pxfm/bessel/
j0f_coeffs.rs

1/*
2 * // Copyright (c) Radzivon Bartoshyk 7/2025. All rights reserved.
3 * //
4 * // Redistribution and use in source and binary forms, with or without modification,
5 * // are permitted provided that the following conditions are met:
6 * //
7 * // 1.  Redistributions of source code must retain the above copyright notice, this
8 * // list of conditions and the following disclaimer.
9 * //
10 * // 2.  Redistributions in binary form must reproduce the above copyright notice,
11 * // this list of conditions and the following disclaimer in the documentation
12 * // and/or other materials provided with the distribution.
13 * //
14 * // 3.  Neither the name of the copyright holder nor the names of its
15 * // contributors may be used to endorse or promote products derived from
16 * // this software without specific prior written permission.
17 * //
18 * // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
19 * // AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
20 * // IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
21 * // DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
22 * // FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
23 * // DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
24 * // SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
25 * // CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
26 * // OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
27 * // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
28 */
29
30/**
31J0 zeros and extremums.
32
33Generated by SageMath:
34```python
35# searching for zeros and extremums
36R120 = RealField(120)
37
38zeros = []
39
40mp.prec = 150
41
42step = mpf("0.001")
43epsilon = mpf("1e-35")
44x = mpf("0.0")
45
46def j0_prime(x):
47    return diff(lambda t: besselj(0, t), x)
48
49previous_zero = R120(0)
50j0_zeros = []
51
52while x < mpf("76.0"):
53    f1 = besselj(0, x)
54    f2 = besselj(0, x + step)
55    if f1 * f2 < 0:
56        zero = findroot(lambda t: j0(t), (x, x + step), solver='bisect', tol=mp.mpf("1e-41"))
57        previous_zero = zero
58        j0_zeros.append(zero)
59    if previous_zero is not None and abs(x - mpf(f'{round(x)}')) < epsilon:
60        zeros.append(previous_zero)
61    x += step
62
63j0_extrema = []
64
65x = mpf("0.0")
66while x < mpf("76.0"):
67    d1 = mp.diff(lambda t: j0(t), x)
68    d2 = mp.diff(lambda t: j0(t), x + step)
69    if d1 * d2 < 0:
70        extremum = findroot(lambda t: mp.diff(lambda u: j0(u), t), (x, x + step), solver='bisect', tol=mp.mpf("1e-41"))
71        j0_extrema.append(extremum)
72    x += step
73
74# Print results
75for i, z in enumerate(j0_zeros):
76    print(f"Zero {i+1}: x ≈ {z}")
77
78print("Extrema (peaks/valleys) of J0(x):")
79for e in j0_extrema:
80    print(f"nExtrema: {e}")
81
82j0_zeros.extend(j0_extrema)
83
84j0_zeros = sorted(j0_zeros)
85
86# Print results
87for i, z in enumerate(j0_zeros):
88    print(f"Peak or zero {i+1}: x ≈ {z}")
89
90print("")
91
92print("pub(crate) static J0_ZEROS: [(u64, u64); 48] = [")
93print(f"(0x0, 0x0),")
94for z in j0_zeros:
95    k = split_double_double(z)
96    hi = double_to_hex(k[1])
97    lo = double_to_hex(k[0])
98    print(f"({lo}, {hi}),")
99
100print("];")
101```
102**/
103pub(crate) static J0_ZEROS: [(u64, u64); 48] = [
104    (0x0, 0x0),
105    (0xbca0f539d7da258e, 0x40033d152e971b40),
