Table I
Comparison of Six Methods for Numerically Solving the Abel Equation Using, as Input, Thirtyone Accurate Data Points and the OffAxis Peak Distribution Given by Eq. (13). The Corresponding Values of the Normalized r and x Are Found by Multiplying m by
${\scriptstyle \frac{1}{30}}$
   i(r) − i_{k}(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, Thirtyone Data Points Which Are Good to Two Decimal Places, and the OffAxis Peak Distribution Given by Eq. (13). The Corresponding Values of the Normalized r and x Are Found by Multiplying m by
${\scriptstyle \frac{1}{30}}$
   i(r) − i_{k}(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 TwoPlace 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 TwoPlace 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 TwoPlace 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 