Write A MATLAB Script To Solve The System Of Linear Equation

Write A Matlab Script That Will Solve the System Of Linear Equation

Write a Matlab script that will solve the system of linear equations shown below. This can be done by solving for the vector of unknowns (X) in the equation AX=B where A is the coefficient matrix and B is the vector on the right-hand side. In the script, also include code to verify that the result is correct. Figure 1: Results (X) and the solution to the equation to AX = B using X (stored in a variable called B2).

B2 is used to verify the results. Now, plot the lines for the equations in part 1. Plot the line for the 1st equation in blue and the line for the 2nd equation in green. Analyze the equations for “x” from 0 to 1 at an interval of 0.01. It should look like the plot shown. Use the data tab to verify your results. Label the “x” and “y” axes and title the figure as shown. Figure 2: Expected plot. The Fibonacci sequence is defined using the following equation: Given the following initial conditions, write a function that generates the first “n” Fibonacci numbers. Figure 3: Example output. HINT: you will have to solve for “y” in each equation of part 1 Use a for-loop.

Paper For Above instruction

The task involves developing multiple MATLAB scripts and functions to address a series of computational problems, including solving systems of linear equations, plotting equations, and generating Fibonacci numbers. Through these tasks, students will demonstrate proficiency in MATLAB programming, numerical analysis, and understanding of mathematical sequences.

Introduction

The first task in this assignment involves solving a system of linear equations using MATLAB. Such systems are commonplace in engineering and scientific computations and solving them efficiently is critical. MATLAB offers robust tools for solving these systems and verifying solutions, as well as visualizing the relationships between variables. The subsequent parts extend the problem-solving skills by involving plotting techniques and sequence generation, reinforcing computational thinking in MATLAB.

Solving the System of Linear Equations

Suppose the system is represented as AX = B, where A is a known coefficient matrix, B is a known right-hand side vector, and X is the vector of unknowns. MATLAB's backslash operator (\) provides an effective method to solve such systems. For example, given a matrix A and vector B, the solution can be obtained as X = A\B. To verify the accuracy, the product A*X can be compared to B; identical results indicate correctness.

Implementing this, the script defines A and B, computes X, and then verifies by calculating the residual norm. This residual measures the difference between A*X and B. A small residual confirms the correctness of the solution.

Plotting the Equations

Using the solutions from the previous step, MATLAB's plotting capabilities allow visualization of lines represented by equations. Plotting the first equation in blue and the second in green over a range of x from 0 to 1 with an interval of 0.01 enables students to see the intersection point visually. Labeling axes and titling the figure clarifies the graph purpose.

Generating Fibonacci Numbers

The Fibonacci sequence is a well-known series where each number is the sum of the two preceding ones, starting with initial values. Creating a function in MATLAB involves implementing a for-loop to generate the sequence up to n terms. Beginning with predefined initial numbers, subsequent Fibonacci numbers are calculated iteratively and stored in an array.

This sequence generation reinforces the understanding of loops, indexing, and recursive sequences in MATLAB.

Conclusion

This assignment develops core MATLAB skills, including solving linear systems, plotting functions, and creating sequences. These skills are fundamental for engineers and scientists, fostering analytical thinking and computational proficiency. Proper verification and visualization enhance understanding, ensuring students grasp both the numerical and graphical aspects of mathematical problems.

References

  • Higham, N. J., & Higham, D. J. (2016). MATLAB Guide (3rd ed.). SIAM.
  • MathWorks. (2023). MATLAB documentation. https://www.mathworks.com/help/matlab/
  • Chapra, S. C., & Canale, R. P. (2015). Numerical Methods for Engineers (7th ed.). McGraw-Hill Education.
  • Burden, R. L., & Faires, J. D. (2010). Numerical Analysis (9th ed.). Brooks Cole.
  • Brown, C. (2017). Introduction to MATLAB Programming. Journal of Computing Sciences in Colleges, 33(2), 152-159.

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