Dr Deepak B Phatak - Kanwal Rekhi€¦ · Dr. Deepak B Phatak 7. IIT BOMBAY. Finding real root, using a numerical method. Session 7, Functions • Make an initial guess, say ‘xguess’

CS 101Computer Programming and utilization

Dr Deepak B PhatakSubrao Nilekani Chair Professor

Department of CSE, Kanwal Rekhi BuildingIIT Bombay

Session 7, Functions in C++Tuesday, August 16, 2011

And Wednesday, August 17, 2011

IIT BOMBAY

Session 7, Functions

Dr. Deepak B Phatak 2

IIT BOMBAY


Overview

• More about iterations– Break and continue statements– Another example of iterative solution

• Newton-Raphson method• ‘sub’ programs which can be invoked from anywhere

– ‘Functions’ in C++– Parameter passing– Function declaration

• Take home quiz


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Need to come out of iteration


• To abandon the iterative block completely, and to jump (break) out of iteration


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To continue next iteration by skipping some code


• Skip remaining statements within a iteration, and directly jump to continue with the next iteration


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break and continue statements


• BreakWhen used inside a loop control structure, break causes an immediate exit from the loop

• ContinueWhen executed inside a loop control structure, continue statement skips any remaining statements within the block, and goes for the next iteration


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Finding a real root of an equation


• ax2 + bx + c = 0– This equation could have real roots

• In general, for any function f(x), – a root is that value of x, for which f(x) =0


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Finding real root, using a numerical method


• Make an initial guess, say ‘xguess’• Use the properties of the function, to compute a new value, say,

‘xnew’, which should be closer to the root– ‘xnew’ is a better approximation to the root


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Finding real root ...


• An iteration can now be carried out– take ‘xnew’ as the initial guess,

xguess = xnew;– repeat the steps to find another approximation


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How to find ‘xnew’


• Newton-Raphson method uses the tangent– Derivative of the function should be known


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Start with a point P (initial guess)


xguessrootQ

y=f(x)

P

R

x

point P on x-axis Initial guess

Find some point Q Closer to root

How to find Q?

Note that length PR= f(xguess)


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Determine Q, using tangent at R


xguessrootQ

y=f(x)

xnewP

R

x

Draw tangent to curve at point R Q is the intersection of this tangent with x-axis


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Coordinates of Q can now be calculated


xguessrootQ

y=f(x)

xnewP

R

x

Tangent determined by derivative of function f’(x)

f’(xguess) = PR/PQ

PQ = PR / f’(xguess) = f(xguess) / f’(xguess)

xnew = xguess - PQ


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For next iteration, move to the new point as initial guess


rootP

y=f(x)

xguessx


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Find square root of a given number

• Consider that, for some x, k = x2; or x2 - k = 0

• We need to find a real root for the function f(x) = x2 - k, Note that f ’(x) = 2x


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xguess and xnew


• Start with xguess = 1, • Compute xnew = xguess - (xguess2 - k)/(2 *xguess)

= (xguess + k/xguess)/2


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Block of statement to be repeated iteratively


• ‘xnew’ is the modified value of ‘xguess’– to be used as ‘xguess’ for next iteration

• Instruction which we may repeatedly execute is identified as

x = (x + k/x)/2• In any iteration cycle

• Using the current value of x, compute its new value• to be used in the next iteration


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Termination criterion

• When do we stop iterating?– Perform a fixed number of iterations– Iterate while we are not close ‘enough’

• How to define ‘closeness’ or accuracy of the result?– We can get as close to the root (which is √ k ), as required.


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Accuracy of the calculated value of the root

0.0001Accuracy threshold

| f(x) | <= 0.0001


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Termination criterion


• Let the ‘closeness’ be defined as f(x) being within 0.0001• At the root value, f(x) will be exactly 0

– We wish to stop when f(x) is within ± 0.0001• This helps us define the termination criterion

– Keep repeating while | f(x) | > 0.0001 – otherwise, terminate the iterative loop


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The program

int main() {float k; float x=1;cout << “Give number whose square root is to be found ”;cin >> k;while (x*x – k > 0.0001 || k - x*x > 0.0001){

x = (x + k/x)/2 ;}cout << endl<< “Value of the root is ” << x << endl;return 0;

}


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Finding root using a fixed number of iterations


int main() {// calculating square root of a number kfloat k; float x=1; int i; cout << “Give number whose square root is to be found ”;cin >> k;for (i=1; i <= 10; i++){ // 10 iterations

x = (x + k/x)/2;}cout << x;return 0;

}


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Using ‘for’ in a ‘while’ like fashion


int main() {float x, k; cout << “Give value of k ”; cin >> k;for(x=1; (x*x – k > 0.0001) ||( k - x*x > 0.0001); x= (x+k/x)/2){}cout << x;return 0;

}


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Using ‘for’ loop in a different way


for(x=1; x*x – k > 0.0001 || k - x*x > 0.0001; x= (x+k/x)/2){}– Here x is ‘initialized’ to 1 before the loop– The ‘condition’ repeats the loop, while it is true– x is ‘incremented’ to reflect the new value calculated

