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The heat flow problem is an ideal program to use to demonstrate how nicely FORTRAN 90 can express regular array programs:
PROGRAM HEATROD
PARAMETER(MAXTIME=200)INTEGER TICKS,I,MAXTIME
REAL*4 ROD(10)ROD(1) = 100.0
DO I=2,9ROD(I) = 0.0
ENDDOROD(10) = 0.0
DO TICKS=1,MAXTIMEIF ( MOD(TICKS,20) .EQ. 1 ) PRINT 100,TICKS,(ROD(I),I=1,10)
ROD(2:9) = (ROD(1:8) + ROD(3:10) ) / 2ENDDO
100 FORMAT(I4,10F7.2)END
The program is identical, except the inner loop is now replaced by a single statement that computes the "new" section by averaging a strip of the "left" elements and a strip of the "right" elements.
The output of this program is as follows:
E6000: f90 heat90.f
E6000:a.out1 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
21 100.00 82.38 66.34 50.30 38.18 26.06 18.20 10.35 5.18 0.0041 100.00 87.04 74.52 61.99 50.56 39.13 28.94 18.75 9.38 0.00
61 100.00 88.36 76.84 65.32 54.12 42.91 32.07 21.22 10.61 0.0081 100.00 88.74 77.51 66.28 55.14 44.00 32.97 21.93 10.97 0.00
101 100.00 88.84 77.70 66.55 55.44 44.32 33.23 22.14 11.07 0.00121 100.00 88.88 77.76 66.63 55.52 44.41 33.30 22.20 11.10 0.00
141 100.00 88.89 77.77 66.66 55.55 44.43 33.32 22.22 11.11 0.00161 100.00 88.89 77.78 66.66 55.55 44.44 33.33 22.22 11.11 0.00
181 100.00 88.89 77.78 66.67 55.55 44.44 33.33 22.22 11.11 0.00E6000:
If you look closely, this output is the same as the red-black implementation. That is because in FORTRAN 90:
ROD(2:9) = (ROD(1:8) + ROD(3:10) ) / 2
is a
single assignment statement. As shown in
[link] , the right side is completely evaluated before the resulting array section is assigned into
ROD(2:9) . For a moment, that might seem unnatural, but consider the following statement:
I = I + 1
We know that if
I starts with 5, it's incremented up to six by this statement. That happens because the right side (5+1) is evaluated before the assignment of 6 into
I is performed. In FORTRAN 90, a variable can be an entire array. So, this
is a red-black operation. There is an "old"
ROD on the right side and a "new"
ROD on the left side!
To really "think" FORTRAN 90, it's good to pretend you are on an SIMD system with millions of little CPUs. First we carefully align the data, sliding it around, and then— wham— in a single instruction, we add all the aligned values in an instant. [link] shows graphically this act of "aligning" the values and then adding them. The data flow graph is extremely simple. The top two rows are read-only, and the data flows from top to bottom. Using the temporary space eliminates the seeming dependency. This approach of "thinking SIMD" is one of the ways to force ourselves to focus our thoughts on the data rather than the control. SIMD may not be a good architecture for your problem but if you can express it so that SIMD could work, a good SPMD environment can take advantage of the data parallelism that you have identified.
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