homework
Post on 17-Jan-2016
212 Views
Preview:
DESCRIPTION
TRANSCRIPT
DREXEL UNIVERSITY
Department of Mechanical Engineering & Mechanics
Applied Engineering Analytical & Numerical Methods I
MEM 591 - Fall 2012
HOMEWORK #3: Due Thursday, October 18 @ 6 PM
1. [30 points]
Suppose is strictly column diagonal dominant, which means that for each k,
| | ∑ .
i) Which is its first pivot?
ii) Perform the first step of the decomposition for and show explicitly the matrix , as well as .
iii) Derive a recursive formula that gives each element of other than the first line, i.e. each element of the
submatrix .
iv) Explain under what condition for , no row interchange would take place if the Gauss Elimination with Partial
Pivoting (GEPP) method was continued.
2. [30 points]
Consider the matrix we showed in class 2 1 1 0 4 3 3 1
8 6
77
9 5 9 8
.
i) [25 points]
Show explicitly what the permutation matrix , as well as and are in the case that partial pivoting begins
from the first row. Confirm your analytical results using appropriately the same command in MATLAB ("lu").
ii) [5 points]
Show why the triangularization part of the method allows for a simpler calculation of the determinant of .
3. [40 points]
Construct the following subroutines:
i) [25 points]
ii) [10 points]
GEPP
iii)[5 points]
Consider the linear system of equations A , where
1 for 1 , 6 and
120211
.
Apply your GEPP code and find the solution. Verify by using the "\" command in MATLAB.
Remark: Recall that you can only use essential built-in commands in MATLAB such as size, length, zeros, eye and
for/if statements. Any multiplication, needs to be in a component form(i.e. A(i,j)*B(j)). Remember to comment
important lines of your code. Modify the following script to prepare your codes:
i) function [P,L,U]=pplu(A); [m m]=size(A); % Number of rows and columns of A U=A; % Set U=A L=eye(m); % Set an initial matrix L P=eye(m); % Set an initial matrix P for i=1:m-1 %% Begin the Partial Pivoting . . . . . %% Partial Pivoting procedure ends and the triangularization procedure %% continues . . . . . end ii) function [U,L,P,x]=gepp(A,b); %% 1st part: Partial Pivoting and triangularization . . . %% 2nd part: Find the solution . . .
top related