FORTRAN vector

The actual program used at NPL was written by N.P. Barry on the basis of the methods described previously. It is written in FORTRAN and has been implemented on IBM 370 and UNIVAC 1100 computers operated by computer bureaux. Vector algebra is employed. The reason why the graphs have double boundaries is that the calculation can be performed for boundaries of any convex polygon of up to 30 sides. This permits calculations to be restricted to the stability range of particular components, for example, that of water or chloride. 

The CFF is known to run or have run on CRAY XMP, Amdahl VPllOO, many IBMs, Siemens, UNISYS, CDC, many VAXes, Ardent Titan. The program is a patchwork prepared over 20 years, written in IBM FORTRAN IV and later cleaned to conform to FORTRAN 77 new routines are written in FORTRAN 77. Development is now done on an Amdahl VPllOO, and vectorization is used where appropriate. 

It is worth pointing out that the vectorised code is written in standard FORTRAN. The CRAY FORTRAN compiler simply recognises the vectorisable loops and translates these into hardware vector orders. An inspection of the machine code thus generated revealed that very little was to be gained by hand coding the kernels into Assembly language. 

The algorithm is based on the calculation of a row of X in vector sequences. The FORTRAN code in Figure 6 gives a picture of the procedure, but must be coded in CAL, with all vectors labeled Vx being directly replaced by V registers - their dimension is always... 

With the advent of vector processors over the last ten years, the vector computer has become the most efficient and in some instances the only affordable way to solve certain computational problems. One such computer, the Texas Instruments Advanced Scientific Computer (ASC), has been used extensively at the Naval Research Laboratory to model atmospheric and combustion processes, dynamics of laser implosions, and other plasma physics problems. Furthermore, vectorization is achieved in these programs using standard Fortran. This paper will describe some of the hardware and software differences which distinguish the ASC from the more conventional scalar computer and review some of the fundamental principles behind vector program design. 

A sample of various kinds of Fortran statements which automatically vectorize on the ASC have been included in Table I. The asterisk which appears next to certain statements indicates VTS required. 

In order to achieve maximum vectorization of array operations there can be no interruptions in execution, such as a conditionality which excludes some members of an array from a particular operation. However, since conditionalities are often required, some means of incorporating them in a vectorizable manner must be found. To illustrate the effect of conditionality and vector length on the ASC consider the following Fortran ... 

Finally, most doubly or triply subscripted array operations can execute as a single vector instruction on the ASC. To demonstrate the hardware capabilities of the ASC,the vector dot product matrix multiplication instruction, which utilizes one of the most powerful pieces of hardware on the ASC, is compared to similar code on an IBM 360/91 and the CDC 7600 and Cyber 174. Table IV lists the Fortran pattern, which is recognized by the ASC compiler and collapsed into a single vector dot product instruction, the basic instructions required and the hardware speeds obtained when executing the same matrix operations on all four machines. Since many vector instructions in a CP pipe produce one result every clock cycle (80 nanoseconds), ordinary vector multiplications and additions (together) execute at the rate of 24 million floating point operations per second (MFLOPS). For the vector dot product instruction however, each output value produced represents a multiplication and an addition. Thus, vector dot product on the ASC attains a speed of 48 million floating point operations per second. 

This integration method can be optimized for the ASC in two steps. The first is to construct the code so that vectorization over each set of equations occurs. Here the main problem is the decision process associated with the application of the "stiff" or "normal" formulas to each equation. If these formulas are implemented in the usual fashion with an IF test in the appropriate DO Loops the smooth flow of contiguous data from core through the CPU will be inhibited and scalar code will result. Optimization of this process can be accomplished by calculating both formulas and applying a multiplicative factor 0 or 1. The following example of Fortran code illustrates this technique. 

The periodic boundary conditions in loop 3 contain 2 branches. Vectorization was achieved by using the FORTRAN callable vector merge function. 

Except for the above gather-scatter operations, simulations are easily vectorizable as defined by the CRAY FORTRAN or the CYBER 200 FORTRAN. Typical vectors have 50 to 500 elements each. 

The remaining obstacle to producing a working vector processor system is the library of FORTRAN callable subroutines. 

Take the case of matrix multiplication, this would usually be implemented in FORTRAN as a direct translation of the expression in Table XI which is the usual "row by column and add" sequence of operations. The evaluation of a single Cjk does not satisfy the criteriof given above because although B k is a vector with unit address increment from component to component Aj, is not and its address increment is N. This of course is a function of the FORTRAN compiler and can be circumvented by storing A in an "unnatural" order (i.e. by rows, giving the transpose of A in the usual method of storage by columns) but this is not usually worthwhile because of the potential for confusion. 

In most intstances however merely being able to arrange the calculation as a series of vector operations, without worrying over the "unit address increment" requirement, makes extremely good if not maximal use of AFPP, VP or AP. As an illustration of this point Table XIII shows "normal" FORTRAN code for a pivotal condensation matrix inverter (the author is unfortunately by now anonymous) and Table XIV shows the vectorized version for the MVP-9500 at about two thirds completion. The VPLIB version is completely (as far as the author can manage at leastl) vectorized and written in assembler. Most of the vectorization is fairly obvious and only the reduction loops contain any obscurity. In order to maintain peak vector efficiency the MVP-9500 reduction loop does a little more work than is strictly necessary an alternative would ruin the vector flow. It is left as an exercise to the determined reader to unravel the full correspondence between Tables XIII and XIV. 

The author has presented details of a cost effective vector processor for use with S-100 microcomputers and produced a library of FORTRAN callable subroutines for general purpose floating point computations. Brief details of the construction of a molecular mechanics program using the vector processor have been given. 

DO-loop range defined in text. 6 CPU time in microseconds. c Using vector order repertoire. FORTRAN H extended plus compiler with OPT = 2. 

M. Guzzi, D. Padua, J. Hoeflinger, and D. Lawrie, /. Supercomput., 3, 37 (1990). Cedar Fortran and Other Vector and Parallel Fortran Dialects. 

R. Vogelsang, M. Schoen, and C. Hoheisel, Comput. Phys. Commun., 30, 235 (1983). Vectorisation of Molecular Dynamics Fortran Programs Using the Cyber 205 Vector Processing Computer. 

G. S. Grest, B. Diinweg, and K. Kremer, Comput. Phys. Commun., 55, 269 (1989). Vectorized Link-Cell All-Fortran Code for Molecular Dynamics Simulations for a Large Number of Particles. 

M. Schoen, Comput. Phys. Commun., 52, 175 (1989). Structure of a Simple Molecular Dynamics Fortran Program Optimized for Cray Vector Processing Computers. 

Ab initio calculations for large organometallic and other compounds. FORTRAN programs designed for high performance vector and parallel processing machines. 

F. lavernaro and D. Trigiante, Discrete conservative vector fields induced by the trapezoidal method, JNAIAM J. Numer. Anal. Indust. Appl. Math., 2006, 1(1), 113-130. L. F. Shampine, P. H. Muir and H. Xu, A user-friendly fortran BVP solver, JNAIAM J. Numer. Anal. Indust. Appl. Math., 2006, 1(2), 201—217. 

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