CSE 20232 Lecture 4 – Algorithms & Decisions

CSE 20232Lecture 4 – Algorithms & Decisions • Algorithms • Basic Control Structures • Comparisons and if (…) statement • What is a function? • Math Library Functions • Character Functions • Reading • Sample Programs

Algorithms • What is an algorithm? • A precise description of the sequence of steps necessary to solve some task • Your programs begin as algorithms • Sample: finding absolute value • 1 – prompt user and get users input value • 2 – if value is negative then negate it • 3 – output result

Converting Algorithms to Code • Place algorithm in program shell as comments #include <iostream> using namespace std; int main () { // 1 - prompt user and get user’s input // 2 - if input value is negative then negate it // 3 - output result return 0; }

Converting Algorithms to Code • Then fill in C++ code for each algorithm step #include <iostream> using namespace std; int main () { int value; // 1 - prompt user and get user’s input cout << “Enter an integer value: “; cin >> value; // 2 - if input value is negative then negate it if (value < 0) value = -value; // 3 - output result cout << “Its absolute value is : “ << value << endl; return 0; }

Basic Control of Execution • There are three primary methods of controlling the flow of program execution • Sequence – this is the default • Selection – here a choice is made to • Perform a step or not • Perform one step or another • Repetition – also known as Iteration • Repeat a step over and over • Recursion also allows us to repeat steps, but we will wait to discuss this until later

Control structure example • Program that repeatedly calculates absolute values #include <iostream> using namespace std; int main () { int value; cout << “Enter an integer value (or q to quit): “; cin >> value; while (! cin.fail()) // <-- repetition { if (value < 0) // <-- selection value = -value; cout << “Its absolute value is : “ << value << endl; cout << “Enter an integer value (or q to quit): “; cin >> value; } return 0; }

Comparisons • Operators • (x < y) less than • (x <= y) less than or equal • (x > y) greater than • (x >= y) greater than or equal • (x == y) equal • (x != y) not equal

Logical (boolean) expressions • Operators • (x && y) x and y • (x || y) x or y –- (inclusive or) • (! x) not x • Precedence (highest to lowest) • ! • < <= > >= == != • && • ||

Comparisons & Logic • Comparisons and logical (boolean) expressions all evaluate to true or false • In C, zero (0) takes place of false and any non-zero value can represent true • Examples:x=2x=12 • (x < 5) true false • (1 <= x) && (x <= 10) true false • (x < 1) || (10 < x) false true • !(x == 5) == (x != 5) true true

Boolean Operation Truth Tables x y x && y x || y ! x ------- ------- ------- ------- ------- false false false false true false true false true true true false false true false true true true true false

If ( … ) statement • if ( … ) allows two methods of choosing what code to execute or not • This form either executes the <statement> or not, depending on the truth of the <expression> if ( <expression> ) <statement> • This form executes <statement 1> if <expression> is true, otherwise it executes <statement 2> if ( <expression> ) <statement 1> else <statement 2>

If ( … ) statement examples cin >> value; if (value < 0) value = -value; // skipped if value not negative if ((x < 1) || (10 < x)) cout << “x is out of range\n”; else cin << “x is in range\n”;

Code Blocks • A block of code is any sequence of statements between open and closed braces { } • A block can take the place of any single statement in C or C++ • A block may also have declarations of variables (objects) that exist only within the block • This is called local scope • Locally declared objects cease to exist when execution leaves the block

Code Block Example cout << “Enter a non-negative value: “; cin >> value; if (value < 0) { // give user a chance to correct the input cout << “That value is negative\n” << “Please enter a non-negative value: “; cin >> value; }

Nesting • Since if ( … ) is a statement itself, it can be nested inside other if statements if (a < 0) cout << “is negative”; else if (a > 0) cout << “is positive”; else cout << “is zero”;

Nesting Warning • The else is always paired with the closest preceding unpaired if • The following does not perform as the indentation would lead you to believe • When A is -15 output is is negative is very negative • When A is -8 output is is negative is non-negative • When A is 10 output is is non-negative if (A < 0) cout << “ is negative”; if (A < -10) cout << “ very negative”; else cout << “ is non-negative”;

Nesting Warning • The code below “fixes” the confusion, associates the else with the first if, and outputs appropriate descriptions of A • When A is -15 output is is negative is very negative • When A is -8 output is is negative • When A is 10 output is is non-negative if (A < 0) { cout << “ is negative”; if (A < -10) cout << “ very negative”; } else cout << “ is non-negative”;

What is a function? • A function is a named, parameterized block of code that computes a value or performs a task • We invoke a function by using its name with appropriate parameters as a statement in our code, or as part of an expression • Example: • double sqrt(double v); // prototype • The square root function from the math library • It computes the square root of the value of the parameter v and returns the result to the calling context • y = sqrt(x+37.5); // invocation

Math Library Functions#include <cmath> • abs(x) – absolute value of x • sqrt(x) – square root of x • pow(x,y) – xy • ceil(x) – smallest integer larger than x • floor(y) – largest integer smaller than x • exp(x) - ex • log(x) – natural log of x – ln(x) • log10(x) – log10(x) • sin(x) – sine of x, range +/- 1.0, x in radians • cos(x) – cosine of x, range +/- 1.0, x in radians • tan(x) – tangent of x, x in radians • asin(x) – arcsine of x, range +/- PI/2 • acos(x) – arccosine of x, range 0 to PI • atan(x) – arctangent of x, range +/- PI/2 • atan2(x,y) – arctangent of x,y, range +/ PI

