Application of the Gauss-Seidel Method to the Chebyshev Rational Approximation Method for Solving Nuclear Fuel Depletion Systems

Figures & data

Fig. 1. Convergence behavior of GS for each CRAM matrix, examined with CRAM-PFD-AO-16, depletion system with 3307 nuclides and a depletion interval of 1 day.

TABLE I DMI Run Times and Results for the 1693-Nuclide Case

AO	Time (ms)	MNARD	ANARD	NNND
4	$357.769\pm 4.963$	$1.16\times {10}^8$	$1.90\times {10}^5$	160
8	$704.775\pm 6.464$	$1.63\times {10}^4$	$4.53\times {10}^1$	75
12	$1049.482\pm 6.112$	$5.47\times {10}^6$	$3.23\times {10}^3$	21
16	$1394.419\pm 9.405$	$3.66\times {10}^0$	$4.71\times {10}^{-3}$	1
20	$1733.940\pm 6.449$	$9.93\times {10}^{-1}$	$1.04\times {10}^{-3}$	0
24	$2079.826\pm 10.398$	$8.84\times {10}^{-1}$	$8.18\times {10}^{-4}$	0
28	$2420.296\pm 9.718$	$6.41\times {10}^0$	$3.95\times {10}^{-3}$	0
32	$2762.308\pm 7.923$	$1.78\times {10}^5$	$1.09\times {10}^2$	0
36	$3069.058\pm 10.340$	$3.08\times {10}^2$	$1.84\times {10}^{-1}$	0
40	$3446.458\pm 13.190$	$1.98\times {10}^2$	$1.17\times {10}^{-1}$	0
44	$3788.330\pm 10.768$	$4.07\times {10}^8$	$2.40\times {10}^5$	1
48	$4077.579\pm 16.048$	—	—	0

TABLE II DMI ARD Results for the 1693-Nuclide Case for Nuclides with NNDs Greater than ${10}^{-10}$ atoms/b∙cm

AO	MNARD	ANARD	NNND
4	$4.89\times {10}^0$	$3.76\times {10}^{-2}$	5
8	$6.17\times {10}^{-3}$	$2.65\times {10}^{-5}$	0
12	$2.71\times {10}^{-6}$	$7.89\times {10}^{-9}$	0
16	$5.00\times {10}^{-10}$	$1.48\times {10}^{-12}$	0
20	$1.07\times {10}^{-12}$	$4.72\times {10}^{-15}$	0
24	$5.72\times {10}^{-5}$	$3.04\times {10}^{-5}$	0
28	$1.07\times {10}^{-12}$	$4.61\times {10}^{-15}$	0
32	$3.42\times {10}^{-13}$	$4.53\times {10}^{-15}$	0
36	$3.25\times {10}^{-12}$	$1.06\times {10}^{-14}$	0
40	$1.07\times {10}^{-12}$	$4.44\times {10}^{-15}$	0
44	$1.96\times {10}^{-12}$	$8.70\times {10}^{-15}$	0
48	-	-	0

Fig. 2. ARDs between various AOs of CRAM to AO-48 applying DMI (a) for all nuclides and (b) for nuclides with NNDs greater than ${10}^{-10}$ atoms/b∙cm and ARDs greater than ${10}^{-4}$.

TABLE III SGE Run Times and Results for the 1693-Nuclide Case

AO	Time (ms)	MNARD	ANARD	NNND
4	$10.973\pm 0.472$	$1.16\times {10}^8$	$1.91\times {10}^5$	160
8	$15.471\pm 0.547$	$1.63\times {10}^4$	$4.54\times {10}^1$	74
12	$20.231\pm 0.962$	$5.82\times {10}^2$	$7.60\times {10}^{-1}$	21
16	$24.380\pm 0.805$	$3.66\times {10}^0$	$3.94\times {10}^{-3}$	1
20	$28.175\pm 0.808$	$1.11\times {10}^{-2}$	$1.09\times {10}^{-5}$	0
24	$33.423\pm 1.022$	$6.80\times {10}^{-5}$	$3.33\times {10}^{-5}$	0
28	$36.585\pm 1.351$	$2.55\times {10}^{-8}$	$2.68\times {10}^{-11}$	0
32	$41.539\pm 1.296$	$2.42\times {10}^{-11}$	$3.25\times {10}^{-14}$	0
36	$45.512\pm 1.648$	$1.17\times {10}^{-14}$	$1.16\times {10}^{-15}$	0
40	$49.883\pm 1.840$	$9.97\times {10}^{-15}$	$6.53\times {10}^{-16}$	0
44	$53.299\pm 1.759$	$1.00\times {10}^{-14}$	$9.30\times {10}^{-16}$	0
48	$58.016\pm 1.428$	—	—	0

TABLE IV SGE ARD Results for the 1693-Nuclide Case for Nuclides Greater than ${10}^{-10}$ atoms/b∙cm

