MATLAB and R Programming (ME-705) - Important Questions

Write a MATLAB function file that accepts an $n \times n$ matrix (which may contain complex entries), extracts its main diagonal as a vector and returns the sum of the diagonal elements. The function should validate input dimensions and handle non-square inputs by returning an error message. Provide the function code and a short explanation of each step.

Write a MATLAB script to generate a random complex vector $z$ of length $100$, compute $\sum z$, and plot $\operatorname{real}(z)$ versus $\operatorname{imag}(z)$. Include proper axis labels and a title. Explain the purpose of each command used.

Using R, create a data frame with one numeric column and one factor column. Show how to append a new row to the data frame, and demonstrate how to subset only the rows where the factor has a specified level. Provide the R commands and brief comments explaining each step.

Explain the differences between MATLAB matrices and R data frames. Provide examples (short code snippets in MATLAB and R) showing a scenario where a MATLAB matrix is preferable and a scenario where an R data frame is preferable. Discuss implications for indexing, heterogenous data and common operations.

Demonstrate plotting in MATLAB: plot the parametric curve $x=\cos(t)$, $y=\sin(2t)$ for $t \in [0,2\pi]$. Label the axes, add a legend and title, and set the plot to have a grid. Provide the script and explain each plotting command.

Write a MATLAB script or function that reads a CSV file containing numeric columns, computes the column-wise mean (ignoring missing values), and writes the resulting vector of means to a new CSV file. Include error handling for non-numeric columns and describe how missing values (NaN) are handled.

Explain how to vectorize loops in MATLAB. Convert the following loop into vectorized form and provide both versions:

for i = 1:n y(i) = x(i)^2 + 3*x(i) + 5; end

Explain why the vectorized version is more efficient.

Write an R function that takes two numeric vectors (predictor and response) and fits a linear model using lm(). The function should return the estimated coefficients, residuals and produce a residual versus fitted values plot. Provide the function code and brief explanation of the outputs.

Describe how to debug MATLAB code using breakpoints and the debugger. Explain how to use the MATLAB profiler to identify performance bottlenecks. Include a short code example where you set a breakpoint and explain what information you inspect while debugging.

Demonstrate how to use MATLAB structures and cell arrays to store heterogeneous student records (name, ID, marks). Show how to create a structure array for three students and access the marks of the second student. Provide commands and brief explanations.

Write a MATLAB function that solves the system $Ax=b$ using LU decomposition (you may use MATLAB's lu() to obtain L and U). Compare the solution and computation time with MATLAB's backslash operator ($A\backslash b$) for a random dense matrix of size $500\times 500$. Present the code and a short analysis of results.

In R, demonstrate how to import data from an Excel file (using readxl or a similar package), identify and handle missing values (e.g., impute or remove), and export the cleaned data to CSV. Provide the R commands and brief explanation of choices for handling missing data.