Numerical Methods,

1
Numerical Methods in Scientific Computation

• Lecture 2
• Programming and Software
• Introduction to error analysis

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Packages vs. Programming

• Packages
• MATLAB
• Excel
• Mathematica
• Maple
• Packages do the work for you
• Most offer an interactive environment

3
MATLAB

• Software product from The MathWorks, Inc.
• Originally focused on matrix manipulations
• Interactive tool to do numerical functions, and
  visualization
• Commands can be saved into user scripts call and
  m-file
• Graphics and graphical user interfaces (GUI) are
  built into program

4
MATLAB

• Base product is extended with a variety of
  toolboxes
• Optimization
• Statistics
• Curve fitting
• Image processing

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MATLAB

• Example Find roots of
• gtgt p 1 -6 -72 -27
• gtgt r roots (p)
• gtgt r
• 12.1229
• -5.7345
• -0.3884

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MATLAB

• Advantages
• Large library of functions to evaluate a wide
  variety of numerical problems
• Interactive tool allows immediate evaluation of
  results
• Disadvantages
• Expensive
• Not as fast as C/C/FORTRAN code

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Programming language concepts

• Interpreted Languages
• MATLAB
• Usually integrated with an interactive GUI
• Allows quick evaluation of algorithms
• Ideal for debugging and one-time analysis and in
  cases where high speed is not critical

8
Programming language concepts

• Compiled Languages
• FORTRAN, C, C
• Source code is created with an editor as a plain
  text file
• Programs then needs to be compiled (converted
  from source code to machine instructions with
  relative addresses and undefined external
  routines still needed).
• The compiled routines (called object modules)
  need then to be linked with system libraries.
• Faster execution than interpreted languages

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Programming concepts

• Data representation
• Constants, variables, type declarations
• Data organization and structures
• Arrays, lists, trees, etc.

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Programming concepts

• Mathematical formulas
• Assignment, priority rules
• Input/Output
• Stdin / Stdout, files, GUI

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Programming concepts

• Structured programming
• Set of rules that prescribe good programming
  style
• Single entry point and exit point to all control
  structures
• Flexible enough to allow creativity and personal
  expression
• Easy to debug, test and update

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Structured programming

• Control structures
• Sequence
• Selection
• Repetition
• Computer code is clearer and easier to follow

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Algorithm expression

• Flowcharts
• Visual representation
• Useful in planning
• Pseudocode
• Simple representation of an algorithm
• Basic program constructs without being
  language-specific
• Easy to modify and share with others

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Flowcharts
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Control structures

• Sequence
• Implement one instruction at a time

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Control structures

• Selection
• Branching
• IF/THEN/ELSE
• CASE/ELSE

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Control structures
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Control structures
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Control structures

• Repetition
• DOEXIT loops (break loop)
• Similar to C while loop
• DOFOR loop (count-controlled)

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Control structures

• pretest
• posttest
• midtest

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Control structures
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Modular programming

• Break complicated tasks into manageable parts
• Independent and self-contained
• Well-defined modules with a single entry point
  and single exit point
• Always a good idea to return a value for status
  or error code

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Modular programming

• Makes logic easier to understand and digest
• Allows black-box development of modules
• Requires well-defined interfaces with no side
  effects
• Isolates errors
• Allows for reuse of modules and development of
  libraries

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Software engineering approach

• Specification A clear statement of the problem,
  the requirements, and any special parameters of
  operation
• Algorithm/Design A flow chart or pseudo code
  representation of how exactly how will the
  problem be solved.

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Software engineering approach

• Implementation Breaking the algorithm into
  manageable pieces that can be coded into the
  language of choice, and putting all the pieces
  together to solve the problem.
• Verification Checking that the implementation
  solves the original specification. In numerical
  problems, this is difficult because most of the
  time you dont know what the correct answer is.

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Example case

• How do you solve the sum of integers problem?
• Simplest
• sum 1 2 3 4 5 .... N
• Coded specifically for specific values of N.

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Possible solutions

• Intermediate solution
• Does not require much thought, takes advantage of
  looping ability of most languages
• C/C
• MATLAB

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Possible solutions

• Analytical solution
• Requires some thought about the sequence remember
  back to one of your math classes.

