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数学研发论坛»论坛 › 【数学研究】 › 难题征解 › 疑似对数螺线的数列
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[原创] 疑似对数螺线的数列

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KeyTo9_Fans
发表于 2019-5-7 16:13:50 | 显示全部楼层 |阅读模式

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下面这个数列疑似对数螺线上的一些离散的点的实部：
  1. 1 0
  2. 2 -1
  3. 3 2
  4. 4 5
  5. 5 4
  6. 6 -5
  7. 7 -19
  8. 8 -26
  9. 9 -9
  10. 10 38
  11. 11 89
  12. 12 83
  13. 13 -38
  14. 14 -251
  15. 15 -388
  16. 16 -185
  17. 17 489
  18. 18 1310
  19. 19 1391
  20. 20 -226
  21. 21 -3419
  22. 22 -5897
  23. 23 -3646
  24. 24 5893
  25. 25 18852
  26. 26 22499
  27. 27 1397
  28. 28 -45850
  29. 29 -88601
  30. 30 -66650
  31. 31 66401
  32. 32 267499
  33. 33 356346
  34. 34 89093
  35. 35 -601156
  36. 36 -1314097
  37. 37 -1158383
  38. 38 667774
  39. 39 3741407
  40. 40 5546110
  41. 41 2295309
  42. 42 -7659985
  43. 43 -19242814
  44. 44 -19424251
  45. 45 5183236
  46. 46 51510283
  47. 47 84994109
  48. 48 47724838
  49. 49 -93962793
  50. 50 -278192026
  51. 51 -316948183
  52. 52 7481795
  53. 53 696584794
  54. 54 1284243205
  55. 55 897124796
  56. 56 -1091185001
  57. 57 -3969137799
  58. 58 -5059320802
  59. 59 -896122801
  60. 60 9223520798
  61. 61 19148102197
  62. 62 15880024999
  63. 63 -11595475198
  64. 64 -55847123195
  65. 65 -79279775196
  66. 66 -27717201805
  67. 67 119005171781
  68. 68 281849271974
  69. 69 269841077191
  70. 70 -103296164762
  71. 71 -773991685711
  72. 72 -1222385870117
  73. 73 -614939096838
  74. 74 1484734623749
  75. 75 4096051276412
  76. 76 4448641619615
  77. 77 -517205183711
  78. 78 -10546632703490
  79. 79 -18574120415809
  80. 80 -11958924148226
  81. 81 17679034154781
  82. 82 58758711422303
  83. 83 71612721831554
  84. 84 7133900402693
  85. 85 -140916567005948
  86. 86 -278421900662501
  87. 87 -216251955890803
  88. 88 195952611374950
  89. 89 831543034934199
  90. 90 1130264280112550
  91. 91 319020589694201
  92. 92 -1838739336727501
  93. 93 -4119567241468454
  94. 94 -3730112774547707
  95. 95 1909173213954044
  96. 96 11597592566697703
  97. 97 17538099368759817
  98. 98 7761446362655774
  99. 99 -23283418786755793
  100. 100 -60180557084869090
  101. 101 -62196684029528891
  102. 102 13505845479440215
  103. 103 159166505380593986
  104. 104 268037901015551749
  105. 105 157562234185046436
  106. 106 -283148017690539517
  107. 107 -867914658271731691
  108. 108 -1010366775781790362
  109. 109 -16866334004565593
  110. 110 2144563117739495974
  111. 111 4039710885797583617
  112. 112 2922380877844443595
  113. 113 -3245026791688070406
  114. 114 -12351681673069593595
  115. 115 -16068746645023550404
  116. 116 -3394419058110330001
复制代码

