Arithmetic matrix

Section 1.5 dealt with the analysis of variance method, which may easily be applied for FUFE analysis of results. Hereby one should take care to transform the FUFE design arithmetic matrix into the table, which is required by the analysis of variance notation. One should also keep in mind the difference in processing designs with... 

V.L. Druskin and L.A. Knizhnerman Krylov subspace approximation of eigen-pairs and matrix functions in exact and computer arithmetics. Num. Lin. Alg. Appl., 2 (1995) 205-217... 

On the other hand, one should not lose sight of the rather paradoxical fact that if C and A are both full (i.e., possessing few or no null elements), then more arithmetic operations are required to form the product CA than to find A l. Hence the matrix C, which is ordinarily constructed in practice, is by no means full, and, moreover, it is easily inverted. Indeed, quite often it is G-1 that is formed explicitly and G by inverting C l. 

Proper evaluation of the necessary actions in solving problem (5) by the matrix elimination method is stipulated, as usual, by the special structures of the matrices involved. Because all the matrices are complete in spite of the fact that C is a tridiagonal matrix, O(iVf) arithmetic operations are required for determination of one matrix on the basis of all of which are known to us in advance. Thus, it is necessary to perform 0 Ni N2) operations in practical implementations with all the matrices j = 1,2,N-2- Further, 0 N ) arithmetic operations are required for determination of one vector with knowledge of and 0 Nf N2) operations for determination of all vectors Pj. 

Chapter 9 dealt with the basic operations of addition of two matrices with the same dimensions, of scalar multiplication of a matrix with a constant, and of arithmetic multiplication element-by-element of two matrices with the same... 

In the foregoing discussion the properties of the incidence matrix and the cycle matrix were illustrated in terms of a cyclic digraph, but the results on the ranks of these matrices actually hold true for any connected digraph with N vertices. For an undirected graph, M and C contain only 0 and 1 (sometimes referred to as binary matrices), mathematical relations of identical form are obtained except that modulo 2 arithmetic2 is used instead of ordinary arithmetic. The ranks of M and defined in terms of modulo 2 arithmetic are JV — 1 and C, as before, and Eqs. (10) and (11) are modified to read... 

Since it is necessary to represent the various quantities by vectors and matrices, the operations for the MND that correspond to operations using the univariate (simple) Normal distribution must be matrix operations. Discussion of matrix operations is beyond the scope of this column, but for now it suffices to note that the simple arithmetic operations of addition, subtraction, multiplication, and division all have their matrix counterparts. In addition, certain matrix operations exist which do not have counterparts in simple arithmetic. The beauty of the scheme is that many manipulations of data using matrix operations can be done using the same formalism as for simple arithmetic, since when they are expressed in matrix notation, they follow corresponding rules. However, there is one major exception to this the commutative rule, whereby for simple arithmetic ... 

Equations 4-4, 4-5, and 4-6 represent the three elements of the matrix product of [A] and [B], Note that each row of this resulting matrix contains only one element, even though each of these elements is the result of a fairly extensive sequence of arithmetic operations. Equations 4-4, 4-5, and 4-7, however, represent the symbolism you would normally expect to see when looking at the set of simultaneous equations that these matrix expressions replace. Note further that this matrix product [A][fl] is the same as the entire left-hand side of the original set of simultaneous equations that we originally set out to solve. 

Let us examine these symbolic transformations with a view toward seeing how they translate into the required arithmetic operations that will provide the answers to the original simultaneous equations. There are two key operations involved. The first is the inversion of the matrix, to provide the inverse matrix. This is an extremely intensive computational task, so much so that it is in general done only on computers, except in the simplest cases for pedagogical purposes, such as we did in our previous chapter. 

In the usual formulation of the extended Hiickel method, the elements of the hamiltonian matrix are computed according to a simple set of arithmetic rules, and do not depend on the molecular orbitals. In this way, there is no need for the iterations required by more sophisticated methods, and in practice the results may be obtained nowadays in a question of seconds for any reasonably sized complex. 

If the data include outliers, it is advisable to use robust versions of centering and scaling. The simplest possibility is to replace the arithmetic means of the columns by the column medians, and the standard deviations of the columns by the median absolute deviations (MAD), see Sections 1.6.3 and 1.6.4, as shown in the following M-code for a matrix X. 

A more robust correlation measure, -y Vt, can be derived from a robust covariance estimator such as the minimum covariance determinant (MCD) estimator. The MCD estimator searches for a subset of h observations having the smallest determinant of their classical sample covariance matrix. The robust location estimator—a robust alternative to the mean vector—is then defined as the arithmetic mean of these h observations, and the robust covariance estimator is given by the sample covariance matrix of the h observations, multiplied by a factor. The choice of h determines the robustness of the estimators taking about half of the observations for h results in the most robust version (because the other half of the observations could be outliers). Increasing h leads to less robustness but higher efficiency (precision of the estimators). The value 0.75n for h is a good compromise between robustness and efficiency. 

For identifying outliers, it is crucial how center and covariance are estimated from the data. Since the classical estimators arithmetic mean vector x and sample covariance matrix C are very sensitive to outliers, they are not useful for the purpose of outlier detection by taking Equation 2.19 for the Mahalanobis distances. Instead, robust estimators have to be taken for the Mahalanobis distance, like the center and... 

Here, L is a lower triangular matrix (not to be confused with L, the Cholesky factor of the matrix of nonlinear parameters A ), and D is a diagonal matrix. The scheme of the solution of the generalized symmetric eigenvalue problem above has proven to be very efficient and accurate in numerous calculations. But the main advantage of this scheme is revealed when one has to routinely solve the secular equation with only one row and one column of matrices H and S changed. In this case, the update of factorization (117) requires only oc arithmetic operations while, in general, the solution from scratch needs oc operations. 

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