I have a function f that is the truncation of the fourier transform of a sequence (i.e. f = sum( cos(a*x) for a in sequence), where sequence has 10,000 terms). I want to find the absolute maximum on an interval.
The documentation suggest getting the maximum of the plot, which works. I would like to know where this maximum occurs, though. As the maximum is 3272.9689885743473, f - 3200 should have a root, but find_root doesn't find a root. I guess this is technically a bug, but not very surprising given the definition of f.
My main question is, how do I find out where this maximum occurs? I can squint at the plot, but this isn't accurate enough.
sequence = [3, 4, 6, 9, 10, 17, 18, 25, 30, 32, 37, 44, 45, 52, 59, 60, 71, 72, 73, 86, 87, 94, 99, 106, 107, 114, 121, 122, 134, 135, 142, 149, 156, 161, 163, 168, 175, 176, 183, 190, 197, 204, 211, 218, 225, 226, 237, 238, 245, 252, 253, 265, 266, 273, 280, 287, 288, 300, 301, 314, 315, 322, 327, 334, 335, 342, 349, 350, 362, 363, 370, 377, 384, 391, 396, 403, 404, 411, 418, 419, 431, 432, 439, 446, 453, 454, 465, 466, 473, 480, 481, 493, 494, 501, 508, 515, 516, 528, 529, 542, 543, 550, 555, 562, 563, 570, 577, 584, 591, 598, 605, 612, 619, 620, 631, 632, 639, 646, 647, 659, 660, 667, 674, 681, 682, 694, 695, 708, 709, 721, 722, 729, 736, 743, 744, 756, 757, 764, 771, 772, 784, 785, 798, 805, 812, 813, 825, 826, 833, 840, 841, 848, 853, 860, 867, 874, 875, 887, 888, 895, 902, 903, 910, 915, 922, 923, 930, 937, 944, 949, 956, 957, 964, 971, 972, 984, 985, 992, 999, 1006, 1013, 1018, 1026, 1033, 1040, 1041, 1053, 1054, 1061, 1068, 1069, 1080, 1081, 1082, 1095, 1102, 1103, 1115, 1116, 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57914, 57915, 57922, 57929, 57942, 57949, 57950, 57962, 57963, 57970, 57977, 57984, 57991, 58004, 58011, 58012, 58019, 58031, 58032, 58039, 58046, 58047, 58059, 58060, 58073, 58074, 58081, 58088, 58093, 58101, 58108, 58109, 58116, 58121, 58128, 58129, 58136, 58143, 58150, 58157, 58170, 58177, 58178, 58185, 58190, 58198, 58205, 58212, 58219, 58226, 58239, 58240, 58247, 58259, 58260, 58267, 58274, 58275, 58287, 58288, 58295, 58302, 58309, 58316, 58321, 58329, 58336, 58337, 58344, 58349, 58356, 58357, 58364, 58371, 58378, 58385, 58398, 58405, 58406, 58413, 58418, 58426, 58433, 58440, 58441, 58453, 58454, 58467, 58468, 58475, 58487, 58488, 58495, 58502, 58503, 58515, 58516, 58523, 58530, 58537, 58544, 58557, 58564, 58565, 58572, 58577, 58584, 58585, 58592, 58599, 58600, 58612, 58613, 58626, 58633, 58634, 58641, 58646, 58654, 58661, 58668, 58669, 58681, 58682, 58689, 58696, 58703, 58710, 58715, 58723, 58730, 58731, 58743, 58744, 58751, 58758, 58765, 58772, 58779, 58792, 58793, 58800, 58812, 58813, 58820, 58827, 58828, 58835, 58840, 58847, 58848, 58861, 58862, 58869, 58882, 58889, 58896, 58897, 58909, 58910, 58917, 58924, 58931, 58938, 58951, 58958, 58959, 58966, 58971, 58978, 58979, 58986, 58993, 59000, 59007, 59020, 59021, 59028, 59035, 59040, 59048, 59055, 59056, 59063, 59068, 59075, 59076, 59089, 59090, 59097, 59104, 59117, 59124, 59125, 59137, 59138, 59145, 59152, 59159, 59166, 59179, 59186, 59187, 59194, 59199, 59206, 59207, 59214, 59221, 59222, 59234, 59235, 59248, 59249, 59256, 59263, 59268, 59276, 59283, 59284, 59291, 59296, 59303, 59304, 59317, 59318, 59325, 59332, 59345, 59352, 59353, 59365, 59366, 59373, 59380, 59387, 59394, 59407, 59414, 59415, 59422, 59434, 59435, 59442, 59449, 59450, 59462, 59463, 59476, 59477, 59484, 59491, 59496, 59504, 59511, 59512, 59519, 59524, 59531, 59532, 59539, 59546, 59553, 59560, 59573, 59580, 59581, 59588, 59593, 59601, 59608, 59615, 59622, 59629, 59642, 59643, 59650, 59662, 59663, 59670, 59677, 59678, 59690, 59691, 59698, 59705, 59712, 59719, 59732, 59739, 59740, 59747, 59752, 59759, 59760, 59767, 59774, 59781, 59788, 59801, 59808]
f(x) = sum(cos(i*x) for i in sequence)
p = plot(f, 5.55, 5.6)
p.ymax()
find_root(f-3200, 5.55, 5.6)
