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- (Sparse matrix–vector product) Recall from Section 3.4.2 that a matrix is said to be sparse if most of its entries are zero. More formally, assume a m × n matrix A has sparsity coefficient γ(A) ≪ 1, where γ(A) ≐ d(A)/s(A), d(A) is the number of nonzero elements in A, and s(A) is the size of A (in this case, s(A) = mn). 1. Evaluate the number of operations (multiplications and additions) that are required to form the matrix– vector product Ax, for any given vector x ∈ Rn and generic, non-sparse A. Show that this number is reduced by a factor γ(A), if A is sparse. 2. Now assume that A is not sparse, but is a rank-one modification of a sparse matrix. That is, A is of the form à + uv⊤, where à ∈ Rm,n is sparse, and u ∈ Rm, v ∈ Rm are given. Devise a method to compute the matrix–vector product Ax that exploits sparsity.23( The matrix below is stored in the Row-Major order in the linear memory, so in what order are the numbers stored. a. 8, 4, 3, 7, 9, -8, 5, 6, -2. b. 8, 6, -8, 7, 3, 4, -5, 9, -2. c. 8, -5, 7, 4, 6, 9, 3, -2 , -8. d. 8, 4, 3, -5, 6, -2, 7, 9, -8.d = (2 4 6 8] dl = [ - 3 - 3 - 3] d2 = [ -1 - 1 ] Create a matrix D by putting d on the main diagonal, dl on the first upper diagonal, and d2 on the second lower diagonal.
- Matrix multiplication plays an important role in a number of applications. Two matrices can only be multiplied if the number of columns of the fi rst matrix is equal to the number of rows in the second.Let’s assume we have an m × n matrix A and we want to multiply it by an n × p matrix B. We can express their product as an m × p matrix denoted by AB (or A ⋅ B). If we assign C = AB, and ci,j denotes the entry in C at position (i, j), then for each element i and j with 1 ≤ i ≤ m and 1 ≤ j ≤ p. Now we want to see if we can parallelize the computation of C. Assume that matrices are laid out in memory sequentially as follows: a1,1, a2,1, a3,1, a4,1, ..., etc.Assume that we are going to compute C on both a single core shared memory machine and a 4-core shared-memory machine. Compute the speedup we would expect to obtain on the 4-core machine, ignoring any memory issues.Repeat above Exercise, assuming that updates to C incur a cache miss due to false sharing when consecutive elements are in a…Solving system of linear equations with two variables using Cramer’s rule. In mathematics the coefficientsof linear equations are represented in the form of matrices. A matrix is a two-dimensional array inprogramming. The steps of Cramer’s rule to solve the system of equations are:Step 1: Determining the coefficient matrix from linear equations e.g. a1x+b1y=d1a2x+b2y=d2Coefficient matrix: D = Step 2: Determining the constant column C = Step 3: Finding determinant |D| = (a1*b2) - (b1*a2) and checking if |D| is not equal to zero then do thefollowing: 1. Determining X-matrix: DX = The coefficients of the x−column are replaced by the constant column a1 b1a2 b2d1d2 d1 b1d2 b2 2. Determining Y-matrix: Dy = The coefficients of the Y−column are replaced by the constant column3. To solve for x: x=|DX|/|D|4. To solve for y: y=|Dy|/|D|Note that if |D|=0 then the matrix is singular and solving the system of equationis not possible. You’re required to automate all these steps by writing a program…Let y be a column vector: y = [1, 2, 3, 4, 5, 6] so that y.shape = (6,1). Reshape the vector into a matrix z using the numpy.array.reshape and (numpy.array.transpose if necessary) to form a new matrix z whose first column is [1, 2, 3], and whose second column is [4, 5, 6]. Print out the resulting array z.
- Exercise 1: Memory Allocation in C a) Write a C program which dynamically allocates memory to define an integer 5 x 5 matrix initialized to zero by using the calloc function. Do not forget to free the memory in the end of your program. b) What happens if you attempt to print your matrix before and after freeing memory?List Variations - Suggest some real-life applications where Sparse Matrices can be implemented. Selecting from non-linked options as well as linked options, discuss your choice in terms of possible row and column operations on the matrix and consider efficiency issues. (dont put code but pseudo) something that not about A thin matrix is outlined to be a matrix containing a high proportion of components that area unit zeros. Sparse matrices of large order are of nice interest and application in science and industry; as an example, electrical networks, structural engineering, power distribution, reactor diffusion, and solutions to differential equations whereas conclusions among this paper ar primarily drawn considering orders of bigger than a thousand, a lot of ~s applicable to distributed matrices of smaller orders within the a whole lot. Because of increasing use of enormous order thin matrices and therefore the tendency to try to unravel larger order issues, great attention must…Matlab screenshot - If x=[ 2 6 12; 15 6 3; 10 11 1], then a) replace the first row elements of matrix x with its average value. b) reshape this matrix into row vector.
- %Encode A1, b1 and x1 as the vector of unknowns. A1 = b1 = syms xv1 = %Check the size of A, set it as m1 and n1 [m1,n1] = %Augment A and b to form AM1 AM1 = %Solve the Reduced Rwo Echelon of AM1. RREFA1 = %Collect the last column and set as bnew1, set the remaining elements as Anew1 bnew1= Anew1 = %Check if Anew is an identity matrix, if it is, bnew is the solution if Anew1 =eye(m1,n1) Root1 = bnew1 else display("No Solution") end %Augment the matrix A1 with the identity Matrix of the same size, set the result as AMI1 AMI1 = %Find the reduced row echelon form of AMI1, set the result as RREFAI1 RREFAI1= %Collect the second half of the matrix as AInew1, set the remaining elements as AIold1 AInew1= AIold1 = %Check if AIold1 is an identity matrix, if it is, AInew1 is the inverse %Encode A2, b2 and xv2 as the vector of unknowns. A2 = b2 = syms xv2 = %Check the size of A2, set it as m2 and n2 [m2,n2] = %Augment A2 and b2 to form AM2 AM2 = %Solve the…A C_++ program for Solving system of linear equations with two variables using Cramer’s rule. In mathematics the coefficientsof linear equations are represented in the form of matrices. A matrix is a two-dimensional array inprogramming. The steps of Cramer’s rule to solve the system of equations are: Step 1: Determining the coefficient matrix from linear equations e.g.a1x+b1y=d1a2x+b2y=d2Coefficient matrix: D =Step 2: Determining the constant column C =Step 3: Finding determinant |D| = (a1*b2) - (b1*a2) and checking if |D| is not equal to zero then do thefollowing:1. Determining X-matrix: DX =The coefficients of the x−column are replaced by the constant columna1 b1a2 b2d1d2d1 b1d2 b22. Determining Y-matrix: Dy =The coefficients of the Y−column are replaced by the constant column3. To solve for x: x=|DX|/|D|4. To solve for y: y=|Dy|/|D|Note that if |D|=0 then the matrix is singular and solving the system of equationis not possible.You’re required to automate all these steps by writing a…Create a MATLAB function file that solves matrices with the following conditions: a. Create a 4 x 4 vector C (range of vector numbers should be multiples of 3) b. Find the sums of columns 1 and 2 of C c. Find a version of C with rows 2 and 4 d. Display the fourth column of transpose C