AO	MNARD	ANARD	NNND
4	$4.89\times {10}^0$	$3.76\times {10}^{-2}$	5
8	$6.17\times {10}^{-3}$	$2.65\times {10}^{-5}$	0
12	$2.71\times {10}^{-6}$	$7.86\times {10}^{-9}$	0
16	$5.00\times {10}^{-10}$	$1.45\times {10}^{-12}$	0
20	$5.06\times {10}^{-14}$	$1.26\times {10}^{-15}$	0
24	$5.72\times {10}^{-5}$	$3.04\times {10}^{-5}$	0
28	$1.00\times {10}^{-14}$	$8.57\times {10}^{-16}$	0
32	$1.00\times {10}^{-14}$	$2.60\times {10}^{-15}$	0
36	$9.64\times {10}^{-15}$	$1.04\times {10}^{-15}$	0
40	$9.64\times {10}^{-15}$	$6.01\times {10}^{-16}$	0
44	$1.00\times {10}^{-14}$	$8.29\times {10}^{-16}$	0
48	—	—	0

Fig. 3. ARDs between various AOs of CRAM to AO-48 applying SGE (a) for all nuclides and (b) for nuclides with NNDs greater than ${10}^{-10}$ atoms/b∙cm and ARDs greater than ${10}^{-4}$.

TABLE V ARDs Between CRAM-SGE-48 and CRAM-DMI-48 When Compared to ADM-25 for All Nuclides

Method	MNARD	ANARD
SGE	$2.40\times {10}^{-2}$	$1.42\times {10}^{-5}$
DMI	$5.17\times {10}^2$	$3.06\times {10}^{-1}$

Fig. 4. ARDs of CRAM-SGE-48 and CRAM-DMI-48 compared to ADM-25 for the 1693-nuclide case.

TABLE VI Nuclides with an ARD Greater than ${10}^{-7}$ Between the ADM-25 and CRAM-DMI-48 Solutions for the 1693-Nuclide Case

Nuclide	Decay Constant (s^–1)	Curium-242 Spontaneous Fission Yield	ADM NND (atom/b∙cm)	ARD
${ }^8\mathrm{Be}$	$8.47\times {10}^{15}$	—	$3.26\times {10}^{-37}$	$2.40\times {10}^{-2}$
${ }^{70m2}\mathrm{Cu}$	$0.105$	$2.71\times {10}^{-9}$	$1.41\times {10}^{-29}$	$3.80\times {10}^{-2}$
${ }^{73m1}\mathrm{Zn}$	$0.120$	$3.58\times {10}^{-8}$	$1.64\times {10}^{-28}$	$1.38\times {10}^{-4}$
${ }^{73m2}\mathrm{Zn}$	$53.3$	$2.03\times {10}^{-7}$	$2.09\times {10}^{-30}$	$4.36\times {10}^{-4}$
${ }^{77m}\mathrm{Zn}$	$0.660$	$6.53\times {10}^{-9}$	$5.42\times {10}^{-30}$	$1.39\times {10}^{-2}$
${ }^{117m}\mathrm{Pd}$	$36.3$	$8.37\times {10}^{-8}$	$1.26\times {10}^{-30}$	$1.82\times {10}^{-1}$
${ }^{114m}\mathrm{Ag}$	$462$	$1.24\times {10}^{-4}$	$1.48\times {10}^{-28}$	$3.17\times {10}^{-3}$
${ }^{155m}\mathrm{Gd}$	$21.7$	$2.50\times {10}^{-8}$	$6.34\times {10}^{-31}$	$5.17\times {10}^2$
${ }^{168m}\mathrm{Ho}$	$5.25\times {10}^{-3}$	$1.76\times {10}^{-7}$	$1.84\times {10}^{-26}$	$2.12\times {10}^{-4}$
${ }^{177m}\mathrm{Yb}$	$0.108$	$2.70\times {10}^{-10}$	$1.37\times {10}^{-30}$	$2.37\times {10}^{-7}$

TABLE VII CRAM-AO-48 with GS with Varying Numbers of Iteration Run Times and Results for the 1693-Nuclide Case

GS Iterations	Time (ms)	MNARD	ANARD	NNND
1	$7.267\pm 0.263$	$1.00\times {10}^0$	$9.27\times {10}^{-1}$	139
2	$13.015\pm 0.410$	$1.02\times {10}^0$	$1.09\times {10}^{-1}$	42
3	$18.537\pm 0.458$	$1.25\times {10}^0$	$4.08\times {10}^{-2}$	25
4	$23.747\pm 0.539$	$1.04\times {10}^0$	$2.48\times {10}^{-2}$	5
5	$29.317\pm 0.610$	$9.95\times {10}^{-1}$	$1.22\times {10}^{-2}$	0
6	$34.343\pm 0.730$	$3.43\times {10}^{-1}$	$3.51\times {10}^{-3}$	0
7	$40.494\pm 0.765$	$9.10\times {10}^{-2}$	$6.34\times {10}^{-4}$	0
8	$45.018\pm 0.912$	$1.22\times {10}^{-2}$	$4.66\times {10}^{-5}$	0
9	$50.632\pm 1.134$	$1.74\times {10}^{-3}$	$2.99\times {10}^{-6}$	0
10	$56.140\pm 1.043$	$2.87\times {10}^{-4}$	$2.56\times {10}^{-7}$	0

