MATLAB functions

Functions in MATLAB

• Functions are blocks of code designed for specific tasks, aiding in program complexity management.
• MATLAB offers built-in functions like sqrt, sin, and plot, and allows users to create custom functions.

Anatomy of a Function

• The function declaration starts with the keyword function.
• Output arguments are variables returned by the function, enclosed in square brackets [] (optional for single output).
• The function name should be descriptive and follow MATLAB conventions.
• Input arguments are values received by the function, enclosed in parentheses ().
• Comments describing the function are displayed with the help command.
• Statements are MATLAB commands executing the function's task.
• Function definition concludes with the keyword end.

Example: Circle Area Function

• A function to calculate the area of a circle takes the radius as input.
• It calculates area as area = pi * radius^2;.
• The function is saved in a file named circle_area.m and called from the command window or script.

Branching and Loops

• if statements, for loops, and while loops can be incorporated to create functions that adapt based on input or repeat actions.

Simulations

• Simulations are computer programs that model real-world systems.
• Systems too complex or dangerous for direct experimentation can be studied using these.

Simple Function: Fahrenheit to Kelvin

• fahr_to_kelvin converts temperature from Fahrenheit (temp_F) to Kelvin (temp_K).
• The conversion formula is $T_K = (T_F - 32) \times \frac{5}{9} + 273.15$.

Number of Days in a Month

• days_in_month calculates days in a given month using an integer input from 1 to 12.
• An if statement determines the number of days based on the month. Months 1, 3, 5, 7, 8, 10, and 12 have 31 days; 4, 6, 9, and 11 have 30 days; and 2 has 28 days (non-leap year).

Projectile Trajectory Simulation

• projectile_trajectory simulates a projectile's path, influenced by gravity.
• Inputs: initial speed v0 (m/s), launch angle theta (degrees), time step dt (s).
• Outputs: vectors x and y for coordinates, and t for time values.
• Convert theta to radians.
• Calculate initial x and y velocities (v0x and v0y) using v0 and theta.
• Initialize x and y coordinates to zero.
• Update position in a while loop at each time step.
• Update x coordinate using $x = v_{0x} \times t$.
• Update y coordinate using $y = v_{0y} \times t - \frac{1}{2} \times g \times t^2$ ($g = 9.81 m/s^2$).
• Stop when y becomes negative (projectile hits ground).
• Plot the trajectory where y is plotted against x, creating a display of the projectile's flight path over time.

Modifying Projectile Trajectory for Air Resistance

• The projectile_trajectory function is updated to incorporate air resistance.
• Air resistance is proportional to the square of the projectile's velocity: $F_D = \frac{1}{2} \times C_D \times \rho \times A \times v^2$
• New inputs: drag coefficient CD, cross-sectional area A. Air density $\rho$ is $1.225 kg/m^3$.
• Update x and y velocities at each time step with air resistance.
• $F_{Dx} = F_D \times \cos(\theta)$ and $F_{Dy} = F_D \times \sin(\theta)$.
• $a_x = \frac{F_{Dx}}{m}$ and $a_y = -g + \frac{F_{Dy}}{m}$ where m is mass (kg) and g is $9.81 m/s^2$.
• Update velocities: $v_x = v_x + a_x \times dt$ and $v_y = v_y + a_y \times dt$.
• Plot trajectory with and without air resistance on the same graph.

Linear Algebra

Definition of Vectors

• Vectors are ordered lists of numerical values.
• They are represented as column matrices: $\vec{a} = \begin{bmatrix} a_1 \ a_2 \ \vdots \ a_n \end{bmatrix}$.
• The $a_i$s are components, and $n$ is the vector space dimension.

