1. Two Dimensional Arrays
Copyright © 2012 Pearson Education, Inc.
Chapter 8
Two Dimensional Arrays

2. Copyright © 2012 Pearson Education, Inc.
Outline
Objectives
Two Dimensional Arrays
Problem Solving Applied: Terrain Navigation
Two Dimensional Arrays and the vector Class
Matrices
Numerical technique: Solution to Simultaneous Equations
Problem Solving Applied: Electrical-Circuit Analysis
Higher Dimensional Arrays
Copyright © 2012 Pearson Education, Inc.

3. Copyright © 2012 Pearson Education, Inc.
Objectives
Develop problem solutions in C++ containing:
Matrix computations
Input from Data Files
Functions to Compute Sums and Averages
Techniques for solving a system of simultaneous equations
Vectors to implement the concept of two-dimensional arrays
Copyright © 2012 Pearson Education, Inc.

4. Two Dimensional Arrays
Copyright © 2012 Pearson Education, Inc.

5. Two Dimensional Arrays
Declaration and Initialization
Computation and Output
Function Arguments
Copyright © 2012 Pearson Education, Inc.

6. Two Dimensional Arrays
A two dimensional array stores data as a logical collection of rows and columns.
Each element of a two-dimensional array has a row position and a column position.
To access an element in a two-dimensional array, you must specify the name of the array followed by:
a row offset
a column offset
Copyright © 2012 Pearson Education, Inc.

7. Declaration and Initialization
The declaration of a two-dimensional array requires a row size and a column size.
A consecutive block of (row size)*(column size) memory locations are allocated.
All array elements must be of the same type.
Elements accessed by two offsets – a row offset and a column offset.
The name of the array holds the address of the first byte of memory
Copyright © 2012 Pearson Education, Inc.

8. Copyright © 2012 Pearson Education, Inc.
Example
//Declaration
int data[2][3];
Memory Snapshot ?
data
Copyright © 2012 Pearson Education, Inc.

9. Copyright © 2012 Pearson Education, Inc.
Example
//Declaration
int data[2][3];
row/column form:
col 0
col 1
col 2 ?
row 0
row 1
Copyright © 2012 Pearson Education, Inc.

10. 2D Array Definition Syntax
data_type identifier[ [row_size] ][column_size] [= initialization_list ];
//row_size and column_size must be integer constants
Examples
int data[2][5]; //allocates consecutive memory for 10 integer values
double t[2][2] = {{3.0,5.0},{2.1,7.2}}; //allocates and initializes
Valid References
cout << data[1][3];
cout << t[1][0];
Invalid References
cout << data[2][5]; //invalid offset.
cout << t[-1][-1]; //invalid offset
Copyright © 2012 Pearson Education, Inc.

11. Initialization Examples
int temp[4][3] = {50, 70, 60, 48, 75, 62, 51, 69, 60, 52, 78, 63}; int temp[4][3] = {{50, 70, 60}, {48, 75, 62}, {51, 69, 60}, {52, 78, 63}}; int t2[7][4] = {{50, 70, 60}, {48, 75, 62}, {51, 69, 60}, {52, 78, 63}}; int temp[][3] = {{50, 70, 60}, {48, 75, 62}, {51, 69, 60}, {52, 78, 63}}; int temp[][3] = {50, 70, 60, 48, 75, 62, 51, 69, 60, 52, 78, 63};
Copyright © 2012 Pearson Education, Inc.

12. Example: Input Using cin
Nested for loops are often used when inputting and assigning values to a two-dimensional array.
Nested loops are generally useful for getting around the 2D arrays…
//Declaration
double table[RSIZE][CSIZE];
for (int i=0; i<RSIZE; ++i) //every row
for (int j=0; j<CSIZE; ++j )//every col
cin >> table[i][j];
Copyright © 2012 Pearson Education, Inc.

13. Copyright © 2012 Pearson Education, Inc.
Example: Assignment
//Declaration
const int RSIZE(3),CSIZE(2);
double v[RSIZE][CSIZE];
for (int i=0; i<RSIZE; ++i) //every row
for (int j=0; j<CSIZE; ++j )//every col
v[i][j] = i+j; V 1 2 3
Copyright © 2012 Pearson Education, Inc.

14. Example: Computations
Compute the average value of an array with n rows and m columns.
double sum(0), average;
for (int i=0; i<n; ++i) //every row
for (int j=0; j<m; ++j )//every col
sum += array[i][j];
average = sum / (n*m);
Copyright © 2012 Pearson Education, Inc.

15. Example: Computations
Compute the average value of the nth row of a 2D array with r rows and c columns.
double sum(0), rowAverage;
for (int j=0; j<c; ++j ) //every col
sum += array[n][j];
average = sum / c;
Copyright © 2012 Pearson Education, Inc.

16. Copyright © 2012 Pearson Education, Inc.
Modify!
Modify the C++ statements on the previous slide to compute the average of the mth column.
Copyright © 2012 Pearson Education, Inc.

17. Copyright © 2012 Pearson Education, Inc.
Outputting 2D Arrays
Two dimensional arrays are often printed in a row by row format, using nested for statements.
When printing the row values of an array, be sure to print:
whitespace between the values in a row.
a newline character at the end of each row.
Copyright © 2012 Pearson Education, Inc.

