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{ |class="wikitable sortable" !#!!result { {subst:#if:1|{ {subst:tbl|square2|8|3}}}} | }

gives:

#	result
1	Expression error: Unrecognised word "expression"
2	Expression error: Unrecognised word "expression"
3	Expression error: Unrecognised word "e"
4	1.85302018885E+15
5	43046721
6	6561
7	81
8	9

{ |class="wikitable sortable" !#!!result { {subst:#if:1|{ {subst:tbl|double2|7|3}}}} | }

gives:

#	result
1	384
2	192
3	96
4	48
5	24
6	12
7	6

{ |class="wikitable sortable" !#!!result { {subst:#if:1|{ {subst:tbl|t2|7|abc}}}} | }

gives:

#	result
1	start-1-middle-start-2-middle-start-3-middle-start-4-middle-start-5-middle-start-6-middle-start-7-middle-abc-end-end-end-end-end-end-end
2	start-2-middle-start-3-middle-start-4-middle-start-5-middle-start-6-middle-start-7-middle-abc-end-end-end-end-end-end
3	start-3-middle-start-4-middle-start-5-middle-start-6-middle-start-7-middle-abc-end-end-end-end-end
4	start-4-middle-start-5-middle-start-6-middle-start-7-middle-abc-end-end-end-end
5	start-5-middle-start-6-middle-start-7-middle-abc-end-end-end
6	start-6-middle-start-7-middle-abc-end-end
7	start-7-middle-abc-end

Analysis of square root iteration[edit]

Take trivial case of sqr(1), to conveniently analyse how the error develops: first the error is approximately divided by two each time; when the approximation is about twice the true value the convergence gets very fast, only a few more steps are needed.

For finding the square root of e.g. 5 the same analysis applies: apart from rounding, after n steps the result is sqr(5) times the nth result of the previous iteration scheme with the starting value divided by sqr(5).

{ |class="wikitable sortable" !#!!result { {subst:#if:1|{ {subst:tbl|iteration step|50|1900000000000}}}} | }

gives:

#	result
1	1
2	1
3	1
4	1
5	1
6	1
7	1.00000001814
8	1.00019051057
9	1.01971120328
10	1.21923798866
11	1.91676309429
12	3.55199419711
13	6.96031677226
14	13.8484231572
15	27.6606939314
16	55.303305761
17	110.597569732
18	221.190618477
19	442.378976449
20	884.756822644
21	1769.51308016
22	3539.02587775
23	7078.05161422
24	14156.1031578
25	28312.2062803
26	56624.4125429
27	113248.825077
28	226497.650149
29	452995.300296
30	905990.60059
31	1811981.20118
32	3623962.40235
33	7247924.8047
34	14495849.6094
35	28991699.2188
36	57983398.4375
37	115966796.875
38	231933593.75
39	463867187.5
40	927734375
41	1855468750
42	3710937500
43	7421875000
44	14843750000
45	29687500000
46	59375000000
47	118750000000
48	237500000000
49	475000000000
50	950000000000

{ |class="wikitable sortable" !#!!result { {subst:#if:1|{ {subst:tbl|iteration step|5|2000000000000}}}} | }

gives:

#	result
1	Expression error: Unrecognised word "expression"
2	Expression error: Unrecognised word "expression"
3	Expression error: Unrecognised word "expression"
4	Expression error: Unrecognised word "e"
5	1E+12

{|class="wikitable sortable"
!#!!result
{{subst:#if:1|{{subst:tbl|square|7}}}}
|}

#	result
1	1
2	4
3	9
4	16
5	25
6	36
7	49

Compare:

4 19 116 125 136 149

Analysis of square root iteration[edit]

See also[edit]