Vector arithmetic

In subsequent sections of this chapter, I will use this simple vector arithmetic to show how to compute phases from various kinds of data. In the next section, I will use complex vectors to derive an equation for electron density as a function of reflection intensities and phases. 

In ref 146 the authors present a non-standard (nonlinear) two-step explicit P-stable method of fourth algebraic order and 12th phase-lag order for solving second-order linear periodic initial value problems of ordinary differential equations. The proposed method can be extended to be vector-applicable for multi-dimensional problem based on a special vector arithmetic with respect to an analytic function. 

Usually p > n + 1, and the preceding equation can be solved only in the least mean square error sense. Using the vector arithmetic, the solution is given by... 

The flow of particles or fluids is described by their flux (see Chapter 17, page 308). To avoid vector arithmetic, let s just consider a flow along a single direction, which we ll choose to be the x-axis. The flux J is defined as the amount of material passing through a unit area per unit time. Sometimes the flux is defined as a quantity that is not divided by a unit area (see Chapter 17, for example), but we do divide by the unit area in this chapter. You are free to define the amount of material in terms of either the number of particles, or their mass, or their volume, whichever is most convenient for the problem at hand. 

Arithmetical operations on complex numbers are performed much as for vectors. Thus, if a j hi and y = c + di, then ... 

Thus the average cost per share for John is the arithmetic mean of pi, po,. . . , pn, whereas that for Mary is the harmonic mean of these n numbers. Since the harmonic mean is less than or equal to the arithmetic mean for any set of positive numbers and the two means are equal only i pi=po = =pn, we conclude that the average cost per share for Mary is less than that for John if two of the prices Pi are distinct. One can also give a proof based on the Gaiichy-Schwarz inequality, To this end, define the vectors... 

In such matters some progress can be achieved by combinations of the decomposition method and the method of separation of variables. For example, this can be done using the method of separation of variables for the reduced system (6) upon eliminating the unknown vectors with odd subscripts j. This trick allows one to solve problem (2) here the expenditures of time are Q 2nin2 og N2 arithmetic operation, half as much than required before in the method of separation of variables. 

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. 

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 ... 

The vectors generated by the Lanczos recursion differ from the Krylov vectors in that the former are mutually orthogonal and properly normalized, at least in exact arithmetic. In fact, the Lanczos vectors can be considered as the Gram-Schmidt orthogonalized Krylov vectors.27 Because the orthogonalization is performed implicitly along the recursion, the numerical costs are minimal. 

To illustrate the principles of SLP, we note that a Lanczos recursion initiated by an arbitrary vector qj can, in exact arithmetic, yield not only the eigenvalues of H, but also overlaps of prespecified vectors with eigenvectors, as shown below ... 

Basically, each variable j can be characterized by its arithmetic mean, xj, variance Vj, and standard deviation, v,- (Figure 2.9). The means x to x, form the mean vector 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. 

Here, x, is an object vector, and the center is estimated by the arithmetic mean vector x, alternatively robust central values can be used. In R a vector d Mahalanobis of length n containing the Mahalanobis distances from n objects in X to the center... 

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... 

The most widely known algorithm for partitioning is the k means algorithm (Hartigan 1975). It uses pairwise distances between the objects, and requires the input of the desired number k of clusters. Internally, the k-means algorithm uses the so-called centroids (means) representing the center of each cluster. For example, a centroid c, of a cluster j = 1,..., k can be defined as the arithmetic mean vector of all objects of the corresponding cluster, i.e.,... 

In Subheading 2.3. the important class of vectors with continuous-valued components is described. A number of issues arise in this case. Importantly, since the objects of concern here are vectors, the mathematical operations employed are those applied to vectors such as addition, multiplication by a scalar, and formation of inner products. While distances between vectors are used in similarity studies, inner products are the most prevalent type of terms found in MSA. Such similarities, usually associated with the names Carbo and Hodgkin, are computed as ratios, where the inner product term in the numerator is normalized by a term in the denominator that is some form of mean (e.g., geometric or arithmetic) of the norms of the two vectors. 

Equation (11.126) shows that the critical angle coc relating base vectors Rx, R2 to eigenvectors Ui, u2 at (TC,PC) is determined by the ratio of geometrical and arithmetic means of the diagonal metric elements. Inserting the specific metric expressions from Table 11.1, we obtain finally... 

Prior to the searching operation, the set of high E reflections is expanded to the full set of reflections, and the Ji vectors are transformed into a set of real integers m-j in such a way as to preserve the arithmetic relationship among the Ji. One such mapping is... 

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