1. 6. LU Factorization 2005010211 Kim Hong Soo 2003010478 Kim Jin Soo
Choi Jang Won
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2. Contents LU Factorization from Gaussian Elimination
LU Factorization of Tridiagonal Matrices
LU Factorization with Pivoting
Direct LU Factorization
Application of LU Factorization
Matlab’s Method
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3. Example. Circuit Analysis Application
Solve by LU Factorization of coefficient matrix
Flow around the left loop 20 ( i - i ) + 10 ( i - i ) = 1 2 1 3
Flow around upper loop 25 i + 10 ( i - i ) + 20 ( i - i ) = 2 2 3 2 1
Flow around lower loop 30 i + 10 ( i - i ) + 10 ( i - i ) =
200 3 3 2 3 1
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4. 6.1 LU Factorization from Gaussian Elimination
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5. LU Factorization from Gaussian Elimination - Example 6.1
Example 6.1 Three-By-Three System , ,
Step 1: , ,
Step 2: , ,
Multiply L by U, and Verify the result: .
LU = =
= A
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6. LU Factorization from Gaussian Elimination - Example 6.2
Example 6.2 Four-By-Four System
Step 1: Row 1 is unchanged, and rows 2-4 are modified to give
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7. LU Factorization from Gaussian Elimination - Example 6.2
Example 6.2 Four-By-Four System
Step 2: Row 1 and 2 are unchanged, row 3 and 4 are transformed, yielding
Step 3: The fourth row is modified to complete the forward elimination stage:
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8. LU Factorization from Gaussian Elimination - Example 6.2
Example 6.2 Four-By-Four System
Multiply L by U to verify the result: . =
L U = A
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9. LU Factorization from Gaussian Elimination - MATLAB Function for LU Factorization
Using Gaussian Elimination
function [L, U] = LU_Factor(A)
% LU factorization of matrix A
% using Gaussian elimination without pivoting
% Input : A  n-by-n matrix
% Output L ( lower triangular ) and
% U (upper triangular )
[n, m] = size(A);
L = eye(n); % initialize matrices
U = A;
for j = 1 : n
for I = j + 1 : n
L( i , j ) = U( i, j ) / U( j, j );
U( i , : ) = U( i, j ) / L( j, j ) * U( j, : );
end
% display L and U L U
% verify results
B = L * U A
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10. LU Factorization from Gaussian Elimination - MATLAB Function for LU Factorization
>> LU_factor(A);
L =
L =
U =
B =
A =
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11. LU Factorization from Gaussian Elimination - Example 6.3
Example 6.3 LU Factorization for Circuit Analysis Example
Step 1: The matrices after the first stage of Gaussian elimination are ,
Step 2: The matrices after the second stage of Gaussian elimination are ,
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12. LU Factorization from Gaussian Elimination - Discussion. =
Here, d12 = ca11+a21, d22=ca12+a22, and d32 = ca13+a32.
We write this as CA = D
We also need the inverse of the matrix C . = , or
C C = I .
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13. LU Factorization from Gaussian Elimination - Discussion. =
M M = I . . =
M M = I . . =
M M = I .