有没有可能根据这些离散的数据，把整个对数螺线还原出来？

也就是把：

$1$、周期
$2$、初始相位
$3$、初始半径
$4$、半径的增长速率

这$4$个参数还原出来即可。
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.·.·.
发表于 2019-5-10 17:01:47 | 显示全部楼层
……
能给一下对数螺线的方程吗？
我感觉或许可以用Fermat小定理在复数上的推广，以及同余，把这个问题解掉
  1. a=Vec([0,-1,2,5,4,-5,-19,-26,-9,38,89,83,-38,-251,-388,-185,489,1310,1391,-226,-3419,-5897,-3646,5893,18852,22499,1397,-45850,-88601,-66650,66401,267499,356346,89093,-601156,-1314097,-1158383,667774,3741407,5546110,2295309,-7659985,-19242814,-19424251,5183236,51510283,84994109,47724838,-93962793,-278192026,-316948183,7481795,696584794,1284243205,897124796,-1091185001,-3969137799,-5059320802,-896122801,9223520798,19148102197,15880024999,-11595475198,-55847123195,-79279775196,-27717201805,119005171781,281849271974,269841077191,-103296164762,-773991685711,-1222385870117,-614939096838,1484734623749,4096051276412,4448641619615,-517205183711,-10546632703490,-18574120415809,-11958924148226,17679034154781,58758711422303,71612721831554,7133900402693,-140916567005948,-278421900662501,-216251955890803,195952611374950,831543034934199,1130264280112550,319020589694201,-1838739336727501,-4119567241468454,-3730112774547707,1909173213954044,11597592566697703,17538099368759817,7761446362655774,-23283418786755793,-60180557084869090,-62196684029528891,13505845479440215,159166505380593986,268037901015551749,157562234185046436,-283148017690539517,-867914658271731691,-1010366775781790362,-16866334004565593,2144563117739495974,4039710885797583617,2922380877844443595,-3245026791688070406,-12351681673069593595,-16068746645023550404,-3394419058110330001],116)
复制代码

a%5可以发现循环节是48
a%7可以发现循环节长度100
感觉这道题同余+复数是可以解的
然而怎么解我就不会了
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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wayne
发表于 2019-5-10 19:45:23 | 显示全部楼层
什么叫离散的点的实部。是 二维点坐标$(x,y)$里的$x$么

点评

KeyTo9_Fans
对数螺线是极坐标上的函数r=f(w)，自变量w是角度，函数值r是半径。极坐标(w,r)可转换成复平面坐标(x,yi)，实部指的是复平面坐标的x。  发表于 2019-5-10 20:45
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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KeyTo9_Fans
 楼主| 发表于 2019-5-10 21:09:31 | 显示全部楼层
我自己测量出来的参数如下：

