Table I
Comparison of Six Methods for Numerically Solving the Abel Equation Using, as Input, Thirty-one Accurate Data Points and the Off-Axis Peak Distribution Given by Eq. (13). The Corresponding Values of the Normalized r and x Are Found by Multiplying m by
| | | i(r) − ik(r) |
---|
| | |
|
---|
m | I(x) | i(r) | Nestor and Olsen Eq. (3) | Maecker Eq. (4) | Ladenburg et al. Eq. (5) | Frie Eq. (8) | This paper, 3rd deg Eq. (10) | This paper, 4th deg Eq. (10) |
---|
0 | 1.1875 | 0.7500 | −0.0021 | −0.0110 | 0.0030 | −0.0002 | −0.0062 | −0.0018 |
1 | 1.1888 | 0.7621 | −0.0036 | −0.0160 | 0.0015 | −0.0005 | −0.0041 | −0.0003 |
2 | 1.1918 | 0.7939 | −0.0050 | −0.0191 | 0.0008 | −0.0008 | −0.0012 | 0.0012 |
3 | 1.1943 | 0.8380 | −0.0062 | −0.0195 | 0.0002 | −0.0010 | −0.0004 | 0.0012 |
4 | 1.1938 | 0.8875 | −0.0069 | −0.0173 | −0.0004 | −0.0013 | −0.0027 | 0.0000 |
5 | 1.1876 | 0.9352 | −0.0071 | −0.0124 | −0.0009 | −0.0015 | −0.0149 | −0.0036 |
6 | 1.1732 | 0.9740 | −0.0064 | −0.0053 | −0.0015 | −0.0018 | −0.0017 | 0.0035 |
7 | 1.1488 | 0.9968 | −0.0039 | 0.0034 | −0.0017 | −0.0017 | 0.0095 | 0.0073 |
8 | 1.1144 | 0.9985 | −0.0020 | 0.0094 | −0.0009 | −0.0008 | 0.0109 | 0.0029 |
9 | 1.0719 | 0.9873 | −0.0011 | 0.0138 | −0.0008 | −0.0006 | 0.0091 | −0.0013 |
10 | 1.0221 | 0.9657 | −0.0002 | 0.0177 | −0.0007 | −0.0005 | 0.0057 | −0.0029 |
11 | 0.9657 | 0.9349 | 0.0005 | 0.0210 | −0.0006 | −0.0003 | −0.0001 | 0.0001 |
12 | 0.9036 | 0.8960 | 0.0012 | 0.0239 | −0.0005 | −0.0002 | 0.0005 | 0.0003 |
13 | 0.8368 | 0.8500 | 0.0019 | 0.0262 | −0.0004 | −0.0001 | 0.0011 | 0.0004 |
14 | 0.7664 | 0.7978 | 0.0025 | 0.0281 | −0.0004 | 0.0000 | 0.0015 | 0.0003 |
15 | 0.6935 | 0.7407 | 0.0030 | 0.0295 | −0.0003 | 0.0000 | 0.0016 | 0.0002 |
16 | 0.6192 | 0.6797 | 0.0034 | 0.0304 | −0.0002 | 0.0001 | 0.0015 | 0.0000 |
17 | 0.5447 | 0.6157 | 0.0038 | 0.0308 | −0.0001 | 0.0002 | 0.0017 | 0.0000 |
18 | 0.4713 | 0.5499 | 0.0042 | 0.0308 | −0.0000 | 0.0003 | 0.0014 | 0.0003 |
19 | 0.4000 | 0.4833 | 0.0045 | 0.0303 | 0.0001 | 0.0003 | 0.0007 | 0.0004 |
20 | 0.3319 | 0.4170 | 0.0047 | 0.0293 | 0.0001 | 0.0004 | 0.0000 | 0.0003 |
21 | 0.2681 | 0.3520 | 0.0049 | 0.0279 | 0.0002 | 0.0005 | −0.0004 | 0.0002 |
22 | 0.2097 | 0.2894 | 0.0049 | 0.0260 | 0.0003 | 0.0005 | −0.0001 | 0.0000 |
23 | 0.1574 | 0.2301 | 0.0049 | 0.0237 | 0.0004 | 0.0006 | 0.0024 | 0.0001 |
