Question

100 GAUSSIAN ELIMINAONO the system Ax = b (where A is n × n and nonsingular), MATLAB uses Gaussian elimination with partial pivoting to solve the system. If you want to see the LU decomposition, use the MATLAB command lu. Exercise 1.8.10 Use MATLAB to check the LU decomposition obtained in Example 1.8.2 Enter the matrix A, then type [L, U, P] = lu (A) . Once you have L, U, and P, you can put them back together by typing P'*L*U. (In MATLAB the prime symbol, applied to a real matrix, means "transpose") You can also try [K,V] = lu (A) , and see what that gives you. Give a matrix equation that relates K to L For a description of the lu command, type help lu or look in MATLAB's help browser

Answer 1

I do not have example 1.8.2. So I am taking A as magic matrix of dimension 3X3.
You just need to substitute that A with the matrix given in example 1.8.2.

Command Window New to MATLAB? See resources for Getting Started. >>A-magic (3) IL,U,P1-lu (A) .0000 0.5000 0.3750 1.0000 0.5441 1.0000 8.0000 6.0000 0 8.5000 -1.0000 5.2941 1.0000

Command Window New to MATLAB? See resources for Getting Started. >> K, V]-lu (A) .0000 0.3750 0.5000 0.5441 1.0000 1.0000 8.0000 6.0000 8.5000 1.0000 5.2941 1.0000 >> NO' I WILL COMPARE L AND K .0000 0.5000 0.3750 1.0000 0.5441 1.0000 .0000 0.3750 0.5000 0.5441 1.0000 1.0000 >> WE CAN EASILY SEE THAT if I INTERCHANGE RO 2 AND ROW 3 OF ANY OF L OR K, THEN I WILL GET OTHER ONE

Command Window > L 1.0000 0.5000 0.3750 1.0000 0.5441 1.0000 1.0000 0.3750 0.5000 0.5441 1.0000 1.0000 ans 1.0000 0.5000 0.3750 1.0000 0.5441 1.0000 乓>> % WE CAN EASILY SEE THAT L-Ped

A Command Window >> help lu lu lu factorization L,U] lu (A) stores an upper triangular matrix in U and a "psychologically lower triangular matrix" (i.e. a product of lower triangular and permutation matrices) in L, so that A = L*U. A can be rectangular [L,U,P] = lu(A) returns unit lower triangular matrix L, upper triangular matrix U, and permutation matrix P so that PA L U [L,U,p] = lu (A,' vector') returns the permutation information as a vector instead of a matrix. That is, p is a row vector such that A(p,:) =L"U . Similarly, [L, U,P] = lu (A, "matrix') returns a permutation matrix P. This is the default behavior Y = lu (A) returns the output from LAPACK'S DGETRE r ZGETRE routine if A is full. If A is sparse, Y contains the strict lower triangle of L embedded in the same matrix as the upper triangle of U. In both full and sparse cases, the permutation information is 1。3t. [L, U, P,Q] = lu (A) returns unit lower triangular matrix L, upper triangular matrix U, a permutation matrix P and a column reordering matrix Q so that P*ALU for sparse non-empty A. This uses UMEPACK and is significantly more time and memory efficient than the other syntaxes, even when used with COLAMD [L, U,P, qlu (A, 'vector') returns two row vectors p and q so that A(p.q) = L*U. Using 'matrix' in place of 'vector' returns permutation matrices [L,U, p, Q,R] = lu (A) returns unit lower triangular matrix L, upper triangular matrix U, permutation matrices P and, and a diagonal scaling matrix R so that P (RIA) *Q L U for sparse non-empty A This uses UMFPACK as well. Typically, but not always, the row-scaling leads to a sparger and more stable factorization. Note that this fact rization is the same as that used by sparse MLD|VIDE when UMFPACK is used

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