旋转周期：$7.12799604$
初始相位：$2.87001717$
初始半径：$1.32333834$
半径增率：$1.46755626$

以上$4$个极坐标参数可以转化成两个复数：

$a=-1.27483729+0.35498486i$，$b=0.93338020+1.13248522i$

于是$a*b^n$的实部（四舍五入到整数）和原始数列很像：
  1. 1 -2
  2. 2 -0
  3. 3 3
  4. 4 6
  5. 5 5
  6. 6 -4
  7. 7 -18
  8. 8 -25
  9. 9 -8
  10. 10 39
  11. 11 90
  12. 12 84
  13. 13 -37
  14. 14 -250
  15. 15 -387
  16. 16 -184
  17. 17 490
  18. 18 1311
  19. 19 1392
  20. 20 -225
  21. 21 -3418
  22. 22 -5896
  23. 23 -3645
  24. 24 5894
  25. 25 18853
  26. 26 22500
  27. 27 1398
  28. 28 -45849
  29. 29 -88600
  30. 30 -66649
  31. 31 66402
  32. 32 267500
  33. 33 356347
  34. 34 89094
  35. 35 -601155
  36. 36 -1314096
  37. 37 -1158382
  38. 38 667775
  39. 39 3741408
  40. 40 5546111
  41. 41 2295310
  42. 42 -7659983
  43. 43 -19242811
  44. 44 -19424249
  45. 45 5183234
  46. 46 51510278
  47. 47 84994104
  48. 48 47724842
  49. 49 -93962774
  50. 50 -278191997
  51. 51 -316948169
  52. 52 7481758
  53. 53 696584695
  54. 54 1284243102
  55. 55 897124821
  56. 56 -1091184724
  57. 57 -3969137332
  58. 58 -5059320534
  59. 59 -896123326
  60. 60 9223519220
  61. 61 19148100387
  62. 62 15880025076
  63. 63 -11595471060
  64. 64 -55847115577
  65. 65 -79279769981
  66. 66 -27717208780
  67. 67 119005147170
  68. 68 281849241034
  69. 69 269841073181
  70. 70 -103296104183
  71. 71 -773991562919
  72. 72 -1222385772444
  73. 73 -614939183279
  74. 74 1484734246296
  75. 75 4096050756568
  76. 76 4448641471847
  77. 77 -517204318788
  78. 78 -10546630752059
  79. 79 -18574118646679
  80. 80 -11958925108915
  81. 81 17679028461979
  82. 82 58758702827788
  83. 83 71612718172421
  84. 84 7133912392296
  85. 85 -140916536431706
  86. 86 -278421869495924
  87. 87 -216251964390579
  88. 88 195952527015913
  89. 89 831542895000450
  90. 90 1130264202099569
  91. 91 319020749926471
  92. 92 -1838738864502789
  93. 93 -4119566705190150
  94. 94 -3730112801742160
  95. 95 1909171987517062
  96. 96 11597590321447872
  97. 97 17538097836555708
  98. 98 7761448402065816
  99. 99 -23283411598403004
  100. 100 -60180548044341760
  101. 101 -62196682783906936
  102. 102 13505828025510008
  103. 103 159166469861063900
  104. 104 268037872489182780
  105. 105 157562258333602500
  106. 106 -283147909898087940
  107. 107 -867914508619239810
  108. 108 -1010366730497772900
  109. 109 -16866576321733632
  110. 110 2144562563532753400
  111. 111 4039710374809710600
  112. 112 2922381130056810000
  113. 113 -3245025200673485800
  114. 114 -12351679236420268000
  115. 115 -16068745547049804000
  116. 116 -3394422322352631800
复制代码

但还不是完全一样。

有没有可能把$a$和$b$调得更准一些，使得$a*b^n$的实部和原始数列完全一样？
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mathe
发表于 2019-5-10 22:14:50 | 显示全部楼层
如果存在a,b,使得$a*b^n$的实部和此数列完全相同，那么我们有${a*b^n+\bar{a}*\bar{b}^n}/2$是此数列.
于是二次多项式$(x-b)(x-\bar{b})$是此数列的特征多项式，也就是数列是一个二阶递推数列，这个应该是不成立的。

点评

KeyTo9_Fans
斐波那契数列$f(n)=f(n-1)+f(n-2)$不就是二阶递推数列吗？前后两项的差值是$f(n-2)$，是一阶的，但不见得是等差数列啊？  发表于 2019-5-10 22:41
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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wayne
发表于 2019-5-10 23:51:38 | 显示全部楼层
KeyTo9_Fans 发表于 2019-5-10 21:09
我自己测量出来的参数如下：
旋转周期：$7.12799604$

最好把拟合的评估参数都计算出来，比对一下

我给出我的：
  1. data = Import["E:/wayne.txt", "Data"];
  2. nlm = NonlinearModelFit[data, E^(a + c n) Cos[b + d n], {a, b, c, d},  n, Method -> NMinimize];
复制代码

拟合的效果：
  1. nlm[{"BestFit", "ParameterTable"}]
复制代码

\[\begin{array}{l|llll}
\text{} & \text{Estimate} & \text{Standard Error} & \text{t-Statistic} & \text{P-Value} \\
\hline
a & 0.245849 & 0.0244837 & 10.0413 & \text{2.6362748336318182$\grave{ }$*${}^{\wedge}$-17} \\
b & 9.85598 & 0.0151787 & 649.33 & \text{4.266114630842456$\grave{ }$*${}^{\wedge}$-202} \\
c & 0.3839 & 0.000214305 & 1791.38 & \text{1.882727337467992$\grave{ }$*${}^{\wedge}$-251} \\
d & -0.882863 & 0.000131467 & -6715.47 & \text{9.99901447$\grave{ }$*${}^{\wedge}$-316} \\
\end{array}\]

