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9 Pages

### 7. Predicates and Loops

Course: ENGR 101, Fall 2010
School: Michigan
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Word Count: 821

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101 ENGR Sec. 100 Fall 10 Predicates and Loops ENGR 101, Lecture 7: 29 Sep 10 Announcements Project 3 due Wed 9PM (29 Sep) Exam #1 Tue 5 Oct Prof. M. Wellman sample exams available this weekend take practice gateway exam for participation credit review session: Monday lecture 1 ENGR 101 Sec. 100 Fall 10 How Many Hours Left to Work? int workHoursLeft(int date, int dueDate, int firstDay) // calculates #...

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101 ENGR Sec. 100 Fall 10 Predicates and Loops ENGR 101, Lecture 7: 29 Sep 10 Announcements Project 3 due Wed 9PM (29 Sep) Exam #1 Tue 5 Oct Prof. M. Wellman sample exams available this weekend take practice gateway exam for participation credit review session: Monday lecture 1 ENGR 101 Sec. 100 Fall 10 How Many Hours Left to Work? int workHoursLeft(int date, int dueDate, int firstDay) // calculates # work hours before project is due, // given month started on firstDay { int hours = 0; while (date <= dueDate) { if (date is not a weekend day) hours += 10; date++; } return hours; } Su M T W Th F Sa 1 2 345 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 1 2 3 4 5 6 0 What Day is date? (firstDay 1 + date) % 7 firstDay is 3 Su M T W Th F Sa 1 2 345 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 1 Prof. M. Wellman 2 3 4 5 6 0 2 ENGR 101 Sec. 100 Fall 10 Testing for the Weekend bool weekendDay(int date, int firstDay) // true if date falls on the weekend, // given that month started on firstDay { int day; day = (firstDay 1 + date) % 7; Su M return (day == 0 or day == 1); } 67 T W Th F Sa 1 2 345 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 1 2 3 4 5 6 0 How Many Hours Left to Work? int workHoursLeft(int date, int dueDate, int firstDay) // calculates # work hours before project is due, // given month started on firstDay { int hours = 0; while (date <= dueDate) { if ( not weekendDay(date,firstDay) ) hours += 10; date++; } return hours; } Prof. M. Wellman 3 ENGR 101 Sec. 100 Fall 10 Predicates Functions that return bool (Boolean) values are called predicates. Used in condition expressions. Write some predicates: bool twothreefive(int x); // true iff x divisible by 2, 3, or 5 bool sorted(double x, double y, double z); // true iff args given in a sorted order Which is Correct? // true iff x divisible by 2, 3, or 5 A bool twothreefive (int x) { return ((x % 2) or (x % 3) or (x % 5)); } B bool twothreefive (int x) { return ((x % 2) and (x % 3) and (x % 5)); } C bool twothreefive (int x) { return (not(x % 2) or not(x % 3) or not(x % 5)); } D bool twothreefive (int x) { return (!(x % 2) and !(x % 3) and !(x % 5)); } Prof. M. Wellman 4 ENGR 101 Sec. 100 Fall 10 Which is Correct? // true iff args given in a sorted order A bool sorted(double x, double y, double z) { return ((x >= y and y >= z) or (x <= y and y <= z)); } B bool sorted(double x, double y, double z) { return ((x >= y or y >= z) or (x <= y or y <= z)); } C double sorted(double x, double y, double z) { return ((x y >= and y >= z) or (x <= y and y <= z)); } D double sorted(double x, double y, double z) { return ((x >= y or y >= z) or (x <= y or y <= z)); } Analyzing Loops Loops solve a problem through a series of repeated steps. Each step represents progress toward goal (solution) accumulating a result over a series of terms processing a sequence of inputs successive approximation Start Prof. M. Wellman ? Goal 5 ENGR 101 Sec. 100 Fall 10 Factorial Revisited // factorial, assumes N is positive int fact( int N ) { int count=1, f=1; while (count < N) { count = count + 1; f = f * count; } return f; } Start loop body moves the data toward the goal f accumulates result over a series of terms Goal f = N! Factorial Loop Invariant // factorial, assumes N is positive Start int fact( int N ) { int count=1, f=1; established at initialization while (count < N) { loop invariant: count = count + 1; f = count ! f = f * count; } preserved each iteration return f; } corresponds to goal at loop termination Prof. M. Wellman Goal f = N! 6 ENGR 101 Sec. 100 Fall 10 Example: Calculate Weeks to Produce PCs Loop Invariant Initialize PCs to make, chips on hand, weeks How many PCs can we make this week? More PCs to make? weeks to produce = weeks so far + weeks to produce remaining quota with chips on hand LOOP Increment weeks Deliver more chips Decrement PCs to make Decrement chips on hand Example: Power xy // Compute x to the y power double power(double x, int y) { double z = 1; What is the loop int n = 0; invariant? while (n < y) { z = z * x; n++; } return z; } Prof. M. Wellman Checklist: initialization preservation in loop termination 7 ENGR 101 Sec. 100 Fall 10 Example: Power xy // Compute x to the y power double power(double x, int y) { double z = 1; int n = 0; power (4,3) while (n < y) { z = z * x; n++; } return z; } x y z n 4 3 1 0 4 3 4 1 4 3 16 2 4 3 64 3 initialization termination n=y Iterative Approximation double squareRoot(double x, double eps) { double guess = 1.0; double residual = abs(guess * guess - x); termination while ( residual > eps ) { Loop invariant guess = newGuess(guess,x); residual = abs(guess * guess - x); } return guess; } Goal: square of return value within eps of x adapted from Holloway Sec. 3.6 Prof. M. Wellman 8 ENGR 101 Sec. 100 Fall 10 Improving the Guess double newGuess(double oldGuess, double x) // improve the guess of sqrt(x) { return (oldGuess + x/oldGuess) / 2 ; } Is this progress? How do we know? adapted from Holloway Sec. 3.6 Next Lecture Prof. M. Wellman Exam Review Then more loops 9
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