computer science 1620 functions. given a number n, the factorial of n, written n!, is computed as...

Computer Science 1620

Functions

Given a number n, the factorial of n, written n!, is computed as follows:

note: 0! = 1 examples:

n! = n x (n-1) x (n-2) x … x 1

3! = 3 x 2 x 1 = 6

5! = 5 x 4 x 3 x 2 x 1 = 120

0! = 1

Write a program that takes a number n from the user, and calculates the factorial of that number.

#include <iostream>#include <iomanip> using namespace std;

int main() {

int n;cout << "Please enter a number:";cin >> n;

double factn = 1.0; for (int i = 1; i <= n; i++) {

factn *= i;}

cout << fixed << setprecision(0);cout << n << "! = " << factn << endl;

return 0;}

Write a program that takes a number from the user, and calculates the factorial of that number.


int main() {



factn *= i;}


return 0;}


We'll use a double to storethe factorial, since factorials are typically very big numbers.


int main() {



factn *= i;}


return 0;}


Example: Given a set of n objects, the number of combinations of those objects taken r at a time can be calculated as:

ie. Given a class of 112 students, the number of possible groups of 5 students is:

)!(!

!

rnr

nC

r

n nr −==⎟⎟

⎠

⎞⎜⎜⎝

⎛

134153712)!5112(!5

!112

5

112 1125 ≈

−==⎟⎟

⎠

⎞⎜⎜⎝

⎛C

Write a program that takes two integers n and r from the user, and calculates Cn

r for those two numbers, and displays the results.


int main() {

int n, r;cout << "Please enter n and r:";cin >> n >> r;


factn *= i;}

double factr = 1.0; for (int i = 1; i <= r; i++) {

factr *= i;}

double factnr = 1.0; for (int i = 1; i <= (n-r); i++) {

factnr *= i;}

double result = factn / (factr * factnr);cout << fixed << setprecision(0);cout << n << " choose " << r << " = " << result << endl;

return 0;}




int main() {



factn *= i;}

double factr = 1.0; for (int i = 1; i <= r; i++) {

factr *= i;}

double factnr = 1.0; for (int i = 1; i <= (n-r); i++) {

factnr *= i;}

double result = factn / (factr * factnr);cout << fixed << setprecision(0);cout << n << " choose " << r << " = " << result << endl;

return 0;}



Here's where we needto calculate Cn

r

Problem Breakdown:

double factn = 1.0; for (int i = 1; i <= n; i++) { factn *= i; } double factn = 1.0; for (int i = 1; i <= n; i++) { factn *= i; } double factn = 1.0; for (int i = 1; i <= n; i++) { factn *= i; } double result = factn / (factr * factnr);

Calculate n!

Calculate r!

Calculate (n-r)!

Combine results into Cn

r

Problem Breakdown:

double factn = 1.0; for (int i = 1; i <= n; i++) { factn *= i; } double factr = 1.0; for (int i = 1; i <= r; i++) { factr *= i; } double factn = 1.0; for (int i = 1; i <= n; i++) { factn *= i; } double result = factn / (factr * factnr);

Calculate n!

Calculate r!

Calculate (n-r)!


r

Problem Breakdown:

double factn = 1.0; for (int i = 1; i <= n; i++) { factn *= i; } double factr = 1.0; for (int i = 1; i <= r; i++) { factr *= i; } double factnr = 1.0; for (int i = 1; i <= n-r; i++) { factnr *= i; } double result = factn / (factr * factnr);

Calculate n!

Calculate r!

Calculate (n-r)!


r


int main() {



cout << fixed << setprecision(0);cout << n << " choose " << r << " = " << result << endl;

return 0;}


r for those two numbers.

Does this solve our problem?Yes

Any problems with previous code?duplication!!! the calculation for factorial is the same, but

the code to calculate is repeated 3 times the only change is the stopping condition value,

and the variable that we set.



double <result> = 1.0; for (int i = 1; i <= <input>; i++) { factn *= i;}


int main() {




return 0;}

Imagine a "factorial machine"one input, for a number none output, for the computed number n!

1205

factorial


10

factorial


x

result

double result = 1.0;for (int i = 1; i <= x; i++) { result *= i;}

120

5

3 questions where does this code go? how do we set the input (x)? how do we get the output (result)?

