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numerical analysis question 3. For the matrix in problem 1, apply the Householder transformation to transform...
3. Given the matrix [ -1 2 -1] A= 3 2 1 10 10 1 Following steps (a)(b) to obtain the LU decomposition of the matrix A with partial piv- oting (a) Apply the Gaussian elimination method with partial pivoting to obtain an upper trian- gular matrix U. Record the corresponding permutation matrix for each pivoting step, and the numbers lik used to eliminate the zeros in column k. (b) Based on (a), express the matrices P, L and U...
ALTSIS AND NUMERICAL ANALYSIS 2. (a) Let A be the matrix 2 -115 8-4 Write down the 3 x 3 permutation matrix P such that PA interchanges the 1st and 3rd rows of A. Find the inverse of P Use Gaussian elimination with partial pivoting to find an upper triangular matix U, permutation matrices Pi and P2 and lower triangular matrices M and M2 of the form 1 0 0 0 1 1 0 0 0 bi 1 with land...
In this exercise, you will work with a QR factorization of an mxn matrix. We will proceed in the way that is chosen by MATLAB, which is different from the textbook presentation. An mxn matrix A can be presented as a product of a unitary (or orthogonal) mxm matrix Q and an upper-triangular m × n matrix R, that is, A = Q * R . Theory: a square mxm matrix Q is called unitary (or orthogona) if -,or equivalently,...
1 2-4 Given A=1 2 6 7 (i) Use Householder transformations to transform A into an upper triangular matrix R. Also transform the vector b = 9 | , th b(1)-H2Hib. (ii) Solve Rx = b(1) for x and check your answer by computing b-Ax. -2 1 8 at is compute -3
1 2-4 Given A=1 2 6 7 (i) Use Householder transformations to transform A into an upper triangular matrix R. Also transform the vector b = 9 |...
(911 (1) (a) Recall that a square matrix A has an LU decomposition if we can write it as the product A = LU of a lower triangular matrix and an upper triangular matrix. Show that the matrix 0 1 21 A= 3 4 5 (6 7 9] does not have an LU decomposition 0 0 Uji U12 U13 O 1 2 Il 21 l22 0 0 U22 U23 = 3 4 5 (131 132 133 0 0 U33 6...
(4.2) Let 4 7 A= 4 7 -2 1 (a) Find the QR decomposition of A. It has to be of the form A QR where Q is a 3 x 3 orthogonal matrix, and R is 3 x 2 upper-triangular. (b) Use part (a) to find the least squares solution to the -6 Ax -4 -2
ce of least squates solutions. Problem III.3 (5 points), Consider matrix B (as in the right). Find the QR factorization of B. That is, find a matrix Q whose columns are orthonormal and an upper triangular square mnatrix R with positive diagonal entries such that B QR. -2 1 24-1 B 3= 243 -2 1 Hìnt. Apply the Gram-Schmidt process. Keep track of the relevant linear combinationas
Below are the results of each step in the transformation of a matrix to row reduced echelon form, using Gaussian elimination. These are the same steps involved in the decomposition PALU - 2 2 0 6 AP 1 1 2 0 0 0 2 -3 1 3 3 3 1 3 3 3 0 2 3 0 0 4 2 2 A o 2 3 0 A0 0 2 -13 0 2-1 0 0 0 -2 0 0 2 -3...
Solve the 3-bus example below with a zero injection at bus 1
using orthogonal decomposition. Instead of developing a method
yourself based on Given’s rotation, simply use the MATLAB function
qr() to obtain the Q and U matrices.
M 1 2#32 M1-0 M32 72 Zero injection system example. ence heimakis 5.0 5.0 H-0 -4.0 7.5 -5.0 The measurement vector in p.u. is 0.32 meas0.72 And the variance matrix is 104 0 0 R- 0 10-4 0 0 10-20
points PooleLinAlg4 5.3.017 1 The columns of Q were obtained by applying the Gram-Schmidt Process to the columns of A. Find the upper triangular matrix R such that A QR 2 10 6 5 A=110 10-3 , Q = Need Help?Read It Talk to a Tutor + -1 points PooleLinAJg4 5.3.018. The columns of Q were obtained by applying the Gram-Schmidt Process to the columns of A. Find the upper triangular matrix R such that A = QR. (Enter sqrt(n)...