18. Copyright © 2012 Pearson Education, Inc.
Example: Printing
for (int i=0; i<n; ++i) {//every row
for (int j=0; j<m; ++j )//every col
cout << array[i][j] << ‘ ‘;
cout << endl; //add end-of-line each row }
Copyright © 2012 Pearson Education, Inc.

19. 2D Arrays as Function Parameters
2-D arrays are always passed by reference.
The column dimension must be specified. The leftmost dimension (row) may be empty [].
Function prototype example:
int rowAverage(int Arr[][COLSIZE], int whichRow);
Array declaration in main:
int table [ROWSIZE][COLSIZE];
Function invocation example:
avg = rowAverage(table, 3);
Copyright © 2012 Pearson Education, Inc.

20. Copyright © 2012 Pearson Education, Inc.
Documentation Style
Well-documented functions always include pre-conditions and post-conditions in the function’s comment header block.
Pre-conditions describe the conditions assumed to be true at the time the function is called.
If pre-conditions are not met, there is no guarantee that the function will work correctly.
Post-conditions describe the changes that will be made to the arguments during the execution of the function.
Copyright © 2012 Pearson Education, Inc.

21. Problem Solving Applied: Terrain Navigation
Copyright © 2012 Pearson Education, Inc.

22. Problem Solving Applied: Terrain Navigation
Copyright © 2012 Pearson Education, Inc.

23. Problem Solving Applied: Terrain Navigation
Copyright © 2012 Pearson Education, Inc.

24. Problem Solving Applied: Terrain Navigation
Copyright © 2012 Pearson Education, Inc.

25. Two Dimensional Arrays and the vector Class
Copyright © 2012 Pearson Education, Inc.

26. 2D Arrays using the STL vector Class
Just as the vector may be used as a generic implementation of the array concept, the vector class may be used to provide an generic for 2D arrays.
Dynamic sizing of array.
Methods for determining and modifying size and capacity dynamically.
Copyright © 2012 Pearson Education, Inc.

27. 2D vector Definition Syntax
vector< vector< type_specifier > > identifier[(row_size,column_size)];
Examples
vector<vector<double> > temp(rows,cols);
vector<vector<int> > table(10,4);
vector<vector<char> > tags;
vector<vector<Point> > image(height,width);
Copyright © 2012 Pearson Education, Inc.

28. Passing vector Instances to Functions
Unlike 2D arrays, objects of vector types are passed by value to functions.
Thus a copy is made of the arguments to the function call, and changes made to the formal parameter will NOT affect the argument value.
Pass vector objects by reference to allow the function to alter the argument.
Pass vector objects by const reference to avoid copying potentially large datasets.
Copyright © 2012 Pearson Education, Inc.

29. Copyright © 2012 Pearson Education, Inc.
Matrices
Copyright © 2012 Pearson Education, Inc.

30. Copyright © 2012 Pearson Education, Inc.
Matrix
A matrix is a set of numbers arranged in a rectangular grid with rows and columns.
A square matrix has the same number of rows as columns.
Row and Column offsets in Matrices are 1-based.
2D Arrays are useful for representing matrices.
Remember that C++ uses 0-based offsets!
Copyright © 2012 Pearson Education, Inc.

31. Matrix Computations and Operations
The determinant of a matrix is a scalar value used in computing matrix inverses and solving systems of simultaneous equations.
The transpose of a matrix is a new matrix in which the rows of the original matrix are the columns of the transpose.
Matrices of the same size may be added or subtracted element-by-element.
Matrix multiplication (M * N) is defined only when the number of columns of M is equal to the number of rows in N.
Result has the size rows(M) x cols(N)
Copyright © 2012 Pearson Education, Inc.

32. Numerical Technique: Solution to Simultaneous Equations
Copyright © 2012 Pearson Education, Inc.

33. Graphical Interpretation
Solving two simultaneous (linear) equations is finding the point at which the lines intersect.
Not all lines intersect, therefore there is not always a solution.
In 2D, parallel lines do not intersect unless they are identical lines.
When the lines are identical, there is no single point at which they intersect.
Copyright © 2012 Pearson Education, Inc.

34. Graphical Interpretation
In high dimensional spaces, analogous situations may occur.
Copyright © 2012 Pearson Education, Inc.

35. Copyright © 2012 Pearson Education, Inc.
Solution Methods
Simultaneous equations may be solved by a number of various methods, each of which has advantages and disadvantages.
Gauss elimination
Matrix Inverse
and others…
Copyright © 2012 Pearson Education, Inc.

36. Problem Solving Applied: Electrical-Circuit Analysis
Copyright © 2012 Pearson Education, Inc.

37. Problem Solving Applied: Electrical-Circuit Analysis
Copyright © 2012 Pearson Education, Inc.

38. Problem Solving Applied: Electrical-Circuit Analysis
Copyright © 2012 Pearson Education, Inc.

39. Problem Solving Applied: Electrical-Circuit Analysis
Copyright © 2012 Pearson Education, Inc.

40. Problem Solving Applied: Electrical-Circuit Analysis
Copyright © 2012 Pearson Education, Inc.

41. Higher Dimensional Arrays
Copyright © 2012 Pearson Education, Inc.

42. Higher Dimensional Arrays
C++ allows arrays to be defined with more than two dimensions. E.g.: int b[3][4][2]; //declare a 3-D array
An offset must be specified for each dimension.
Same ‘rules’ apply as for 1D and 2D arrays.
Offsets are unchecked by C++.
Only left-most dimension may be omitted when passing a multidimensional array to a function.
Copyright © 2012 Pearson Education, Inc.