TABLE VIII CRAM-AO-48 with GS with Varying Numbers of Iteration Results for the 1693-Nuclide Case for Nuclides with NNDs Greater than ${10}^{-10}$ atoms/b∙cm

GS Iterations	MNARD	ANARD	NNND
1	$1.00\times {10}^0$	$9.05\times {10}^{-1}$	90
2	$9.93\times {10}^{-1}$	$7.09\times {10}^{-2}$	0
3	$1.46\times {10}^{-1}$	$1.02\times {10}^{-3}$	0
4	$3.50\times {10}^{-4}$	$1.29\times {10}^{-6}$	0
5	$7.81\times {10}^{-8}$	$1.08\times {10}^{-9}$	0
6	$8.83\times {10}^{-10}$	$6.37\times {10}^{-12}$	0
7	$3.77\times {10}^{-11}$	$1.12\times {10}^{-13}$	0
8	$9.64\times {10}^{-15}$	$7.77\times {10}^{-16}$	0
9	$9.64\times {10}^{-15}$	$6.87\times {10}^{-16}$	0
10	$9.64\times {10}^{-15}$	$6.87\times {10}^{-16}$	0

Fig. 5. ARDs between various numbers of GS iterations for CRAM-AO-48 to CRAM-SGE-AO-48 (a) for all nuclides and (b) for nuclides with NNDs greater than ${10}^{-10}$ atoms/b∙cm and ARDs greater than ${10}^{-4}$.

TABLE IX CRAM-AO-12 with GS with Varying Numbers of Iteration Run Times and ARDs to CRAM-SGE-AO-48 for the 1693-Nuclide Case

GS Iterations	Time (ms)	MNARD	ANARD	NNND
1	$2.780\pm 0.147$	$1.33\times {10}^2$	$1.07\times {10}^0$	149
2	$4.351\pm 0.215$	$5.71\times {10}^2$	$8.93\times {10}^{-1}$	111
3	$5.576\pm 0.231$	$5.73\times {10}^2$	$7.89\times {10}^{-1}$	88
4	$7.084\pm 0.252$	$5.61\times {10}^2$	$9.02\times {10}^{-1}$	57
5	$8.357\pm 0.362$	$5.30\times {10}^2$	$9.31\times {10}^{-1}$	51
6	$9.776\pm 0.337$	$6.59\times {10}^2$	$1.49\times {10}^0$	38
7	$11.351\pm 0.327$	$6.92\times {10}^2$	$1.50\times {10}^0$	30
8	$12.653\pm 0.341$	$7.14\times {10}^2$	$1.12\times {10}^0$	25
9	$13.869\pm 0.349$	$5.82\times {10}^2$	$7.63\times {10}^{-1}$	22
10	$15.153\pm 0.393$	$5.82\times {10}^2$	$7.60\times {10}^{-1}$	21

TABLE X CRAM-AO-12 with GS with Varying Numbers of Iteration ARD to CRAM-SGE-AO-48 for the 1693-Nuclide Case for Nuclides with NNDs Greater than ${10}^{-10}$ atoms/b∙cm

GS Iterations	MNARD	ANARD	NNND
1	$1.00\times {10}^0$	$9.05\times {10}^{-1}$	72
2	$1.08\times {10}^0$	$8.79\times {10}^{-2}$	1
3	$2.30\times {10}^{-1}$	$2.30\times {10}^{-3}$	0
4	$1.80\times {10}^{-3}$	$8.35\times {10}^{-6}$	0
5	$7.38\times {10}^{-6}$	$5.25\times {10}^{-8}$	0
6	$2.81\times {10}^{-6}$	$8.74\times {10}^{-9}$	0
7	$2.71\times {10}^{-6}$	$7.89\times {10}^{-9}$	0
8	$2.71\times {10}^{-6}$	$7.86\times {10}^{-9}$	0
9	$2.71\times {10}^{-6}$	$7.86\times {10}^{-9}$	0
10	$2.71\times {10}^{-6}$	$7.86\times {10}^{-9}$	0

Fig. 6. ARDs between various numbers of GS iterations for CRAM-AO-12 to CRAM-SGE-AO-48 (a) for all nuclides and (b) for nuclides with NNDs greater than ${10}^{-10}$ atoms/b∙cm and ARDs greater than ${10}^{-4}$.