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14. 6.2 LU Factorization of Tridiagonal Matrix
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15. LU Factorization of Tridiagonal Matrix
LU Factorization of a tridiagonal matrix T
less computation
less computer memory
6.2.1 Matlab function for LU Factorization of
a Tridiagonal Matrix
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16. LU Factorization of Tridiagonal Matrix
function [dd, bb] = LU_tridiag(a, d, b)
% LU factorization of a tridiagonal matrix T
% Input
% a vector of elements above main diagonal, a(n) = 0
% d diagonal of matrix T
% b vector of elements below main diagonal, b(1) = 0
% The factorization of T consists of
% Lower bidiagonal matrix,
% 1’s on main diagonal; lower diagonal is bb
% Upper bidiagonal matrix,
% main diagonal is dd; upper diagonal is a
N = length(d)
bb(1) = 0;
dd(1) = d(1);
for i = 2 : n
bb(i) = b(i) / dd(i-1);
dd(i) = d(i) – bb(i) * a(i-1);
end
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17. LU Factorization of Tridiagonal Matrix Example 6.4
Example 6.4 LU Factorization of Tridigonal System
four-by-four tridigonal matrix
Consider again the four-by-four tridigonal matrix from Example 3.7, , , ,
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18. LU Factorization of Tridiagonal Matrix Example 6.4
For i = 2, dd1 = d1 = 2, a1 = -1
bb2 = b2/dd1 = -1/2
dd2 = d2 – bb2a1 = 2-(-1/2)(-1) = 3/2
For i = 3, a2 = -1
bb3 = b3/dd2 = -1/(3/2) = -2/3
dd3 = d3 - bb3a2 = 2-(-2/3)(-1) = 4/3
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19. LU Factorization of Tridiagonal Matrix Example 6.4
For i = 4, a3 = -1
bb4 = b4/dd2 = -1/(4/3) = -3/4
dd4 = d4 - bb4a3 = 2-(-3/4)(-1) = 5/4
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20. 6.3 LU Factorization with pivoting
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21. 6.3.1 Why pivoting?
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22. 6.3.2 Theory for pivoting 1/2 한 쪽에 A행렬, 다른 한쪽에 I행렬을 놓는다.
(A: LU 분해 할 NxN행렬, I: NxN 단위행렬)
A행렬을 가우스 소거법을 이용해 상삼각 행렬로 변환하면서 A행렬에 가하는 행렬 연산을 기록한다. (Nx: 소거, Px: 교환)
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23. 6.3.2 Theory for pivoting 2/2 Nx 행렬을 좌측으로 빼면서 N’x로 변환한다.
좌측 N’x들의 역행렬을 I행렬에 곱한다.
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24. 6.3.3 pivoting example 1/2
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25. 6.3.3 pivoting example 2/2
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26. 6.3.4 pivoting procedure
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27. 6.3.5 pivoting code Function [L, U, P] = LU_pivot(A)
[n, n1] = size(A);
L=eye(n); P=eye(n); U=A;
For j = 1:n
[pivot m] = max(abs(U(j:n, j))); m = m+j-1;
if m ~= j
% interchange rows m and j in U
% interchange rows m and j in P
if j >= 2
% interchange rows m and j in columns 1:j-1 of L
end
for i = j+1:n
L(i, j) = U(i, j) / U(j, j);
U(i, :) = U(i, :) – L(i, j)*U(j, :);
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28. 6.4 Direct LU Factorization
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29. 6.4.1 Theory for Direct LU factorization
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30. 6.4.2 Type of Direct LU factorization
변수의 수= N(N+1)/2 + N(N+1)/2 = N^2+N
Doolittle 방식
L의 대각 원소가 1
Crout 방식
U의 대각 원소가 1
Cholesky 방식
L과 U의 같은 위치 대각 원소끼리 같음
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31. 6.4.3 Doolittle example
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32. 6.4.4 Doolittle code Function [L, U] = Doolittle(A) [n, m] = size(A);
U = zeros(n, m); L = eye(n);
for k = 1:n
U(k, k) = A(k, k) – L(k, 1:k-1)*U(1:k-1, k);
for j = k+1:n
U(k, j) = A(k, j) – L(k, 1:k-1)*U(1:k-1, j);
L(j,k)= (A(j,k)– L(j, 1:k-1)*U(1:k-1, k))/U(k,k);
end
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33. 6.4.5 Cholesky example
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34. 6.4.6 Cholesky code Function [L, U] = Cholesky(A)
% A is assumed to be symmetric.