106    (0xbca60155a9d1b256, 0x400ea75575af6f09),
107    (0x3c975054cd60a517, 0x4016148f5b2c2e45),
108    (0xbc9b226d9d243827, 0x401c0ff5f3b47250),
109    (0xbcb51970714c7c25, 0x40214eb56cccdeca),
110    (0x3cc02610a51562b6, 0x402458d0d0bdfc29),
111    (0x3cb444fd5821d5b1, 0x40279544008272b6),
112    (0x3cb2bce7fd18e693, 0x402aa5baf310e5a2),
113    (0xbcc9796609364e85, 0x402ddca13ef271d2),
114    (0xbcdd2a68e88ab317, 0x4030787b360508c5),
115    (0xbcd165fd108f46ff, 0x403212313f8a19f6),
116    (0xbcd21830197e9e86, 0x40339da8e7416ca4),
117    (0x3cc1d2dfa1c3b5a8, 0x4035362dd173f792),
118    (0xbcc1bf33afef88f1, 0x4036c294e3d4d8ac),
119    (0x3cd0847c620015e0, 0x40385a3b930156dd),
120    (0x3cc1a2686480d882, 0x4039e7570dcea106),
121    (0x3cdd2b3714972b28, 0x403b7e54a5fd5f11),
122    (0xbcb42ce39ec976fb, 0x403d0bfcf471fccc),
123    (0xbcc36bbabc1c9f31, 0x403ea27591cbbed2),
124    (0xbcdbe3a1cd066b66, 0x404018476e6b2bf0),
125    (0x3cda326cf4307839, 0x4040e34e13a66fe6),
126    (0xbced5fbbff045068, 0x4041aa890dc5e97c),
127    (0xbcd0b6068f861c6f, 0x404275637a9619ec),
128    (0x3cc9eafeca0ca4fd, 0x40433cc523d5cb69),
129    (0xbcb34c86f4e27936, 0x4044077a7ed6293a),
130    (0x3cc489bd556e510a, 0x4044cefcf1734b62),
131    (0x3cced48fe99f45ef, 0x40459992c65d0d8d),
132    (0x3ce4f716f3179d90, 0x404661315d6b133f),
133    (0xbcd05a7a0525058f, 0x40472bac0f810810),
134    (0xbcef3950a842db79, 0x4047f36312028ad6),
135    (0x3ce575dc7f8a031a, 0x4048bdc6293f0657),
136    (0x3ce85d7bdb30baf1, 0x404985928f96d51e),
137    (0xbcdfa16a338bbaee, 0x404a4fe0ee444c7b),
138    (0x3ce3d41e041caa68, 0x404b17c038c2018c),
139    (0xbce43e4a90356acf, 0x404be1fc41a4c607),
140    (0x3cda139ce2cd08ac, 0x404ca9ec5a82324b),
141    (0x3ce0f4b1c9544480, 0x404d74180c9e41f6),
142    (0xbcb12e6ef2e594e1, 0x404e3c1731d64f1e),
143    (0x3ceff10a69607aab, 0x404f06343d0971d4),
144    (0x3cdfd1ee8286358a, 0x404fce40efb1156e),
145    (0xbcf8991ca07c84c0, 0x40504c28621f11e6),
146    (0x3ced3cacfc720419, 0x4050b034dde75b42),
147    (0xbcee90a52cffc26f, 0x40511536cb22d72b),
148    (0xbcee669304bfe748, 0x40517948db63675c),
149    (0xbcf5328276c045f2, 0x4051de4554a1c2dd),
150    (0x3cf8eb4a94b63936, 0x4052425c7dcacdf6),
151    (0x3cff05f585843675, 0x4052a753fa820480),
152];
153
154/**
155Roots and values for extremums and J0 zeros.
156
157Generated by MPFR:
158```text
159let mut arr = vec![];
160for zeros in J0_ZEROS.iter() {
161    let mpfr = Float::with_val(107, f64::from_bits(zeros.1)).j0();
162    arr.push(mpfr.to_f64().to_bits());
163}
164println!(
165    "arr: [{}]",
166    arr.iter()
167        .map(|x| format!("0x{:016x}", x))
168        .collect::<Vec<_>>()
169        .join(", ")
170);
171```
172**/
173pub(crate) static J0_ZEROS_VALUE: [u64; 48] = [
174    0x3ff0000000000000,
175    0xbc919b7921f03c8e,
176    0xbfd9c6cf582cbf7f,
177    0xbc7fbb40985f6e34,
178    0x3fd33518b3874e8a,
179    0xbc96e8eeb22e5818,
180    0xbfcff654544ebcd1,
181    0xbc92d8ed368e0843,
182    0x3fcbf3337873a7d8,
183    0xbca50be2ef09843e,
184    0xbfc925c6fca08f55,
185    0x3caa2122af76659e,
186    0x3fc70c511227d5aa,
187    0x3c98b4b912da2218,
188    0xbfc5664e13b70622,
189    0xbca55e059345b430,
190    0x3fc40f90793605bb,
191    0x3cb1c17abe35eaae,
192    0xbfc2f2072e638cf4,
193    0x3c966608ac164dbd,
194    0x3fc1ff5eec6a01cd,
195    0x3cacc62c8409daa7,
