疑似对数螺线的数列
下面这个数列疑似对数螺线上的一些离散的点的实部：1 0
2 -1
3 2
4 5
5 4
6 -5
7 -19
8 -26
9 -9
10 38
11 89
12 83
13 -38
14 -251
15 -388
16 -185
17 489
18 1310
19 1391
20 -226
21 -3419
22 -5897
23 -3646
24 5893
25 18852
26 22499
27 1397
28 -45850
29 -88601
30 -66650
31 66401
32 267499
33 356346
34 89093
35 -601156
36 -1314097
37 -1158383
38 667774
39 3741407
40 5546110
41 2295309
42 -7659985
43 -19242814
44 -19424251
45 5183236
46 51510283
47 84994109
48 47724838
49 -93962793
50 -278192026
51 -316948183
52 7481795
53 696584794
54 1284243205
55 897124796
56 -1091185001
57 -3969137799
58 -5059320802
59 -896122801
60 9223520798
61 19148102197
62 15880024999
63 -11595475198
64 -55847123195
65 -79279775196
66 -27717201805
67 119005171781
68 281849271974
69 269841077191
70 -103296164762
71 -773991685711
72 -1222385870117
73 -614939096838
74 1484734623749
75 4096051276412
76 4448641619615
77 -517205183711
78 -10546632703490
79 -18574120415809
80 -11958924148226
81 17679034154781
82 58758711422303
83 71612721831554
84 7133900402693
85 -140916567005948
86 -278421900662501
87 -216251955890803
88 195952611374950
89 831543034934199
90 1130264280112550
91 319020589694201
92 -1838739336727501
93 -4119567241468454
94 -3730112774547707
95 1909173213954044
96 11597592566697703
97 17538099368759817
98 7761446362655774
99 -23283418786755793
100 -60180557084869090
101 -62196684029528891
102 13505845479440215
103 159166505380593986
104 268037901015551749
105 157562234185046436
106 -283148017690539517
107 -867914658271731691
108 -1010366775781790362
109 -16866334004565593
110 2144563117739495974
111 4039710885797583617
112 2922380877844443595
113 -3245026791688070406
114 -12351681673069593595
115 -16068746645023550404
116 -3394419058110330001
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。 ……
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
a=Vec(,116)
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 什么叫离散的点的实部。是 二维点坐标$(x,y)$里的$x$么 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 -2
2 -0
3 3
4 6
5 5
6 -4
7 -18
8 -25
9 -8
10 39
11 90
12 84
13 -37
14 -250
15 -387
16 -184
17 490
18 1311
19 1392
20 -225
21 -3418
22 -5896
23 -3645
24 5894
25 18853
26 22500
27 1398
28 -45849
29 -88600
30 -66649
31 66402
32 267500
33 356347
34 89094
35 -601155
36 -1314096
37 -1158382
38 667775
39 3741408
40 5546111
41 2295310
42 -7659983
43 -19242811
44 -19424249
45 5183234
46 51510278
47 84994104
48 47724842
49 -93962774
50 -278191997
51 -316948169
52 7481758
53 696584695
54 1284243102
55 897124821
56 -1091184724
57 -3969137332
58 -5059320534
59 -896123326
60 9223519220
61 19148100387
62 15880025076
63 -11595471060
64 -55847115577
65 -79279769981
66 -27717208780
67 119005147170
68 281849241034
69 269841073181
70 -103296104183
71 -773991562919
72 -1222385772444
73 -614939183279
74 1484734246296
75 4096050756568
76 4448641471847
77 -517204318788
78 -10546630752059
79 -18574118646679
80 -11958925108915
81 17679028461979
82 58758702827788
83 71612718172421
84 7133912392296
85 -140916536431706
86 -278421869495924
87 -216251964390579
88 195952527015913
89 831542895000450
90 1130264202099569
91 319020749926471
92 -1838738864502789
93 -4119566705190150
94 -3730112801742160
95 1909171987517062
96 11597590321447872
97 17538097836555708
98 7761448402065816
99 -23283411598403004
100 -60180548044341760
101 -62196682783906936
102 13505828025510008
103 159166469861063900
104 268037872489182780
105 157562258333602500
106 -283147909898087940