% L is computed, and U = L’
[n, m] = size(A);
L = zeros(n, n);
for k = 1:n
L(k, k) = sqrt( A(k,k) – L(k, 1:k-1)*L(k, 1:k-1)’ );
for i = k+1:n
L(i, k) = (A(i,k) – L(i, 1:k-1)*L(k, 1:k-1)’)/L(k, k);
end
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35. 6.5 Applications of LU Factorization
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36. Contents 6.5 Applications of LU Factorization 6.6 MATLAB’s Methods
Linear Equations
Tridiagonal System
Determinant of a Matrix
Inverse of a Matrix
6.6 MATLAB’s Methods
lu, chol, det, inv
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37. 6.5.1 Solving Systems of Linear Equations
Ax = b → LUx = b → Ly = b
A = LU, Ux = y
Solve the system Ly = b for y
Forward substitution
y1 → y2 → …… yn
Solve the system Ux = y for x
Backward substitution
xn → xn-1 → …… x1
If pivoting has been used
Ax=b → PAx=Pb, where PA=LU, Pb=c
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38. Ex 6.8 Solving Electrical Circuit for Several Voltages
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39. Ex 6.8 Solving Electrical Circuit for Several Voltages
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40. MATLAB Function to Solve the Linear System LUx=b
function x = LU_Solve(L, U, b)
% Function to solve the equation L U x = b
% L --> Lower triangular matrix (with 1's on diagonal)
% U --> Upper triangular matrix
% b --> Right-hand side vector
[n m] = size(L); z = zeros(n,1); x = zeros(n,1);
% Solve L z = b using forward substitution
z(1) = b(1);
for i = 2:n
z(i) = b(i) - L(i, 1:i-1) * z(1:i-1);
end
% Solve U x = z using back substitution
x(n) = z(n) / U(n, n);
for i = n-1 : -1 : 1
x(i) = (z(i) - U(i,i+1:n) * x(i+1:n)) / U(i, i);
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41. 6.5.2 Solving a Tridiagonal System
Ex 6.9
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42. MATLAB Function for Solving a Tridiagonal System
function x = LU_tridiag_solve(a, d, b, r)
% Function to solve the eqquation A x = r
% LU factorization of A is expressed as
% Lower bidiagonal matrix:
% diagonal is 1s, lower diagonal is b; b(1) = 0.
% Upper bidiagonal matrix: diagonal is d, upper diagonal is a
% Right-hand-side vector is r
% Solve L z = r using forward substitution
n = length(d);
z(1) = r(1);
for i = 2:n
z(i) = r(i) - b(i) * z(i-1);
end
% Solve U x = z using back substitution
x(n) = z(n) / d(n);
for i = n-1 : -1 : 1
x(i) = (z(i) - a(i) * x(i+1)) / d(i);
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43. 6.5.3 Determinant of a Matrix
Ex 6.10
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44. The inverse of an n-by-n matrix A Axi=ei (i=1, … ,n)
6.5.4 Inverse of a Matrix
The inverse of an n-by-n matrix A Axi=ei (i=1, … ,n)
ei=[0 0 … 1 … 0 0]’, where the 1 appears in the ith position
X whose columns are the solution vectors x1, … , xn is A-1
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45. Ex 6.11 Finding a Matrix Inverse Using LU Factorization
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46. Ex 6.11 Finding a Matrix Inverse Using LU Factorization
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47. MATLAB Function function x = LU_Solve_Gen(L, U, B)
% Function to solve the equation L U x = B
% L --> Lower triangular matrix (1's on diagonal)
% U --> Upper triangular matrix
% B --> Right-hand-side matrix
[n n2] = size(L); [m1 m] = size(B);
% Solve L z = B using forward substitution
for j = 1:m
z(1, j) = b(1, j);
for i = 2 : n
z(i,j) = B(i,j) - L(i, 1:i-1) * z(1:i-1, j);
end
% Solve U x = z using back substitution
x(n, j) = z(n, j) / U(n, n);
for i = n-1 : -1 : 1
x(i,j) = (z(i,j)-U(i,i+1:n)*x(i+1:n,j)) / U(i,i);
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48. 6.6 MATLAB’s Method
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49. 4 built-in functions 6.6 MATLAB’s Method
LU decomposition of a square matrix A
[L, U] = lu(A), [L, U, P] = lu(A)
Cholesky factorization
chol
Determinant of a matrix
det
Inverse of a matrix
inv
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