1. 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.

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

less than ten simultaneous linear equations.

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

  is

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

 is

4. 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 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

  300

  1500

  7500

  30000

5. The algorithm in solving [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

                    for i from 2 to n do

                     sum = 0

                        for j from 1 to i do

                            sum = sum +

                        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 +

                        end do

                    z_i = (c_i – sum) / l_ii

                   end do

                    for i from 2 to n do

                        for j from 1 to (i-1) do

                            sum = sum +

                        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 +

                       end do

                      z_i = (c_i – sum) / l_ii

                    end do

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

, , ,

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