24 | 0.1121 | 0.1754 | 0.0048 | 0.0210 | 0.0005 | 0.0006 | 0.0010 | 0.0004 |
25 | 0.0743 | 0.1262 | 0.0046 | 0.0179 | 0.0006 | 0.0007 | −0.0006 | 0.0004 |
26 | 0.0444 | 0.0836 | 0.0043 | 0.0145 | 0.0006 | 0.0008 | −0.0015 | 0.0001 |
27 | 0.0226 | 0.0486 | 0.0038 | 0.0107 | 0.0007 | 0.0007 | −0.0010 | −0.0003 |
28 | 0.0085 | 0.0223 | 0.0031 | 0.0067 | 0.0007 | 0.0011 | 0.0011 | −0.0003 |
29 | 0.0016 | 0.0058 | 0.0018 | 0.0027 | 0.0011 | 0.0000 | 0.0041 | 0.0004 |
30 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
Table II
Standard Deviations Obtained by Using Exact Values of I(x) Obtained from Eqs. (1) and (11) as Test Values
Number of intervals | Nestor and Olsen Eq. (3) | Maecker Eq. (4) | Ladenburg et al. Eq. (5) | Frie Eq. (8) | This paper, 3rd deg Eq. (10) | This paper, 4th deg Eq. (10) |
---|
10 | 0.0125 | 0.0504 | 0.00326 | 0.00237 | 0.349 | 0.180 |
20 | 0.00456 | 0.0260 | 0.000773 | 0.000647 | 0.00113 | 0.000395 |
30 | 0.00251 | 0.0176 | 0.000336 | 0.000296 | 0.000706 | 0.000127 |
40 | 0.00164 | 0.0133 | 0.000187 | 0.000169 | 0.000565 | 0.0000775 |
50 | 0.00118 | 0.0107 | 0.000119 | 0.000109 | 0.000492 | 0.0000652 |
Table III
Standard Deviations Obtained by Using Exact Values of I(x) Obtained from Eqs. (1) and (12) as Test Values
Number of intervals | Nestor and Olsen Eq. (3) | Maecker Eq. (4) | Ladenburg et al. Eq. (5) | Frie Eq. (8) | This paper, 3rd deg Eq. (10) | This paper, 4th deg Eq. (10) |
---|
10 | 0.0133 | 0.0522 | 0.00351 | 0.00307 | 0.345 | 0.183 |
20 | 0.00487 | 0.0271 | 0.000842 | 0.000787 | 0.00256 | 0.00153 |
30 | 0.00270 | 0.0183 | 0.000369 | 0.000354 | 0.00124 | 0.000915 |
40 | 0.00177 | 0.0139 | 0.000206 | 0.000200 | 0.000908 | 0.000759 |
50 | 0.00127 | 0.0111 | 0.000131 | 0.000129 | 0.000786 | 0.000702 |
Table IV
Standard Deviations Obtained by Using Exact Values of I(x) Obtained from Eqs. (1) and (13) as Test Values
Number of intervals | Nestor and Olsen Eq. (3) | Maecker Eq. (4) | Ladenburg et al. Eq. (5) | Frie Eq. (8) | This paper, 3rd deg Eq. (10) | This paper, 4th deg Eq. (10) |
---|
10 | 0.0194 | 0.0559 | 0.00907 | 0.00643 | 0.385 | 0.208 |
20 | 0.00731 | 0.0302 | 0.00209 | 0.00170 | 0.0101 | 0.00345 |
30 | 0.00406 | 0.0207 | 0.000887 | 0.000754 | 0.00459 | 0.00184 |
40 | 0.00267 | 0.0158 | 0.000492 | 0.000433 | 0.00264 | 0.00133 |
50 | 0.00192 | 0.0128 | 0.000308 | 0.000276 | 0.00182 | 0.00110 |
