2.2 Errors

1
2.2 Errors
2
Why Study Errors First?

• Nearly all our modeling is done on digital
  computers (aside what would a non-digital analog
  computer look like?)

3
Analog Computer for Fitting a Line to a Set of
Points
4
Why Study Errors First?

• Nearly all our modeling is done on digital
  computers (aside what would a non-digital analog
  computer look like?)
• A better set of terms is discrete vs. continuous
  (recall the discrete/continuous distinction from
  1.2).
• Discrete devices have finite precision - we
  cannot represent all the different values wed
  like.
• Finite precision leads to errors

5
Finite Precision with Bits

• Bit value of 0 or 1 (aside why base 2 ?)
• With n bits, we can distinguish 2n different
  values
• E.g., with three bits we get eight possible
  values.
• These values can correspond to anything we want
  to represent

000 0 A Greg A 001 1 B Wade B 010 2 C Sam C 01
1 3 D Jordan I 100 4 E Mark E 101 5 F Jason
F 110 6 G Gaurav G 111 7 H Alex H
6
Floating-Point Numbers

• History origins of computers in finance
  (banking) and military (physics).
• Financial calculations are (usually) fixed-point
  the decimal place is fixed in one position, two
  digits from right 129.95, 0.59, etc.
• In physical calculations, we need the decimal
  place to float 3.14159, 0.333, 6.0221415
  1023 , etc.

7
Exponential Notation

• Format Mantissa X 10Exponent, or a X 10n
• Abbreviated a e n or a E n for example,
  6.0221415e23
• Mantissa (a.k.a. significand, a.k.a. fractional
  part) is a floating-point number exponent is an
  integer.
• So someone has to decide the precision
• IEEE 754 standard with 64-bit numbers, use 52
  bits for mantissa, 11 for exponent, 1 for sign (
  or -)
• Rough conversion 3 bits ? 1 decimal digit (why?)
  E.g., 53 bits ? 16 decimal digits of precision

8
Normalized Notation

• Normalization is a common operation e.g.,
  percentage normalizes everything to 100
• 2/4 75/50 50
• Scientists usually prefer to normalize to 1
• 2/4 75/150 0.5
• So a normalized number in exponential notation
  has the decimal place immediately before the
  first nonzero digit
• .314159e1, .60221415e24, etc.
• Significant digits of a normalized floating-point
  number are all digits except leading zeros.

9
Precision and Magnitude

• Precision is the number of significant digits.
• Magnitude is the power of 10.

. 3 1 4 1 5 9 e 1
6 digits of precision
Magnitude 1
10
Quick Review Question 1

• For the number 0.0004500, give
• the significand in normalized exponenent notation
• the magnitude in normalized exponenent notation
• the precision

11
Absolute and Relative Error

• If correct is the answer and result is the result
  obtained, then

absolute error correct - result
relative error (absolute error)
correct
(absolute error)
X 100
correct
12
Truncation

• To truncate a normalized number to k significant
  digits, eliminate all digits of the significand
  beyond the kth digit.
• E.g., .314159 truncated to 5 significant digits
  .?????

13
Truncation

• To truncate a normalized number to k significant
  digits, eliminate all digits of the significand
  beyond the kth digit.
• E.g., .314159 truncated to 5 significant digits
  .31415
• Relative error correct - result

X 100
correct
14
Truncation

• To truncate a normalized number to k significant
  digits, eliminate all digits of the significand
  beyond the kth digit.
• E.g., .314159 truncated to 5 significant digits
  .31415
• Relative error 3.14159 - 31415

X 100
.31415
15
Truncation

• To truncate a normalized number to k significant
  digits, eliminate all digits of the significand
  beyond the kth digit.
• E.g., .314159 truncated to 5 significant digits
  .31415
• Relative error .00009

X 100
.31415
16
Truncation

• To truncate a normalized number to k significant
  digits, eliminate all digits of the significand
  beyond the kth digit.
• E.g., .314159 truncated to 5 significant digits
  .31415
• Relative error 0.028649

17
Rounding

• To round a normalized number to precision k,
  consider the (k1)th significant digit d. If it
  is less than 5, round down the normalized number
  by truncating the significand to k significant
  digits. If d is greater than or equal to 5,
  round up the normalized number by truncating the
  significand to k significant digits and then
  adding 1 to the kth significant digit of the
  significand, carrying as necessary to digits on
  the left.