TABLE XI DMI Run Times and Results for the 299-Nuclide Case

AO	Time (ms)	MNARD	ANARD	NNND
4	$3.414\pm 0.212$	$4.41\times {10}^7$	$1.84\times {10}^5$	29
8	$6.641\pm 0.351$	$1.63\times {10}^4$	$1.68\times {10}^2$	32
12	$9.895\pm 0.357$	$3.74\times {10}^2$	$2.54\times {10}^0$	18
16	$12.960\pm 0.366$	$2.08\times {10}^0$	$1.20\times {10}^{-2}$	1
20	$16.037\pm 0.454$	$5.74\times {10}^{-3}$	$2.99\times {10}^{-5}$	0
24	$19.278\pm 0.367$	$6.78\times {10}^{-5}$	$3.93\times {10}^{-5}$	0
28	$22.072\pm 0.394$	$1.15\times {10}^{-8}$	$5.33\times {10}^{-11}$	0
32	$25.294\pm 0.438$	$1.04\times {10}^{-11}$	$4.92\times {10}^{-14}$	0
36	$28.351\pm 0.509$	$8.74\times {10}^{-15}$	$1.81\times {10}^{-15}$	0
40	$31.359\pm 0.599$	$8.74\times {10}^{-15}$	$5.37\times {10}^{-16}$	0
44	$34.244\pm 0.571$	$9.16\times {10}^{-15}$	$1.81\times {10}^{-15}$	0
48	$37.076\pm 0.486$	—	—	0

TABLE XII DMI Results for the 299-Nuclide Case for Nuclides with NNDs Greater than ${10}^{-10}$ atoms/b∙cm

AO	MNARD	ANARD	NNND
4	$7.45\times {10}^0$	$6.41\times {10}^{-2}$	2
8	$8.57\times {10}^{-3}$	$5.25\times {10}^{-5}$	0
12	$1.73\times {10}^{-6}$	$1.10\times {10}^{-8}$	0
16	$4.10\times {10}^{-11}$	$5.89\times {10}^{-13}$	0
20	$5.73\times {10}^{-14}$	$9.81\times {10}^{-16}$	0
24	$5.30\times {10}^{-5}$	$3.43\times {10}^{-5}$	0
28	$8.74\times {10}^{-15}$	$6.90\times {10}^{-16}$	0
32	$9.16\times {10}^{-15}$	$2.49\times {10}^{-15}$	0
36	$8.74\times {10}^{-15}$	$1.81\times {10}^{-15}$	0
40	$8.74\times {10}^{-15}$	$6.33\times {10}^{-16}$	0
44	$9.16\times {10}^{-15}$	$1.94\times {10}^{-15}$	0
48	—	—	0

Fig. 7. ARDs between various AOs of CRAM to AO-48 applying DMI (a) for all nuclides and (b) for nuclides with NNDs greater than ${10}^{-10}$ atoms/b∙cm and ARDs greater than ${10}^{-4}$.

TABLE XIII SGE Run Times and Results for the 299-Nuclide Case

AO	Time (ms)	MNARD	ANARD	NNND
4	$1.190\pm 0.094$	$4.41\times {10}^7$	$1.84\times {10}^5$	29
8	$1.898\pm 0.121$	$1.63\times {10}^4$	$1.68\times {10}^2$	32
12	$2.579\pm 0.172$	$3.74\times {10}^2$	$2.54\times {10}^0$	18
16	$3.259\pm 0.226$	$2.08\times {10}^0$	$1.20\times {10}^{-2}$	1
20	$4.004\pm 0.252$	$5.74\times {10}^{-3}$	$2.99\times {10}^{-5}$	0
24	$4.716\pm 0.257$	$6.78\times {10}^{-5}$	$3.93\times {10}^{-5}$	0
28	$5.399\pm 0.249$	$1.15\times {10}^{-8}$	$5.33\times {10}^{-11}$	0
32	$6.161\pm 0.323$	$1.04\times {10}^{-11}$	$4.89\times {10}^{-14}$	0
36	$6.720\pm 0.319$	$9.16\times {10}^{-15}$	$1.89\times {10}^{-15}$	0
40	$7.435\pm 0.380$	$9.16\times {10}^{-15}$	$9.52\times {10}^{-16}$	0
44	$8.089\pm 0.323$	$8.94\times {10}^{-15}$	$9.34\times {10}^{-16}$	0
48	$8.698\pm 0.285$	—	—	0

TABLE XIV SGE Run Times and Results for the 299-Nuclide Case for Nuclides with NNDs Greater than ${10}^{-10}$ atoms/b∙cm