  1. E^{a + b I, c + d I} /. nlm["BestFitParameters"]
复制代码

即得：

\[a=-1.16166 - 0.534453 i，b=0.932094 - 1.13412 i\]

跟KeyTo9_Fans 的计算结果大致相同，但又不尽相同。

点评

wayne
哈哈哈，我找到原因了  发表于 2019-5-11 08:24
KeyTo9_Fans
$b$比较接近，但$a$好像差得有点远？  发表于 2019-5-11 07:16
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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wayne
发表于 2019-5-11 08:25:59 | 显示全部楼层
跟KeyTo9_Fans答案不一致的原因找到了，原因是前面的拟合函数报了warning，没关注，设置更高的计算精度后就跟KeyTo9_Fans完全一样。
nlm = NonlinearModelFit[data, E^(a + c n) Cos[b + d n], {a, b, c, d},  n, WorkingPrecision -> 50]
我来秀一下：
这个是Fans的结果：
$a=-1.27483729+0.35498486i$，$b=0.93338020+1.13248522i$

这个是我的：
\[ a = -1.2748373416548137601939656027607260609539677534973 +
0.354984875064525930534409910078275152662157966757 I\]
\[ b =0.9333801995869310465628603640680169309656399643975 +
1.1324852222262455048500390346150772548588801289209 I\]