107 -867914508619239810
108 -1010366730497772900
109 -16866576321733632
110 2144562563532753400
111 4039710374809710600
112 2922381130056810000
113 -3245025200673485800
114 -12351679236420268000
115 -16068745547049804000
116 -3394422322352631800
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？ 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。 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：
data = Import["E:/wayne.txt", "Data"];
nlm = NonlinearModelFit, {a, b, c, d},n, Method -> NMinimize];
拟合的效果：
nlm[{"BestFit", "ParameterTable"}]
\[\begin{array}{l|llll}
\text{} & \text{Estimate} & \text{Standard Error} & \text{t-Statistic} & \text{P-Value} \\
\hline
a & 0.245849 & 0.0244837 & 10.0413 & \text{2.6362748336318182$\grave{ }$*${}^{\wedge}$-17} \\
b & 9.85598 & 0.0151787 & 649.33 & \text{4.266114630842456$\grave{ }$*${}^{\wedge}$-202} \\
c & 0.3839 & 0.000214305 & 1791.38 & \text{1.882727337467992$\grave{ }$*${}^{\wedge}$-251} \\
d & -0.882863 & 0.000131467 & -6715.47 & \text{9.99901447$\grave{ }$*${}^{\wedge}$-316} \\
\end{array}\]
E^{a + b I, c + d I} /. nlm["BestFitParameters"]
即得：
\
跟KeyTo9_Fans 的计算结果大致相同，但又不尽相同。 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, {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】。
ans = nlm["FitResiduals"];
Table[{data[], Round], N], 10]}, {i, Length}] // Column
{{1,0},-2,1.591923058}
{{2,-1},0,-0.7739055108}
{{3,2},3,-1.006494478}
{{4,5},6,-1.099349367}
{{5,4},5,-0.9108724352}
{{6,-5},-4,-1.031123080}
{{7,-19},-18,-1.014406800}
{{8,-26},-25,-0.9730619182}
{{9,-9},-8,-1.016659604}
{{10,38},39,-0.9980678063}
{{11,89},90,-0.9939394848}
{{12,83},84,-1.006150156}
{{13,-38},-37,-0.9974832616}
{{14,-251},-250,-0.9993258140}
{{15,-388},-387,-1.001752911}
{{16,-185},-184,-0.9987936791}
{{17,489},490,-1.000231692}
{{18,1310},1311,-1.000359288}
{{19,1391},1392,-0.9995810055}
{{20,-226},-225,-1.000196346}
{{21,-3419},-3418,-1.000024360}
{{22,-5897},-5896,-0.9998877202}
{{23,-3646},-3645,-1.000086009}
{{24,5893},5894,-0.9999775836}
{{25,18852},18853,-0.9999794945}
{{26,22499},22500,-1.000028182}
{{27,1397},1398,-0.9999850943}
{{28,-45850},-45849,-0.9999998346}
{{29,-88601},-88600,-1.000007064}
{{30,-66650},-66649,-0.9999939538}
{{31,66401},66402,-1.000001961}
{{32,267499},267500,-1.000001108}
{{33,356346},356347,-0.9999981293}
{{34,89093},89094,-1.000001107}
{{35,-601156},-601155,-0.9999999091}
{{36,-1314097},-1314096,-0.9999995648}
{{37,-1158383},-1158382,-1.000000419}
{{38,667774},667775,-0.9999998385}
{{39,3741407},3741408,-0.9999999452}
{{40,5546110},5546111,-1.000000122}
{{41,2295309},2295310,-0.9999999193}
{{42,-7659985},-7659984,-1.000000013}
{{43,-19242814},-19242813,-1.000000026}
{{44,-19424251},-19424250,-0.9999999714}
{{45,5183236},5183237,-1.000000013}
{{46,51510283},51510284,-1.000000002}
{{47,84994109},84994110,-0.9999999918}
{{48,47724838},47724839,-1.000000005}
{{49,-93962793},-93962792,-0.9999999986}
{{50,-278192026},-278192025,-0.9999999991}
{{51,-316948183},-316948182,-1.000000003}
{{52,7481795},7481796,-0.9999999999}
{{53,696584794},696584795,-0.9999999990}
{{54,1284243205},1284243206,-0.9999999969}
{{55,897124796},897124797,-0.9999999949}
{{56,-1091185001},-1091185000,-0.9999999990}
{{57,-3969137799},-3969137798,-1.000000008}
{{58,-5059320802},-5059320801,-1.000000016}
{{59,-896122801},-896122800,-1.000000014}
{{60,9223520798},9223520799,-0.9999999918}
{{61,19148102197},19148102198,-0.9999999553}
{{62,15880024999},15880025000,-0.9999999350}
