This document discusses solutions to creative problems using functions. It presents four creative questions that involve using a black box function to multiply matrices in different ways. The black box functions take matrices as input and output their product or square. The questions demonstrate how to use the black box functions to multiply two arbitrary matrices by decomposing the matrices in ways compatible with the black box functions. It also discusses general concepts of using functions to break down large tasks, hide implementation details, and build on existing functionality through modular programming.

1. Solutions to Creative
Problems
Lecture 16
Feb 12, 2008

2. Creative Question
• Given a Black box that takes as input
two symmetric matrices and outputs the
product of the same, use the black box
to multiple two arbitrary matrices.

3. a b
x
c d
0 0 a b
0 0 c d
a c 0 0
b d 0 0
e g
f h
x
=
p q
r s
0 0 e f
0 0 g h
e g 0 0
f h 0 0
p
r
0
0
q
s
0
0
0 0
0 0
u v
wx

4. Creative Question
• Given a Black box that takes as input a
matrix and outputs the square of the
same, use the black box to multiple two
arbitrary matrices.

5. a b
x
c d
0 0 a b
0 0 c d
e g 0 0
f h 0 0
e g
f h
x
=
p q
r s
0 0 a b
0 0 c d
e g 0 0
f h 0 0
p
r
0
0
q
s
0
0
0 0
0 0
u v
wx

6. Creative Question
• Given a Black box that takes as input a
lower triangular matrix and an upper
triangular matrix and outputs the
product of the same, use the black box
to multiple two arbitrary square
matrices.

7. a b
x
c d
0 0 0 0
0 0 0 0
a b 0 0
c d 0 0
e g
f h
x
=
p q
r s
0 0 e g
0 0 f h
0 0 0 0
0 0 0 0
0
0
0
0
0
0
0
0
0
0
p
r
0
0
q
s

8. Creative Question
• Given a Black box that takes as input
two lower triangular matrices and
outputs the product of the same, use
the black box to multiple two arbitrary
square matrices.

9. a b
x
c d
000000
000000
000000
000000
00ab00
00cd00
x
e g
f h
=
000000
000000
eg0000
f h0000
000000
000000
p q
r s
=
000000
000000
000000
000000
pq0000
r s0000

10. Functions = outsourcing
• Break large computing tasks into small ones
• Helps you to build on what others have done
• You and others write functions
• When you want to build a program, find out how to use
the function and use it.
• Use standard functions provided by the
library.
• You are hidden from the implementation
• Example – you don’t have to worry about how pow(m,n)
is implemented
• As engineers from different disciplines you
will use and develop different set of functions

11. Modular Programming
Subprograms
functions in C, C++, procedures and functions in
Pascal
facilitate modular programming
Overall task is divided into modules
Each module - a collection of subprograms
a subprogram may be invoked at several points
A commonly used computation
hiding the implementation
incorporating changes
easier

12. Example of function sets
• String manipulation
• Mathematical
• Finite Element Method
• Used in structural analysis by Mechanical, Civil, Aero,
etc. for stress calculations etc.
• Most function libraries cost a lot
• Business opportunity – identify functions that are useful
to you area of study, create libraries.
• Functions for use in different software.
• Say, functions for web services

13. Basics
• Function is a part of your program.
• It cannot be a part of any other function
• Main() is a function: it is the main function. Execution starts
there or the control flow starts there
• From there it can flow from one function to another, return after
a computation with some values, probably, and then flow on.
• Transfer of control is affected by calling a function
•
•
•
•
•
•
•
With a function call, we pass some parameters
These parameters are used within the function
A value is computed
The value is returned to the function which initiated the call
The calling function can ignore the value returned
It could use it in some other computation
A function could call itself, these are called recursive function
calls

14. Add function to your Program
• A program was a set of variables, and
assignments to variables
• Now add function to it.
•
•
•
•
Set of variables
Some functions including main()
Communicating values to each other
Computing and returning values for each other
• Instead of one long program, we now write
structured program composed of functions

15. Features
• C program -- a collection of functions
– function main ( ) - mandatory - program starts here.
• C is not a block structured language
– a function can not be defined inside another function
– only variables can be defined in functions / blocks
• Variables can be defined outside of all functions
– global variables - accessible to all functions
– a means of sharing data between functions - caution
• Recursion is possible
– a function can call itself - directly or indirectly

16. Function template
Return-type function-name(argument
declarations)
{
declaration and statements
return expression;
}

17. Function Definition in C
return-type function-name (argument declarations)
{
variable/constant declarations and
No function
statements }
declarations here!
Arguments or parameters:
the means of giving input to the function
type and name of arguments are declared
names are formal - local to the function
Return Value: for giving the output value
return ( expression ); -- optional
Invoking a function: funct-name(exp1,exp2,…,expn)
Matching the
number and type
of arguments

18. Function Prototype
• defines
– the number of parameters, type of each
parameter,
– type of the return value of a function
• used by the compiler to check the usage
– prevents execution-time errors
• function prototype of power function
– int power ( int, int );
– no need for naming the parameters
• function prototypes - given in the beginning

19. Power Function
#include <stdio.h>
function prototype
int power (int, int);
-- Computes the nth
power of base.
main () {
for ( int i = 0; i < 20; i ++ )
printf(“%d %d %dn”, i, power(3,i), power(-4,i);}
int power (int base, int n) {
int i, p = 1;
for ( i = 1; i <= n ; i ++)
p = p ∗ base;
return p;
}
A block
Invocation with
arguments

20. Calling Power Function with i=3
printf(“%d %d %dn”, i, power(3,i), power(-4,i);}
27
int power (int base, int n) {
int i, p = 1;
for ( i = 1; i <= n ; i ++)
p = p ∗ base;
return p;
}
-64
int power (int base, int n) {
int i, p = 1;
for ( i = 1; i <= n ; i ++)
p = p ∗ base;
return p;
}

21. Thank You
• For Exam
– Bring BOTH your Institute ID card and RFID card - else you shall not be permitted to
write the exam.
• All the Best