Elements of Floating-point Arithmetic

140 Pages·2011·2.57 MB·English

by  Sanzheng Qiao

Floating-pointNumbers SourcesofErrors StabilityofanAlgorithm SensitiviyofaProblem Fallacies Summary
Elements of Floating-point Arithmetic
SanzhengQiao
DepartmentofComputingandSoftware
McMasterUniversity
September,2011
Floating-pointNumbers SourcesofErrors StabilityofanAlgorithm SensitiviyofaProblem Fallacies Summary
Outline
1 Floating-pointNumbers
Representations
IEEEFloating-pointStandards
UnderflowandOverflow
Correctly RoundedOperations
2 SourcesofErrors
RoundingError
TruncationError
Discretization Error
3 Stability ofanAlgorithm
4 Sensitiviy ofa Problem
5 Fallacies
Floating-pointNumbers SourcesofErrors StabilityofanAlgorithm SensitiviyofaProblem Fallacies Summary
Outline
1 Floating-pointNumbers
Representations
IEEEFloating-pointStandards
UnderflowandOverflow
Correctly RoundedOperations
2 SourcesofErrors
RoundingError
TruncationError
Discretization Error
3 Stability ofanAlgorithm
4 Sensitiviy ofa Problem
5 Fallacies
Floating-pointNumbers SourcesofErrors StabilityofanAlgorithm SensitiviyofaProblem Fallacies Summary
Two ways of representing floating-point
On paperwe write afloating-pointnumberin theformat:
d .d d βe
1 2 t
± ··· ×
0 < d < β, 0 d < β (i > 1)
1 i
≤
t: precision
β: base(or radix), almostuniversally2,other
commonlyusedbasesare10 and16
e: exponent,integer
Floating-pointNumbers SourcesofErrors StabilityofanAlgorithm SensitiviyofaProblem Fallacies Summary
Two ways of representing floating-point (cont.)
Examples:
1.0 10−1
×
t = 2 (the lastzero counts), β = 10,e = 1
−
Floating-pointNumbers SourcesofErrors StabilityofanAlgorithm SensitiviyofaProblem Fallacies Summary
Two ways of representing floating-point (cont.)
Examples:
1.0 10−1
×
t = 2 (the lastzero counts), β = 10,e = 1
−
1.234 102
×
t = 4,β = 10,e = 2
Floating-pointNumbers SourcesofErrors StabilityofanAlgorithm SensitiviyofaProblem Fallacies Summary
Two ways of representing floating-point (cont.)
Examples:
1.0 10−1
×
t = 2 (the lastzero counts), β = 10,e = 1
−
1.234 102
×
t = 4,β = 10,e = 2
1.10011 2−4
×
t = 6,β = 2 (binary), e = 4
−
Floating-pointNumbers SourcesofErrors StabilityofanAlgorithm SensitiviyofaProblem Fallacies Summary
Two ways of representing floating-point (cont.)
Examples:
1.0 10−1
×
t = 2 (the lastzero counts), β = 10,e = 1
−
1.234 102
×
t = 4,β = 10,e = 2
1.10011 2−4
×
t = 6,β = 2 (binary), e = 4
−
Theprecision t, the baseβ, andtherangeofthe exponente
determinea floating-pointnumbersystem.
Floating-pointNumbers SourcesofErrors StabilityofanAlgorithm SensitiviyofaProblem Fallacies Summary
In memory,a floating-pointnumberis stored in three
consecutive fields:
sign (1 bit)
exponent(dependsonthe range)
fraction (dependsonthe precision)
Floating-pointNumbers SourcesofErrors StabilityofanAlgorithm SensitiviyofaProblem Fallacies Summary
In memory,a floating-pointnumberis stored in three
consecutive fields:
sign (1 bit)
exponent(dependsonthe range)
fraction (dependsonthe precision)
In orderfora memoryrepresentationto beuseful,there must
be astandard.