Problem Solving

1. Problem Solving • We now have enough tools to start solving some problems. • For any problem, BEFORE you start writing a program, determine: • What are the input values that are needed from the user? • What results (outputs) do we need to determine? • What other values are needed? • math constants: , e, etc. • physical constants • What temporary values might need to be calculated? • How do we calculate the results?

2. Example 1 • Calculate the area of a circle from its radius • what input values do we need? • what other values do we need? • what is our output? • how do we calculate the result?

3. Example 2 • Calculate the distance travelled by an object under constant acceleration, given a specified time. • What input values do we need? • What other values do we need? • What is the result? • How do we calculate the result?

4. Example 3: • Carbon-14 dating • General radioactive decay equation: • Q0: initial quantity of radioactive substance • Q(t): quantity of radioactive substance at time t (in years) • : radioactive decay constant for a specific element • Q(t) = Q0e–t is the rate of decay equation • Carbon-14 is continuously taken in by a living organism, and the amount of material at death is assumed to be known. • Once the intake stops, the carbon-14 decays at the above rate, where  = 0.00012097 / year • Measure the percentage of carbon-14 remaining

5. Example 3 continued: • Solve for t to determine the age: • Another useful thing to know: if half of the Carbon-14 remains, the sample is about 5730 years old. • Create a Java program to determine the age: • what are the inputs? • what are the outputs? • what other values do we need? • how do we calculate the formula?

6. Expressions with mixed data types • [See text, sections 2.4.4 and 2.4.5] • In general, you cannot mix data types in Java for either variables or constants. • However, for numeric data types (int, long, float, double), there are some permitted exceptions. • Example: double aVariable = 1 + 2.3; • The int constant 1 is temporarily converted to the double value 1.0, and aVariable is assigned the value of 1.0 + 2.3, namely 3.3. • In a mixed expression, the value for a more restrictive data type (e.g. int) is temporarily converted to a less restrictive data type (e.g. double) • You cannot assign a value from a less restrictive data type to a more restrictive data type (example: double to int is not permitted)

7. Mixed expression examples • Be careful! What are the values of the following, and what is the type of the result?

8. Problem Solving and Program Design • [now at chapter 3 in the text] • Requirements • Design • Coding • Testing and/or verification • Maintenance • All of the above should be documented as you go along.

9. Requirements • Requirements are the elements of the problem related to a user. • Challenges with requirements: • Need to describe what is the problem, NOT how to solve it. • Requirements are often described imprecisely, or are incomplete. • Such requirements often lead to problems later on because the program designer made an assumption that was different from the users. • Often described using scenarios (sample user sessions)

10. Requirements • Two types of requirements: • Functional: requirements directly related to the problem at hand • example: determine age of sample given carbon-14 percentage • Non-functional: other indirect requirements • examples: requirements related to usability, performance, reliability, use of specific hardware • Requirements should be verifiable: it should be possible to determine clearly whether or not the requirement has been met. • Bad example: program should be “fast” • Good example: program should run scenario X in no more than 2 seconds.

11. Design • This is where the problem is solved. • The solution has to take into account: • Data: structure and format of data related to the problem and solution • Algorithm: sequencing of steps needed to solve the problem • Events: occurrences to which the program must react • Constraints: limitations of resources that are available (e.g. amount of memory, processor speed) • Several potential solutions may be available, and the choice would be made in terms of memory efficiency, speed, ease of implementation, etc.

12. Coding • This is where the program is created. • Once you are familiar with a programming language, only about 10-15% of the time involved in problem solving should be used for coding. • The real thinking should be during the design phase and translating a design to program code should be almost mechanical.

13. Testing and Verification • Testing: running a set of experiments to see if the program works • Run the program with input data, and see if it produces the expected output • Does not prove that the program always works • Verification: determining that the program will run in all cases • Extremely difficult to do; usually only attempted for small critical sections of software (e.g. nuclear plant control, flight control) • Requires either a mathematical proof, or model-checking techniques

14. Testing • Can be a large part of time and cost of developing a program. • Testing the program is separate from finding the actual cause of any errors that are found. • “Debugging”: determining what caused a problem • Two types of testing: • “Black-box”: the testing assumes no knowledge of how the program works; inputs are given to the program and the output is checked. • “White-box”: testing is based on knowledge of the program; objective is to ensure that all parts of the program have been used

15. Maintenance • Once a program has been developed, the program will often be modified. • Resolving problems found by users. • Additional features • Performance improvements • Moved to different system • Difficulties with maintenance: • Often not done by original designer • Can be done in a hurry, when related to an urgent user problem. • Care must be taken that changes made do not break other parts of the program that were working.

16. Selection of alternatives • An important control structure in any programming language is the selection of alternatives. • The selection is normally based on the result of a Boolean expression. • Example: Sell a person an adult movie ticket if their age is at least 18 and less than 65. • A Boolean test will have two possible outcomes: true or false. The program may have different statements to execute for each of the two alternatives.