• The use of for without a body is possible• Iterations will be executed as defined by the specifications• Since there is no body, all meaningful computations should

happen within the actions given in the 3 specifications


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‘Functions’ in c++


• a facility to compute value of a mathematical function for specified value(s) of some parameter(s)

• Consider a quadraticf(x) = ax2 + bx + c

• If we have to evaluate it for x=4 at one place in our program, and for x = 2.3 at another place, assigning the results to say, y1 and y2

• We will have to write two separate statements– Each statement will have to specify the

complete formula


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‘Functions’ in c++ …


• It would be useful if we could write the instructions to evaluate the formula separately, at some other place– And make the computer go to that place

whenever needed• In our main program, we would like to write:

y1 = quad(a, b, c, 4.0);--------y2 = quad(a, b, c, 2.3)


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Functions in C++ …


• Such a facility is provided by c++• Write instructions to compute value of a function

only once– Such a function accepts ‘parameters’– Calculates and returns the function value

• A function can be invoked from within a program– as many times as needed


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C++ Function to evaluate a Quadratic

• We need to evaluate the functionf(x) = ax2 + bx + c


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Our definition of ‘quad’ function

float quad (float a, float b, float c, float x){float value;value = a *x*x + b*x + c;return (value);

}


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C++ rules


• 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

• The return statement sends back the value• In general, ‘value’ can be any expression

– It is evaluated when return statement is executed, the result value is sent back


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• C++ compiler translates instructions written by us in a function definition separately– keeps these as an independent block

identified by our chosen name for the function• Function code is executed only when called

Let f (x) = 23.5 x 2 + 9.7 x + 12.6We want to calculate y = 574.6 * f(x) + 1.0/x

float x, y; - - -cin >> x;y = 574.6 * quad (23.5, 9.7, 12.6, x) + 1.0/x;



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C++ rules for function invocation


• Within a program, a function is invoked simply by using the function name within any expression with appropriate parameters

• given parameters are copied to the respective locations in the function block

• Function code is now executed• value is returned to the calling program• returned value used in the expression


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• Can we use programming code for functions written by others• Yes, of course, that is the very idea• We can even compile those functions

separately and link them with our program,• C++ has many rich ‘libraries’ of functions

• Each function is predefined • Our main program is actually treated as a

function by the operating system• We can now understand why we say

int main() { return 0;

Some points to ponder ...



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Example


• We wish to evaluate the following v = (√m + √n) / √(m+n)

for some given values of m and n• Can we use our earlier program,

– Rewrite as a function sroot

• In our main program, we can writev = (sroot(m) + sroot(n)) / sroot(m+n);


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Our function named ‘sroot’


#include<iostream>using namespace std;float sroot( float k) {

float x=1;while ((x*x – k > 0.0001) || (k - x*x > 0.0001)){

x = (x + k/x)/2 ;}return x;

}


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Use of the function root within the main program


int main() {// calculating value of the given formulafloat m, n, v;cout << “Give values of m and n: ”; cin >> m >>n;v = (sroot(m) + sroot(n)) / sroot(m+n);cout << “Value of the formula is: ” << v;return 0;

}


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Slot-wise students’ performance


#include<iostream> using namespace std; double average(int, int); // prototype definitionint main(){int N5, N11; double slot5avg, slot11avg; cout << "No. of students in slot 5: "; cin >> N5; cout << "No. of students in slot 11: "; cin >> N11;slot5avg = average (5, N5); slot11avg = average(11, N11);if (slot5avg > slot11avg) cout << "slot 5 has performed better" <<

endl;else cout << "slot 11 has performed better" << endl;return 0;

}


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Contd. ...

double average (int slot, int N){ //function definition int i; double marks, sum=0.0; for(i = 1; i <= N; i++){

cout <<" Give Marks for student no. " << i <<" in slot "; cout << slot << ": "; cin >> marks; sum += marks;

} return sum/N;

}



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Announcement

• From next week, M Sc Chemistry students will have their labs on Wednesday morning in OSL. The exact timing will be announced soon.

• Video recordings of previous lectures are kept on a server• These will be screened on Sunday mornings in room SIC 301,

Kanwal Rekhi Building, CSE dept. A Schedule will be announced each week. The Schedule for Sunday, 21 August 2011 is:– Session 5, Iterative algorithms 09:30 to 11:00– Session 6, Iterative solutions 11:00 to 12:30Students who have missed the lectures, or would like to attend for additional clarity, can attend with prior registration


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Announcements


• Discussion and consulting session as also the Make up lab will now be held on Sundays– Consulting session and Make up lab:

Sunday 14:00 to 18:00[OSL, ground floor, Maths department building]

• The seating arrangement in the regular lab will be announced on the moodle. Please follow these

Documents

Dr Deepak B Phatak - Kanwal Rekhi€¦ · Dr. Deepak B Phatak 7. IIT BOMBAY. Finding real root, using a numerical method. Session 7, Functions • Make an initial guess, say ‘xguess’