Recursion equations

Before equations (9.99) can be run, and initial value of P(/c//c) is required. Ideally, they should not be close to the final value, so that convergence can be seen to have taken place. In this instance, P(/c//c) was set to an identity matrix. Figure 9.16 shows the diagonal elements of the Kalman gain matrix during the first 20 steps of the recursive equation (9.99). 

Using the recursive equations (9.29) and (9.30), solve, in reverse time, the Riccati equation commencing with P(A ) = 0. 

The LST, on the other hand, explicitly takes into account all correlations (up to an arbitrary order) that arise between different cells on a given lattice, by considering the probabilities of local blocks of N sites. For one dimensional lattices, for example, it is simply formulated as a set of recursive equations expressing the time evolution of the probabilities of blocks of length N (to be defined below). As the order of the LST increases, so does the accuracy with which the LST is able to predict the statistical behavior of a given rule. 

Behavioral Classification We will shortly see that many rules may in fact give rise to the same set of recursive equations of a given order. This suggests that the LST provides an alternative behavioral classification scheme to the four (thus far... 

In this chapter we start the discussion of an alternative model for programs, designed to reflect recursive properties of programming languages. We shall see that this model does indeed represent an augmentation of the flowchart model we have been studying up to now. One topic of concern will be when recursion equations can be translated into flowchart form - when recursion schemes are flowchartable. 

DEFINITION A recursion scheme is a finite set of recursion equations and a designated initial defined function letter F such that ... 

The definition of computation in a recursion scheme is a little more complicated than for a flowchart scheme. Computations are defined from the inside out when the equation has nested defined function letters. It has been shown by B. Rosen that evaluating recursion equations from the inside out produces a system with the... 

First let us extend the definition of recursion equation as we did the definition of WHILE scheme. Let a Boolean expression be any expression involving predicate terms P(t, ...,tm) where each is a terminal term (not necessarily a variable), and the connectives AND, OR, and NOT. We define a recursion expression inductively, by saying that first any term is a recursion expression, and then that any statement of the form IF Q THEN ELSE E2 is a recursion expression if Q is a Boolean predicate and E and E2 are recursion expressions. 

We now turn to a restriction, rather than an extension, of recursion equations. First, let us permit an equation ... 

For convenience in juggling notation we assume that the recursion equations in our linear recursion scheme S define in order defined functions Fq,F, ... to some F the equation defining F is called equation i. We let the variables in X = (x, ...,xn) and Y = (y, ...,y ) be all distinct and assume that the start equation defines Fq(X) while equation i for i 0 defines F (Y) this allows us to store automatically the input in separate locations. We can without loss of generality assume that all defined functions are m-placed except perhaps for FQ. ... 

Strong, H. R., "Translating Recursion Equations into Flow Charts," J. Computer System Sciences, 5 (1971) 254-285. 

This result is generalized into the following recursive equation (Equation 7-11) and a generalized equation (Equation 7-12) for the oxygen concentration at the end of the /th pressure cycle ... 

H. Liang and Y. Liu, Recursion equations in predicting band with under gradient elution. J. ChromatogrA 1040 (2004) 19-31. 

An automated and optimal subtraction procedure was previously developed for Eq. (8). The algorithm takes one spectrum after another and optimally subtracts previous signals. The criterion used is minimization of the signal entropy (see Section 4.5.2). The recursive equation used appears in Eq. (9), where is an experimental spectrum, is a set of reference spectra, and x are scalar coefficients. The values of x are easily determined. 

After inserting these expansions in the Schrodinger equation (8.11), we obtain the following recursive equations ... 

The recursive equation can now be solved step by step and yields the final result... 

The CUSUM control chart was originally developed using a graphical approach based on V masks. However, for computer calculations, it is more convenient to use an equivalent algebraic version that consists of two recursive equations... 

We now want to find a single power series in t to replace the expression on the right-hand side of (B.32), and its terms will then be equivalent with those of the left-hand side and yield the desired sn. This is done by splitting (B.32) into partial fractions. An example follows here, namely the case k = 2 from (B.25) above. We now solve the recursive equations for sn-i, for all n > 2. We have... 

This leads to the well known Smith-Ewart recursion equation which has been solved by Stockmayer (14) extended by O Toole (15) and further by Ugelstad et alia (IST to the case where radicaT reabsorption occurs. This expression has been solved in detail by computer for this paper in the manner presented by Ugelstad et alia (16). In particular the average number of radicals per par-ticle, TT, has been related to the quantities m and a, defined below, for the case where no aqueous termination or growth takes place... 

Figure 11. Solutions of the Smith-Ewart recursion equation for the case of no aqueom propagation or termination. Dotted line m = 0 (Smith-Ewart Case II). Curve 1 (m = 10 ) depicts typical styrene-like polymerization. Curve 2(m = 0.01) depicts radiation initiated emulsion polymerization of vinyl chloride. Curve 3 (m > 1.0) depicts chemically initiated emulsion polymerization of vinyl chloride.

Combining relations (3.287), (3.288) and (3.289) results in a set of recursive equations, which are called the Kalman filter equations ... 

In Section 4.2 we have shown that stochastic models present a good adaptability to numerical solving. In the opening line we asserted that it is not difficult to observe the simplicity of the numerical transposition of the models based on poly stochastic chains (see Section 4.1.1). As far as recursion equations describe the model, the numerical transposition of these equations can be written directly, without any special preparatives. 

When a stochastic model is described by a continuous polystochastic process, the numerical transposition can be derived by the classical procedure that change the derivates to their discrete numerical expressions related with a space discretisation of the variables. An indirect method can be used with the recursion equations, which give the links between the elementary states of the process. 

Stockmayer (1957) was the first to present a general solution to the Smith-Ewart recursion equation for . This solution is reproduced below for the case m which free radical escape from particles is not possible. 

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