Exercise 8

Added Exercise 8 to repository

Informatik_I/Exercise_8/Task_4/README.org

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+#+title: Task 4: Dual Numbers
+
+#+author: JirR02
+* Dual Numbers
+:PROPERTIES:
+:CUSTOM_ID: dual-numbers
+:END:
+In this task, you are required to implement a basic version of /dual
+number/. Dual numbers are conceptually similar to complex number.
+
+A dual numbers \(n\) is of the following form:
+
+\[
+z = a + \epsilon b
+\]
+
+with the following identity:
+
+\[
+\epsilon ^2 = 0
+\]
+
+Adding two dual numbers is done component-wise:
+
+\[
+(a + \epsilon b) + (c + \epsilon d) + (a+c) + \epsilon (b + d)
+\]
+
+Subtracting two dual numbers is also done component-wise:
+
+\[
+(a + \epsilon b) - (c + \epsilon d) = (a + \epsilon b) + (-c - \epsilon d) = (a - c) + \epsilon(b - d)
+\]
+
+Using the identity above, multiplying two dual numbers gives
+
+\[
+(a + \epsilon b)(c + \epsilon d) = ac + \epsilon(ad + bc)
+\]
+
+** Tasks
+:PROPERTIES:
+:CUSTOM_ID: tasks
+:END:
+You are required to extend the provided implementation of dual numbers
+by adding the implementation for various arithmetic operators in
+=DualNumber.cpp=.
+
+*** Task 1: Addition operator
+:PROPERTIES:
+:CUSTOM_ID: task-1-addition-operator
+:END:
+Implement the addition operator
+
+#+begin_src shell
+DualNumber operator+(const DualNumber& dn1, const DualNumber& dn2)
+#+end_src
+
+in file =DualNumber.cpp=.
+
+*** Task 2: Subtraction operator
+:PROPERTIES:
+:CUSTOM_ID: task-2-subtraction-operator
+:END:
+Implement the subtraction operator
+
+#+begin_src shell
+DualNumber operator-(const DualNumber& dn1, const DualNumber& dn2)
+#+end_src
+
+in file =DualNumber.cpp=.
+
+*** Task 3: Multiplication operator
+:PROPERTIES:
+:CUSTOM_ID: task-3-multiplication-operator
+:END:
+Implement the multiplication operator
+
+#+begin_src shell
+DualNumber operator*(const DualNumber& dn1, const DualNumber& dn2)
+#+end_src
+
+in file =DualNumber.cpp=.
+
+* Solution
+:PROPERTIES:
+:CUSTOM_ID: solution
+:END:
+#+begin_src cpp
+#include "DualNumber.h"
+
+DualNumber operator+(const DualNumber &dn1, const DualNumber &dn2) {
+  DualNumber res;
+  double a = dn1.a + dn2.a;
+  double b = dn1.b + dn2.b;
+  res.a = a;
+  res.b = b;
+  return res;
+}
+
+DualNumber operator-(const DualNumber &dn1, const DualNumber &dn2) {
+  DualNumber res;
+  double a = dn1.a - dn2.a;
+  double b = dn1.b - dn2.b;
+  res.a = a;
+  res.b = b;
+  return res;
+}
+
+DualNumber operator*(const DualNumber &dn1, const DualNumber &dn2) {
+  DualNumber res;
+  double a = dn1.a * dn2.a;
+  double b = (dn1.a * dn2.b) + (dn1.b * dn2.a);
+  res.a = a;
+  res.b = b;
+  return res;
+}
+#+end_src
+
+--------------
+
+Made by JirR02 in Switzerland 🇨🇭