Functiona subprograma piece of code that can be re-usedcalled from other places in the programexample: code to write out my name and

location

cout << ”Robert" << endl;cout << "Lethbridge" << endl;

Function written outside of

main every function has

a: return type name parameter list

you call a function by typing its name, followed by args

#include <iostream> using namespace std;

void whoAmI() {cout << ”Robert" << endl;cout << "Lethbridge" << endl;

}

int main() {

whoAmI(); whoAmI(); return 0;}



}

int main() {


RobertLethbridgeRobertLethbridge

Output:

Start Here



}

int main() {



Output:

At this point, we move tothe first line in the function whoAmI


void whoAmI() {cout << "Robert" << endl;cout << "Lethbridge" << endl;

}

int main() {



Output:

Perform output



}

int main() {



Output:

Function ended, start again right after function call



}

int main() {



Output:

At this point, we move tothe first line in the function whoAmI



}

int main() {



Output:

Perform output



}

int main() {



Output:

Function ended, start again right after function call



}

int main() {


RobertLethbridgeRobertLethbridge ...

Output:

At this point, program terminates

Example: Write a program with a function called factorial that computes the factorial of 5 and writes the value to output. Call that function from the main function.


void factorial() { double result = 1.0; for (int i = 1; i <= 5; i++) {

result *= i; } cout << fixed << setprecision(0); cout << result;}

int main() {

cout << "The factorial of 5 is ";factorial();cout << endl;

return 0;}

Write a program with a function called factorial that computes the factorial of 5 and writes the value to output. Call that function from the main function.


x

result


120

5


Function Inputsending data to the functionaccomplished through a list of parameters

a list of variablescontained in parameter list

we set these variables in the call using arguments

a list of expressions

Function Inputs declared as parameters function can have 0 or

more parameters specify the type and the

variable name Function arguments

the value of each argument is assigned to corresponding parameter variable

void square(int x) { cout << x*x;}

int main() {

cout "2 x 2 = "; square(2); cout "(2+1) x (2+1) = "; square(2+1); return 0; }



int main() {

cout "2 x 2 = "; square(2); cout "\n(2+1) * (2+1) = "; square(2+1); return 0; }

2 x 2 = 4(2+1) x (2+1) = 9

Output:

Start Here



int main() {


2 x 2 = 4(2+1) x (2+1) = 9

Output:

Perform Output



int main() {


2 x 2 = 4(2+1) x (2+1) = 9

Output:

At this point, we move tothe first line in the function squareThe parameter x gets the value of the expression in the call, which is2



int main() {


2 x 2 = 4(2+1) x (2+1) = 9

Output:

Perform Output.Note that x hasvalue 2.



int main() {


2 x 2 = 4(2+1) x (2+1) = 9

Output:

Function ended, startright after function call.



int main() {


2 x 2 = 4(2+1) x (2+1) = 9

Output:

Perform Output



int main() {


2 x 2 = 4(2+1) x (2+1) = 9

Output:

At this point, we move tothe first line in the function squareThe parameter x gets the value of the expression in the call, which is2+1 = 3



int main() {


2 x 2 = 4(2+1) x (2+1) = 9

Output:

Perform Output.Note that x hasvalue 3.



int main() {


2 x 2 = 4(2+1) x (2+1) = 9

Output:

Function ended, startright after function call.



int main() {


2 x 2 = 4(2+1) x (2+1) = 9

Output:

Program terminates.

Example: Write a program with a function called factorial that computes the factorial of x (x is an inputted number) and writes the value to output. Call that function from the main function with the value 5.


void factorial(int x) { double result = 1.0; for (int i = 1; i <= x; i++) {

result *= i; } cout << fixed << setprecision(0); cout << result;}

int main() {

cout << "The factorial of 5 is ";factorial(5);cout << endl;

return 0;}

Write a program with a function called factorial that computes the factorial of x and writes the value to output. Call that function from the main function with the value 5.


x

result


120

5


Function Output receiving data from a functionaccomplished through a return statement recall that an expression in C++ has a value

2; // the value of this expression is 22 + 3; // the value of this expression is 52 > 3; // the value of this expression is false (0)

i = 2; // the value of this expression is 2

a function call is an expression with a valuesquare(2);square(2+1);

what is the value ofthese expressions?

