Nested loop program

Typically, an algorithm will be specified in the form of a nested loop program (NLP) [3]. It is useful to include the specification of a nested loop program as an integral part of the design trajectory, mainly because it can be specified in an easily comprehensible syntax and because such a program often can be found in a library. 

Figure 2 The nested loop program in pseudo-MATLAB code for the Floyd-Steinberg algorithm.

We now consider a structure node in which the set of nodes is partitioned in a number of subsets. Nodes in a subset have identical behavior. These subsets are called node domains. Let D be a domain and iV be a type of node. Then a node domain V)vd is characterized by the pair (iV, D) and consists of an indexed set of nodes Vnd = i I D. The node domain of the Floyd-Steinberg algorithm is given in figure 5. The boundary planes of the domain are derived from the upper and lower bound expressions of the FOR-loop statements of its nested loop program. 

We will now provide the procedure for converting a nested loop program (NLP) into a DG node by extracting the fiow dependencies between the operations (iterations) of the NLP. In general, this is a very difficult problem and the procedure we are going to explain will only work for so-called static and linear NLPs. More specifically, the NLP may only contain statements that are of the following type ... 

J. Bu, E. Deprettere, and L. Thiele. Systolic array implementation of nested loop programs. Proc. Int. Conf Application Specific Array Pro-cessing, Vol. 4, pages 31-42, Sep 1990. 

This generic procedure affords the powder EPR absorption spectrum, which should be differentiated to get the powder EPR spectrum. Note that the whole procedure consists of three nested loops with the computation of an exponential (Equation 4.8) within the inner loop. Coded in a higher language (C, FORTRAN95) and run on a standard PC, this program will generate the EPR spectrum of a simple S = 1/2 or an effective S = 1/2 system in a split second (of the order of 10 ms or less). It is, however, useful to think about ways to make it as fast as possible, because extending... 

As posed here, the problem is a nonlinear programming one and involves nested loops of calculations, the outer loop of which is Equation (j) subject to Equations (a) through (i), and subject to the inequality constraints. If capital costs are to be included in the objective function, refer to Frey and colleagues (1997). 

There are two nested loops in the program (see Figure 5) o Composition of the recycle gas o Octane number of the C5+ reformate... 

For a discussion and an example on how to program an arbitrary number of nested loops using a single loop algorithm, allowing to realize an arbitrary number of nested sums, see references [45b,c]. 

Once the lattice is created, the bonds are randomly destroyed in a proportion specified at the command line. There is a linked list of sheets in the structure, and each sheet involves a linked list of bonds thus, all bonds are processed using a double nested loop. For each bond, a random number between 0 and 1 is generated and compared to the specified probability (which is also between 0 and 1) if the random number is smaller, the bond is deleted. When a bond is deleted, it acquires the property deleted , which is used later in the program. 

We will assume here that the (non-uniform) dependence graph of the APP has already been extracted from the initial textual specification. In practice, this step can be performed automatically from applicative languages in a relatively simple way, as explained briefly in chapter 6 or from procedural nested loop notations expressed in any standard programming language, with more complex procedures, as discussed in chapters 4 and 5. 

The extension to higher DF QS measures not only becomes the straightforward task of adding two more do loops for one QS measure order, but because of this modular structure of the Algorithm TDQS, the nested do loop programming technique [49-51] can be applied for the purpose of generating such a code, which could be associated to any kind of QS computation level and becomes completely adapted to parallel machines. 

In standard high level language programming the dimension of the NSS n, signals the number of nested do loops which are necessary to reproduce the structure in a computational environment. But the mathematical usefulness of this entity can be easily recognized when the particular characteristic of this symbolic unit is analyzed the involved vector parameters could be chosen with arbitrary and variable dimensions. There are many scientific and mathematical formulae which will benefit of this property, when written in a paper or computationally implemented. 

There are two nested convergence loops in the program (see Figure... 

Nested summation symbolism is very convenient for practical purposes, because successive generation of k vector elements can be programmed in a general way imder any high level language, using a unique do or for loop irrespective of the number of sums, or binomial expressions of type (3.4) entering the product (3.5). 

A4. In a multiconponent simulation program for distillation the loops are nested. The outermost loop is mole fractions, next is flow rates and the innermost loop is tenperature. 

Memory hierarchy management Every modem computer has a cache hierarchy, designed to reduce the average amount of time required to retrieve a value from memory. To improve the performance of programs, many compilers transform loop nests to improve data locality—i.e., they reorder memory references in a way that increases the likelihood that data elements will found in the cache when they are needed. To accomplish this, the compiler must transform the loops so that they repeatedly iterate over blocks of data that are small enough to fit in the cache. 

The second section of the program is devoted to the propagation of the solution and to the calculation of the final quantities for an array of energies. This section contains four nested DO loops. The outer DO loop runs over the specified energy values. The next internal DO loop runs over the quantum number /. The innermost DO loop runs over the various sectors to perform the propagation. Inside this DO loop several calls to the routines performing the different matrix manipulations (add, multiply, transpose, invert) are made. 

We have obtained increasing speedups over the PDG-based method when the source code contains loops that are nested deeper. For example, we analyzed a program for loop nesting depths one to five, obtaining speedups 2.4, 6.3, 14.7, 27.3, 48.2. When we compared the execution times of our SLV-based method with those of CodeSurfer, we obtained the speedups -, 80.0, 96.0, 92.0, 92.5. (Here, the speedup for the first item could not be computed since the time resolution of CodeSurfer gave zero execution time.)... 

This example is slow and tedious with a spreadsheet, because the multiple trial and error calculations must be done one at a time. It is shown that way because it is clear what is happening. The calculation is much easier in a Fortran program with nested DO loops. 

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