Recurrence formulas

Thus, we see that this problem has a unique solution if the value y is given for some i. For the sake of simplicity let be known in advance for i = 0. With this, one can determine all the values y-, y, . . by the recurrence formula just established. In the case qi z= q = const and ipi = 0 this provides support for the view that the whole collection of yi constitutes a geometric progression. If qi = q and (pi qb 0 then... 

The requirement a << 1 necessitates making some modifications for stability of this or that difference scheme. As a final result of minor changes, the recurrence formulae have the representations... 

All of the calculations are best conducted by recurrence formulas, whose algorithm is certainly stable and is called through execution algorithm. 

The determinant det A of the square matrix A xn is most simply defined by the row-wise recurrence formula... 

A brief summary will be given of the Newmark numerical integration procedure, which is commonly used to obtain the time history response for nonlinear SDOF systems. It is most commonly used with either constant-average or linear acceleration approximations within the time step. An incremental solution is obtained by solving the dynamic equilibrium equation for the displacement at each time step. Results of previous time steps and the current time step are used with recurrence formulas to predict the acceleration and velocity at the current time step. In some cases, a total equilibrium approach (Paz 1991) is used to solve for the acceleration at the current time step. 

Recurrence formulae for the Hermite polynomials follow directly from the defining relation (38.1). If we differentiate both sides of that equation with respect to we obtain the relation... 

The evaluation of more complicated integrals can be effected by combining Ibis result with the recurrence formulae wc have already established for the 1 Termite functions. For instance, it follows from equation (40.3) that... 

For N > 4 the recurrence procedure becomes rather cumbersome. Instead of the algorithm (7)-(10) a closed recurrence formula for the energy of m-body interactions can be obtained ... 

In the general case, the energy of m-body interactions in a PT approach is expressed through the energy of k-body (fc < m - 1) ones by the following recurrence formula... 

When the component balances are solved by recurrence formulas, the flow rate of a component on the extremes of the mixture boiling range often approaches zero. This leads to computer underflow and divide-by-zero errors. Protection against such errors should be included in the recurrence calculations to trap such errors and set the component flow rate to zero on that stage. 

Thanks to efficient recurrence formulae, multipole moments and multipole moment derivatives can be calculated at very high order with a low computational cost. The calculation of reaction field factors, however, may become computationally expensive at high order due to the increasing number of linear equations to be solved. Thus, in practice, the multipole moment expansion is cut off at a maximum value of f (/max), usually taken around 6. In order to get an order of magnitude of the error introduced by the truncation, let us consider Kirkwood s equations [5] for the free energy of a charge distribution of charges q, and r, in a spherical cavity of radius a ... 

For a well known upwards recurrence formula is used [42,53] ... 

After Smith and Ewart ( ), we define the kinetics of emulsion polymerization in terms of the recurrence formula,... 

To solve equation (18), subject to the boundary conditions given in equations (19)-(21) and to the additional requirement that y be bounded, is a straightforward mathematical problem. By the method of separation of variables, for example, it can be shown (through the use of recurrence formulae and other known properties of Bessel functions of the first kind) that... 

The quantities A and B may be obtained by means of the fundamental recurrence formulas... 

In general the approximants can be obtained using the fundamental recurrence formulas (2.3), which in our specific case are... 

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