1Dipartimento di Elettronica Applicata and CNISM, Università degli Studi “Roma Tre,” Via della Vasca Navale 84, I-00146 Rome, Italy (borghi@uniroma3.it)
A simple computational approach is proposed for the evaluation of cuspoid diffraction catastrophes, which are fundamental tools for the description of wave fields in proximity to stable caustics. In particular, on representing the Pearcey and the swallowtail functions in terms of convergent series expansions, the action of the Weniger transformation on them is numerically studied for both real and complex values of the cuspoid arguments. Numerical experiments, aimed at giving evidence of the effectiveness and implementative ease of the proposed method, together with some quantitative comparisons to previously published results, are also presented.
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Action of the WT on the Sequence of the Partial Sum When the Pearcey Function for Is Evaluateda
n
Weniger
Rel. Error
2
0.789143196736191+i0.752244260976698
3
0.788924613329184+i0.752104718658744
4
0.788922837761597+i0.752103958459505
5
0.788922837595800+i0.752103959748944
6
0.788922837596925+i0.752103959759210
7
0.788922837596969+i0.752103959759243
0
First column, sequence index; second column, Weniger-transformed partial sums sequence; third column, relative error with respect to the exact value provided in Ref. [16]. The Set Precision command has been set to 16. Apparent convergence is indicated by the underscore.
Table 2
Action of the WT on the Sequence of the Partial Sum When the Pearcey Function at Is Evaluateda
n
Weniger
Rel. Error
2
0.49536515219377812782+i0.44772875288867395878
3
0.49536645883190588444+i0.44772662697697731673
4
0.49536645430004872922+i0.44772662507675031687
5
0.49536645430069339117+i0.44772662507720659519
6
0.49536645430069323699+i0.44772662507720490246
7
0.49536645430069323558+i0.44772662507720490422
0
The relative error is evaluated with respect to the exact value provided in Ref. [19]. The Set Precision command has been set to 21.
Table 3
Action of the WT on the Sequence of the Partial Sum When the Percey Function P at , Is Evaluateda
n
Weniger
Rel. Error
2
0.30635851681039519036–i0.75455767282699299105
4
0.33281685595479998177–i0.70371574454167668969
6
0.33281451559348601422–i0.70372161361775682640
8
0.33281451555690072066–i0.70372161356841540881
11
0.33281451555689852113–i0.70372161356841618142
0
The relative error is evaluated with respect to the exact value provided in [19]. The Set precision command has been set to 22.
Table 4
Action of the WT on the Sequence of the Partial Sum When the Pearcey Function P at , , Is Evaluateda
n
Weniger
Rel. Error
2
7.0167200427854761916–i1.3171182346425539127
17
4
26.033788429732012225–i12.918762038019051347
68
6
9.1178477421368429646–i8.9674170368625276335
29
8
5.2975850464634091414–i2.3745194070431739257
14
10
0.09888346381729293522+i1.09590777904603107155
1.6
12
0.03923031298735608531+i0.42537289450020610506
0.054
14
–0.06224979955594184568+i0.42618635148582673904
16
–0.06244308315612694702+i0.42630921207660752726
18
–0.06244390091247327288+i0.42630912873694486937
20
–0.06244389966171575882+i0.42630912362538171452
22
–0.06244389962656160469+i0.42630912362404120427
24
–0.06244389962649910633+i0.42630912362420167343
26
–0.06244389962649959403+i0.42630912362420209448
28
–0.06244389962649959566+i 0.42630912362420209354
0
The relative error is evaluated with respect to the exact value provided in [19]. The Set Precision command has been set to 28.
Table 5
Evaluation of the Swallowtail Function on a 3D Real grida
x
y
z
WT Order
x
y
z
WT Order
0.0
23
0.0
0.0
9
0.0
0.0
20
0.0
0.0
0.0
7
0.0
4.0
24
0.0
0.0
4.0
9
0.0
8.0
29
0.0
0.0
8.0
10
4.0
34
0.0
4.0
14
4.0
0.0
32
0.0
4.0
0.0
10
4.0
4.0
31
0.0
4.0
4.0
15
4.0
8.0
35
0.0
4.0
8.0
15
8.0
46
0.0
8.0
21
8.0
0.0
46
0.0
8.0
0.0
19
8.0
4.0
46
0.0
8.0
4.0
22
8.0
8.0
46
0.0
8.0
8.0
26
0.0
12
4.0
0.0
10
0.0
0.0
9
4.0
0.0
0.0
8
0.0
4.0
12
4.0
0.0
4.0
10
0.0
8.0
17
4.0
0.0
8.0
13
4.0
20
4.0
4.0
16
4.0
0.0
16
4.0
4.0
0.0
15
4.0
4.0
18
4.0
4.0
4.0
15
4.0
8.0
20
4.0
4.0
8.0
17
8.0
30
4.0
8.0
23
8.0
0.0
24
4.0
8.0
0.0
20
8.0
4.0
28
4.0
8.0
4.0
20
8.0
8.0
34
4.0
8.0
8.0
20
For each real triplet , with x ∊ , y ∊ , and z ∊ , the minimum value of the WT order needed to retrieve the value S within the same accurancy (of six digits) as in [14] is reported.
