Q1. The LU decomposition method is computationally more efficient than Naïve Gauss elimination method for solving

a single set of simultaneous linear equations

multiple sets of simultaneous linear equations with different coefficient matrices and the same right hand side vectors.

multiple sets of simultaneous linear equations with the same coefficient matrix but different right hand sides.

less than ten simultaneous linear equations.

Q2. The lower triangular matrix [L] in the [L][U] decomposition of the matrix given below

is

Q3. The upper triangular matrix [U] in the [L][U] decomposition of the matrix given below

is

Q4. For a given 20002000 matrix [A], assume that it takes about 15 seconds to find the inverse of [A] by use of the [L][U] decomposition method, that is, finding the [L][U] once, and then doing forward substitution and back substitution 2000 times using the 2000 columns of the identity matrix as the right hand side vector.  The approximate time, in seconds, that it will take to find the inverse if found by repeated use of the Naive Gauss elimination method, that is, doing forward elimination and back substitution 2000 times by using the 2000 columns of the identity matrix as the right hand side vector is most nearly

  300

  1500

  7500

  30000

Q5. The algorithm for solving the set of n equations [A][X] = [C], where [A] = [L][U] involves solving [L][Z] = [C] by forward substitution.  The algorithm to solve [L][Z]=[C] is given by

  z₁=c₁/l₁₁                              

                    for i from 2 to n do

                     sum = 0

                        for j from 1 to i do

                            sum = sum + l_ij*z_j

                        end do

                    z_i = (c_i – sum) / l_ii

                   end do

      z₁=c₁/l₁₁                         

                    for i from 2 to n do

                     sum = 0

                        for j from 1 to (i-1) do

                            sum = sum + l_ij*z_j

                        end do

                    z_i = (c_i – sum) / l_ii

                   end do

         z₁=c₁/l₁₁                    

                    for i from 2 to n do

                        for j from 1 to (i-1) do

                            sum = sum + l_ij*z_j

                        end do

                   z_i = (c_i – sum) / l_ii

                  end do

          for i from 2 to n do

                      sum = 0

                        for j from 1 to (i-1) do

                            sum = sum + l_ij*z_j

                       end do

                      z_i = (c_i – sum) / l_ii

                    end do

Q6. To solve boundary value problems, the finite difference methods are used resulting in simultaneous linear equations with tridiagonal coefficient matrices.  These are solved using the specialized [L][U] decomposition method.  The set of equations in matrix form with a tridiagonal coefficient matrix for

, , ,

using the finite difference method with a second order accurate central divided difference method and a step size of  is