196    0xbfc12dd57bf18ada,
197    0x3ca18e7b1f5c77ae,
198    0x3fc076826cc2c191,
199    0xbc8376cc57901722,
200    0xbfbfa8b41711c83a,
201    0xbc9df12b66b26573,
202    0x3fbe8727daa3daed,
203    0xbc9eab3b2bf5813d,
204    0xbfbd8293aa55d18f,
205    0xbcb3795064667200,
206    0x3fbc96700bf039e2,
207    0xbcabd526bce38ac7,
208    0xbfbbbf246914235f,
209    0x3cb14dbc2a692434,
210    0x3fbaf9cb49c4f935,
211    0x3cac341b9f831174,
212    0xbfba4407e04298d1,
213    0xbcb9e23f9c9ca91e,
214    0x3fb99be744018c90,
215    0xbcc3725f95922088,
216    0xbfb8ffc9bd24fe08,
217    0x3cb79a2187735244,
218    0x3fb86e51be0a9153,
219    0xbcc0012df1d3ef73,
220    0xbfb7e656efb009ae,
221    0xbcc6ecd27843db59,
222];
223
224/**
225Taylor coefficients for J0 [0; 74.62]
226
227Generated by SageMath with Sollya:
228```python
229def compute_intervals(zeros):
230    intervals = []
231    for i in range(0, len(zeros)):
232        if i == 0:
233            a = (zeros[i]) / 2 - 0.05 - zeros[i]
234            b = (zeros[i] + zeros[i + 1]) / 2 + 0.05 - zeros[i]
235            intervals.append((RealField(18)(a), RealField(18)(b), RealField(110)(zeros[i])))
236        elif i + 1 > len(zeros) - 1:
237            a = (zeros[i - 1] + zeros[i]) / 2 - 0.05 - zeros[i]
238            b = (zeros[i]) + 0.83 + 0.05 - zeros[i]
239            intervals.append((RealField(18)(a), RealField(18)(b), RealField(110)(zeros[i])))
240        else:
241            a = (zeros[i - 1] + zeros[i]) / 2 - zeros[i] - 0.05
242            b = (zeros[i] + zeros[i + 1]) / 2 + 0.05  - zeros[i]
243            intervals.append((RealField(18)(a), RealField(18)(b), RealField(110)(zeros[i])))
244    return intervals
245
246intervals = compute_intervals(j0_zeros)
247# print(intervals)
248
249def build_sollya_script(a, b, zero, deg):
250    return f"""
251prec = 500;
252bessel_j0 = library("./notes/bessel_sollya/cmake-build-release/libbessel_sollya.dylib");
253f = bessel_j0(x + {zero});
254d = [{a}, {b}];
255pf = remez(f, {deg}, d);
256for i from 0 to degree(pf) do {{
257    write(coeff(pf, i)) >> "coefficients.txt";
258    write("\\n") >> "coefficients.txt";
259}};
260"""
261
262def load_coefficients(filename):
263    with open(filename, "r") as f:
264        return [RR(line.strip()) for line in f if line.strip()]
265
266def call_sollya_on_interval(a, b, zero, degree=12):
267    sollya_script = build_sollya_script(a, b, zero, degree)
268    with open("tmp_interval.sollya", "w") as f:
269        f.write(sollya_script)
270    import subprocess
271    if os.path.exists("coefficients.txt"):
272        os.remove("coefficients.txt")
273    try:
274        result = subprocess.run(
275            ["sollya", "tmp_interval.sollya"],
276            check=True,
277            capture_output=True,
278            text=True
279        )
280    except subprocess.CalledProcessError as e:
281        return
282
283degree = 13
284
285print(f"pub(crate) static J0F_COEFFS: [[u64;{degree + 1}]; {len(intervals)}] = [")
286for i in range(0, len(intervals)):
287    interval = intervals[i]
288    call_sollya_on_interval(interval[0], interval[1], interval[2], degree)
289    coeffs = load_coefficients(f"coefficients.txt")
290    print("[")
291    for c in coeffs:
292        print(double_to_hex(c) + ",")
293    print("],")
294print("];")
295```
296**/
297pub(crate) static J0F_COEFFS: [[u64; 14]; 47] = [