Tables (5)
Table 1
Action of the WT on the Sequence of the Partial Sum When the Pearcey Function for Is Evaluateda
n
Weniger
Rel. Error
2
0.789143196736191+i0.752244260976698
3
0.788924613329184+i0.752104718658744
4
0.788922837761597+i0.752103958459505
5
0.788922837595800+i0.752103959748944
6
0.788922837596925+i0.752103959759210
7
0.788922837596969+i0.752103959759243
0
First column, sequence index; second column, Weniger-transformed partial sums sequence; third column, relative error with respect to the exact value provided in Ref. [16]. The Set Precision command has been set to 16. Apparent convergence is indicated by the underscore.
Table 2
Action of the WT on the Sequence of the Partial Sum When the Pearcey Function at Is Evaluateda
n
Weniger
Rel. Error
2
0.49536515219377812782+i0.44772875288867395878
3
0.49536645883190588444+i0.44772662697697731673
4
0.49536645430004872922+i0.44772662507675031687
5
0.49536645430069339117+i0.44772662507720659519
6
0.49536645430069323699+i0.44772662507720490246
7
0.49536645430069323558+i0.44772662507720490422
0
The relative error is evaluated with respect to the exact value provided in Ref. [19]. The Set Precision command has been set to 21.
Table 3
Action of the WT on the Sequence of the Partial Sum When the Percey Function P at , Is Evaluateda
n
Weniger
Rel. Error
2
0.30635851681039519036–i0.75455767282699299105
4
0.33281685595479998177–i0.70371574454167668969
6
0.33281451559348601422–i0.70372161361775682640
8
0.33281451555690072066–i0.70372161356841540881
11
0.33281451555689852113–i0.70372161356841618142
0
The relative error is evaluated with respect to the exact value provided in [19]. The Set precision command has been set to 22.
Table 4
Action of the WT on the Sequence of the Partial Sum When the Pearcey Function P at , , Is Evaluateda
n
Weniger
Rel. Error
2
7.0167200427854761916–i1.3171182346425539127
17
4
26.033788429732012225–i12.918762038019051347
68
6
9.1178477421368429646–i8.9674170368625276335
29
8
5.2975850464634091414–i2.3745194070431739257
14
10
0.09888346381729293522+i1.09590777904603107155
1.6
12
0.03923031298735608531+i0.42537289450020610506
0.054
14
–0.06224979955594184568+i0.42618635148582673904
16
–0.06244308315612694702+i0.42630921207660752726
18
–0.06244390091247327288+i0.42630912873694486937
20
–0.06244389966171575882+i0.42630912362538171452
22
–0.06244389962656160469+i0.42630912362404120427
24
–0.06244389962649910633+i0.42630912362420167343
26
–0.06244389962649959403+i0.42630912362420209448
28
–0.06244389962649959566+i 0.42630912362420209354
0
The relative error is evaluated with respect to the exact value provided in [19]. The Set Precision command has been set to 28.
Table 5
Evaluation of the Swallowtail Function on a 3D Real grida
x
y
z
WT Order
x
y
z
WT Order
0.0
23
0.0
0.0
9
0.0
0.0
20
0.0
0.0
0.0
7
0.0
4.0
24
0.0
0.0
4.0
9
0.0
8.0
29
0.0
0.0
8.0
10
4.0
34
0.0
4.0
14
4.0
0.0
32
0.0
4.0
0.0
10
4.0
4.0
31
0.0
4.0
4.0
15
4.0
8.0
35
0.0
4.0
8.0
15
8.0
46
0.0
8.0
21
8.0
0.0
46
0.0
8.0
0.0
19
8.0
4.0
46
0.0
8.0
4.0
22
8.0
8.0
46
0.0
8.0
8.0
26
0.0
12
4.0
0.0
10
0.0
0.0
9
4.0
0.0
0.0
8
0.0
4.0
12
4.0
0.0
4.0
10
0.0
8.0
17
4.0
0.0
8.0
13
4.0
20
4.0
4.0
16
4.0
0.0
16
4.0
4.0
0.0
15
4.0
4.0
18
4.0
4.0
4.0
15
4.0
8.0
20
4.0
4.0
8.0
17
8.0
30
4.0
8.0
23
8.0
0.0
24
4.0
8.0
0.0
20
8.0
4.0
28
4.0
8.0
4.0
20
8.0
8.0
34
4.0
8.0
8.0
20
For each real triplet , with x ∊ , y ∊ , and z ∊ , the minimum value of the WT order needed to retrieve the value S within the same accurancy (of six digits) as in [14] is reported.