Table V
Comparison of Six Methods of Numerically Solving the Abel Equation Using, as Input, Thirty-one Data Points Which Are Good to Two Decimal Places, and the Off-Axis Peak Distribution Given by Eq. (13). The Corresponding Values of the Normalized r and x Are Found by Multiplying m by
| | | i(r) − ik(r) |
---|
| | |
|
---|
m | I(x) | i(r) | Nestor and Olsen Eq. (3) | Maecker Eq. (4) | Ladenburg et al. Eq. (5) | Frie Eq. (8) | This paper, 3rd deg Eq. (10) | This paper, 4th deg Eq. (10) |
---|
0 | 1.1900 | 0.7500 | −0.0469 | −0.0504 | −0.0450 | −0.0463 | −0.0323 | −0.0390 |
1 | 1.1900 | 0.7621 | −0.0381 | −0.0437 | −0.0378 | −0.0379 | −0.0270 | −0.0311 |
2 | 1.1900 | 0.7939 | −0.0168 | −0.0256 | −0.0130 | −0.0144 | −0.0155 | −0.0133 |
3 | 1.1900 | 0.8380 | 0.0078 | −0.0053 | 0.0032 | 0.0040 | −0.0038 | 0.0043 |
4 | 1.1900 | 0.8875 | 0.0233 | 0.0035 | 0.0473 | 0.0440 | 0.0024 | 0.0123 |
5 | 1.1900 | 0.9352 | −0.0330 | −0.0310 | −0.0405 | −0.0395 | −0.0129 | −0.0014 |
6 | 1.1700 | 0.9740 | 0.0111 | 0.0065 | 0.0276 | 0.0262 | −0.0001 | 0.0050 |
7 | 1.1500 | 0.9968 | −0.0240 | −0.0101 | −0.0306 | −0.0299 | 0.0112 | 0.0084 |
8 | 1.1100 | 0.9985 | 0.0127 | 0.0219 | 0.0187 | 0.0183 | 0.0130 | 0.0040 |
9 | 1.0700 | 0.9873 | 0.0031 | 0.0188 | −0.0049 | −0.0041 | 0.0114 | 0.0000 |
10 | 1.0200 | 0.9657 | 0.0179 | 0.0292 | 0.0306 | 0.0301 | 0.0080 | −0.0015 |
11 | 0.9700 | 0.9349 | −0.0241 | 0.0040 | −0.0359 | −0.0351 | 0.0008 | 0.0008 |
12 | 0.9000 | 0.8960 | 0.0222 | 0.0388 | 0.0230 | 0.0232 | −0.0007 | −0.0013 |
13 | 0.8400 | 0.8500 | −0.0041 | 0.0191 | 0.0006 | 0.0006 | −0.0010 | −0.0022 |
14 | 0.7700 | 0.7978 | −0.0186 | 0.0128 | −0.0305 | −0.0298 | −0.0004 | −0.0023 |
15 | 0.6900 | 0.7407 | 0.0144 | 0.0373 | 0.0183 | 0.0183 | 0.0007 | −0.0015 |
16 | 0.6200 | 0.6797 | −0.0092 | 0.0222 | −0.0182 | −0.0177 | 0.0022 | 0.0002 |
17 | 0.5400 | 0.6157 | 0.0163 | 0.0410 | 0.0129 | 0.0132 | 0.0051 | 0.0045 |
18 | 0.4700 | 0.5499 | 0.0077 | 0.0333 | 0.0063 | 0.0065 | 0.0057 | 0.0046 |
19 | 0.4000 | 0.4833 | 0.0011 | 0.0285 | −0.0072 | −0.0068 | 0.0047 | 0.0030 |
20 | 0.3300 | 0.4170 | 0.0132 | 0.0352 | 0.0123 | 0.0125 | 0.0029 | 0.0006 |
21 | 0.2700 | 0.3520 | −0.0005 | 0.0235 | −0.0079 | −0.0076 | 0.0008 | −0.0019 |
22 | 0.2100 | 0.2894 | 0.0066 | 0.0256 | 0.0055 | 0.0057 | −0.0007 | −0.0038 |
