Numeric Processing

1
Numeric Processing

• Chapter 6,
• Exploring the Digital Domain

2
Digital Number Representations

• Integers
• infinite discrete subset of the number line
• are represented with a limited range
• Decimal numbers (real numbers)
• infinite and continuous
• are represented with limited range and limited
  precision

3
Integer Storage

• All integers between two values (one negative and
  one positive) are stored with exact precision
• The specific values marking the range limits
  depend on the particular computer system being
  used
• If calculations with integers give rise to
  numbers outside the allowable range, we say that
  an integer overflow error has occurred

4
Real (Decimal) Number Storage

• Real numbers are stored in floating point
  representation
• a sign
• an exponent
• a mantissa (normalized decimal fraction)
• no digits to the left of the decimal
• first digit to the right of the decimal is
  nonzero
• Limited precision because most real numbers have
  an infinite decimal expansion (this holds no
  matter what number base is used in the
  representation)

5
Real Number StorageLimited Range and Precision

• There are three categories of numbers left out
  when floating point representation is used
• numbers out of range because their absolute value
  is too large (similar to integer overflow)
• numbers out of range because their absolute value
  is too small (numbers too near zero to be stored
  given the precision available
• numbers whose binary representations require
  either an infinite number of binary digits or
  more binary digits than the bits available

6
Real Number StorageLimited Range and Precision
Illustrated
7
Limited Range and PrecisionSome Consequences

• Limited range will invalidate certain
  calculations
• If integers are involved, this can often be
  avoided by switching to real numbers
• For real number calculations, this problem arises
  infrequently and in those cases can sometimes be
  handled by special methods. It is not a common
  occurrence in non-scientific work.
• Limited precision for real numbers is very
  pervasive
• Assume that most decimal calculations will, in
  fact, be in error!
• Evaluate and use computer calculations with this
  in mind

8
Social ThemesRisks in Numerical Computing

• Almost all computer calculations involve roundoff
  error (limited precision error)
• If not monitored and planned for carefully, such
  errors can lead to unexpected and catastrophic
  results
• Arianne 5 Rocket Failure
• Patriot Missile Failure during Gulf War

9
Software for Numerical Work

• Software Libraries
• Spreadsheets
• Mathematical Software
• symbolic manipulation
• data analysis
• data visualization

10
Spreadsheets Brief History

• First spreadsheets appeared in early 1980s
• ran on personal computers
• helped popularize personal computers
• Basic Organization Hasnt Changed
• computations organized on a two-dimensional
  worksheet
• both built-in and user-supplied formulas used to
  facilitate computation

11
Spreadsheets Basic Features

• Worksheet Organization
• rectangular grid of cells
• cells are identified by the row (indicated by an
  integer) and column (indicated by a letter) in
  which they appear
• Entering Data
• one cell is active at a time (called the current
  cell)
• a separate data entry bar is associated with the
  current cell
• text, numerical data, dates, and formulas can be
  entered into the current cell through the entry
  bar

12
Worksheet Organization Illustrated
13
Spreadsheets Basic Features (contd)

• Formatting Data
• variety of formats depending on the type of data
• data format is associated with the cell and can
  be changed later
• Using Formulas in a Worksheet
• cell addresses (column, row) can be used like
  variable names in formulas
• formulas begin with an or other special symbol
  (like _at_)

14
Using Spreadsheet FormulasAn Example
15
Spreadsheets Basic Features (contd)

• Copying (Replicating) Formulas
• formulas can be copied to perform repetitive
  calculations
• especially useful when similar calculations take
  place on a group (block,row, column) of
  contiguous data
• Cell Referencing
• cell references will be adjusted automatically
  when formulas are copied if relative cell
  addressing is used in formulas
• cell references will remain the same if absolute
  cell addressing is used in formulas
• relative cell addressing is the default
  addressing scheme

16
Copying Spreadsheet FormulasAn Example
The formula in cell B12 has been copied to cells
C12 and D12. Note how the cell references are
automatically adjusted.
17
SpreadsheetsAdditional Features

• Using absolute references and problem parameters
• Using built-in functions
• Using logical functions
• Displaying data in charts
• Spreadsheets as decision support tools

18
Spreadsheet ModelsUsing Problem Parameters

• Most spreadsheet models will require
  modifications over time as underlying assumptions
  and important problem values change
• Good spreadsheet design will minimize the danger
  of making errors when updating spreadsheet models
• Problem parameters are important problem values
  that are subject to change over time
• Separating problem parameters, placing them in
  clearly identified cells, then referring to them
  by absolute references in formulas within the
  model makes changing them relatively error-proof

19
Spreadsheet ModelsDecision Support

• Modeling problem parameters as clearly identified
  separate entities enables convenient what if
  analysis in spreadsheet models
• What if analysis involves observing modified
  calculations and results when problem parameters
  change. In other words, the model is examined in
  response to questions of the form What if
  changes to ?

20
Spreadsheet ModelsUsing Built-in Functions

• Built-in functions act on arguments to produce
  resulting values
• A user of a built-in function need not know all
  the details of how the function does its
  calculation
• The user needs to understand only what input
  arguments are required and what type of result is
  produced
• An extensive library of financial and statistical
  functions adds powerful problem-solving
  capabilities for the average user

21
Spreadsheet ModelsUsing Built-in Logical
Functions

• Logical functions allow spreadsheet models to
  make decisions during calculations
• Such processing is called conditional processing
  and is a fundamental property of programming
  languages
• For example, the IF function acts on a logical
  (or Boolean) expression to take one of two
  actions, as shown here

22
Spreadsheet ModelsDisplaying Data in Charts

• Graphical display and summary of data is often
  easier to interpret than the raw data itself
• Spreadsheets typically provide several types of
  charts
• bar charts
• pie charts

23
Bar ChartAn Example
24
Pie ChartAn Example
25
Summary

• Integers are stored with limited range real
  numbers (decimals) with limited range and limited
  precision
• Almost all decimal calculations involve roundoff
  errors because of limited precision
• Spreadsheets provide powerful, yet easy to use,
  calculational environments
• formulas and relative referencing provide for
  easy calculation extension
• absolute referencing and problem parameters
  enable modeling
• built-in functions (arithmetic and logical)
• displaying data in charts
• decision support tool