Download Floating-point Arithmetic

Chapter 2
Floating-point
Computation
Prof. Chuan-Ming Liu
MCSE Lab, NTUT
TAWIAN
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Outline
• Positional Number System
• Floating-point Arithmetic
• Condition of Problems and Stability
of Algorithms
• Ill-Conditioned Problems
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Positional Number System
•
• Example:
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Positional Number System
•
• Example: Decimal system: (0.4213)10
numbers represented
real-world number set
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Positional Number System
• Conversion between number systems
– Example 1: binary to decimal
– Consider a polynomial
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Positional Number System
– Horner’s rule for polynomial:
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Positional Number System
– Example 2: decimal to binary
– Two methods:
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Positional Number System
• Method 1.
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Positional Number System
• Method 2.
2
2
2
418
0
0
209
1
10
0
010
0
0010
0
00010
104
2
52
2
26
2
13
2
5
2
3
1
100010
0
0100010
1
110100010
1
The whole process is the “reverse” of the Horner’s rule
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Positional Number System
– Not all decimal fractions are exactly
represented in the binary system.
– Example 1: (decimal to binary)
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Positional Number System
– Example 2: (binary to decimal)
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Positional Number System
– Terminating an infinite binary fraction.
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Positional Number System
• Fixed-Point Notation:
– Present a number by the positions
– E.g. (101.011)2=1*22+0+1*20+0*2-1+2-2+2-3
– Difficulty occurs when presenting very
large or very small numbers.
– The difficulty can be overcome by
scientific notation.
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Positional Number System
– E.g. 0.00000056 = 5.6 * 10-7
scale factor
Significant digits
– One key part to consider with is how
many digits to retain during computation.
– E.g.
• 1.23 + 15.6 = 16.83
• if using 3-digit computation
• 1.23 + 15.6 = 16.8
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Floating-point Arithmetic
•
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Floating-point Arithmetic
• In above floating-point representatoin
– δ1≠0 implies it is a normalized floatingpoint number
– e is the exponent.
– δ1 δ2…δt is the mantissa (or fraction).
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Floating-point Arithmetic
• Usually,
• The floating-point numbers are not
uniformly distributed on the real line.
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Floating-point Arithmetic
•
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Floating-point Arithmetic
– The floating-point numbers in a given
group βe are
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Floating-point Arithmetic
– The number of floating-point numbers in
a group is
• By counting
(+ or -)
t-1
• By spacing
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Floating-point Arithmetic
– The total number of floating-point
numbers in the system is
• The number with largest magnitude is
• The number with smallest magnitude is
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Floating-point Arithmetic
• Example:
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Floating-point Arithmetic
• How to represent an arbitrary real
number in the floating-point system
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Floating-point Arithmetic
– chopping
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Floating-point Arithmetic
– rounding
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Floating-point Arithmetic
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Floating-point Arithmetic
– hence,
–
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Floating-point Arithmetic
• Example:
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Floating-point Arithmetic
• Example:
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Floating-point Arithmetic
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Floating-point Arithmetic
Conclusion:
largest magnitude number add last.
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Condition of Problems and
Stability of Algorithms
• Poor accuracy depends on either the
problem or the algorithm.
– The problem can be ill-conditioned :
small perturbations in the input could
lead to large perturbations in the output.
– The algorithm may be poorly designed:
unstable.
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Condition of Problems and
Stability of Algorithms
– Example: (using 3-digit with chopping)
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Condition of Problems and
Stability of Algorithms
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Condition of Problems and
Stability of Algorithms
– Compare the result with the exact
solution!!
The error comes from that we cancel all
the significant digits and have a result
heavily contaminated with rounding
errors.
– Example: (using 4-digit rounding)
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Condition of Problems and
Stability of Algorithms
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Condition of Problems and
Stability of Algorithms
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Condition of Problems and
Stability of Algorithms
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Condition of Problems and
Stability of Algorithms
• Cancellation
– Error-producing involves the
cancellation of significant digits due to
the substraction of nearly equal
numbers.
– Recall how to avoid cancellation we can
use rationalization to remedy the error
for the smaller root.
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Condition of Problems and
Stability of Algorithms
– The smaller root can be expressed
alternatively by
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Condition of Problems and
Stability of Algorithms
– No method can remedy the solution for
the previous example. (using chopping)
– A stable algorithm for computing the
roots of a quadratic ax2+bx+c=0 may be
outlined as
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Ill-Conditioned Problems
• the ill-conditioning of a problem is
measured by the condition number K.
(the larger K is the more illconditioned is the problem.)
•
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Ill-Conditioned Problems
• From Taylor series,
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Ill-Conditioned Problems
• Example:
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Ill-Conditioned Problems
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Ill-Conditioned Problems
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Ill-Conditioned Problems
• Ill-conditioned problem
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Ill-Conditioned Problems
• A stable algorithm for an illconditioned problem is an algorithm
for which the computed solution is
near the exact solution of another
slightly perturbed problem.
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