298    [
299        0xbca69de51bcc0120,
300        0xbfe09cdb3655127d,
301        0x3fbba1deea029925,
302        0x3facfae8643687e7,
303        0xbf81bb1cbe1caeb6,
304        0xbf61f992590d3f89,
305        0x3f315382bbf1a06b,
306        0x3f06ed3ba591e6da,
307        0xbed232c8d8aba5cc,
308        0xbea1cceb52db45ee,
309        0x3e68006b6b5682ca,
310        0x3e329cc1da0793f0,
311        0xbdf5dbb0053da32b,
312        0xbdc04e14e5d96539,
313    ],
314    [
315        0xbfd9c6cf582cbf7f,
316        0x3cb1d608597b5d8f,
317        0x3fc9c6cf582cbf55,
318        0xbf91f06d14e12b29,
319        0xbf8b589d1da0f7e4,
320        0x3f50f9103d00ae93,
321        0x3f38644561ce9290,
322        0xbefa2a034c749192,
323        0xbed83a0686fd1dd2,
324        0x3e96a5085d7844b5,
325        0x3e6ebfcaacd49b87,
326        0xbe2964b9932b57f3,
327        0xbdfad73cf4701511,
328        0x3db5acbea609d3e5,
329    ],
330    [
331        0xbc6a1105f9dbcdd5,
332        0x3fd5c6e60a097826,
333        0xbf9f8f72e7a8471e,
334        0xbfab2150cb41ece4,
335        0x3f72f7ffe901b59c,
336        0x3f627e31fe9ddafb,
337        0xbf26f641f369bbbd,
338        0xbf0863f485fc7e89,
339        0x3ecad77cbbb32bcd,
340        0x3ea32e705af41964,
341        0xbe62d9d2bb447e35,
342        0xbe341f0e4f4dc44a,
343        0x3df198b51eee9749,
344        0x3dbec8f104a030ca,
345    ],
346    [
347        0x3fd33518b3874e8a,
348        0xbca7f56ee546a569,
349        0xbfc33518b3874e3b,
350        0x3f7d34125d59fb96,
351        0x3f880c83bdee71b4,
352        0xbf4483c20f2abbd1,
353        0xbf36ffa5fc0ccc0f,
354        0x3ef2ccf7bbc1d620,
355        0x3ed796a6caa245fc,
356        0xbe91e85738cfdd42,
357        0xbe6e69b581df4be3,
358        0x3e2550287122730e,
359        0x3dfab73cc2e87ac7,
360        0xbdb250809ed630ba,
361    ],
362    [
363        0x3c684670459725e1,
364        0xbfd15f7977a772d5,
365        0x3f900f7fcf183c65,
366        0x3fa68b984ec64b1d,
367        0xbf648e63600c54a1,
368        0xbf60e0d6038721a4,
369        0x3f1d7960512fb682,
370        0x3f07800bc7410c66,
371        0xbec3324799a7e0fb,
372        0xbea30e8df2325ca1,
373        0x3e5cecefe3d75abb,
374        0x3e3458cbaa06ef8d,
375        0xbdec4dd6cf551d07,
376        0xbdbefa0975b65713,
377    ],
378    [
379        0xbfcff654544ebcd1,
380        0x3ca0ffc636e7373a,
381        0x3fbff654544ebc20,
382        0xbf70c17ff72b1352,
383        0xbf84b0c5d5d9e7c9,
384        0x3f394154be849b76,
385        0x3f34e12c2fdccc3a,
386        0xbee9f32fcfbab067,
387        0xbed63c53e4cc158b,
388        0x3e8adbb88b7826d7,
389        0x3e6d5f825ac22ee0,
390        0xbe20f17a50351084,
391        0xbdfa326458f60fce,
392        0x3dae0e0c6fa040ee,
393    ],
394    [
395        0xbc648b5c145d146f,
396        0x3fcdc13e66ac2e78,
397        0xbf842ff0cdc58194,
398        0xbfa38d1dd8992ef5,
399        0x3f5a55e9b344ea02,
400        0x3f5e2e16f97e7f5f,
401        0xbf13dfc37714f92e,
402        0xbf05ce7f4a51d917,
403        0x3ebbb177bc7895b4,
404        0x3ea2346df84ab988,
405        0xbe56125b44798bd5,
406        0xbe33d7cc75fbdd4e,
407        0x3de68bc84d67bb0a,
408        0x3dbe749161c75b7f,
409    ],
410    [
411        0x3fcbf3337873a7d8,
412        0xbc999eb31028b75f,
413        0xbfbbf3337873a725,
414        0x3f66604d91f94ba7,
415        0x3f8251858010ffd8,
416        0xbf314bc11a41832e,