18
Quick Review Question 3

• Round each of the following so that the
  significand has a precision of 2
• 0.93742e-5
• 0.93472e-5
• 0.93572e-5

19
Assignment

• An assignment statement causes the computer to
  store the value of an expression in a memory
  location associated with a variable. In most
  programming languages the assignment statement
  has a format similar to the following, with the
  expression always appearing on the right amd the
  varoable getting the value always being on the
  left of an assignment operator, here an equal
  sign
• variable expression

20
Assignment

• An assignment statement causes the computer to
  store the value of an expression in a memory
  location associated with a variable. In most
  programming languages the assignment statement
  has a format similar to the following, with the
  expression always appearing on the right amd the
  varoable getting the value always being on the
  left of an assignment operator, here an equal
  sign
• variable expression

R-value
L-value
21
Assignment

• An assignment statement causes the computer to
  store the value of an expression in a memory
  location associated with a variable. In most
  programming languages the assignment statement
  has a format similar to the following, with the
  expression always appearing on the right amd the
  varoable getting the value always being on the
  left of an assignment operator, here an equal
  sign
• X X 1

22
Round-off Error

• Round-off error is the problem of not having
  enough bits (or digits) to store an entire
  floating-point number and approximating the
  result to the nearest number that can be
  represented.
• E.g., 1/3 0.333 ? 0.3 ? 0.33 ? 0.333 ? etc.

23
Avoidnig Round-off Error

• Some values (1/3) will always result in roundoff
  however, we can help reduce error
• When adding numbers whose magnitudes are
  drastically different, accumulate smaller number
  before combining them with larger ones.
• When multiplying and dividing, perform all
  multiplcations in the numerator before dividing
  by the denominator.

24
Avoiding Round-off Error

• E.g. consider computing (x/y)z on a machine with
  three significant digits of precision, where
• x 2.41 y 9.75 z 1.54
• (x / y)z (2.41 / 9.75)(1.54) (0.247)(1.54)
  0.380
• But (x / y)z (xz) / y
• (xz) / y ((2.41) )(1.54)) / 9.75 3.71 / 9.75
  0.381

25
Overflow and Underflow

• Overflow is an error condition that occurs when
  there are not enough bits to express a value in a
  computer.
• Underflow is an error condition that occurs when
  the result of a computation is too small for the
  computer to represent.

26
Overflow and Underflow in Excel
27
Looping and Error Propagation

• A loop is a segment of code computer
  instructions that is executed repeatedly.
• Accumulating values in a loop (repeatedly) can
  produce error
• E.g., Patriot Missile failure in 1991 Gulf War

28
Patriot Missile Failure

• Patriots internal clock measured time in 1/10 of
  a second.
• Each tick of the clock added 1/10 s to current
  time.
• But 1/10 cannot be represented in a finite number
  of bits (just as 1/3 cannot be represented in a
  finite number of decimal digits)

29
Representing Fractions in Binary

• Consider ordinary Base-10 notation what does
  e.g. 1205.91410 mean?
• 1 2 0 5 . 9 1 4
• 103 102 101 100 . 10-1 10-2 10-3

• Binary works the same way e.g., what is 101.012?
• 1 0 1 . 0 1 22 21
  20 . 2-1 2-2
• 4 1 1/4 5.025

30
Patriot Missile Failure

• Each one-tenth increment produced an error of
  about .000000095 seconds.
• After ten hours of running Patriots computer

• Scud travels 1,676 meters / sec. So error

1676 meters 0.34 sec 1 sec
570 meters

• Result Patriot missed Scud , 28 Americans died

31
Error in the Text!

• During that third of a second, a Scud flew about
  1,676 meters.
• Shiflet Shiflet p. 24
• A Scud travels at about 1,676 meters per second
• http//www.ima.umn.edu/arnold/disasters/patriot
  .html

(FYI speed of sound at sea level 340 m / s)
32
Looping and Error Propagation in Excel
Highlight click corner drag downward
See apparent uniform sequence
Double click for actual values
Enter values
Courtesy of Bob Panoff
33
Avoiding Error Propagation in Excel
Enter first value highlight rest
Do Fill/Series
Courtesy of Bob Panoff