Character Functions#include <cctype> • toupper(ch) – returns upper case equiv. of ch • tolower(ch) – returns lower case equiv. of ch • isalpha(ch) – true if ch is a letter a..z, A..Z • isdigit(ch) – true if ch is a digit 0..9 • There are more …

Reading (revised) • Continue reading in Deitel … • Week 2 • Sections 2.5-2.7, 6.1-6.6 (also Ch4 sections below) • Appendices B, C, E.1-E.2 • Week 3 • Sections 4.1-4.12, 15.1-15.8 • Appendices F.1-F.3

Trajectories General Position Calculations x = x0 + vx0 * t y = y0 + vy0 * t + 0.5 * g * t2 vy = vy0 + g * t Height = vy0 * tmid + 0.5 * g * tmid2 vy(tmid) = 0 = vy0 + g * tmid So, tmid = -vy0 / g v0 vy0 g = -32.0 ft/sec2 vx0 tof = 2 * tmid Range = vx0 * tof

Calculating Range of a Projectile • This is a parabolic trajectory under the influence of Earth gravity, but no air • Algorithm • 1 – prompt user and read launch angle in degrees • 2 – prompt user and read initial velocity in ft/sec • 3 – calculate vx & vy (initial velocity components) • 4 – calculate time of flight (no air, gravity -32 ft/s/s) • 5 – calculate range • 6 – output results

Calculating Range of a Projectile // range.cpp – JHS – 8/29/06 – CSE Notre Dame #include <iostream> #include <iomanip> #include <cmath> using namespace std; int main () { const double g = -32.0; const double rads_per_degree = 2 * 3.14159 / 360.0; double thetaD, thetaR, v0, vx0, vy0, t, time_of_flight; // 1 – prompt user and read launch angle in degrees cout << “ Enter angle of barrel above horizon (in degrees): “; cin >> thetaD; thetaR = thetaD * rads_per_degree; // convert to radians // 2 – prompt user and read initial velocity in ft/sec cout << “ Enter initial velocity of projectile (in ft/sec): “; cin >> v0;

Calculating Range of a Projectile // 3 – calculate vx & vy (initial velocity components) vx0 = v0 * cos(thetaR); vy0 = v0 * sin(thetaR); // 4 – calculate time of flight (no air, gravity -32 ft/s/s) t = -vy0 / g; time_of_flight = 2.0 * t; // 5 – calculate range, etc, and output results cout << fixed << setprecision(2); cout << “Projectile firing: Initial conditions\n”; cout << setw(10) << thetaD << “ degrees above horizon\n”; cout << setw(10) << v0 << “ ft/sec muzzle velocity\n”; cout << “Trajectory characteristics:\n”; cout << setw(10) << time_of_flight << “ seconds in flight\n”; cout << setw(10) << (vx0*time_of_flight) << “ feet down range\n”; cout << setw(10) << (vy0*t + 0.5*g*t*t) << “ feet max height\n”; return 0; }

Quadratic Polynomial Roots • Find solutions (roots) to 0 = ax2 + bx + c • There are several different cases • 1: a == 0, b != 0, there is one real root x = -c/b • 2: a == 0, b == 0, c != 0, nonsense equation with no roots • 3: a == 0, b == 0, c == 0, every value of x is a root • 4: a != 0, x = (-b +/- sqrt(b2 - 4ac))/2a, potentially two roots • (A) (b2 - 4ac) < 0, roots are complex not real • (B) (b2 - 4ac) == 0, there is one real root x = -b/2a • (C) (b2 - 4ac) > 0, there are two real roots • x1 = (-b + sqrt(b2 - 4ac))/2a • x2 = (-b - sqrt(b2 - 4ac))/2a

Quadratic Polynomial Roots • Algorithm: • 1 - prompt user and read three coeficients (a,b,c) • 2 - determine which case above (1,2,3,4) • 3 - in cases 1, 2, & 3 computer root and output result or output appropriate message • 4 - in case 4, calculate the value of b2 - 4ac and determine which case applies (A,B,C) • 5 - in case A output message • 6 - in cases B & C calculate root(s) and output results

Quadratic Polynomial Roots // roots.cpp – JHS - 2003 #include <iostream> #include <cmath> using namespace std; int main( ) { double a,b,c; // coefficients of quadratic equation double rad; // will be the value under the square root double root1,root2; cout << "Enter the values of a, b, and c for" << " the quadratic equation\n"; cout << " (ax^2 + bx + c) = 0\n"; cout << "> "; cin >> a >> b >> c; cout << "\nFor the equation (" << a << "x^2 + " << b << "x + " << c << " = 0)\n";

Quadratic Polynomial Roots if (a == 0.0) { if (b == 0.0) if (c == 0.0) cout << "Any value is a root of this equation\n"; else cout << "There is no root of this equation\n"; else { root1 = -c / b; cout << “One real root (x = " << root1 << ")\n"; } } else { rad = b * b - 4 * a * c;

Quadratic Polynomial Roots if (rad < 0.0) cout << " There are only complex roots\n"; else if (rad == 0.0) { root1 = -b / (2 * a); cout << “One real root (x = " << root1 << ")\n"; } else { root1 = (-b - sqrt(rad)) / (2 * a); root2 = (-b + sqrt(rad)) / (2 * a); cout << “Two real roots (x = " << root1 << " and x = " << root2 << ")\n"; } } return 0; }

CSE 20232 Lecture 4 – Algorithms & Decisions