APPENDIXThere are several way.s in which one can arrange to carry out a particular calculation in various modes of arithmetic. The following particularly convenient way was suggested to us by Pi'ofes.sor W. Kahan.Most of the programming was written in FORTRAN II for the IBJM 7090 and 7094. All of the ste]:)s ia the program which were to be part of the calculation procedure under study were put on cards with a D ia column 1. As a result, these steps are compiled as double precision calculations. Double precision computations are performed in FORTRAN II by referencing a FAP subroutine which ia turn carries out the double precisioa operatioas. For exaaiple, if a double precisioa addition is to be executed, a reference is made to the entry point (DFAD) in the FAP subroutine. Similarly double precision subtraction, multiplication and division cause references to (DFSB), (DFJN'IP) and (DFDP). The arguments for these references are supplied in a standard manner.Normally, references to a subroutine will cause ihe loader to look for that subroutine ia the source deck. If the subroutine is not with the source deck the loader expects to find it in the hbrary. Thus, to provide for different modes of arithmetic oae need only .supply, along with the source deck, a modified version of the subroutine which would ordinarily be used to perform the double precision operations. Such a subroutine was wi'itten, and it was given the ability to perform any one of the five desired modes of arithmetic.In order to specify the particular mode of operation desired, one uses the FORTRAN statement CALL ARITH(I) where / = 1, 2, 3, 4 or 5 depending on which mode is required. APJTH i.s another entry point in the same subroutine. Eatry at this point causes a switch to be set which determines which mode will be used in subsequent entries to the arithmetic parts of the subroutine.The advantage of this way of implementing the various modes of calculatioa is that any calculation can easily be treated ia whatever mode oae wishes. It is necessary only to write each step of the calculation ia FORTRAN, with a D in column 1, and then to precede the whole calculation with the appropriate CALL ARITH statement. Of course the value of I can be changed for subsequent calculatioas, aad the same calculatioa can be carried out any number of times in any aiode of arithmetic.It is clear that the technique can be easily extended so that further values of I could correspond to other types of arithmetic, such as interval arithmetic or sigailicance arithmetic. The only restriction is that the speeial type of arithmetic must not use more than two memory words for each item.The above technique cannot be implemented in FOR-TRAN IV on the 7094 because double precision operations with this system use hardware iastcad of subroutines.Volume 9 / Number 2 / Febnuirv, 1%6

This paper tackles the problem of the region of stability of the fourth order Runge-Kutta method for the solution of systems of differential equations. Such a region can be characterized by means of linear transformation but can not be given in a closed form. In the present case, the region has been determined using the electronic digital computer Z22.

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In this paper we consider the modified Helmholtz type equation governing interior two-dimensional boundary value problems (BVPs) for anisotropic functionally graded materials (FGMs) with Dirichlet and Neumann boundary conditions. Persistently spatially changing diffusivity and leakage factor coefficients are involved in the governing equation. Both the anisotropic diffusivity and leakage factor coefficients vary according to an exponential gradation function. We use a technique of transforming the variable coefficient governing equation to a constant coefficient equation for deriving a boundary integral equation. And from the boundary integral equation obtained a standard boundary element method (BEM) is constructed to find numerical solutions to the BVPs. In order to illustrate the application of the BEM, some particular examples of BVPs are solved. The results show the convergence, accuracy, consistency between the scattering and flow solutions and efficiency (less computation time) of the BEM solutions. The results also show the impact of the inhomogeneity and anisotropy of the material on the solutions.

Practically, predictor-corrector methods for the solution of differential equations are widely used. In these numerical methods, the question of stability is most important. For a single differential equation Crane and Lambert (1962) studied the stability of a fourth-order generalized corrector formula; Chase (1962) and subsequently Emanuel (1963) demonstrated for particular methods that the overall integration procedure does not generally possess the same stability properties as the corrector except when the step of integration tends to zero. In the present paper it is shown that it is possible to develop a necessary and sufficient condition for the stability of general predictor-corrector methods for the solution of systems of differential equations.KEY WORDS AND PHRASES: stability, predictor-corrector, system of differential equations, Hermitian form, eigenvalues, Jacobian matrix, system of differential equations CR CATEGORIES: 5.17The system of m ordinary differential equations can be written in vector form as follows:with the initial value y' ~-~ -= f(x; y), x0, xe] and the column vectorsDefine the matrices Y = (y.+¢) and F = (f,+j), j = -q ( 1 ) 0 , and the column vectors a = (aj), b -~ (be), a = (dj), g = (~j), j = --q(1)0.THEOREM. The general predictor-corrector method P.+i = Ya "4-hFb, m,~+l = Pn+l "4-am(p,, --on),(2) cn+~ = Y£ + hFf~ + hblf(x,~+l ; ms+l), y.+1 = c,~+I "4-a~(p~+l -ca+i), (ay -am = 1),