AO	MNARD	ANARD	NNND
4	$7.45\times {10}^0$	$6.41\times {10}^{-2}$	2
8	$8.57\times {10}^{-3}$	$5.25\times {10}^{-5}$	0
12	$1.73\times {10}^{-6}$	$1.10\times {10}^{-8}$	0
16	$4.10\times {10}^{-11}$	$5.88\times {10}^{-13}$	0
20	$5.85\times {10}^{-14}$	$1.33\times {10}^{-15}$	0
24	$5.30\times {10}^{-5}$	$3.43\times {10}^{-5}$	0
28	$8.74\times {10}^{-15}$	$7.61\times {10}^{-16}$	0
32	$8.81\times {10}^{-15}$	$2.09\times {10}^{-15}$	0
36	$9.16\times {10}^{-15}$	$1.95\times {10}^{-15}$	0
40	$9.16\times {10}^{-15}$	$1.13\times {10}^{-15}$	0
44	$8.71\times {10}^{-15}$	$9.72\times {10}^{-16}$	0
48	—	—	0

Fig. 8. ARDs between various AOs of CRAM to AO-48 applying SGE (a) for all nuclides and (b) for nuclides with NND greater than ${10}^{-10}$ atoms/b∙cm and ARDs greater than ${10}^{-4}$.

TABLE XV CRAM-AO-48 with GS with Varying Numbers of Iteration Run Times and Results for the 299-Nuclide Case

GS Iterations	Time (ms)	MNARD	ANARD	NNND
1	$1.251\pm 0.079$	$1.00\times {10}^0$	$8.78\times {10}^{-1}$	0
2	$2.266\pm 0.134$	$1.00\times {10}^0$	$1.13\times {10}^{-1}$	0
3	$3.241\pm 0.187$	$1.02\times {10}^0$	$7.03\times {10}^{-2}$	3
4	$4.181\pm 0.181$	$9.74\times {10}^{-1}$	$3.91\times {10}^{-2}$	0
5	$5.268\pm 0.173$	$6.28\times {10}^{-1}$	$1.78\times {10}^{-2}$	0
6	$6.215\pm 0.231$	$3.10\times {10}^{-1}$	$7.10\times {10}^{-3}$	0
7	$7.286\pm 0.252$	$1.42\times {10}^{-1}$	$2.60\times {10}^{-3}$	0
8	$8.090\pm 0.273$	$5.82\times {10}^{-2}$	$8.60\times {10}^{-4}$	0
9	$9.081\pm 0.259$	$2.06\times {10}^{-2}$	$2.50\times {10}^{-4}$	0
10	$10.082\pm 0.330$	$5.82\times {10}^{-3}$	$6.08\times {10}^{-5}$	0

TABLE XVI CRAM-AO-48 with GS with Varying Numbers of Iteration Results for the 299-Nuclide Case for Nuclides with NNDs Greater than ${10}^{-10}$ atoms/b∙cm

GS Iterations	MNARD	ANARD	NNND
1	$1.00\times {10}^0$	$8.87\times {10}^{-1}$	0
2	$3.49\times {10}^{-4}$	$1.56\times {10}^{-5}$	0
3	$2.50\times {10}^{-9}$	$6.81\times {10}^{-11}$	0
4	$9.38\times {10}^{-15}$	$7.68\times {10}^{-16}$	0
5	$9.16\times {10}^{-15}$	$8.28\times {10}^{-16}$	0
6	$9.16\times {10}^{-10}$	$8.28\times {10}^{-12}$	0
7	$9.16\times {10}^{-11}$	$8.28\times {10}^{-13}$	0
8	$9.16\times {10}^{-15}$	$8.28\times {10}^{-16}$	0
9	$9.16\times {10}^{-15}$	$8.28\times {10}^{-16}$	0
10	$9.16\times {10}^{-15}$	$8.28\times {10}^{-16}$	0

Fig. 9. ARDs between various numbers of GS iterations for CRAM-AO-48 to CRAM-SGE-AO-48 (a) for all nuclides and (b) for nuclides with NNDs greater than ${10}^{-10}$ atoms/b∙cm and ARDs greater than ${10}^{-4}$.

TABLE XVII CRAM-AO-12 with GS with Varying Numbers of Iteration Run Times and ARDs to CRAM-SGE-AO-48 for the 299-Nuclide Case

GS Iterations	Time (ms)	MNARD	ANARD	NNND
1	$0.464\pm 0.028$	$3.99\times {10}^1$	$1.08\times {10}^0$	4
2	$0.721\pm 0.048$	$4.00\times {10}^1$	$3.49\times {10}^{-1}$	5
3	$0.960\pm 0.063$	$4.00\times {10}^1$	$3.20\times {10}^{-1}$	25
4	$1.179\pm 0.082$	$4.00\times {10}^1$	$3.09\times {10}^{-1}$	19
5	$1.459\pm 0.091$	$4.00\times {10}^1$	$5.85\times {10}^{-1}$	20
6	$1.703\pm 0.097$	$6.74\times {10}^1$	$1.35\times {10}^0$	19
7	$1.941\pm 0.116$	$3.11\times {10}^2$	$2.91\times {10}^0$	19
8	$2.152\pm 0.143$	$2.39\times {10}^2$	$2.28\times {10}^0$	22
9	$2.387\pm 0.150$	$1.14\times {10}^2$	$2.05\times {10}^0$	21
10	$2.584\pm 0.155$	$1.32\times {10}^2$	$2.66\times {10}^0$	21