23 | 0.1600 | 0.2301 | −0.0077 | 0.0138 | −0.0134 | −0.0132 | 0.0005 | −0.0057 |
24 | 0.1100 | 0.1754 | 0.0021 | 0.0199 | −0.0033 | −0.0031 | −0.0002 | −0.0018 |
25 | 0.0700 | 0.1262 | 0.0086 | 0.0223 | 0.0042 | 0.0043 | −0.0004 | 0.0022 |
26 | 0.0400 | 0.0836 | 0.0121 | 0.0215 | 0.0065 | 0.0067 | 0.0005 | 0.0046 |
27 | 0.0200 | 0.0486 | 0.0122 | 0.0166 | 0.0129 | 0.0129 | 0.0027 | 0.0046 |
28 | 0.0100 | 0.0223 | −0.0030 | 0.0024 | −0.0076 | −0.0075 | 0.0062 | 0.0025 |
29 | 0.0000 | 0.0058 | 0.0058 | 0.0058 | 0.0058 | 0.0058 | 0.0094 | −0.0005 |
30 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
Table VI
Standard Deviations Obtained by Using Two-Place Values of I(x) Obtained from Eqs. (1) and (11) as Test Values
Number of intervals | Nestor and Olsen Eq. (3) | Maecker Eq. (4) | Ladenburg et al. Eq. (5) | Frie Eq. (8) | This paper, 3rd deg Eq. (10) | This paper, 4th deg Eq. (10) |
---|
10 | 0.0139 | 0.0505 | 0.0104 | 0.0839 | 0.351 | 0.177 |
20 | 0.0113 | 0.0268 | 0.0144 | 0.0125 | 0.00742 | 0.00672 |
30 | 0.0162 | 0.0209 | 0.0233 | 0.0208 | 0.00322 | 0.00371 |
40 | 0.0185 | 0.0187 | 0.0279 | 0.0254 | 0.00300 | 0.00595 |
50 | 0.0194 | 0.0179 | 0.0255 | 0.0249 | 0.00634 | 0.00565 |
Table VII
Standard Deviations Obtained by Using Two-Place Values of I(x) Obtained from Eqs. (1) and (12) as Test Values
Number of intervals | Nestor and Olsen Eq. (3) | Maecker Eq. (4) | Ladenburg et al. Eq. (5) | Frie Eq. (8) | This paper, 3rd deg Eq. (10) | This paper, 4th deg Eq. (10) |
---|
10 | 0.0143 | 0.0526 | 0.0105 | 0.00805 | 0.347 | 0.180 |
20 | 0.0123 | 0.0285 | 0.0164 | 0.0141 | 0.00509 | 0.00422 |
30 | 0.0138 | 0.0211 | 0.0193 | 0.0167 | 0.00662 | 0.00706 |
40 | 0.0182 | 0.0198 | 0.0253 | 0.0231 | 0.00425 | 0.00695 |
50 | 0.0187 | 0.0179 | 0.0237 | 0.0232 | 0.00404 | 0.00534 |
Table VIII
Standard Deviations Obtained by Using Two-Place Values of I(x) Obtained from Eqs. (1) and (13) as Test Values
Number of intervals | Nestor and Olsen Eq. (3) | Maecker Eq. (4) | Ladenburg et al. Eq. (5) | Frie Eq. (8) | This paper, 3rd deg Eq. (10) | This paper, 4th deg Eq. (10) |
---|
10 | 0.0250 | 0.0590 | 0.0142 | 0.0162 | 0.382 | 0.205 |
20 | 0.0159 | 0.0341 | 0.0146 | 0.0150 | 0.0141 | 0.0102 |
30 | 0.0176 | 0.0253 | 0.0220 | 0.0216 | 0.00972 | 0.0100 |
40 | 0.0187 | 0.0224 | 0.0227 | 0.0225 | 0.0120 | 0.00986 |
50 | 0.0164 | 0.0192 | 0.0189 | 0.0189 | 0.00922 | 0.0113 |