Numerical Computing Final Exam Prep

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Questions and Answers

What type of method is Gaussian elimination considered?

A direct method with finite precision in theory.

What does LU factorization decompose a matrix into?

Lower and upper triangular parts.

Which strategy can resolve numerical issues in LU factorization?

Pivoting strategies.

What impedes performing naive Gaussian elimination?

Having a row with all zero coefficients. Signup and view all the answers

How many cubic polynomials are used to construct a cubic spline with n data points?

n - 1. Signup and view all the answers

What is the primary use of a cubic spline?

Interpolation. Signup and view all the answers

What condition must the spline satisfy at each data point in cubic spline interpolation?

Both first and second derivatives must be continuous. Signup and view all the answers

Which of the following matrices is a permutation matrix?

P1. Signup and view all the answers

Explain why SVD is beneficial for image compression and identify the primary reason.

SVD is beneficial for image compression because it separates image information into components with varying importance, allowing for the removal of less significant components to reduce storage requirements. Signup and view all the answers

What condition must a square matrix meet to be considered invertible?

A square matrix is invertible if and only if its columns (or rows) are linearly independent. Signup and view all the answers

Is it true that a matrix with a determinant of 0 is always invertible? Explain your reasoning.

No, a matrix with a determinant of 0 is not invertible, as it indicates that the matrix's rows or columns are linearly dependent. Signup and view all the answers

Describe the significance of reducing computational cost in image storage.

Reducing computational cost in image storage is significant because it optimizes the use of memory and processing resources, allowing for faster access and manipulation of images. Signup and view all the answers

What is the relationship between the number of rows and columns in a matrix for it to be considered for inversion?

For a matrix to be considered for inversion, it must have an equal number of rows and columns. Signup and view all the answers

What is the primary objective of least squares approximation?

To minimize the sum of the squares of the residuals. Signup and view all the answers

What does the failure of Cholesky factorization of a matrix A indicate?

A is not positive definite. Signup and view all the answers

Which iterative method can be utilized for solving linear systems?

Jacobi method. Signup and view all the answers

Which of the following methods is known for numerical stability?

LU factorization. Signup and view all the answers

What is a crucial factor in deciding when pivoting is needed?

Condition number. Signup and view all the answers

Which method converges faster, Jacobi or Gauss-Seidel?

Gauss-Seidel converges faster. Signup and view all the answers

Under what conditions can the pseudo-inverse of a matrix be the same as its classical inverse?

Conditionally True. Signup and view all the answers

What are the two matrices that QR factorization decomposes a matrix A into?

Orthogonal matrix and upper triangular matrix. Signup and view all the answers

What is the purpose of the swap operations in the given code segment?

The swap operations are used to interchange rows of matrices U and P to maintain the correct row order during LU decomposition. Signup and view all the answers

In the context of Gauss-Seidel iteration, how does the approximate solution evolve typically?

The approximate solution evolves by using the most recent values to compute the next estimate, leading to potentially faster convergence. Signup and view all the answers

What is the significance of the norm calculation ∥x(2) − x(1)∥2 in iterative methods?

The norm calculation assesses the difference between successive approximations, indicating the convergence of the iterative method. Signup and view all the answers

What does the line 'np.fill_diagonal(L, 1)' imply about the matrix L?

It implies that the diagonal elements of matrix L are set to 1, making L a unit lower triangular matrix. Signup and view all the answers

What essential conditions must be fulfilled to apply the Gauss-Seidel method successfully?

The matrix A must be diagonally dominant or symmetric positive definite for the Gauss-Seidel method to converge reliably. Signup and view all the answers

How does the loop 'for j in range(i + 1, n)' contribute to the LU decomposition process?

This loop iterates through the rows below the current row i, updating matrix L and adjusting matrix U to eliminate variables. Signup and view all the answers

Why is it necessary to swap rows in the matrices L and U during LU factorization?

Swapping rows is necessary to maintain the correct structure of L and U and to ensure numerical stability during the factorization. Signup and view all the answers

How do you extract the Red channel from an image in the given code?

You can extract the Red channel using <code>Red[:, :, 0] = photo[:, :, 0]</code>. Signup and view all the answers

What would be the effect of not including the condition 'if i > 0' in the swapping logic for L?

Not including that condition could lead to unnecessary swaps involving the first row, potentially disrupting the structure of L. Signup and view all the answers

What is the purpose of performing SVD on each color channel?

The purpose of performing SVD is to retrieve the corresponding U, S, and V components for each channel, enabling compression. Signup and view all the answers

What does the variable `k` represent in the context of the code?

<code>k</code> represents the number of singular values to be used for compression. Signup and view all the answers

How are the compressed components constructed for the Red channel?