TABLE XVIII CRAM-AO-12 with GS with Varying Numbers of Iteration ARDs to CRAM-SGE-AO-48 for the 299-Nuclide Case for Nuclides with NNDs Greater than ${10}^{-10}$ atoms/b∙cm

GS Iterations	MNARD	ANARD	NNND
1	$1.00\times {10}^0$	$8.87\times {10}^{-1}$	0
2	$2.12\times {10}^{-3}$	$6.56\times {10}^{-5}$	0
3	$1.73\times {10}^{-6}$	$1.22\times {10}^{-8}$	0
4	$1.73\times {10}^{-6}$	$1.10\times {10}^{-8}$	0
5	$1.73\times {10}^{-6}$	$1.10\times {10}^{-8}$	0
6	$1.73\times {10}^{-6}$	$1.10\times {10}^{-8}$	0
7	$1.73\times {10}^{-6}$	$1.10\times {10}^{-8}$	0
8	$1.73\times {10}^{-6}$	$1.10\times {10}^{-8}$	0
9	$1.73\times {10}^{-6}$	$1.10\times {10}^{-8}$	0
10	$1.73\times {10}^{-6}$	$1.10\times {10}^{-8}$	0

Fig. 10. ARDs between various numbers of GS iterations for CRAM-AO-12 to CRAM-SGE-AO-48 (a) for all nuclides and (b) for nuclides with NNDs greater than ${10}^{-10}$ atoms/b∙cm and ARDs greater than ${10}^{-4}$.

TABLE XIX DMI Run Times and Results for the 37-Nuclide Case

AO	Time (ms)	MNARD	ANARD	NNND
4	$0.068\pm 0.007$	$2.37\times {10}^2$	$9.22\times {10}^0$	6
8	$0.118\pm 0.008$	$1.52\times {10}^0$	$4.87\times {10}^{-2}$	1
12	$0.164\pm 0.012$	$1.55\times {10}^{-3}$	$4.61\times {10}^{-5}$	0
16	$0.211\pm 0.014$	$5.94\times {10}^{-7}$	$1.70\times {10}^{-8}$	0
20	$0.257\pm 0.018$	$1.10\times {10}^{-10}$	$3.22\times {10}^{-12}$	0
24	$0.301\pm 0.017$	$6.05\times {10}^{-5}$	$3.78\times {10}^{-5}$	0
28	$0.346\pm 0.022$	$4.93\times {10}^{-15}$	$4.88\times {10}^{-16}$	0
32	$0.387\pm 0.020$	$9.80\times {10}^{-15}$	$2.98\times {10}^{-15}$	0
36	$0.436\pm 0.025$	$5.83\times {10}^{-15}$	$1.60\times {10}^{-15}$	0
40	$0.478\pm 0.022$	$5.69\times {10}^{-15}$	$1.23\times {10}^{-15}$	0
44	$0.521\pm 0.025$	$5.75\times {10}^{-15}$	$1.73\times {10}^{-15}$	0
48	$0.562\pm 0.028$	—	—	0

TABLE XX DMI Results for the 37-Nuclide Case for Nuclides with NNDs Greater than ${10}^{-10}$ atoms/b∙cm

AO	MNARD	ANARD	NNND
4	$1.14\times {10}^0$	$8.56\times {10}^{-2}$	1
8	$8.92\times {10}^{-4}$	$4.20\times {10}^{-5}$	0
12	$4.74\times {10}^{-8}$	$3.50\times {10}^{-9}$	0
16	$3.21\times {10}^{-11}$	$1.66\times {10}^{-12}$	0
20	$5.69\times {10}^{-15}$	$6.42\times {10}^{-16}$	0
24	$4.90\times {10}^{-5}$	$3.39\times {10}^{-5}$	0
28	$2.76\times {10}^{-15}$	$2.97\times {10}^{-16}$	0
32	$9.80\times {10}^{-15}$	$3.26\times {10}^{-15}$	0
36	$5.69\times {10}^{-15}$	$1.68\times {10}^{-15}$	0
40	$5.69\times {10}^{-15}$	$1.40\times {10}^{-15}$	0
44	$5.75\times {10}^{-15}$	$1.88\times {10}^{-15}$	0
48	—	—	0

Fig. 11. ARDs between various AOs of CRAM to AO-48 applying DMI (a) for all nuclides and (b) for nuclides with NNDs greater than ${10}^{-10}$ atoms/b∙cm and ARDs greater than ${10}^{-4}$.

