Two-week ISTE workshop onEffective teaching/learning of computer

programming

Dr Deepak B PhatakSubrao Nilekani Chair Professor

Department of CSE, Kanwal Rekhi BuildingIIT Bombay

Lecture 4, Functions

Wednesday 30 June 2010

Overview

Iterative Solution (Contd.)• Finding roots of a given function• Different ways of prescribing iteration

Functions• Need, definition and usage

Workshop Projects

Lecture 4 Functions

Lecture 4 Functions

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.

Lecture 4 Functions

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 = √kthen x2 = kand x2 – k = 0

Lecture 4 Functions

How to get better xi+1 given xi

Point A =(xi,0) known.

f’ (xi) = AB/AC = f(xi)/(xi - xi+1) xi+1

= (xi- f(xi)/f’ (xi))

Calculate f(xi ).

Point B=(xi,f(x

i)) is now known

Approximate f by tangentC= intercept on x axis C=(x

i+1,0)

f(x)x

ix

i+1

A

B

C

Lecture 4 Functions

Square root of k

xi+1

= (xi- f(xi)/f’ (xi))

f(x) = x2 - k, f’ (x) = 2x

xi+1 = xi - (xi2 - k)/(2xi) = (xi + k/xi)/2

Starting with x0=1, 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 ?

Lecture 4 Functions

Program segment

// calculating square root of a number kfloat k;cin >> k;float xi=1; // Initial guess. Known to work.for (int i=0; i < 10; i++){ // 10 iterations xi = (xi + k/xi)/2; }cout << xi;

Lecture 4 Functions

Another way

float xi, k; cin >> k; for( xi = 1 ;

// Initial guess. Known to work.

xi*xi – k > 0.001 || k - xi*xi > 0.001 ;

//until error in the square is at most 0.001

xi = (xi + k/xi)/2);

cout << xi;

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’

Lecture 4 Functions

Lecture 4 Functions

Yet Another wayfloat k; cin >> k;float xi=1;While (xi*xi – k > 0.001 || k - xi*xi > 0.001){ xi = (xi + k/xi)/2 ; } cout << xi;

Lecture 4 Functions

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.

Lecture 4 Functions

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’.

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

Lecture 4 Functions

Example of functions

float myfunction (float a, float b, float c, float x){

float value;

value = a *x*x + b*x + c;

return (value);

}

float myderivative(float a, float b, float x){

float value;

value = 2*a*x + b;

return (value);

}

Lecture 4 Functions

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

Lecture 4 Functions

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

Lecture 4 Functions

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

xi+1 = (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)

Lecture 4 Functions

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 i=0; i < 10; i++){ x = (x - myfunction(a,b,c,x) /myderivative(a,b,x)); } ...}

Lecture 4 Functions

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.

Lecture 4 Functions

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

Lecture 4 Functions

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

Lecture 4 Functions

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?

Question, From GEC_Thrissur:

How to return to value’s from a function at a time for example two roots of quadratic equation

Question, From NIT_Jalandhar

Can the main function return a float value