Function Output given by the return

statement Function call

the value of the function call is the value that is returned

the type of the value returned by a function call is given by its return type

int square(int x) { int result = x * x; return result;}

int main() {

cout "2 x 2 = "; cout << square(2); cout "(2+1) x (2+1) = "; cout << square(2+1); return 0; }



int main() {


2 x 2 = 4(2+1) x (2+1) = 9

Output:

Start here



int main() {


2 x 2 = 4(2+1) x (2+1) = 9

Output:

Perform Output



int main() {


2 x 2 = 4(2+1) x (2+1) = 9

Output:

At this point, we move tothe first line in the function squareThe parameter x gets the value of the expression in the call, which is2



int main() {


2 x 2 = 4(2+1) x (2+1) = 9

Output:

Do calculation.Note that x hasvalue 2.



int main() {


2 x 2 = 4(2+1) x (2+1) = 9

Output:

Returns valuestored in result

The value of the expressionsquare(2) is the return value (4).



int main() {


2 x 2 = 4(2+1) x (2+1) = 9

Output:

Perform output.



int main() {


2 x 2 = 4(2+1) x (2+1) = 9

Output:

At this point, we move tothe first line in the function squareThe parameter x gets the value of the expression in the call, which is(2+1) = 3



int main() {


2 x 2 = 4(2+1) x (2+1) = 9

Output:

Do calculation.Note that x hasvalue 3.



int main() {


2 x 2 = 4(2+1) x (2+1) = 9

Output:

Returns valuestored in result

The value of the expressionsquare(3) is the return value (9).



int main() {


2 x 2 = 4(2+1) x (2+1) = 9

Output:

Perform Output:



int main() {


2 x 2 = 4(2+1) x (2+1) = 9

Output:

Terminate Program!

Example: Write a program with a function called factorial that computes the factorial of x (x is an inputted number) and returns that value. Call that function from the main function with the value 5.


double factorial(int x) { double result = 1.0; for (int i = 1; i <= x; i++) {

result *= i; } return result;}

int main() {

cout << fixed << setprecision(0);cout << "The factorial of 5 is ";cout << factorial(5);cout << endl;

return 0;}

Write a program with a function called factorial that computes the factorial of x and returns that value. Call that function from the main function with the value 5.


x

result

double result = 1.0;for (int i = 1; i <= x; i++) { result *= i;}cout << result; 120

5



int main() {




return 0;}



Rewrite this programusing the factorial function.


double factorial(int x) { double result = 1.0; for (int i = 1; i <= x; i++) result *= i; return result;}

int main() {




return 0;}





int main() {


double factn = factorial(n); double factr = 1.0; for (int i = 1; i <= r; i++) { factr *= i; } double factnr = 1.0; for (int i = 1; i <= n-r; i++) { factnr *= i; } double result = factn / (factr * factnr);


return 0;}





int main() {


double factn = factorial(n); double factr = factorial(r); double factnr = 1.0; for (int i = 1; i <= n-r; i++) { factnr *= i; } double result = factn / (factr * factnr);


return 0;}





int main() {


double factn = factorial(n); double factr = factorial(r); double factnr = factorial(n – r); double result = factn / (factr * factnr);


return 0;}





int main() {


double result = factorial(n) / (factorial(r) * factorial(n – r));


return 0;}





int main() {


cout << fixed << setprecision(0);cout << n << " choose " << r << " = "

<< factorial(n) / (factorial(r) * factorial(n – r)) << endl;

return 0;}



The anatomy of a function:


function header(declaration)

function body(definition)

returntype

functionname

parameterlist

Functions with no return value sometimes, we do not need a value back from our

function

we indicate this with the void return type return keyword in void function simply exits function

void square(int x) { int result = x * x; cout << result;}

void inverse(int x) { if (x == 0) return; double result = 1.0 / x; cout << result;}

Functions as expressions the value of a function call is its return value return value can be used in other

expressionsint square(int x) { int result = x * x; return result;}

int main() { cout << square(2); // outputs 4 cout << square(2) + square(3); // outputs 13 cout << square(square(2)); // outputs 16 return 0;}

Functions with multiple inputs functions can have as many parameters as you like first argument aligns with first parameter, second with second,

etc …

int sum(int x, int y) { return x + y;}

int main() { cout << sum(2, 3); // outputs 5 return 0;}

More than one return statement a function may have more than one return statement function terminates as soon as return statement is reached

int max(int x, int y) { if (x > y) return x; return y;}

int main() { cout << max(2, 3) << endl; // outputs 3 cout << max(3, 2) << endl; // outputs 3 return 0;}

Where have we seen functions?