拟合效果很好，误差全部都在1左右【怀疑是某人有意引入的常熟噪声】。
  1. ans = nlm["FitResiduals"];
  2. Table[{data[[i]], Round[nlm[i]], N[ans[[i]], 10]}, {i, Length[data]}] // Column
复制代码
  1. {{1,0},-2,1.591923058}
  2. {{2,-1},0,-0.7739055108}
  3. {{3,2},3,-1.006494478}
  4. {{4,5},6,-1.099349367}
  5. {{5,4},5,-0.9108724352}
  6. {{6,-5},-4,-1.031123080}
  7. {{7,-19},-18,-1.014406800}
  8. {{8,-26},-25,-0.9730619182}
  9. {{9,-9},-8,-1.016659604}
  10. {{10,38},39,-0.9980678063}
  11. {{11,89},90,-0.9939394848}
  12. {{12,83},84,-1.006150156}
  13. {{13,-38},-37,-0.9974832616}
  14. {{14,-251},-250,-0.9993258140}
  15. {{15,-388},-387,-1.001752911}
  16. {{16,-185},-184,-0.9987936791}
  17. {{17,489},490,-1.000231692}
  18. {{18,1310},1311,-1.000359288}
  19. {{19,1391},1392,-0.9995810055}
  20. {{20,-226},-225,-1.000196346}
  21. {{21,-3419},-3418,-1.000024360}
  22. {{22,-5897},-5896,-0.9998877202}
  23. {{23,-3646},-3645,-1.000086009}
  24. {{24,5893},5894,-0.9999775836}
  25. {{25,18852},18853,-0.9999794945}
  26. {{26,22499},22500,-1.000028182}
  27. {{27,1397},1398,-0.9999850943}
  28. {{28,-45850},-45849,-0.9999998346}
  29. {{29,-88601},-88600,-1.000007064}
  30. {{30,-66650},-66649,-0.9999939538}
  31. {{31,66401},66402,-1.000001961}
  32. {{32,267499},267500,-1.000001108}
  33. {{33,356346},356347,-0.9999981293}
  34. {{34,89093},89094,-1.000001107}
  35. {{35,-601156},-601155,-0.9999999091}
  36. {{36,-1314097},-1314096,-0.9999995648}
  37. {{37,-1158383},-1158382,-1.000000419}
  38. {{38,667774},667775,-0.9999998385}
  39. {{39,3741407},3741408,-0.9999999452}
  40. {{40,5546110},5546111,-1.000000122}
  41. {{41,2295309},2295310,-0.9999999193}
  42. {{42,-7659985},-7659984,-1.000000013}
  43. {{43,-19242814},-19242813,-1.000000026}
  44. {{44,-19424251},-19424250,-0.9999999714}
  45. {{45,5183236},5183237,-1.000000013}
  46. {{46,51510283},51510284,-1.000000002}
  47. {{47,84994109},84994110,-0.9999999918}
  48. {{48,47724838},47724839,-1.000000005}
  49. {{49,-93962793},-93962792,-0.9999999986}
  50. {{50,-278192026},-278192025,-0.9999999991}
  51. {{51,-316948183},-316948182,-1.000000003}
  52. {{52,7481795},7481796,-0.9999999999}
  53. {{53,696584794},696584795,-0.9999999990}
  54. {{54,1284243205},1284243206,-0.9999999969}
  55. {{55,897124796},897124797,-0.9999999949}
  56. {{56,-1091185001},-1091185000,-0.9999999990}
  57. {{57,-3969137799},-3969137798,-1.000000008}
  58. {{58,-5059320802},-5059320801,-1.000000016}
  59. {{59,-896122801},-896122800,-1.000000014}
  60. {{60,9223520798},9223520799,-0.9999999918}
  61. {{61,19148102197},19148102198,-0.9999999553}
  62. {{62,15880024999},15880025000,-0.9999999350}
  63. {{63,-11595475198},-11595475197,-0.9999999736}
  64. {{64,-55847123195},-55847123194,-1.000000086}
  65. {{65,-79279775196},-79279775195,-1.000000212}
  66. {{66,-27717201805},-27717201804,-1.000000210}
  67. {{67,119005171781},119005171782,-0.9999999461}
  68. {{68,281849271974},281849271975,-0.9999994672}
  69. {{69,269841077191},269841077192,-0.9999991377}
  70. {{70,-103296164762},-103296164761,-0.9999995225}
  71. {{71,-773991685711},-773991685710,-1.000000903}
  72. {{72,-1222385870117},-1222385870116,-1.000002628}
  73. {{73,-614939096838},-614939096837,-1.000002938}
  74. {{74,1484734623749},1484734623750,-0.9999999669}
  75. {{75,4096051276412},4096051276413,-0.9999939224}
  76. {{76,4448641619615},4448641619616,-0.9999890064}
  77. {{77,-517205183711},-517205183710,-0.9999924168}
  78. {{78,-10546632703490},-10546632703489,-1.000008637}
  79. {{79,-18574120415809},-18574120415808,-1.000031128}
  80. {{80,-11958924148226},-11958924148225,-1.000038933}
  81. {{81,17679034154781},17679034154782,-1.000007427}
  82. {{82,58758711422303},58758711422304,-0.9999345878}
  83. {{83,71612721831554},71612721831555,-0.9998665814}
  84. {{84,7133900402693},7133900402694,-0.9998907161}
  85. {{85,-140916567005948},-140916567005947,-1.000071188}
  86. {{86,-278421900662501},-278421900662500,-1.000347949}
  87. {{87,-216251955890803},-216251955890802,-1.000484478}
  88. {{88,195952611374950},195952611374951,-1.000176844}
  89. {{89,831543034934199},831543034934200,-0.9993527179}
  90. {{90,1130264280112550},1130264280112551,-0.9984870565}
  91. {{91,319020589694201},319020589694202,-0.9985699039}
  92. {{92,-1838739336727501},-1838739336727500,-1.000424856}
  93. {{93,-4119567241468454},-4119567241468453,-1.003566762}
  94. {{94,-3730112774547707},-3730112774547706,-1.005524463}
  95. {{95,1909173213954044},1909173213954045,-1.002882415}
  96. {{96,11597592566697703},11597592566697704,-0.9944231374}
  97. {{97,17538099368759817},17538099368759818,-0.9845957242}
  98. {{98,7761446362655774},7761446362655775,-0.9834961411}
  99. {{99,-23283418786755793},-23283418786755792,-1.000202792}
  100. {{100,-60180557084869090},-60180557084869089,-1.031362254}
  101. {{101,-62196684029528891},-62196684029528890,-1.054257476}
  102. {{102,13505845479440215},13505845479440216,-1.036373343}
  103. {{103,159166505380593986},159166505380593987,-0.9642556037}
  104. {{104,268037901015551749},268037901015551750,-0.8739259150}
  105. {{105,157562234185046436},157562234185046437,-0.8486311619}
  106. {{106,-283148017690539517},-283148017690539516,-0.9611227490}
  107. {{107,-867914658271731691},-867914658271731690,-1.186396681}
  108. {{108,-1010366775781790362},-1010366775781790361,-1.366501964}
  109. {{109,-16866334004565593},-16866334004565592,-1.305411751}
  110. {{110,2144563117739495974},2144563117739495975,-0.9635325647}
  111. {{111,4039710885797583617},4039710885797583618,-0.5664322777}
  112. {{112,2922380877844443595},2922380877844443595,-0.4212084115}
  113. {{113,-3245026791688070406},-3245026791688070405,-0.5076415375}
  114. {{114,-12351681673069593595},-12351681673069593595,-0.3547032737}
  115. {{115,-16068746645023550404},-16068746645023550404,0.2155862028}
  116. {{116,-3394419058110330001},-3394419058110330001,-0.2155862028}
复制代码