{{63,-11595475198},-11595475197,-0.9999999736}
{{64,-55847123195},-55847123194,-1.000000086}
{{65,-79279775196},-79279775195,-1.000000212}
{{66,-27717201805},-27717201804,-1.000000210}
{{67,119005171781},119005171782,-0.9999999461}
{{68,281849271974},281849271975,-0.9999994672}
{{69,269841077191},269841077192,-0.9999991377}
{{70,-103296164762},-103296164761,-0.9999995225}
{{71,-773991685711},-773991685710,-1.000000903}
{{72,-1222385870117},-1222385870116,-1.000002628}
{{73,-614939096838},-614939096837,-1.000002938}
{{74,1484734623749},1484734623750,-0.9999999669}
{{75,4096051276412},4096051276413,-0.9999939224}
{{76,4448641619615},4448641619616,-0.9999890064}
{{77,-517205183711},-517205183710,-0.9999924168}
{{78,-10546632703490},-10546632703489,-1.000008637}
{{79,-18574120415809},-18574120415808,-1.000031128}
{{80,-11958924148226},-11958924148225,-1.000038933}
{{81,17679034154781},17679034154782,-1.000007427}
{{82,58758711422303},58758711422304,-0.9999345878}
{{83,71612721831554},71612721831555,-0.9998665814}
{{84,7133900402693},7133900402694,-0.9998907161}
{{85,-140916567005948},-140916567005947,-1.000071188}
{{86,-278421900662501},-278421900662500,-1.000347949}
{{87,-216251955890803},-216251955890802,-1.000484478}
{{88,195952611374950},195952611374951,-1.000176844}
{{89,831543034934199},831543034934200,-0.9993527179}
{{90,1130264280112550},1130264280112551,-0.9984870565}
{{91,319020589694201},319020589694202,-0.9985699039}
{{92,-1838739336727501},-1838739336727500,-1.000424856}
{{93,-4119567241468454},-4119567241468453,-1.003566762}
{{94,-3730112774547707},-3730112774547706,-1.005524463}
{{95,1909173213954044},1909173213954045,-1.002882415}
{{96,11597592566697703},11597592566697704,-0.9944231374}
{{97,17538099368759817},17538099368759818,-0.9845957242}
{{98,7761446362655774},7761446362655775,-0.9834961411}
{{99,-23283418786755793},-23283418786755792,-1.000202792}
{{100,-60180557084869090},-60180557084869089,-1.031362254}
{{101,-62196684029528891},-62196684029528890,-1.054257476}
{{102,13505845479440215},13505845479440216,-1.036373343}
{{103,159166505380593986},159166505380593987,-0.9642556037}
{{104,268037901015551749},268037901015551750,-0.8739259150}
{{105,157562234185046436},157562234185046437,-0.8486311619}
{{106,-283148017690539517},-283148017690539516,-0.9611227490}
{{107,-867914658271731691},-867914658271731690,-1.186396681}
{{108,-1010366775781790362},-1010366775781790361,-1.366501964}
{{109,-16866334004565593},-16866334004565592,-1.305411751}
{{110,2144563117739495974},2144563117739495975,-0.9635325647}
{{111,4039710885797583617},4039710885797583618,-0.5664322777}
{{112,2922380877844443595},2922380877844443595,-0.4212084115}
{{113,-3245026791688070406},-3245026791688070405,-0.5076415375}
{{114,-12351681673069593595},-12351681673069593595,-0.3547032737}
{{115,-16068746645023550404},-16068746645023550404,0.2155862028}
{{116,-3394419058110330001},-3394419058110330001,-0.2155862028}
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，:lol
看好多项差-1，添加一个常数项是不是可以拟合的更好？ 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。
\[ a =-1.274837341654813765190289447055370310544537080945648638754038328341341302126524398552661093109976821+0.35498487506452593571589117846616744751685390734168983643383164654387061552581493053486270533805867 i\]\
除了第一个值，其他项目误差都在$0.2$以内 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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