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Let s look at each assignment The cast of (x / y) to int results in the truncation of the fractional component, and information is lost No loss of information occurs when b is assigned the value 255 because a byte can hold the value 255 However, when the attempt is made to assign b the value 257, information loss occurs because 257 exceeds a byte s range In both cases the casts are needed because there is no implicit conversion from int to byte When the short variable s is assigned the value 32,000 through the uint variable u, no data is lost because a short can hold the value 32,000 However, in the next assignment, u has the value 64,000, which is outside the range of a short, and data is lost In both cases the casts are needed because there is no implicit conversion from uint to short Next, u is assigned the value 64,000 through the long variable l In this case, no data is lost because 64,000 is within the range of a uint However, when the value 12 is assigned to u, data is lost because a uint cannot hold negative numbers In both cases the casts are needed because there is no implicit conversion from long to uint Finally, no information is lost, but a cast is needed when assigning a byte value to a char

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higher order implicit methods for stiff equations have been made in an effort to avoid step size limitations due to stability considerations Both collocation and the implicit RungeKutta (Butcher, 1964) methods have been investigated for such cases (Wright, 1970) It has been shown that the collocation method for the general rst-order system yields results that are equivalent to those obtained by implicit Runge-Kutta methods (Weiss, 1974; Wright, 1970) However, the ease and generality of application of the collocation method make it very suitable for practical, dif cult (stiff) ordinary differential equations (Shampine and Gear, 1979; Hulme and Daniel, 1974) Further, the solution requires only algebraic operations to nd the unknown coef cients In addition, for higher order differential equations the solution can be obtained without reducing the equation to a system of rst-order equations (Cerutti, 1974) In applying the method the domain of interest is divided into small elements (subdomains) Also it has been suggested that the roots (Gaussian points) of the Pth Legendre orthogonal polynomial be chosen as p collocation points within each of these elements (de Boor and Swartz 1973; Stroud and Secrest, 1986) When this is done, the resulting global truncation error in the solution of a qth order differential equation will be of order 0(d p+q), where d is the maximum length of an element At the ends of each element, the approximation solution and its rst q - 1 derivatives have errors of order 0(d 2P) As is the case in the synthesis of output motions, B-splines are used as the known functions Evaluation of B-splines and their derivatives, using recurrence relations was discussed earlier Because both the spline interpolation used in the output motion synthesis and spline collocation use B-splines as basis functions and because both require the solution of linear systems, much of the same software can be used for both procedures In the collocation procedure, the solution (U(t)) of a general differential equation is represented as a linear combination of B-splines, written as follows:.

You Try It: Solve the equation 4x 32x = 7 [Hint: Take the logarithm of both sides See also Example 622 below]

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