogeneous simultaneous equations. 

Linearly dependent sets of homogeneous simultaneous equations, for example. 

In what immediately follows, we will obtain eigenvalues i and 2 for //v / = Ei ) from the simultaneous equation set (6-38). Each eigenvalue gives a n-election energy for the model we used to generate the secular equation set. In the next chapter, we shall apply an additional equation of constr aint on the minimization parameters ai, 2 so as to obtain their unique solution set. 

Using equation 3.12, we write the following simultaneous equations... 

Substituting known values for the two experiments into equation 7.6 gives the following pair of simultaneous equations... 

Solving the simultaneous equations, which is left as an exercise, gives the concentration of indium as 0.606 ppm and the concentration of cadmium as 0.205 ppm. 

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