Numerical Computing Final Exam Prep

A numerical method used to solve systems of linear equations by iteratively refining an initial guess. It updates each unknown variable using the latest values of the other variables in the system, until a specified error tolerance is met.

In the context of numerical methods like Gauss-Seidel, the error tolerance is a threshold for the difference between successive iterations of a solution. It determines how accurate the approximation must be to stop the iterative process.

A numerical method for LU factorization, which decomposes a matrix into two triangular matrices (Lower Triangular - L and Upper Triangular - U). This helps solve linear systems iteratively.

A vector containing the unknown variables in a system of linear equations. It is the solution that satisfies the equations.

A matrix representing the coefficients of the variables in a system of linear equations.

A vector containing the constant terms (right-hand side) on the equations in a system of linear equations. It represents the target values the system aims to satisfy.

The 2-norm of a vector is a measure of its length. It calculates the square root of the sum of squared elements in the vector.

The algorithm updates each unknown variable in the system using the latest values of the other variables, iteratively. It aims to find a solution that meets a specified error tolerance.

The Gram-Schmidt process is an algorithm used to orthogonalize a set of vectors. It takes an input set of linearly independent vectors and outputs a set of orthonormal vectors (vectors that are both orthogonal and unit length). This process is used to find a basis (set of linearly independent vectors) for the vector subspace spanned by the input set of vectors.

The orthonormal vectors u1, u2, u3 are computed by applying the Gram-Schmidt process to the given vectors x1, x2, and x3. This procedure involves projecting each vector onto the previously orthogonalized vectors and subtracting the projections.

The orthonormal vectors v1, v2, and v3 are calculated by normalizing the corresponding orthogonal vectors u1, u2, and u3. Normalization involves dividing each vector by its magnitude, resulting in vectors with a length of 1.

Importing the imageio library is necessary to handle image read operations. This library provides functions for reading and writing image files in various formats.

The numpy library, imported with the alias np, provides a wide range of mathematical functions and tools, including array operations for manipulating image data.

The numpy.linalg as npl line imports the linear algebra submodule of numpy, which provides operations specifically relevant to linear algebra problems, including matrix inversions.

The code reads the image file named

The image 'Newton.jpg' is read-in using imread function and stored in the variable 'photo'.

Singular Value Decomposition (SVD) is a powerful matrix factorization technique used in linear algebra. It breaks down a matrix into three matrices: U (left singular vectors), S (singular values), and V (right singular vectors). This decomposition helps in analyzing data, finding hidden patterns, and performing dimensionality reduction.

In SVD, singular values represent the importance of each dimension in the original data. Larger singular values indicate more significant dimensions.

The left singular vectors (U) in SVD represent the directions of the original data. Each column in U corresponds to a direction.

The right singular vectors (V) in SVD represent the directions of the latent variables. They reveal how your data projects onto the singular value space.

Image compression using SVD involves reducing the number of singular values used to represent the image. This reduces the amount of data needed to store the image. The more singular values you keep, the better the quality of the compressed image.

In image compression using SVD, we extract each color channel (Red, Green, Blue) from the image and perform SVD on each channel separately. Then, we choose a specific number of singular values (k) to represent each channel, discarding the less important ones.

The quality of compressed image using SVD depends on the number of singular values kept (k). Higher values lead to better image quality with larger file sizes, while lower values result in lower image quality but smaller file sizes.

Reconstruction of a compressed image using SVD involves recombining the compressed components by performing matrix multiplication. This combines the information from the decomposed image to generate the final result.

A direct method for solving systems of linear equations by systematically eliminating variables. It involves a series of elementary row operations on the augmented matrix.

A technique used in LU factorization to improve numerical stability by ensuring that the pivot element (the diagonal element used for elimination) is non-zero and as large as possible.

A numerical method for solving systems of linear equations that involves repeatedly substituting values for variables until a solution is reached. It's a common method for iterative solutions.

A type of function that interpolates data points using a series of cubic polynomials. It creates a smooth curve that passes through all given points.

A mathematical representation of the solution to a system of linear equations. It's a vector whose entries represent the values of the unknown variables that satisfy the equations.

A type of matrix used in LU factorization to represent a permutation of rows. It's essential for pivoting strategies.

A system of linear equations where the coefficients are arranged in a matrix (A), the variables in a vector (x), and the constant terms in a vector (b). The goal is to find the solution vector (x) that satisfies the equation.

SVD breaks down an image into components of varying importance. This allows for efficient compression by removing less essential information.

A square matrix is invertible if and only if its columns (or rows) are linearly independent. This means no column (or row) can be expressed as a linear combination of the others.

The Gram-Schmidt process takes a set of linearly independent vectors and converts them into a set of orthonormal vectors. These orthonormal vectors form a basis for the space spanned by the original vectors. It involves projecting each vector onto the previously orthogonalized vectors and subtracting the projections.

In a system of linear equations, the solution vector (x) contains the values of the unknown variables. This vector, when substituted into the equations, satisfies all the equations in the system.

The coefficient matrix (A) represents the coefficients of the unknown variables in a system of linear equations. Each row of (A) corresponds to an equation in the system, and each column represents a variable.

The goal is to find the line of best fit that minimizes the sum of the squared distances between the data points and the line.

Cholesky factorization is a method for decomposing a matrix into the product of a lower triangular matrix and its transpose. If the matrix is not positive definite, the factorization cannot be performed.

Numerical stability refers to how well a numerical method performs in the presence of rounding errors. A stable method produces accurate results even when rounding errors are present.

Pivoting is a technique used to improve numerical stability in Gaussian elimination by swapping rows to ensure the diagonal elements are non-zero. The condition number measures how sensitive the solution is to small changes in the input, suggesting when pivoting might be beneficial.

Jacobi method updates each variable using values from the previous iteration, while Gauss-Seidel uses the most recently updated values. Gauss-Seidel converges faster, but Jacobi is better suited for parallel computing.

The pseudo-inverse generalizes the concept of the inverse for non-square matrices. When a matrix is invertible, its pseudo-inverse is the same as its classical inverse.