The compressed components are constructed using <code>U_r_c</code>, <code>V_r_c</code>, and <code>S_r_c</code> by selecting the first <code>k</code> dimensions. Signup and view all the answers

What function is used to perform matrix multiplication to reconstruct each channel back?

The function <code>np.dot()</code> is used for matrix multiplication to reconstruct each channel. Signup and view all the answers

What does the code `comp_img[comp_img < 0] = 0` accomplish?

This code clips any values in the <code>comp_img</code> matrix that are less than 0, setting them to 0. Signup and view all the answers

Describe how the final computed image is assembled from the channel components.

The final image is assembled by assigning each processed color channel to the corresponding index in <code>comp_img</code>. Signup and view all the answers

What does the command `plt.imshow(comp_img)` do?

The command <code>plt.imshow(comp_img)</code> displays the reconstructed image using the computed image matrix. Signup and view all the answers

What is the purpose of using the Gram-Schmidt process in vector analysis?

The Gram-Schmidt process is used to generate an orthogonal or orthonormal set of vectors from a given set of vectors, which helps in simplifying calculations in linear algebra. Signup and view all the answers

In the provided Python code, what is the function of 'np.tril(A)'?

'np.tril(A)' extracts the lower triangular part of the matrix 'A'. Signup and view all the answers

How can the convergence of the solution in the iterative method be assessed from the provided code?

Convergence is assessed by checking if the error 'err' is less than the tolerance 'tol'. Signup and view all the answers

What is the output of the Gram-Schmidt process applied to the given vectors x1, x2, and x3?

The output contains orthogonal vectors u1, u2, and u3 derived from x1, x2, and x3. Signup and view all the answers

What does the Python function 'np.copy()' accomplish in the provided code?

'np.copy()' creates a copy of the input array, which prevents modifications to the original array. Signup and view all the answers

Explain the significance of setting 'maxit' in the iteration loop of the provided code.

'maxit' limits the number of iterations to prevent endless looping in cases where convergence is not achieved. Signup and view all the answers

Why is it essential to compute orthonormal vectors from orthogonal vectors?

Computing orthonormal vectors from orthogonal vectors ensures that the vectors have unit length, which is crucial for applications in numerical methods and algorithms. Signup and view all the answers

Describe the role of 'np.linalg.inv(M)' in the iterative method provided in the code.

'np.linalg.inv(M)' computes the inverse of the matrix M, which is necessary for updating the approximation in the iterative method. Signup and view all the answers

Study Notes

Numerical Computing Exam Notes

Final Exam: 3 hours, 84 marks, 4 questions
Date: May 21, 2024
Instructors: Mukhtar Ullah, Muhammad Ali, Imran Ashraf, Almas Khan

Question 1 (a)

Task: Perform LU factorization manually to find P, L, and U matrices
Matrix: Given problem matrix in the question
Solution Method: Row operations for Upper Triangular matrix, Collect multipliers for Lower Triangular matrix
Details: Includes the row operations, resulting in the upper triangular matrix
Note: Pivoting not necessary

Question 1 (b)

Task: Write Python code for LU decomposition for general Matrix A
Method: Uses partial pivoting
Python Instructions: Python code provided, includes the code components for LU decomposition.

Question 2

Task: Approximate solution of linear system using Gauss-Seidel iterative method
Input Data: Matrix A, Vector b, Initial guess x(0).
Solving steps: Show the calculation steps for each iteration
Convergence: Calculate the difference between consecutive approximations until convergence is reached (determined by a tolerance).

Question 2 (b)

Task: Calculate the Euclidean norm of the difference between the solution in iteration 2 and 1

Question 3 (a) i

Task: Use Gram-Schmidt process to get orthogonal vectors (u1, u2, u3) from given vectors, x1, x2, x3.

Question 3 (a) ii

Task: From the orthogonal vectors, calculate orthonormal vectors (v1, v2, v3)

Question 3 (b)

Python Code Comments: Comments (numbered 1 to 10) are present in the code for proper understanding of the code.

Question 4

Gaussian Elimination: A (direct) method that can be performed with finite precision, used for solving linear systems
Iterative Method: Methods like Gauss-Seidel that perform multiple iterations converge toward a solution (finite vs infinite precision)
LU Factorization: Matrix decomposition into lower (L) and upper (U) triangular matrices for solving linear systems
Pivoting Strategy: A method used to resolve numerical instabilities during LU factorization, involving swapping rows.

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Description

Prepare for your Numerical Computing final with this comprehensive quiz covering LU factorization, Python implementation for LU decomposition, and the Gauss-Seidel iterative method. Review the key concepts and practice your coding skills before the exam on May 21, 2024.