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Verification of algorithm

• What can fail in the above algorithms. (To know
  all the possible failure modes requires knowledge
  of how computers work). Some examples of failures
  are
• For (b) This algorithm is fairly robust.
• But, when N is large execution will be slow
  compared to (c)

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Algorithm (c)

• This is most common algorithm for this type of
  problem, and it has many potential failure modes.
  For example
• (c.1) What if N is less than zero?
• Still returns an answer but not what would be
  expected. (What happens in (b) if N is
  negative?).

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Algorithm (c)

• (c.2) In which order will the multiplication and
  division be done.
• For all values of N, either N or N1 will be even
  and therefore N(N1) will always be even but
  what if the division is done first? Algorithm
  will work half of the time.
• If division is done first and N1 is odd, the
  algorithm will return a result but it will not be
  correct. This is a bug. For verification, it
  means the algorithm works sometimes but not
  always.

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Algorithm (c)

• (c.3) What if N is very large?
• What is the maximum value N(N1) can have?
• There are maximum values that integers can be
  represented in a computer and we may overflow.
  What happens then? Can we code this to handle
  large values of N?

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Verification

• By breaking the program into small modules, each
  of which can be checked, the sum of the parts is
  likely to be correct but not always..
• Problems can be that the program only works some
  of the time or it may not work on all platforms.
• The critical issue to realize all possible cases
  that might occur.

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Introduction to error analysis

• Error is the difference between the exact
  solution and a numerical method solution
• In most cases, you dont know what the exact
  solution is, so how do you calculate the error
• Error analysis is the process of predicting what
  the error will be even if you dont know what the
  exact solution
• Errors can also be introduced by the fact that
  the numerical algorithm has been implemented on a
  computer

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Significant digits

• Can a number be used with confidence?
• How accurate is the number?
• How many digits of the number do we trust?

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Significant digits

• Can the speed be estimated to one decimal place?

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Significant digits

• The significant digits of a number are those that
  can be used with confidence
• The digits that are known with certainty plus one
  estimated digit
• zeros are not always significant digits
• 0.0001845, 0.001845, 0.01845
• 4.53x104, 4.530x104, 4.5300x104
• Exact numbers vs. measured numbers
• Exact numbers have an infinite number of
  significant digits
• ? is an exact number but usually subject to
  round-off

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Significant digits
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Accuracy and precision

• Accuracy is how close a computed or measured
  value is to the true value
• Precision is how close individual computed or
  measured values agree with each other
• Reproducibility
• Inaccuracy/Bias vs. Imprecision/Uncertainty
• Inaccuracy systematic deviation from the truth
• imprecision magnitude of the scatter

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Accuracy and precision
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Accuracy and precision

• The level of accuracy and precision required
  depend on the problem
• Error to represent both the inaccuracy and
  imprecision of predictions

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Error definitions

• Two general types of errors
• Truncation errors due to approximations of
  exact mathematical functions
• Round-off errors due to limited significant
  digit representation of exact numbers
• Et true value approximation

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Error definitions

• Et does not capture the order of magnitude of
  error
• 1V error probably doesnt matter if youre
  measuring line voltage, but it does matter if
  youre measuring the voltage supply to a VLSI
  chip
• Therefore, its better to normalize the error
  relative to the value

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Error definitions

• Example
• Line voltage
• Chip supply voltage

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Error definitions

• What if we dont know the true value?
• Use an approximation of the true value
• How do we calculate the approximate error?
• Use an iterative approach
• Approximate error current approximation
  previous approximation
• Assumes that the iteration will converge

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Error Definitions

• For most problems, we are interested in keeping
  the error less than specified error tolerance
• How many iterations do you do, before youre
  satisfied that the result is correct to at least
  n significant digits?

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Example

• Infinite series expansion of ex
• As we add terms to the expansion, the expression
  becomes more exact
• Using this series expansion, can we calculate
  e0.5 to three significant digits?

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Example

• Calculate the error tolerance
• First approximation
• Second approximation
• Error approximation

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Example

• Third approximation
• Error approximation

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Example
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Next class

• HW1, due 9/5
• Chapra Canale 6th Edition
• 1.8 (typo dy/dx -gt dy/dt), 1.12, 2.5 (choose
  order n6), 2.14
• Next class
• Continue on error analysis