Tests

We conducted certain tests to measure the efficiency of AUTO_DERIV. We have calculated the elapsed time during the computation of the value and the derivatives of functions for which the corresponding analytic expressions were known. In each test, we use a version of the FORTRAN 90 code with the derivatives computed analytically, ``by hand'', and another incorporating AUTO_DERIV.

The first mathematical function we used is the Potential Energy Surface for the HCP molecule, described in [12]. It is a realistic example of a function depending on three variables and it is sufficiently complex for the purpose of exhibiting the capabilities of AUTO_DERIV. Another test was made with the potential for the HF-dimer [13], a function of six variables. For this we where able to calculate analytically the first derivatives only.

In Table 1 we present the elapsed time, averaged over 1000 evaluations, during the computation of each potential and for the two versions (analytic derivatives and using AUTO_DERIV). The results are tabulated for the compilers we had available. The standard DATE_AND_TIME subroutine was employed.

Table 1: Elapsed times during the calculations of the derivatives of Potential Energy Surfaces for the molecular systems HCP and HF-dimer. Time quoted in ms.

Potential	Compiler	Order of derivatives
		0	1	2
HCP (analytic)	NAG	0.626	1.155	2.612
HCP (AUTO_DERIV)	NAG	8.465	38.147	83.975
HCP (analytic)	PGI	0.712	1.217	2.451
HCP (AUTO_DERIV)	PGI	9.105	355.795	807.796
HCP (analytic)	Fujitsu	0.690	1.270	2.703
HCP (AUTO_DERIV)	Fujitsu	10.279	66.731	142.812
HCP (analytic)	Absoft	0.667	1.288	2.973
HCP (AUTO_DERIV)	Absoft	16.075	68.494	156.458
HF dimer (analytic)	NAG	0.071	0.752	-
HF dimer (AUTO_DERIV)	NAG	1.807	5.302	14.250
HF dimer (analytic)	PGI	0.152	0.518	-
HF dimer (AUTO_DERIV)	PGI	1.142	39.513	103.397
HF dimer (analytic)	Fujitsu	0.075	0.596	-
HF dimer (AUTO_DERIV)	Fujitsu	1.903	9.335	23.231
HF dimer (analytic)	Absoft	0.086	1.315	-
HF dimer (AUTO_DERIV)	Absoft	3.174	10.434	22.812

The tests were performed on an Intel Pentium II processor at 450 MHz, running the GNU/Linux operating system. We have also compiled and run the programs on IBM RS/6000 and on HP-PA 2.0 processors with the system's compilers; these results are not presented here. In the test we used the F90/95 compilers shown in Table 2; the options chosen should be the optimal ones.

Table: FORTRAN 90/95 compilers used for the test runs.

COMPILER	OPTIONS
$\parbox{5cm}{NAGWare f95 Rel. 4.0(185)\\ gcc 2.95.1}$	$\parbox{8cm}{$-$Ounroll=1 $-$O4\\ $-$Wc,$-$funroll-loops,$-$O3,$-$fforce-mem\\ $-$Wc,$-$fforce-addr,$-$march=i686}$
Portland Group, Inc. pgf90 v3.0	-fast -tp p6
Fujitsu F90	-O3 -Kfast,eval,PENTIUM_PRO -AR
Absoft Pro Fortran 6.0 f90 v2.1	-B100 -O

The required memory during the tests, naturally, depends directly on the specific function differentiated. As we use a TYPE (func) variable holding $1 + n + n\,(n+1)/2$ REAL(dpk) numbers for each REAL(dpk) mathematical variable in the original code we expect a roughly proportional increase in memory taken by the program.