Given a square matrix A of dimension n, which has only non-null elements in the inputs Aj1, Aji, Aji+1, Anı and Ann for all i e {1,2, .n – 1} Make a function that solves a system A, = b by the Jacobi method that uses of A and receives as parameters the Inputs: A; b; The starting point; The maximum required the structure tolerance; Maximum number of iterations. (Octave/ Matlab) Explain the steps pls.
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- If there is a non-singular matrix P such as P-1AP=D, matrix A is called a diagonalizable matrix. A, n x n square matrix is diagonalizable if and only if matrix A has n linearly independent eigenvectors. In this case, the diagonal elements of the diagonal matrix D are the eigenvalues of the matrix A. A=({{1, -1, -1}, {1, 3, 1}, {-3, 1, -1}}) : 1 -1 -1 1 3 1 -3 1 -1 a)Write a program that calculates the eigenvalues and eigenvectors of matrix A using NumPy. b)Write the program that determines whether the D matrix is diagonal by calculating the D matrix, using NumPy. #UsePythonLets say there is a flow network called G and a flow in G called f, we say that f saturates an edge e if the flow value on that edge is equal to its capacity. The flow sat problem is: given a flow network G and a positive integer k, determine if there exists a flow f in G such that f saturates at least k edges of G. Prove that Flow sat is NP-complete.Consider newton method on a convex function under lower and upper bounds on the eigenvalues of the Hessian and a Lipschitz Hessian. In its region of quadratic convergence, breaktracking will always select pure step sizes. True or False?
- Given a matrix of dimension m*n where each cell in the matrix can have values 0, 1 or 2 which has the following meaning: 0: Empty cell 1: Cells have fresh oranges 2: Cells have rotten oranges So we have to determine what is the minimum time required so that all the oranges become rotten. A rotten orange at index [i,j] can rot other fresh orange at indexes [i-1,j], [i+1,j], [i,j-1], [i,j+1] (up, down, left and right). If it is impossible to rot every orange then simply return -1. Examples: Input: arr[][C] = { {2, 1, 0, 2, 1}, {1, 0, 1, 2, 1}, {1, 0, 0, 2, 1}}; Output: All oranges cannot be rotten. Below is algorithm. 1) Create an empty Q. 2) Find all rotten oranges and enqueue them to Q. Also enqueue a delimiter to indicate beginning of next time frame. 3) While Q is not empty do following 3.a) While delimiter in Q is not reached (i) Dequeue an orange from queue, rot all adjacent oranges. While rotting the adjacents, make sure that time frame is incremented only once. And time frame is…Given a directed graph with positive edge lengths and two distinct vertices uand v in the graph, the “all-pairs uv-constrained shortest path problem” is the problemof computing for each pair of vertices i and j the length of the shortest path from i toj that goes through the vertex u or through the vertex v. If no such path exists, theanswer is ∞. Describe an algorithm that takes a graph G = (V, E) and vertices u and v asinput parameters and computes values L(i, j) that represent the length of uv-constrainedshortest path from i to j for all 1 ≤ i, j ≤ |V|, i ! = u, j ! = u, i != v, j ! = v. Provide clearpseudocode solution. Prove your algorithm correct. Your algorithm must have runningtime in O(|V| ^2).You are given an adjacency list representation of a weighted directed graph G = (V, E) with n vertices, m edges, and no negative cycles. Furthermore, you are given two designated vertices s, t ∈ V and a subset P ⊆ E of the edges are labeled as “premium” edges. Describe a O(mn)-time algorithm to find the shortest path from s to t that uses at most one premium edge (along with any number of non-premium edges).
- An agent is trying to eat all the food in a maze that contains obstacles, but he now has the help of his friends! An agent cannot occupy a squarethat has an obstacle. There are initially k pieces of food (represented by dots), at positions (f1,...,fk). Thereare also n agents at positions (p1,...,pn). Initially, all agents start at random locations in the maze. Consider a search problem in which all agents move simultaneously;that is, in each step each agent moves into some adjacent position (N, S, E, or W, or STOP). Note that any number of agents may occupy the same position. Figure 1: A maze with 3 agents Give a search formulation to the problem of looking for both gold and diamondin a maze (wirte step with detail)? Knowing that you have M squares in the maze that do not have an What is the maximum size of the state space.Type in Latex **Problem**. Let $$A = \begin{bmatrix} .5 & .2 & .3 \\ .3 & .8 & .3 \\ .2 & 0 & .4 \end{bmatrix}.$$ This matrix is an example of a **stochastic matrix**: its column sums are all equal to 1. The vectors $$\mathbf{v}_1 = \begin{bmatrix} .3 \\ .6 \\ .1 \end{bmatrix}, \mathbf{v}_2 = \begin{bmatrix} 1 \\ -3 \\ 2 \end{bmatrix}, \mathbf{v}_3 = \begin{bmatrix} -1 \\ 0 \\ 1\end{bmatrix}$$ are all eigenvectors of $A$. * Compute $\left[\begin{array}{rrr} 1 & 1 & 1 \end{array}\right]\cdot\mathbf{x}_0$ and deduce that $c_1 = 1$.* Finally, let $\mathbf{x}_k = A^k \mathbf{x}_0$. Show that $\mathbf{x}_k \longrightarrow \mathbf{v}_1$ as $k$ goes to infinity. (The vector $\mathbf{v}_1$ is called a **steady-state vector** for $A.$) **Solution**. To prove that $c_1 = 1$, we first left-multiply both sides of the above equation by $[1 \, 1\, 1]$ and then simplify both sides:$$\begin{aligned}[1 \, 1\, 1]\mathbf{x}_0 &= [1 \, 1\, 1](c_1\mathbf{v}_1 +…Consider a graph G in the following Find a path from a to g in the graph G using each search strategy of depth-first search, breadth-first search, least-cost search, best-first search, and A* search. Is the returned solution path an optimal one?
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