The main function: a function called by operating system can have parameters (more on this later) has a return value

0 – program terminated normally nonzero – program terminated abnormally

int main() {

return 0;}

Functions can call other functionsExample: write a program that takes an

integer n, and outputs a table of all values Cn

r for all values 0 <= r <= n

Without functions:#include <iostream>#include <iomanip> using namespace std;

int main() {

int n;cout << "Please enter n:";cin >> n;

cout << " r n choose r" << endl; cout << "--- ----------" << endl;

for (int r = 0; r <= n; r++) { double factn = 1.0; for (int i = 1; i <= n; i++) { factn *= i; } double factr = 1.0; for (int i = 1; i <= r; i++) { factr *= i; } double factnr = 1.0; for (int i = 1; i <= n-r; i++) { factnr *= i; } double result = factn / (factr * factnr);

cout << fixed << setprecision(0); cout << setw(3) << r << setw(15) << result << endl; }return 0;

}

With function factorial:#include <iostream>#include <iomanip> using namespace std;

double factorial(int x) { double result = 1.0; for (int i = 1; i <= x; i++)

result *= i; return result;}

int main() {



for (int r = 0; r <= n; r++) { double factn = factorial(n); double factr = factorial(r); double factnr = factorial(n-r); double result = factn / (factr * factnr);

cout << fixed << setprecision(0); cout << setw(3) << r << setw(15) << result << endl; }return 0;

}

With function factorial:#include <iostream>#include <iomanip> using namespace std;



int main() {



for (int r = 0; r <= n; r++) { double result = factorial(n) / (factorial(r) * factorial(n-r)); cout << fixed << setprecision(0); cout << setw(3) << r << setw(15) << result << endl; }return 0;

}

With factorial, looks much betterwhat is a good candidate for a function

in this code?calculating n choose r

write a function called choose that takes two values n and r, and returns n choose r inputs: n (int), r(int)output: an integer

write a function called choose that takes two values n and r, and calculates and returns n choose r inputs: n (int), r(int) output: an integer

int choose(int n, int r) { double result = factorial(n) / (factorial(r) * factorial(n-r)); return static_cast<int>(result);}


int choose(int n, int r) { double factn = 1.0; for (int i = 1; i <= n; i++) { factn *= i; } double factr = 1.0; for (int i = 1; i <= r; i++) { factr *= i; } double factnr = 1.0; for (int i = 1; i <= n-r; i++) { factnr *= i; } double result = factn / (factr * factnr); double result

}


note: this assumes that we have the factorial function defined in our program!!!


With function factorial and choose:#include <iostream>#include <iomanip> using namespace std;




int main() {



for (int r = 0; r <= n; r++) { cout << setw(3) << r << setw(15) << choose(n, r) << endl; }return 0;

}

Function Placement

2 key ideas1. The engine of a C++ program is the main method

allocates tasks to other functions program functionality can usually be inferred from a

well written main method, as opposed to a poorly written one ie. suppose someone asks: "What does your program

do?" "program outputs n choose r for all 0 <= r <= n"

2. the user is typically interested in what function does, not how it does it

ex. square root

Main function code is more interesting to user than helper functions thus, it is typically placed near the top of

your file, before other functionshowever, if we place a function after the

main function, we can't call it from mainC++ must know the parameter types and return

type of a function before it can be called, for type checking


int main() {




}





int main() {




}




This will not compile.

Function Prototypea function declaration (header) without a

definition (body) tells the compiler what the parameters and

return type of the function will be, when it is defined

allows C++ to do type checking prior to function definition

Function Prototype function header, without bodyappended with semicolon

{ double result = 1.0; for (int i = 1; i <= x; i++) result *= i; return result;}

function header(declaration)

function body(definition)

double factorial(int x);


double factorial(int x);int choose(int n, int r);

int main() {int n;cout << "Please enter n:";cin >> n;



}




This will compile, sincemain can check types.

Function Prototypesince we are only type checking, parameter

names are of no use

can also be written as (more common):

double factorial(int x)

double factorial(int)


double factorial(int);int choose(int, int);

int main() {int n;cout << "Please enter n:";cin >> n;



}




A good (and common) format for your .cc files to follow:

preprocessor directives

function prototypes

int main() {

}

function definitions

Functions - Examples

Write a function called odd that takes an integer number, and returns true if that number is odd, or false if the number is even.

bool odd(int x) { if (x % 2 == 1) {

return true; } return false; }


bool odd(int x) { return (x % 2 == 1); }


Functions - Examples

Write a function called power that returns a real value b raised to a positive power e.

double power(double b, int e) { double result = 1.0; for (int i = 1; i <= e; i++) result *= b; return result; }

Write a function called power that returns a real value b raised to a positive power e.

computer science 1620 functions. given a number n, the factorial of n, written n!, is computed as...

Documents

n double factn

int int main

int n cout n cout

n r double factn

int result

n r cout n cout

factorial of n

number n