由此可以至少得出两个结论，
1）这些数据是经过Fans的执果索因，精心设计和模拟出来的。
2）Fans的计算很准确，只是不够精确而已，应该用高精度计算。这个算是回答了根本问题了吧，



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mathe
发表于 2019-5-11 09:13:19 来自手机 | 显示全部楼层
看好多项差-1，添加一个常数项是不是可以拟合的更好？

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wayne
除了第一个值，其他全部都在0.2以内  发表于 2019-5-11 09:35
wayne
嗯嗯，正有此意，刚好调整了代码，计算出来了，嘿嘿  发表于 2019-5-11 09:26
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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wayne
发表于 2019-5-11 09:22:31 | 显示全部楼层
加入一个常数，，重新拟合一下，得到这个常数是$-0.975357214569165709037574213890439880336881935506908632....$,然后其他主体的参数$a,b$基本变动不大。
\[ a =-1.274837341654813765190289447055370310544537080945648638754038328341341302126524398552661093109976821+0.35498487506452593571589117846616744751685390734168983643383164654387061552581493053486270533805867 i\]\[b = 0.933380199586931046497097554613876356212373785871960376967523940176927507732465163073428502953898103+1.132485222226245504825383030531058952909060928913946663651736146678585330346194031312524266628788471 i\]
除了第一个值，其他项目误差都在$0.2$以内
毋因群疑而阻独见　　毋任己意而废人言
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mathe
发表于 2019-5-11 09:40:43 | 显示全部楼层
应该是去掉第一项以后，递推式为a(n+5)=2a(n+4)-2a(n+3)+a(n)
对应特征多项式为$x^5-2x^4+2x^3-1$,
具有五个特征根
$1, -0.43338019958693104649543603124735974176 - 0.52582717295145441973270628149009351418*I, -0.43338019958693104649543603124735974176 + 0.52582717295145441973270628149009351418*I, 0.93338019958693104649543603124735974176 - 1.1324852222262455048247600868452498278*I, 0.93338019958693104649543603124735974176 + 1.1324852222262455048247600868452498278*I$

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wayne
漂亮，跟拟合结果里出现的$b$很match ！！  发表于 2019-5-11 09:48
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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