Method of chords

This method is known hy several names method of chords, linear interpolation, false position reguia falsi). Its simplicity of calculation (no need for evaluating derivatives of die function) gave it its popularity in the early days of numerical computations. However, its accuracy and speed of convergence are hampered by the choice of x, which forms the pivot point for all subsequent iterations. 

One method of constructing tangents to curves is known as the method of chords. With a compass placed at the point in question on the curve, strike off two arcs on either side of the point, as shown in Fig. 12-5. Next, draw chords through the intersection of the arcs with the curve and extend die chords until they intersect. A line drawn from the intersection of the two chords to the point on the curve is a good approximation of the tangent to the curve at that point. 

On millimeter graph paper (using an expanded scale), plot the curve y = x -3jc jc+l from X = — 1 to X = 1. Using the method of chords, find the slope of the curve at x = 0. Compare the slope found by this method to that found by differentiation. 

In the preceding example, the chords have been taken for equal intervals, because the curve changes slope only gradually and the data are given at integral temperatures at equal intervals. Under these circumstances, the method of numerical differentiation is actually preferable. In many cases, however, the intervals will not be equal nor will they occur at whole numbers. For the latter cases, the chord-area method of differentiation may be necessary, although considerable care is required to avoid numerical errors in calculations. 

Computational methods combined with a novel approach in the application of scattering physics were recently employed by Barbi et al. in a synchrotron SAXS study of the nanostructure of Nafion as a function of mechanical load. A new method of multidimensional chord-distribution function (CDF) analysis was used to visualize the multiphase nano-... 

The conventional design is the default method of the ICPD tray computer programs. All conventional designs have straight, chord-type downcomer areas. (Compare Figs. 3.2 and 3.3.) Note that the nonconventional downcomer areas may have complex weir configurations as compared to the simple, straight weir runs of the conventional downcomers. 

Kinetic processes cause a change in a quantity over time. If this quantity is measured at evenly spaced time intervals, it is relatively easy to find the rate by computing a numerical derivative (Pollard, 1977). The simplest method of numerical differentiation is based on the observation that the chord of a graph has a slope that closely approximates the tangent at an intermediate point. 

An alternative to the polynomial method is the chord method. The chord method is recommended for experiments that reach larger extents of reaction (Casado et al., 1986 Piscitelle, 1990 Waley, 1981). This method extracts... 

Different meaning of chord AB in approximating d a/ dr in classic, Crank-Nicolson or Laasonen method (lower abscissa scale). 

In [170] the authors discussed the numerical solution of Ordinary Differential Equations (ODEs) by using two approached the well known BDF formulae and the Piecewise-Linearized Methods. In the case of BDF method a Chord-Shamanskii iteration procedure is used for computing the nonlinear system which is produced when the BDF formula is applied. In the case of Piecewise-Linearized Methods the computation of the numerical solution at each time step is obtained using a block-oriented method based on diagonal Pade approximation. 

The calculation of escape probabilities is often most conveniently carried out by means of the chord method developed by Dirac. For this analysis we refer to Fig. 7.10, which shows an arbitrary convex volume V. As before, n is the inward directed normal at dA and s(Q) is the chord length from dA in the direction O. Let us assume now that the number of chords of length between s and s + ds m the solid angle y... 

McGarey and Richards also found some evidence for a shoulder in their SANS studies from cross-poly(dimethylsiloxane)-inrer-cross-polystyrene sequential IPN s corresponding to a wavelength of 2500 A. This may have been the result of the very low vinyl content in their poly(dimethyl-siloxane). By other methods of analysis, they estimated chord lengths of the order of 200-500 A. Examination of the older literature, both by Sperling s group as well as others, suggests that spinodal decomposition plays an important part in domain formation of many IPN systems. 

The isotropic chord length distribution (CLD) is of limited practical value if soft matter with only short-range order is studied. Nevertheless, the related notions have been fruitful for the development of new methods for topology visualization from SAXS data. 

The second method is more elegant, because it only involves the numerical computation of moments (cf. Sect. 1.3) of the smeared CLDg2 (rn) followed by moment arithmetics [200], The first step is the computation of the Mellin transform102 of the analytical function gc (rn) which we have selected to describe the needle diameter shape. This is readily accomplished by Mathematica [205], Because the Mellin transform is just a generalized moment expansion, we retrieve for the moments of the normalized chord distribution of the unit-disc103... 

If AH is plotted against m at constant 2, a graphical differentiation by the chord-area method will yield L ii as a function of composition. Alternatively, the data could be fitted to a polynomial and the derivative of that pol5momial then could be computed. Differentiation of Equation (18.12) with respect to 2 at constant m yields... 

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