Vector Operations

• Given vectors $\vec{a}$ and $\vec{b}$ of dimension $n$, and scalar $c$:
  • Addition: $\vec{a} + \vec{b} = \begin{bmatrix} a_1 + b_1 \ a_2 + b_2 \ \vdots \ a_n + b_n \end{bmatrix}$
  • Scalar Multiplication: $c \cdot \vec{a} = \begin{bmatrix} c \cdot a_1 \ c \cdot a_2 \ \vdots \ c \cdot a_n \end{bmatrix}$

Dot Product

• For vectors $\vec{a}$ and $\vec{b}$ of dimension $n$:
  • $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + \dots + a_nb_n = \sum_{i=1}^{n} a_i b_i$

Vector Norms

• Euclidean Norm (L2 Norm): $||\vec{a}||2 = \sqrt{a_1^2 + a_2^2 + \dots + a_n^2} = \sqrt{\sum{i=1}^{n} a_i^2}$
• Manhattan Norm (L1 Norm): $||\vec{a}||1 = |a_1| + |a_2| + \dots + |a_n| = \sum{i=1}^{n} |a_i|$

Definition of Matrices

• Matrices are rectangular arrays of numbers, symbols, or expressions, arranged in rows and columns.
• They are represented as: $A = \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \ a_{21} & a_{22} & \dots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & \dots & a_{mn} \end{bmatrix}$.
• $a_{ij}$ is the element in the $i$-th row and $j$-th column. $m$ is the number of rows, and $n$ is the number of columns.

Matrix Operations

• Given matrices $A$ and $B$ of compatible dimensions and a scalar $c$:
  • Addition: For $m \times n$ matrices $A$ and $B$, $(A + B){ij} = A{ij} + B_{ij}$.
  • Scalar Multiplication: $(cA){ij} = c \cdot A{ij}$.
  • Matrix Multiplication: For matrix $A$ ($m \times n$) and matrix $B$ ($n \times p$), $(AB){ij} = \sum{k=1}^{n} A_{ik} B_{kj}$

Special Matrices

• Identity Matrix: A square matrix with ones on the main diagonal and zeros elsewhere, denoted as $I$.
• Transpose of a Matrix: The transpose of matrix $A$, denoted as $A^T$, is obtained by interchanging rows and columns, such that $(A^T){ij} = A{ji}$.

Matrix Properties

• Commutative Property: For addition, $A + B = B + A$.
• Associative Property: For addition, $(A + B) + C = A + (B + C)$. For multiplication, $(AB)C = A(BC)$.
• Distributive Property: $A(B + C) = AB + AC$ and $(A + B)C = AC + BC$.

System of Linear Equations

• A set of linear equations can be represented as:
  • $\begin{cases} a_{11}x_1 + a_{12}x_2 + \dots + a_{1n}x_n = b_1 \ a_{21}x_1 + a_{22}x_2 + \dots + a_{2n}x_n = b_2 \ \vdots \ a_{m1}x_1 + a_{m2}x_2 + \dots + a_{mn}x_n = b_m \end{cases}$

Matrix Representation

• The system of linear equations can be represented in matrix form as: $Ax = b$.
  • $A$ is the coefficient matrix.
  • $x$ is the vector of variables.
  • $b$ is the vector of constants.

Solving Linear equations

• Gaussian Elimination: Solving linear equations by transforming the augmented matrix into row-echelon form or reduced row-echelon form.
• Inverse Matrix Method: If A is invertible, the solution to $Ax = b$ is $x = A^{-1}b$.

Definition of Eigenvalues and Eigenvectors

• Given a square matrix $A$, the eigenvector $\vec{v}$ and its corresponding eigenvalue $\lambda$ satisfy: $A\vec{v} = \lambda\vec{v}$. $\vec{v}$ is a non-zero vector and $\lambda$ is a scalar.

Characteristic Equation

• For finding eigenvalues, one must solve the characteristic equation: $\text{det}(A - \lambda I) = 0$, where $I$ is the identity matrix and $\text{det}$ is the determinant.

Properties of Eigenvalues/Eigenvectors

• The sum of the eigenvalues equals the trace of matrix $A$.
• The product of the eigenvalues equals the determinant of matrix $A$.

Applications of Linear Algebra

• Linear Regression: Used to solve problems, finding the best-fit line or hyperplane for data points.
• Principal Component Analysis (PCA): A dimensionality reduction technique that uses eigenvectors to find the principal data components of the data.
• Image Processing: Matrices represent images and linear algebra techniques are applied for transformations, filtering, and compression.
• Graph Theory: Adjacency and incidence matrices represent graphs. Analyzing graph properties utilizes linear algebra techniques.