Inverse matrices, calculation

Since the dimension of the principal propagator matrix may be large, it is impractical to calculate the inverse matrix in eq. (10.115) directly. In practice the propagator is therefore calculated in two steps, by first solving for an intermediate vector X (corresponding to U in eq. (10.50)). 

The use of Equation (A. 17) for inversion is conceptually simple, but it is not a very efficient method for calculating the inverse matrix. A method based on use of row operations is discussed in Section A.3. For matrices of size larger than 3 X 3, we recommend that you use software such as MATLAB to find A 1. 

After the required sums have been obtained and normalized they become the elements of a matrix, which must be inverted. The resultant inverse matrix is the basis for the derivation of the final regression equation and testing of its significance. These last steps are accomplished in part through matrix-by-vector multiplications. Anyone who has attempted the inversion of a high-order matrix will appreciate the difficulty of performing this operation through hand calculation. 

Standard computer subroutines are available to obtain the inverse matrix and perform the final calculations of the J,. The heat-transfer rate at each ith surface having an area At is then calculated from... 

Rk] It is important to observe the subtraction of R >0 considered in the definition of the disturbance vector. If R >0, Kg always exists. However, if we accept that R is not necessarily positive, we can have problems making the inverse matrix necessary to calculate Kg. ... 

If the matrix A is a square N x N) matrix, then U, Q and V are all square matrices of the same size. In this case we can easily calculate the inverse matrix ... 

AF/Aflj. This process is repeated for each of the k regression coefficients. Then the cross-products (dF/dai) 8F/8aj) are computed for each of the N data points and the 2(5f/5flj)(5f/5ay) terms obtained. The Pij matrix of I (8F/8ai) 8F/Sap terms is constructed and inverted. The terms along the main diagonal of the inverse matrix are then used with equation 12-11 to calculate the standard deviations of the coefficients. This method may be applied to either linear or non-linear systems. 

Values of the e.s.d.s of parameters can be obtained, as shown in Figure 10.13, in the least-squares refinement from values of the diagonals of the inverse matrix. Similarly, any correlations between parameters, such as is often found to occur between occupancy and atomic displacement parameters, can be identified and taken into account in the description of the resulting molecular structure. The e.s.d.s for the refined parameters can then be used to calculate e.s.d.s of derived parameters, such as distances, angles, and torsion angles. ... 

Although only four figures are obtained in the experimental characteristic composition, we shall make the characteristic vectors self-consistent to six figures since the accuracy of the method for obtaining the inverse matrix given in Section II,B,2,c depends on the self-consistency of the characteristic vectors. In addition, the use of six figures will reduce the accumulation of errors caused by the computation procedure. Using Eq. (85) to calculate Xi from Eq. (134), we have... 

The coeificients A, B,. are not mutually independent and the calculation of the standard errors therefore requires a knowledge of the appropriate covariance (inverse) matrix (for details, see Davies, 1961a). A simpler procedure becomes possible is we define the following variables for each experimental point, ... 

Therefore, it is easy to calculate, say, the inverse or the square root of matrix A. For instance, we obtain for the inverse matrix (F= r) ... 

These calculations can be completed using the spreadsheet functions. The iteration is controlled using a Visual Basic MACRO. The inverse matrix gives the errors and covariances. The errors in the parameters will be obtained from the inverse matrix. They are (sa)2 = daa S, where S is / o/ o /(u — 2) for the two parameters that are determined from the data. Likewise, (sb)2 = clhh S and sab, the covariance term, is dab S. 

We study the dielectric and energy loss properties of diamond via first-principles calculation of the (0,0)-element ( head element) of the frequency and wave-vector-dependent dielectric matrix eg.g CQ, The calculation uses all-electron Kohn-Sham states in the integral of the irreducihle polarizahility in the random phase approximation. We approximate the head element of the inverse matrix hy the inverse of the calculated head element, and integrate over frequencies and momenta to obtain the electronic energy loss of protons at low velocities. Numerical evaluation for diamond targets predicts that the band gap causes a strong nonlinear reduction of the electronic stopping power at ion velocities below 0.2 a.u. 

As shown in Section II, we wish to calculate the poles and residues of P Q . However, even using moderately large operator manifolds, the inverse matrix becomes so large that we cannot evaluate all elements of it equally well. We therefore wish to treat one part of it better than the rest. Which part we choose will be directed by the physics of the problem. In order to do so it is convenient to partition (Lowdin, 1963) the inverse matrix, for instance in the following way (Nielsen et al, 1980), letting hf, = h —... 

Also for your information in 1962, when Bauman wrote his book, neither personal computers nor spreadsheets were available, andhe commented on page 411 of his book that calculating the inverse matrix A-1 represents roughly an hour and a quarter of work, including checking . Thank you, personal computer thank you, spreadsheet. 

It should be noted that the calculation of an inverse matrix requires considerable computational time, and hence various methods have been proposed for the solution of a set of linear equations that take advantage of the structure of the matrix A. The methods may be divided into two classes, namely, direct and iterative methods. 

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