Holistic Numerical Methods

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MULTIPLE CHOICE TEST

(All Tests)

LU DECOMPOSITION

(More on LU Decomposition)

SIMULTANEOUS LINEAR EQNS

(More on Simultaneous Linear Equations)

 

Pick the most appropriate answer.


Q1. The LU decomposition method is computationally more efficient than Nave 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

                                            

                    for i from 2 to n do

                     sum = 0

                        for j from 1 to i do

                            sum = sum +

                        end do

                    zi = (ci sum) / lii

                   end do

 

                                     

                    for i from 2 to n do

                     sum = 0

                        for j from 1 to (i-1) do

                            sum = sum +

                        end do

                    zi = (ci sum) / lii

                   end do

                                                

                    for i from 2 to n do

                        for j from 1 to (i-1) do

                            sum = sum +

                        end do

                   zi = (ci sum) / lii

                   end do

                for i from 2 to n do

                      sum = 0

                        for j from 1 to (i-1) do

                            sum = sum +

                       end do

                      zi = (ci sum) / lii

                    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

 

 

 

  

 

  

 

 

 

 

 


Complete Solution

 

Multiple choice questions on other topics


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Copyrights: University of South Florida, 4202 E Fowler Ave, Tampa, FL 33620-5350. All Rights Reserved. Questions, suggestions or comments, contact kaw@eng.usf.edu  This material is based upon work supported by the National Science Foundation under Grant# Creative Commons License0126793, 0341468, 0717624,  0836981, 0836916, 0836805, 1322586.  Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.  Other sponsors include Maple, MathCAD, USF, FAMU and MSOE.  Based on a work at http://mathforcollege.com/nm.  Holistic Numerical Methods licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.

 

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