Two-week ISTE workshop on Effective teaching/learning of computer programming

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Two-week ISTE workshop onEffective
teaching/learning of computer programming

• Dr Deepak B Phatak
• Subrao Nilekani Chair Professor
• Department of CSE, Kanwal Rekhi Building
• IIT Bombay
• Lecture 4, Functions
• Wednesday 30 June 2010

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Overview

• Iterative Solution (Contd.)
• Finding roots of a given function
• Different ways of prescribing iteration
• Functions
• Need, definition and usage
• Workshop Projects

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Newton Raphson method

• Method to find the root of f(x),
• i.e. x such that f(x)0.
• Method works if
• f(x) and f '(x) can be easily calculated.
• and a good initial guess is available.
• Example To find square root of k.
• use f(x) x2 - k. f (x) 2x.
• f(x), f (x) can be calculated easily.
• only few arithmetic operations needed
• Initial guess x0 1
• It always works! can be proved.

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Newton Raphson method

• Method to find the root of f(x),
• i.e. x such that f(x)0.
• Method works if
• f(x) and f '(x) can be easily calculated.
• and a good initial guess is available.
• Example To find square root of k.
• use f(x) x2 - k. f (x) 2x.
• f(x), f (x) can be calculated easily.
• only few arithmetic operations needed
• Initial guess x0 1
• It always works! can be proved.

Let x vk then x2 k and x2 k 0
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How to get better xi1 given xi
Point A (xi,0) known.
Calculate f(xi ). Point B(xi,f(xi)) is now known
Approximate f by tangent C intercept on x axis
C(xi1,0)
f (xi) AB/AC f(xi)/(xi - xi1) ? xi1
(xi- f(xi)/f (xi))
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Square root of k

• xi1 (xi- f(xi)/f (xi))
• f(x) x2 - k, f (x) 2x
• xi1 xi - (xi2 - k)/(2xi) (xi k/xi)/2
• Starting with x01, we compute x1, then x2, and
  so on
• Each successive value of xi will be closer to the
  root
• We can get as close to sqrt(k) as required by
  carrying out these iterations many times
• Errors in floating point computations ?

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Program segment

• // calculating square root of a number k
• float k
• cin gtgt k
• float xi1
• // Initial guess. Known to work.
• for (int i0 i lt 10 i)
• // 10 iterations
• xi (xi k/xi)/2
• cout ltlt xi

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Another way

• float xi, k
• cin gtgt k
• for( xi 1
• // Initial guess. Known to work.
• xixi k gt 0.001 k - xixi gt 0.001
• //until error in the square is at most 0.001
• xi (xi k/xi)/2)
• cout ltlt xi

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Special ways of using for

• for (xxx yyy zzz)
• www
• In the alternate way we saw, the computations
  required for each iteration are all specified as
  part of the specifications of for statement
  itself.
• Thus the body of statements (www) is missing,
  because it is not required
• A special way of using for
• for ( ) www
• This specifies an infinite iteration, the loop
  must be broken by some condition within www

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Yet Another way

• float k
• cin gtgt k
• float xi1
• While (xixi k gt 0.001 k - xixi gt 0.001)
• xi (xi k/xi)/2
• cout ltlt xi

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While statement

• while (condition)
• loop body
• check condition, if true then execute loop body.
  Repeat.
• If loop body is a single statement, then need not
  use . Always putting braces is recommended
  if we later insert a statement, we may forget to
  put them, so we should do it at the beginning.

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for and while

• If there is a control variable with initial
  value, update rule, and whose value distinctly
  defines each loop iteration, use for.
• Also, if loop executes fixed number of times, use
  for.

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Functions

• Consider a quadratic function
• f(x) ax2 bx c
• f(x) 2ax b
• It would be nice, if we had separate blocks of
  instructions to calculate these for different
  values of x
• A function in c is such a separate block
• It takes one or more parameters and returns a
  single value of a specified type

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Example of functions

• float myfunction (float a, float b, float c,
  float x)
• float value
• value a xx bx c
• return (value)
• float myderivative(float a, float b, float x)
• float value
• value 2ax b
• return (value)

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Syntax

• int myfunction (float a, )
• First word tells the type of the value which will
  be returned.
• Next is the name of the function, which we choose
  appropriately
• This is followed by one or more parameters whose
  values will come from the calling instruction
• Note the return statement
• return (value)
• this says what value is to be sent back. In
  general, it can be an expression which is
  evaluated when return statement is executed

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Function in our model

• We had thought of our computer as a dumbo, so
  imagine each such function to be evaluated by a
  separate assistant dumbo
• Any time a function is invoked within an
  instruction which is executing, the given
  parameters are handed over to the assistant dumbo
• Assistant dumbo calculates the function value and
  returns the same to main dumbo
• Our main dumbo then onwards carries on from
  exactly where he left, using that returned value
  in place of the reference to the function

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Invoking a function (function call)

• Within a program, a function is invoked simply by
  using the function name (with appropriate
  parameters) within any expression
• In the Newton Raphson method, we have a value xi,
  and we calculate next value using
• xi1 (xi- f(xi)/f (xi))
• Suppose our function was
• f(x) ax2 bx c
• Then we could design our program using the two
  functions which we have written
• (myfunction and myderivative)

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Newton Raphson using function calls

• int main()
• float x, a, b, c, root
• // read a, b, c
• ...
• x 1.0
• // This is the initial guess for x
• for (int i0 i lt 10 i)
• x (x - myfunction(a,b,c,x)
• /myderivative(a,b,x))
• ...

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Invocation rules

• x (x - myfunction(a,b,c,x)/
• myderivative(a,b,x))
• When dumbo encounters myfunction while
  evaluating the expression,
• it suspends execution of the program,
• goes over to the defined function with the
  available values of the parameters
• calculates the value executing given instructions
  within that function
• then returns back to the main program, replaces
  the reference to function by the returned value,
  and continues evaluation of the remaining
  expression.

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Some points to ponder

• We see that the calculations pertaining to our
  function evaluation have been separated out,
  perhaps resulting a better structured or
  modular program
• Why is this important
• Suppose we wish to modify this same program to
  calculate a root of another function, then it is
  far easier to replace code only in that part
  where functions are defined.
• Otherwise we may have to search our entire code
  to find which lines we should change

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Some points to ponder ...

• Can I use programming code for functions written
  by ohers
• Yes, of course, that is the very idea
• We can even compile those functions separately
  and link them with our program, but we need to
  include prototype definitions of these functions
  within the program
• We can now understand why we say
• int (main) and return 0
• Our entire program is actually treated as a
  function by the operating system

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Question, From PSG Coimbatore

• If we declare an int variable , the largest value
  it holds varies from system to system. What is
  the reason behind this?

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Question, From GEC_Thrissur

• How to return to values from a function at a
  time for example two roots of
  quadratic equation

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Question, From NIT_Jalandhar

• Can the main function return a float value