Excel cell formula - realize Ackerman function calculation

Catalog

One · brief introduction

Two · Program

The whole program

Program parsing

3、 ... and · Usage method

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One · brief introduction

Ackermann function （ACKERMANN）

Ackermann function (Ackermann) Is an example of a non primitive recursive function . It requires two natural numbers as input values , Output a natural number . Its output value grows very fast , Only for (4,3) The output of is too large to be calculated accurately .

Roughly realized ：【 Pseudo code 】

func Ackermann(m,n){
if m=0 -> return n+1;
else if n=0 -> return Ackermann(m-1,1);
else -> return Ackermann(m-1,Ackermann(m,n-1));
}

That's about it , Its growth rate is very fast , It can be solved with a two-dimensional table ：

notes ： Vertical column is m, The horizontal column is n, First act n=0, The first column is m=0

Null value means that it cannot be calculated due to insufficient parameters

Ackermann

1	2	3	5
2	3	5	13
3	4	7
4	5	9
5	6	11
6	7	13
7	8	15
8	9	17
9	10	19
10	11
11	12
12	13
13	14
14	15
15	16
16	17
17	18
18	19
19	20
20

Two · Program

The whole program

Slide right to see the complete program

=IF(IF(COLUMN()=1,ROW(),IF(ROW()=1,INDIRECT(ADDRESS(2,COLUMN()-1)),INDIRECT(ADDRESS(INDIRECT(ADDRESS(ROW()-1,COLUMN()))+1,COLUMN()-1))))=0,"",IF(COLUMN()=1,ROW(),IF(ROW()=1,INDIRECT(ADDRESS(2,COLUMN()-1)),INDIRECT(ADDRESS(INDIRECT(ADDRESS(ROW()-1,COLUMN()))+1,COLUMN()-1)))))

Program parsing

first IF Indicates whether the judgment result is 0; if 0（ The minimum is 1） It means that it cannot be calculated , Fill with "", Empty cell （ Filling in others will cause errors in secondary calculation ）.

This is the main calculation ：

IF(COLUMN()=1,ROW(),IF(ROW()=1,INDIRECT(ADDRESS(2,COLUMN()-1)),INDIRECT(ADDRESS(INDIRECT(ADDRESS(ROW()-1,COLUMN()))+1,COLUMN()-1))))

the second IF If the column number is 1（ Starts 1）, said m=0, Then the result is line number plus 1.

Here's an explanation , because Excel The row and column number starts with 1, Our table starts with 0, It means that all row and column numbers are larger than they should be 1.

rest

IF(ROW()=1,INDIRECT(ADDRESS(2,COLUMN()-1)),INDIRECT(ADDRESS(INDIRECT(ADDRESS(ROW()-1,COLUMN()))+1,COLUMN()-1)))

The next judgment is whether the line number is 1, If yes, execute ：

INDIRECT(ADDRESS(2,COLUMN()-1))

ADDRESS Represents the coordinate corresponding to the row and column number .INDIRECT obtain ADDRESS Value . It is expressed as the second （ Line mark -1） The value of the column .

If not, execute ：

INDIRECT(ADDRESS(INDIRECT(ADDRESS(ROW()-1,COLUMN()))+1,COLUMN()-1))

Said to get （ The first （ Column mark ） In the column （ Line mark -1） It's worth it +1 Mark the new line ,（ Column mark -1） Label the new column ） Value .

3、 ... and · Usage method

Copy the program to A1 grid , The drop-down cells are copied to A List the grid you want ;

Select the entire A Column , Right click to copy to the column you want .

The next step is Ackermann Table of function values .

（ It is possible to display a null value after copying , That's because it can't be calculated according to the existing data , Only need to A Just drop down a few more rows of column values ）

This is my result ：

1	2	3	5	13
2	3	5	13
3	4	7	29
4	5	9	61
5	6	11	125
6	7	13
7	8	15
8	9	17
9	10	19
10	11	21
11	12	23
12	13	25
13	14	27
14	15	29
15	16	31
16	17	33
17	18	35
18	19	37
19	20	39
20	21	41
21	22	43
22	23	45
23	24	47
24	25	49
25	26	51
26	27	53
27	28	55
28	29	57
29	30	59
30	31	61
31	32	63
32	33	65
33	34	67
34	35	69
35	36	71
36	37	73
37	38	75
38	39	77
39	40	79
40	41	81
41	42	83
42	43	85
43	44	87
44	45	89
45	46	91
46	47	93
47	48	95
48	49	97
49	50	99
50	51	101
51	52	103
52	53	105
53	54	107
54	55	109
55	56	111
56	57	113
57	58	115
58	59	117
59	60	119
60	61	121
61	62	123
62	63	125
63	64	127
64	65	129
65	66	131
66	67	133
67	68	135
68	69	137
69	70	139
70	71	141
71	72	143
72	73	145
73	74	147
74	75	149
75	76	151
76	77	153
77	78	155
78	79	157
79	80	159
80	81	161
81	82
82	83
83	84
84	85
85	86
86	87
87	88
88	89
89	90
90	91
91	92
92	93
93	94
94	95
95	96
96	97
97	98
98	99
99	100
100	101
101	102
102	103
103	104
104	105
105	106
106	107
107	108
108	109
109	110
110	111
111	112
112	113
113	114
114	115
115	116
116	117
117	118
118	119
119	120
120	121
121	122
122	123
123	124
124	125
125	126
126	127
127	128
128	129
129	130
130	131
131	132
132	133
133	134
134	135
135	136
136	137
137	138
138	139
139	140
140	141
141	142
142	143
143	144
144	145
145	146
146	147
147	148
148	149
149	150
150	151
151	152
152	153
153	154
154	155
155	156
156	157
157	158
158	159
159	160
160	161
161