Numerical Methods Contents - SAM

B e min−1
0
spacing B e min−m
spacing B e min−m+1
Gap partly filled with non-normalized numbers
Example 2.4.2 (IEEE standard 754 for machine numbers).
spacing B e min−m+2
→ [30], → link
1 >> format long ;
Code 2.4.6: input errors and roundoff errors
2 >> a = 4 / 3 ; b = a−1; c = 3∗b ; e = 1−c
3 e = 2.220446049250313e−16
4 >> a = 1012/113; b = a−9; c = 113∗b ; e = 5+c
5 e = 6.750155989720952e−14
6 >> a = 83810206/6789; b = a−12345; c = 6789∗b ; e = c−1
7 e = −1.607986632734537e−09
No surprise:
for modern computers B = 2 (binary system)
single precision : m = 24 ∗ ,E ∈ {−125,...,128}
➣ 4 bytes
double precision : m = 53 ∗ ,E ∈ {−1021,...,1024} ➣ 8 bytes
∗: including bit indicating sign
Remark 2.4.3 (Special cases in IEEE standard).
1 >> x = exp(1000) , y = 3 / x , z = x∗sin ( pi ) , w = x∗log ( 1 )
2 x = I n f
3 y = 0
4 z = I n f
w = NaN
E = e max , M ≠ 0 ˆ= NaN = Not a number → exception
E = e max , M = 0 ˆ= Inf = Infinity → overflow
E = 0 ˆ= Non-normalized numbers → underflow
E = 0, M = 0 ˆ= number 0
✸
Ôº½¼ ¾º
!
△
Notation: floating point realization of ⋆ ∈ {+, −, ·, /}: ˜⋆
correct rounding:
For any reasonable M:
Realization of ˜+, ˜−,˜·,˜/:
rd(x) = arg min |x − ˜x|
˜x∈M
(if non-unique, round to larger (in modulus) ˜x ∈ M: “rounding up”)
small relative rounding error
∃eps ≪ 1:
| rd(x) − x|
|x|
Ôº½½½ ¾º
≤ eps ∀x ∈ R . (2.4.1)
⋆ ∈ {+, −, ·,/}: x˜⋆ y := rd(x ⋆ y) (2.4.2)
✸
Example 2.4.4 (Characteristic parameters of IEEE floating point numbers (double precision)).
☞
MATLAB always uses double precision
1 >> format hex ; realmin , format long ; realmin
2 ans = 0010000000000000
3 ans = 2.225073858507201e−308
4 >> format hex ; realmax , format long ; realmax
5 ans = 7 f e f f f f f f f f f f f f f
6 ans = 1.797693134862316e+308
✬
Assumption 2.4.2 (“Axiom” of roundoff analysis).
There is a small positive number eps, the machine precision, such that for the elementary
arithmetic operations ⋆ ∈ {+, −, ·, /} and “hard-wired” functions ∗ f ∈ {exp, sin, cos, log, ...}
holds
with |δ| < eps.
✫
x˜⋆ y = (x ⋆ y)(1 + δ) , ˜f(x) = f(x)(1 + δ) ∀x,y ∈ M ,
∗: this is an ideal, which may not be accomplished even by modern CPUs.
✩
✪
Example 2.4.5 (Input errors and roundoff errors).
✸
Ôº½½¼ ¾º
relative roundoff errors of elementary steps in a program bounded by machine precision !
Ôº½½¾ ¾º
Example 2.4.7 (Machine precision for MATLAB). (CPU Intel Pentium)