417        0xbf32e7decc933381,
418        0x3ee293b4d39e1d9e,
419        0x3ed4a66fbc1f8e7c,
420        0xbe843cb02661be36,
421        0xbe6bdd61e41af9d3,
422        0x3e1ab24db7abbdd7,
423        0x3df940cc3ecafe33,
424        0xbda870435d8d1863,
425    ],
426    [
427        0x3c6110461d393f0c,
428        0xbfca701d0f967501,
429        0x3f7c54b930fef3e4,
430        0x3fa17798aa09f194,
431        0xbf52a21514062f70,
432        0xbf5b541f829ca99f,
433        0x3f0cc0bd9fd11443,
434        0x3f041f3b06c79efa,
435        0xbeb4b22fc3292c98,
436        0xbea1223e5773d27c,
437        0x3e5118e5cf14a591,
438        0x3e32ffb6478c2281,
439        0xbde20a2b34bd7e9f,
440        0xbdbd77afc051bc6c,
441    ],
442    [
443        0xbfc925c6fca08f54,
444        0x3c941bba0a9908a3,
445        0x3fb925c6fca08ea3,
446        0xbf6049377403d97d,
447        0xbf809463bbcfb70d,
448        0x3f297b35471ddd4a,
449        0x3f314dd43fc4bda8,
450        0xbedbf665e54737fe,
451        0xbed32caf8123dd12,
452        0x3e7f510484fa37ba,
453        0x3e6a42a3b8b1effd,
454        0xbe1544b8d56d77a8,
455        0xbdf81cfb72892ddb,
456        0x3da3f5749b01c5c3,
457    ],
458    [
459        0xbc5c64a3611bf4cb,
460        0x3fc8077f56c9b782,
461        0xbf75467eb535dac4,
462        0xbf9fd7c3ad6f5a3e,
463        0x3f4c1b47c806f9c6,
464        0x3f59166c7d3ee8f7,
465        0xbf05f01522f72552,
466        0xbf02aa939ff2e34a,
467        0x3eb0129642c0eedc,
468        0x3ea01716d7274ce0,
469        0xbe4b21ada69ec1ec,
470        0xbe320ffbd6f367e8,
471        0x3ddd441699eaa4df,
472        0x3dbc46351a30e033,
473    ],
474    [
475        0x3fc70c511227d5aa,
476        0xbc903dd711e6b7a7,
477        0xbfb70c511227d4fe,
478        0x3f5910ebe1f1fa45,
479        0x3f7e7dc08e6ff244,
480        0xbf23bd7d15b0a88f,
481        0xbf300357a101d595,
482        0x3ed5ec733cb20d5b,
483        0x3ed1e5d1fa4386ae,
484        0xbe78f423a07fedeb,
485        0xbe68c089aeb3a360,
486        0x3e1143a2a87b28be,
487        0x3df6f2d06548607b,
488        0xbda07d7d6f49ddb9,
489    ],
490    [
491        0x3c57d1baaf487c60,
492        0xbfc62d93aa9d05bb,
493        0x3f70ba9ce88926ac,
494        0x3f9d7073daebb038,
495        0xbf462813c7f32d76,
496        0xbf574a948d055ffc,
497        0x3f01695764ad01fd,
498        0x3f0170ab5ed77aa3,
499        0xbea9c8e2cba34328,
500        0xbe9e4d88f9ebcadb,
501        0x3e460ec3e9bbfca2,
502        0x3e312751ad8cf2c9,
503        0xbdd825b2017389c1,
504        0xbdbb0b4f2fdaca95,
505    ],
506    [
507        0xbfc5664e13b70621,
508        0x3c8adfee580612ab,
509        0x3fb5664e13b7057b,
510        0xbf540ee3940b2f9d,
511        0xbf7c5e1ad9fa40ad,
512        0x3f1fb8a98f136a0c,
513        0x3f2de9be57a255f8,
514        0xbed1bec95e73b04b,
515        0xbed0cf25bf302fd8,
516        0x3e746784408bf531,
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1037        0xbf44486c0b011f0f,
1038        0xbf8f848eec0e0c40,
1039        0x3f1b077fae02763b,
1040        0x3f49310d6e6682f5,
1041        0xbed597a5bccfeb73,
1042        0xbef32a855d113180,
1043        0x3e8069b8fa206f26,
1044        0x3e9100c89784eeac,
1045        0xbe1d14a175c009c5,
1046        0xbe23bbc9bff27ef1,
1047        0x3db0950fae2cbe6f,
1048        0x3dafc6492b94025c,
1049    ],
1050];
pxfm/bessel/j0f_coeffs.rs

pxfm/bessel/
j0f_coeffs.rs
