1. Slides by: Diana Inkpen and Alan Williams
ITI Lab #10
Slides by: Diana Inkpen and Alan Williams

2. Introduction to matrices
A matrix is a two dimensional rectangular grid of numbers:
The dimensions of the matrix are the numbers of rows and columns (in the above case: row dimension 3, column dimension 3).
A value within a matrix is referred to by its row and column indices, in that order.
Math: number rows and columns from 1, from upper left corner
In math notation, M1,2 = 2
For algorithms (and Java), we will use indices starting from 0, to correspond to the array numbering.
In algorithm notation, M[0][1] = 2

3. Matrix element processing
To visit every element of an array, we had to use a loop.
To visit every element of a matrix, we need to use a loop inside a loop:
Outer loop: go through each row of a matrix
Inner loop: go through each column within one row of a matrix.
Recursion could be used in place of each loop

4. line_index <= column_index
Matrix example
Write an algorithm that finds the sum of the upper triangle of a square matrix (i.e. the diagonal and up).
M = 1 2 3 4
How do we know if an element of a square matrix is on or above the main diagonal?
line_index <= column_index

5. Matrix example – cont’d
GIVENS:
M (the matrix)
N (the number of rows and columns in the matrix)
RESULT:
Sum (the sum of all elements in the upper triangle)
INTERMEDIATES:
R (row index in the matrix)
C column index in the matrix)
HEADER:
Sum  CalculateUpperTriangle(M,N)

6. Matrix example – cont’d
BODY:
SUM  0
R  0
true
false
R < N ?
C  0
C < N ?
true
false
false
R  C ?
true 
SUM  SUM + M[R][C]
R  R + 1
C  C + 1

7. Files needed for this lab
WindChill.java
For example 1.
ITI1120.java
For reading in arrays for example 1.
MatrixLib.java
For reading and printing matrices

8. Matrix Example 1: Objective
Design an algorithm that will create a matrix of wind chill values for various temperatures and wind speeds.
The set of temperatures will be stored in an array
The set of wind speeds will be stored in a second array
An algorithm is already available that calculates the wind chill for a specific temperature and wind speed.
Implement the algorithm in Java.
For those of you who haven’t yet experienced an Ottawa winter, you may want to keep the results for future use 

9. Given algorithm: Wind Chill calculation
Suppose we are given the following algorithm:
Chill  WindChill( Temperature, WindSpeed )
that calculates the wind chill for a given temperature in °C, and wind speed in km/hr.
Restrictions:
WindSpeed  5.0 km/hr
Temperature should be between -50.0°C and +5.0°C.
More info:

10. Wind Chill Table Design the following algorithm GIVENS:
TempArray (an array of temperatures in °C)
NT (length of TempArray)
SpeedArray (an array of wind speeds in km/hr)
NS (length of SpeedArray)
RESULTS:
ChillTable (a matrix of wind chill values, with NT rows
and NS columns)
INTERMEDIATES
(to be determined)
HEADER:
ChillTable  WindChillTable( TempArray, NT, SpeedArray, NS)

11. Implementation in Java
Get the following files from the course webpage for lab 10
WindChill.java
ITI1120.java
The file WindChill.java already has a main( ) method as well as an implementation of the method
windChill( temperature, windSpeed )
The file WindChill.java contains the method
printTable( temperatureArray, speedArray, chillTable )
used to print the matrix.

12. Sample output Lowest value during winters 2003 and 2004 in Ottawa.
ITI 1120 Fall 2007, Lab 10, Example 1
Name: Diana Inkpen, Student#
Enter a set of temperatures between -50 and +5:
Enter a set of wind speeds:
Wind Chill Table
Speed:
Temp. |
5.0 |
0.0 |
-5.0 |
-10.0 |
-15.0 |
-20.0 |
-25.0 |
-30.0 |
-35.0 |
-40.0 |
Lowest value
during winters
2003 and 2004
in Ottawa.
Jan. 1994:
this value was
achieved in
Ottawa! (coldest
winter since 1948)

13. Example 2: Matrix Transpose
Write an algorithm that will take a matrix of integers A and transpose the matrix to produce a new matrix AT. The transpose of the matrix is such that element arc in the original matrix will be element aTcr in the transposed matrix. The number of rows in A becomes the number of columns in AT, and the number of columns in A becomes the number of rows in AT.
For example:

14. Translate to Java Translate your matrix transpose algorithm to Java
The header of the method should be as follows:
public static int[][] transpose( int[][] a )
Get the following file from the virtual campus for lab 10
MatrixLib.java
The file MatrixLib.java has the following useful methods:
Method to read a matrix of integers, of specified dimension.
public static int[][] readIntMatrix( int numRows, int numCols )
Method to print a matrix of integers with nice formatting.
public static void printMatrix( int[][] matrix )

15. Example 3: Matrix Multiplication
Suppose that A is an m × n matrix and B is an n × p matrix. The element at row i and column j in A is denoted by aij.
Let C = A × B. Then, C will be an m × p matrix, and for 0 ≤ i < m, and 0 ≤ j < p,
Write an algorithm to multiply two given matrices.

16. Translate to Java
Translate your matrix multiplication algorithm to Java
The header of the method should be as follows:
public static int[][] product( int[][] a, int[][] b )
Use the methods from MatrixLib.java to read and print the matrices.

17. [solutions and extra information follow this slide]

18. The Wind Chill calculation
Here is the complete wind chill algorithm, for your information. It has already been implemented in WindChill.java.
[Source: Environment Canada]
GIVENS:
Temperature (a temperature in °C between -50 and +5)
WindSpeed (a wind speed in km/hr – must be  5 km/hr)
RESULTS:
Chill (the wind chill value)
INTERMEDIATES
VFactor (the wind speed raised to the power 0.16)
HEADER:
Chill  WindChill( Temperature, WindSpeed )
BODY:
VFactor  (WindSpeed) 0.16
Chill  * Temperature * VFactor +
* Temperature * VFactor

19. Wind Chill Algorithm (page 1)
GIVENS:
TempArray (an array of temperatures in °C)
NT (length of TempArray)
SpeedArray (an array of wind speeds in km/hr)
NS (length of SpeedArray)
RESULTS:
ChillTable (a matrix of wind chill values, with NT rows
and NS columns)
INTERMEDIATES
T (index into temperature array)
S (index into speed array)
HEADER:
ChillTable  WindChillTable( TempArray, NT, SpeedArray, NS)

20. Wind Chill Algorithm (page 2)
ChillTable  MakeNewArray(NT)
T  0
T < NT
true
false
ChillTable[T]  MakeNewArray(NS)
S  0
true
false
S < NS
ChillTable[T][S]  WindChill(TempArray[T], SpeedArray[S])
S  S + 1
T  T + 1

21. Transpose Algorithm (page 1)
GIVENS:
A (an array)
NRows (number of rows in A)
NCols (number of columns in A)
RESULTS:
AT (the transposed array)
INTERMEDIATES
TRow (row index in transposed array)
TCol (column index in transposed array)
HEADER:
AT  Transpose( A, NRows, NCols )

22. Transpose Algorithm (page 2)
AT  MakeNewArray(NCols)
Row  0
TRow < NCols
true
false
AT[TRow]  MakeNewArray(NRows)
TCol  0
false
TCol < NRows
true
AT[TRow][TCol]  A[TCol][TRow]
TCol  TCol + 1
TRow  TRow + 1

23. Multiplication Algorithm (page 1)
GIVENS:
A (an array of numbers)
B (another array of numbers)
NARows (number of rows in matrix A)
NACols (number of columns in matrix A)
NBRows (number of rows in matrix B)
NBCols (number of columns in matrix B)
RESULTS:
C (a matrix that is the product of A and B, with
NARows rows and NBCols columns)
INTERMEDIATES
Row (row index for C)
Col (column index for C)
Index (summation index for matrix product)
HEADER:
C  Multiply( A, NARows, NACols, B, NBRows, NBCols)

24. Multiplication Algorithm (page 2)
C  MakeNewArray(NARows)
Row  0
Row < NARows
true
C[Row]  MakeNewArray(NBCols)
Col  0
false
Col < NBCols
true
false
Sum  0
Index  0
Index < NACols
true
false
C[Row][Col]  Sum
Col  Col + 1
Sum  Sum + A[Row][Index]×B[Index][Col]
Index  Index + 1
Row  Row + 1