TABLE XXI SGE Run Times and Results for the 37-Nuclide Case

AO	Time (ms)	MNARD	ANARD	NNND
4	$0.058\pm 0.006$	$2.37\times {10}^2$	$9.22\times {10}^0$	6
8	$0.085\pm 0.005$	$1.52\times {10}^0$	$4.87\times {10}^{-2}$	1
12	$0.113\pm 0.009$	$1.55\times {10}^{-3}$	$4.61\times {10}^{-5}$	0
16	$0.143\pm 0.009$	$5.94\times {10}^{-7}$	$1.70\times {10}^{-8}$	0
20	$0.173\pm 0.012$	$1.10\times {10}^{-10}$	$3.22\times {10}^{-12}$	0
24	$0.201\pm 0.014$	$6.05\times {10}^{-5}$	$3.78\times {10}^{-5}$	0
28	$0.228\pm 0.014$	$2.18\times {10}^{-15}$	$5.32\times {10}^{-16}$	0
32	$0.254\pm 0.014$	$5.75\times {10}^{-15}$	$2.94\times {10}^{-15}$	0
36	$0.284\pm 0.018$	$5.83\times {10}^{-15}$	$1.30\times {10}^{-15}$	0
40	$0.313\pm 0.016$	$4.47\times {10}^{-15}$	$8.33\times {10}^{-16}$	0
44	$0.346\pm 0.021$	$5.75\times {10}^{-15}$	$1.80\times {10}^{-15}$	0
48	$0.367\pm 0.019$	—	—	0

TABLE XXII SGE Results for the 37-Nuclide Case for Nuclides with NNDs Greater than ${10}^{-10}$ atoms/b∙cm

AO	MNARD	ANARD	NNND
4	$1.14\times {10}^0$	$8.56\times {10}^{-2}$	1
8	$8.92\times {10}^{-4}$	$4.20\times {10}^{-5}$	0
12	$4.74\times {10}^{-8}$	$3.50\times {10}^{-9}$	0
16	$3.21\times {10}^{-11}$	$1.66\times {10}^{-12}$	0
20	$5.69\times {10}^{-15}$	$6.42\times {10}^{-16}$	0
24	$4.90\times {10}^{-5}$	$3.39\times {10}^{-5}$	0
28	$2.76\times {10}^{-15}$	$2.97\times {10}^{-16}$	0
32	$9.80\times {10}^{-15}$	$3.26\times {10}^{-15}$	0
36	$5.69\times {10}^{-15}$	$1.68\times {10}^{-15}$	0
40	$5.69\times {10}^{-15}$	$1.40\times {10}^{-15}$	0
44	$5.75\times {10}^{-15}$	$1.88\times {10}^{-15}$	0
48	—	—	0

Fig. 12. ARDs between various AOs of CRAM to AO-48 applying SGE (a) for all nuclides and (b) for nuclides with NNDs greater than ${10}^{-10}$ atoms/b∙cm and ARDs greater than ${10}^{-4}$.

TABLE XXIII CRAM-AO-48 with GS with Varying Numbers of Iteration Run Times and Results for the 37-Nuclide Case

GS Iterations	Time (ms)	MNARD	ANARD
1	$0.079\pm 0.006$	$1.00\times {10}^0$	$3.37\times {10}^{-1}$
2	$0.144\pm 0.011$	$1.63\times {10}^{-3}$	$9.01\times {10}^{-5}$
3	$0.204\pm 0.013$	$5.23\times {10}^{-8}$	$2.81\times {10}^{-9}$
4	$0.264\pm 0.015$	$3.68\times {10}^{-12}$	$1.73\times {10}^{-13}$
5	$0.323\pm 0.017$	$2.18\times {10}^{-15}$	$2.90\times {10}^{-16}$
6	$0.384\pm 0.020$	$2.18\times {10}^{-15}$	$2.90\times {10}^{-16}$
7	$0.450\pm 0.024$	$2.18\times {10}^{-15}$	$2.90\times {10}^{-16}$
8	$0.516\pm 0.025$	$2.18\times {10}^{-15}$	$2.90\times {10}^{-16}$
9	$0.572\pm 0.028$	$2.18\times {10}^{-15}$	$2.90\times {10}^{-16}$
10	$0.643\pm 0.030$	$2.18\times {10}^{-15}$	$2.90\times {10}^{-16}$

TABLE XXIV CRAM-AO-48 with GS with Varying Numbers of Iteration Results for the 37-Nuclide Case for Nuclides with NNDs Greater than ${10}^{-10}$ atoms/b∙cm

