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Re: roundoff in ?quadrature

At 08:18 AM 2/1/00 +0100, William J. Larson wrote:
>
OK my triple-layered Kevlar body armor is on. :-)

Better add a couple layers of Nomex :-)

Errors should be added in "quadrature" i.e. as the square root
of the sum of the squares. Leigh's method over estimates
the total error because both (all) of the errors are unlikely to
be at their biggest (or smallest) extremes simultaneously.

I agree that *sometimes* they add in quadrature. But please don't bet your
life on it.

Example: Suppose I have a stack of 100 index cards. I measure the
thickness of each card to one sig dig. I assign two students (Moe and Joe)
to calculate the height of the stack by adding up all the
contributions. (The cards are not all the same, so I need to use addition
not multiplication.) I only want to know the answer to one sig dig. So
far so good.
-- Moe decides to keep intermediate results to *two* sig digs (thinking
that will provide one guard digit). One hundred /random/ contributions,
each of which is less than or equal to half a count in the second digit,
should be added in /quadrature/, and hence should add up to less than half
a count in the leading digit, consistent with my desired final
accuracy. (Note: the foregoing statement is not reliably true; the
questionable assumptions are indicated with /.../ marks.)
-- Joe decides to keep intermediate results to *three* sig digs because
he doesn't assume errors add in quadrature.

I used a spreadsheet to calculate the results shown below. You will notice
that starting at item 30, Moe suffers from roundoff errors that don't add
in quadrature. As a result, Moe's result is absurd.

The details of this example are contrived, and it's true that Moe probably
would notice the problem if the calculations were carried out by
hand. OTOH errors of exactly this sort are common when the calculations
are done by computer. Numerical integration routines are particularly
vulnerable.


running_totals
item thickness Moe Joe
1 0.03 0.03 0.03 (all measurements in cm)
2 0.04 0.07 0.07
3 0.03 0.10 0.10
4 0.04 0.14 0.14
5 0.04 0.18 0.18
6 0.04 0.22 0.22
7 0.03 0.25 0.25
8 0.03 0.28 0.28
9 0.03 0.31 0.31
10 0.04 0.35 0.35
11 0.03 0.38 0.38
12 0.03 0.41 0.41
13 0.03 0.44 0.44
14 0.03 0.47 0.47
15 0.04 0.51 0.51
16 0.03 0.54 0.54
17 0.03 0.57 0.57
18 0.03 0.60 0.60
19 0.04 0.64 0.64
20 0.03 0.67 0.67
21 0.04 0.71 0.71
22 0.03 0.74 0.74
23 0.03 0.77 0.77
24 0.03 0.80 0.80
25 0.04 0.84 0.84
26 0.04 0.88 0.88
27 0.04 0.92 0.92
28 0.03 0.95 0.95
29 0.03 0.98 0.98
30 0.04 1.0 1.02
31 0.04 1.0 1.06
32 0.03 1.0 1.09
33 0.03 1.0 1.12
34 0.03 1.0 1.15
35 0.03 1.0 1.18
36 0.04 1.0 1.22
37 0.03 1.0 1.25
38 0.03 1.0 1.28
39 0.03 1.0 1.31
40 0.04 1.0 1.35
41 0.04 1.0 1.39
42 0.03 1.0 1.42
43 0.03 1.0 1.45
44 0.03 1.0 1.48
45 0.04 1.0 1.52
46 0.03 1.0 1.55
47 0.04 1.0 1.59
48 0.03 1.0 1.62
49 0.03 1.0 1.65
50 0.03 1.0 1.68
51 0.04 1.0 1.72
52 0.04 1.0 1.76
53 0.03 1.0 1.79
54 0.03 1.0 1.82
55 0.03 1.0 1.85
56 0.03 1.0 1.88
57 0.04 1.0 1.92
58 0.03 1.0 1.95
59 0.03 1.0 1.98
60 0.03 1.0 2.01
61 0.03 1.0 2.04
62 0.03 1.0 2.07
63 0.03 1.0 2.10
64 0.04 1.0 2.14
65 0.03 1.0 2.17
66 0.03 1.0 2.20
67 0.03 1.0 2.23
68 0.03 1.0 2.26
69 0.03 1.0 2.29
70 0.04 1.0 2.33
71 0.03 1.0 2.36
72 0.03 1.0 2.39
73 0.03 1.0 2.42
74 0.03 1.0 2.45
75 0.03 1.0 2.48
76 0.03 1.0 2.51
77 0.04 1.0 2.55
78 0.03 1.0 2.58
79 0.03 1.0 2.61
80 0.03 1.0 2.64
81 0.03 1.0 2.67
82 0.03 1.0 2.70
83 0.04 1.0 2.74
84 0.03 1.0 2.77
85 0.03 1.0 2.80
86 0.04 1.0 2.84
87 0.04 1.0 2.88
88 0.03 1.0 2.91
89 0.03 1.0 2.94
90 0.03 1.0 2.97
91 0.04 1.0 3.01
92 0.03 1.0 3.04
93 0.04 1.0 3.08
94 0.04 1.0 3.12
95 0.03 1.0 3.15
96 0.04 1.0 3.19
97 0.03 1.0 3.22
98 0.04 1.0 3.26
99 0.03 1.0 3.29
100 0.03 1.0 3.32
^^^^^ ^^^^^^
rounded to 2 digs: 1.0 3.3
rounded to 1 dig: 1 3