17. Branching Control Structure Test false true program code program code • With a branch, you are choosing one of two alternatives. • After doing one of the alternatives, the flow of control of the program comes back together.

18. Test false true block 1 block 2 Branches in Java • Java : if ( /* put test here */ ) { // block2 } else { // block1 }

19. Blocks • Whenever we need to identify a group of statements in Java, the statements are enclosed in braces. This is called a “block”. • A block can be used anywhere in Java where a single statement is appropriate. • Blocks are particularly used as: • alternatives in branches • sections of code to repeat, when we see how to do this.

20. Indentation of block statements • When starting a new block, the convention is to indent all code inside the block, so that the reader can easily identify which statements are included in the block: if ( /* put test here */ ) { // block 1, statement 1 // block 1, statement 2 // block 1, statement 3 } else { // block 2, statement 1 // block 2, statement 2 }

21. Expressions in Tests • The TEST in a branch may be any Boolean expression: • Boolean variable • Negation of a Boolean expression • NOT (Java: ! ) • Comparison between two values • Java operators: == != < > <= >= • The data being compared may not necessarily beboolean, but the result of the comparison isboolean • Join two Boolean expressions • AND (Java: && ) • OR (Java: ||)

22. Expressions in Tests • Watch out for • confusing = with == • confusing AND with OR • e.g. test if X is in the range 12-20: X >= 12 && X <= 20

23. Example: Maximum of 2 values • How would we write some code to find the maximum of two int values, x and y? The maximum value is to be stored in variable max. int x; int y; int max; System.out.println(“Enter value for x: “); x = Keyboard.readInt(); System.out.println(“Enter value for y: “); y = Keyboard.readInt();

24. Example: Solution of Quadratic Equations • We want to solve for real values of x, for given values of a, b, and c. • The value of x can be determined from the “well-known” formula: • Depending on the value of the discriminantwe have 3 cases: • two real values for x • one real value for x • no real values for x

25. Flow diagram b2 – 4ac > 0 True False b2 – 4ac < 0 False True x1 = –b/2a No solutions

26. Flow diagram, version 2 d = b2 – 4ac False True d > 0 d < 0 False True x1 = –b/2a No solutions

27. Adding the Complex case • Suppose we now want to solve for real or complex values of x, for given values of a, b, and c. • In this case, we can still use the formula but when we have we will need to be careful to avoid taking a square root of a negative number. • If we take the square root of the absolute value of the discriminant, we have the complex coefficient

28. Flow diagram d = b2 – 4ac False d > 0 True d < 0 True False x1 = –b/2a

29. Example: Solution of Quadratic Equations • We want to solve for real values of x, for given values of a, b, and c. • The value of x can be determined from the “well-known” formula: • Depending on the value of the discriminantwe have 3 cases: • two real values for x • one real value for x • no real values for x

30. Flow diagram b2 – 4ac > 0 True False b2 – 4ac < 0 False True x1 = –b/2a No solutions

31. Flow diagram, version 2 d = b2 – 4ac False True d > 0 d < 0 False True x1 = –b/2a No solutions

32. Structure of nested if statements discriminant = b * b - 4.0 * a * c; if ( discriminant > 0 ) { // We have two real solutions } else { if ( discriminant < 0 ) { // We have no real solutions. } else { // We have one real solution. } }

33. Adding the Complex case • Suppose we now want to solve for real or complex values of x, for given values of a, b, and c. • In this case, we can still use the formula but when we have we will need to be careful to avoid taking a square root of a negative number. • If we take the square root of the absolute value of the discriminant, we have the complex coefficient

34. Flow diagram d = b2 – 4ac False d > 0 True d < 0 True False x1 = –b/2a

35. Testing for Equality withFloating Point Values • Because of the possibility of round-off error, testing for equality (or inequality) of float or double values is NOT recommended. • Instead, test that the value is “close enough” to the desired value by using a tolerance value, and checking that the absolute value of the difference is less than the tolerance: double EPSILON = 1.0e-10; double x; ... if ( x == 37.0 ) // not recommended ... if ( Math.abs( x – 37.0 ) < EPSILON ) // better

36. The switch statement • Allows for multi-way branches • You can only use case statements when the test is on an integer, Boolean, or character variable. • Idea: • List various “cases” for specific values of the tested variable, plus one default case for when there is no match. • Provide a statement block for each case. • You MUST end each statement block with break; or the program will go to the next following case.

37. The switch statement true case 1 statement block 1 break; false true case 2 statement block 2 break; false true case n statement block n break; false defaultstatement block

38. Example System.out.println(“Enter an integer from 1 to 3: “); int a = Keyboard.readInt(); switch( a ) { case 1: { System.out.println(“You entered a 1.”); break; } case 2: { System.out.println(“You entered a 2.”); break; } case 3: { System.out.println(“You entered a 3.”); break; } default: { System.out.println(“Incorrect input.”); break; } System.out.println(“After switch.”); }

39. Missing break statement true case 1 statement block 1 false true case 2 statement block 2 break; false true case n statement block n break; false defaultstatement block