GS Iterations	MNARD	ANARD
1	$1.00\times {10}^0$	$3.85\times {10}^{-1}$
2	$1.63\times {10}^{-3}$	$6.30\times {10}^{-5}$
3	$6.75\times {10}^{-10}$	$2.61\times {10}^{-11}$
4	$2.18\times {10}^{-15}$	$3.40\times {10}^{-16}$
5	$2.18\times {10}^{-15}$	$3.91\times {10}^{-16}$
6	$2.18\times {10}^{-15}$	$3.91\times {10}^{-16}$
7	$2.18\times {10}^{-15}$	$3.91\times {10}^{-16}$
8	$2.18\times {10}^{-15}$	$3.91\times {10}^{-16}$
9	$2.18\times {10}^{-15}$	$3.91\times {10}^{-16}$
10	$2.18\times {10}^{-15}$	$3.91\times {10}^{-16}$

TABLE XXV CRAM-AO-12 with GS with Varying Numbers of Iteration Run Times and Results for the 37-Nuclide Case

GS Iterations	Time (ms)	MNARD	ANARD
1	$0.030\pm 0.002$	$1.00\times {10}^0$	$3.37\times {10}^{-1}$
2	$0.047\pm 0.003$	$5.94\times {10}^{-3}$	$3.75\times {10}^{-4}$
3	$0.063\pm 0.007$	$1.55\times {10}^{-3}$	$4.62\times {10}^{-5}$
4	$0.075\pm 0.006$	$1.55\times {10}^{-3}$	$4.61\times {10}^{-5}$
5	$0.090\pm 0.006$	$1.55\times {10}^{-3}$	$4.61\times {10}^{-5}$
6	$0.106\pm 0.008$	$1.55\times {10}^{-3}$	$4.61\times {10}^{-5}$
7	$0.123\pm 0.010$	$1.55\times {10}^{-3}$	$4.61\times {10}^{-5}$
8	$0.135\pm 0.009$	$1.55\times {10}^{-3}$	$4.61\times {10}^{-5}$
9	$0.148\pm 0.010$	$1.55\times {10}^{-3}$	$4.61\times {10}^{-5}$
10	$0.166\pm 0.011$	$1.55\times {10}^{-3}$	$4.61\times {10}^{-5}$

TABLE XXVI CRAM-AO-12 with GS with Varying Numbers of Iteration Results for the 37-Nuclide Case for Nuclides with NNDs Greater than ${10}^{-10}$ atoms/b∙cm

GS Iterations	MNARD	ANARD
1	$1.00\times {10}^0$	$3.85\times {10}^{-1}$
2	$5.94\times {10}^{-3}$	$2.29\times {10}^{-4}$
3	$4.74\times {10}^{-8}$	$3.90\times {10}^{-9}$
4	$4.74\times {10}^{-8}$	$3.50\times {10}^{-9}$
5	$4.74\times {10}^{-8}$	$3.50\times {10}^{-9}$
6	$4.74\times {10}^{-8}$	$3.50\times {10}^{-9}$
7	$4.74\times {10}^{-8}$	$3.50\times {10}^{-9}$
8	$4.74\times {10}^{-8}$	$3.50\times {10}^{-9}$
9	$4.74\times {10}^{-8}$	$3.50\times {10}^{-9}$
10	$4.74\times {10}^{-8}$	$3.50\times {10}^{-9}$

Fig. 13. ARDs between various numbers of GS iterations for CRAM-AO-48 to CRAM-SGE-AO-48 (a) for all nuclides and (b) for nuclides with NNDs greater than ${10}^{-10}$ atoms/b∙cm and ARDs greater than ${10}^{-4}$.

Fig. 14. CRAM-AO-12 with GS with varying numbers of iteration ARDs to CRAM-SGE-AO-48 for the 37-nuclide case for nuclides with NNDs greater than ${10}^{-10}$ atoms/b∙cm.

TABLE XXVII Shortest-Lived Radionuclides for Each Nuclide Library Tested

Nuclide Case	Nuclide Name	Decay Constant (s^–1)
1693	${ }^8\mathrm{Be}$	$8.47\times {10}^{15}$
299	${ }^{105}\mathrm{Ru}$	$4.34\times {10}^{-5}$
37	${ }^{242}\mathrm{Am}$	$1.20\times {10}^{-5}$

TABLE XXVIII Run Times of CRAM Configurations Using DMI, SGE, and GS-3

CRAM Option	1693-Nuclide Time (ms)	299-Nuclide Time (ms)	37-Nuclide Time (ms)
DMI-AO-48	$4077.579\pm 16.048$	$37.076\pm 0.486$	$0.562\pm 0.028$
DMI-AO-12	$1049.482\pm 6.112$	$9.895\pm 0.357$	$0.164\pm 0.012$
SGE-AO-48	$58.016\pm 1.428$	$8.698\pm 0.285$	$0.367\pm 0.019$
SGE-AO-12	$20.231\pm 0.962$	$2.579\pm 0.172$	$0.113\pm 0.009$
GS-3-AO-48	$18.537\pm 0.458$	$3.241\pm 0.187$	$0.204\pm 0.013$
GS-3-AO-12	$5.576\pm 0.231$	$0.960\pm 0.063$	$0.063\pm 0.007$