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Algorithms Basic Control Structures Comparisons and if (…) statement What is a function? Math Library Functions Character Functions Reading Sample Programs

CSE 20232CSE 20232Lecture 4 – Algorithms & Decisions Lecture 4 – Algorithms & Decisions

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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

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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;}

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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;}

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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

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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;}

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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

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Logical (boolean) expressions

Operators (x && y) x and y (x || y) x or y –- (inclusive or) (! x) not x

Precedence (highest to lowest) ! < <= > >= == != && ||

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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=2 x=12 (x < 5) true false (1 <= x) && (x <= 10) true false (x < 1) || (10 < x) false true !(x == 5) == (x != 5) true true

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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

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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>

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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”;

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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

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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;

}

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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”;

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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”;

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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”;

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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

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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

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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 …

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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

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Trajectories

Range = vx0 * tof

vy0

vx0

v0

Height = vy0 * tmid + 0.5 * g * tmid2

vy(tmid) = 0 = vy0 + g * tmid

So, tmid = -vy0 / g

g = -32.0 ft/sec2

General Position Calculations

x = x0 + vx0 * t

y = y0 + vy0 * t + 0.5 * g * t2

vy = vy0 + g * t

tof = 2 * tmid

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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

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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;

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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;}

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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

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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

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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";

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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;

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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;}