Numerical Methods Contents - SAM

numerically equivalent ˆ= same result when executed with the same machine arithmetic
This introduces a new and important aspect in the study of numerical algorithms!
The above statement means that whenever we study the impact of roundoff errors on LUfactorization
it is safe to consider only the basic version without pivoting, because we can
always assume that row swaps have been conducted beforehand.
“Computers use floating point numbers (scientific notation)”
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Example 2.4.1 (Decimal floating point numbers).
3-digit normalized decimal floating point numbers:
2.4 Supplement: Machine Arithmetic
valid: 0.723 · 10 2 , 0.100 · 10 −20 , −0.801 · 10 5
invalid: 0.033 · 10 2 , 1.333 · 10 −4 , −0.002 · 10 3
A very detailed exposition and in-depth discussion of the all the material in this section can be found
in [24]:
• [24, Ch. 1]: excellent collection of examples concerning the impact of roundoff errors.
• [24, Ch. 2]: floating point arithmetic, see Def. 2.4.1 below and the remarks following it.
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General form of m-digit normalized decimal floating point number:
never = 0 !
x = ± 0 . 1 1 1 1 1 ... 1 1 · 10
E
} {{ }
m digits of mantissa exponent ∈ Z
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Computer = finite automaton ➢ can handle only finitely many numbers, not R
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machine numbers, set M
Essential property:
M is a discrete subset of R
M not closed under elementary arithmetic operations +, −, ·,/.
roundoff errors (ger.: Rundungsfehler) are inevitable
The impact of roundoff means that mathematical identities may not carry over to the computational
realm. Putting it bluntly,
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Computers cannot compute “properly” !
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Of course,
computers are restricted to a finite range of exponents.
Definition 2.4.1 (Machine numbers).
Given ☞ basis B ∈ N \ {1},
☞ exponent range {e min , ...,e max }, e min ,e max ∈ Z, e min < e max ,
☞ number m ∈ N of digits (for mantissa),
the corresponding set of machine numbers is
M := {d · B E : d = i · B −m , i = B m−1 ,...,B m − 1, E ∈ {e min ,...,e max }}
never = 0 !
1 1 . .. 1 1
} {{ }
digits for exponent
machine number ∈ M : x = ± 0 . 1 1 1 1 1 . .. 1 1 · B
} {{ }
m digits for mantissa
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numerical computations
≠
analysis
linear algebra
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Largest machine number (in modulus) : x max = (1 − B −m ) · B e max
Smallest machine number (in modulus) : x min = B −1 · B e min
Distribution of machine numbers:
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