Round-Off and Truncation Errors

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CHAPTER 4

• Round-Off and Truncation Errors

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Numerical Accuracy

• Truncation error Method dependent
• Errors which result from using an approximation
  rather than an exact procedure
• Round-off error Machine dependent
• Errors which result from not being able to
  adequately represent the true value
• Result from using an approximate number to
  represent exact number

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Taylor Series Expansion

• Construction of finite-difference formula
• Numerical accuracy discretization error

x
a
Base point x a
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Taylor series expansions
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Taylor Series and Remainder

• Taylor series (base point x a)
• Remainder

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Truncation Error

• Taylor series expansion
• Example (higher-order terms truncated)

(xi 0, h x ? xi1 x)
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Power series Polynomials
The function becomes more nonlinear as m increases
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A MATLAB Script

• Filename fun_exp.m

function sum exp(x) Evaluate exponential
function exp(x) by Taylor series expansion
f(x)1 x x2/2! x3/3! xn/n! clear
all x input(enter the value of x ) n
input(enter the order n ) term 1 sum
term for i 1 n term termx/i sum
sum term end
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MATLAB For Loops

• Filename fun_exp2.m

function sum exp(x) Evaluate exponential
function exp(x) by Taylor series expansion
f(x)1 x x2/2! x3/3! xn/n! x
input(enter the value of x ) n input(enter
the order n ) term(1) 1 sum(1)
term(1) for i 1 n term(i1)
term(i)x/i sum(i1) sum(i)
term(i1) end Display the results disp(i
term(i) sum(i)) a 1n1 a term sum
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Truncation Error
n term sum
n term sum
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Truncation Error
n term sum
n term sum
How to reduce error?
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Round-off Errors

• Computers can represent numbers to a finite
  precision
• Most important for real numbers - integer math
  can be exact, but limited
• How do computers represent numbers?
• Binary representation of the integers and real
  numbers in computer memory

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32 bits (23, 8, 1)
28 256
MATLAB uses double precision
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Order of operation
Addition problem
exact result
with 3-digit arithmetic
Round-off error
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Cancellation error
If b is large, r is close to b
Difference of two numbers very close to each
other ? potential for greater error!
Rationalize
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Try b 97
(r 96.9794)
x2
(3 sig. figs.)
exact 0.01031 standard 0.01050 rationalized 0.
01031
Corresponding to cancellation, critical
arithmetic
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Significant Figures
48.9 mph? 48.95 mph?
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Significant Digits

• The places which can be used with confidence
• 32-bit machine 7 significant digits
• 64-bit machine 17 significant digits
• Double precision reduce round-off error, but
  increase CPU time

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False Significant Figures
3.25/1.96 1.65816326530162... (from
MATLAB) But in practice only report 1.65
(chopping) or 1.66 (rounding)! Why??
Because we dont know what is beyond the second
decimal place
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Accuracy and precision

• Accuracy - How closely a measured or computed
  value agrees with the true value
• Precision - How closely individual measured or
  computed values agree with each other
• Accuracy is getting all your shots near the
  target.
• Precision is getting them close together.

More Accurate
More Precise
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Numerical Errors
The difference between the true value and the
approximation
Approximation true value true error Et
true value ? approximation x ? x or in
percent
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Approximate Error

• But the true value is not known
• If we knew it, we wouldnt have a problem
• Use approximate error

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Number Systems

• Base-10 (Decimal) 0,1,2,3,4,5,6,7,8,9
• Base-8 (Octal) 0,1,2,3,4,5,6,7
• Base-2 (Binary) 0,1 off/on, close/open,
  negative/positive charge
• Other non-decimal systems
• 1 lb 16 oz, 1 ft 12 in, ½, ¼, ..

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Decimal System (base 10)
Binary System (base 2)
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Integer Representation

• Signed magnitude method
• Use the first bit of a word to indicate the sign
  0 negative (off), 1 positive (on)
• Remaining bits are used to store a number

1 0 1 0 0 1 0 1 1 0
Sign Number
off / on, close / open, negative / positive
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Integer Representation

• 8-bit word
• /- 0000000 are the same, therefore we may use
  -0 to represent -128
• Total numbers 28 256 (-128 ?127)

Sign Number

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Integer Representation

• 16-bit word
• Range -32,768 to 32,767
• Overflow gt 32,767 (cannot represent 43,000 AM
  students)
• Underflow lt -32,768 (magnitude too large)
• 32-bit word
• Range -2,147,483,648 to 2,147,483,647
• 9 significant digits
• Overflow world population ?6 billion
• Underflow budget deficit -100 billion

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Integer Operations

• Integer arithmetic can be exact as long as you
  don't get remainders in division
• 7/2 3 in integer math
• or overflow the maximum integer
• For a 8-bit computer max 128 (or -127)
• So 123 45 overflow
• and -74 2 underflow

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Floating-Point Representation

• Real numbers (also called floating-point numbers)
  are represented differently
• For fraction or very large numbers
• Store as
• sign is 1 or 0 for negative or positive
• exponent is maximum value (positive or negative)
  of base
• mantissa contains significant digits

sign signed exponent mantissa
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Floating-Point Representation

sign of number
signed exponent mantissa

• m mantissa
• B Base of the number system
• e signed exponent
• Note the mantissa is usually normalized if the
  leading digit is zero

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Integer representation
Floating-point number representation
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Decimal Representation

• 8-bit word

sign signed exponent number

10951467 (base B 10) mantissa m
-(110-1 410-2 610-3 710-4 )
-0.1467 signed exponent e (9101 5100)
95
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Floating-Point Representation

• 8-bit word (without normalization)

sign signed exponent number

01110101 (base B 2) mantissa m (02-1
12-2 02-3 12-4 ) 5/16 signed exponent
e - (121 120) -3
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Normalization
(Less accurate) (Normalization)

• Remove the leading zero by lowering the exponent
  (d1 1 for all numbers)
• if m lt 1/2, multiply by 2 to remove the leading 0
• floating-point allow fractions and very large
  numbers to be represented, but take up more
  memory and CPU time

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Binary Representation

• 8-bit word (with normalization)

sign signed exponent number

10111001 (base B 2) mantissa m -(12-1
02-2 02-3 12-4 ) -9/16 signed
exponent e (121 120) 3
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Single Precision

• A real variable (number) is stored in four words,
  or 32 bits (64 bits for Supercomputers)
• bit (binary digit) 0 or 1
• byte 4 bits, 24 16 possible values
• word 2 bytes 8 bits, 28 256 possible values

23 for the digits 32 bits 8
for the signed exponent 1 for
the sign
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Double Precision

• A real variable is stored in eight words, or 64
  bits
• 16 words, 128 bits for supercomputers
• signed exponent ? 210 ? 1024

52 for the digits 64 bits 11
for the signed exponent 1 for
the sign
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Round-off Errors

• Floating point characteristics contribute to
  round-off error (limited bits for storage)
• Limited range of quantities can be represented
• A finite number of quantities can be represented
• The interval between numbers increases as the
  numbers grow
• Example - three significant digits

0.0100 0.0101 0.0102 0.0999 (0.0001
increment) 0.100 0.101 0.102 .
0.999 (0.001 increment) 1.00 1.01
1.02 . 9.99 (0.01 increment)
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MATLAB

• Finite number of real quantities (integers, real
  numbers or text) can be represented
• For 8-bit, 28 256 quantities
• For 16-bit, 216 65536 quantities
• MATLAB uses double precision
• 4 bytes 64 bits
• more than 1019 (264) quantities