Embed Size (px)
DESCRIPTION
Lecture 10 - Nonlinear gradient techniques and LU Decomposition. CVEN 302 June 24, 2002. Lecture’s Goals. Nonlinear Gradient technique LU Decomposition Crout’s technique Doolittle’s technique Cholesky’s technique. Nonlinear Equations. - PowerPoint PPT Presentation
Citation preview
Lecture 10 - Nonlinear gradient Lecture 10 - Nonlinear gradient techniques and LU Decompositiontechniques and LU Decomposition
CVEN 302
June 24, 2002
Lecture’s GoalsLecture’s Goals
• Nonlinear Gradient technique
• LU Decomposition– Crout’s technique – Doolittle’s technique– Cholesky’s technique
Nonlinear EquationsNonlinear Equations
• The nonlinear equations can be solved using a gradient technique.
• The minimization technique calculates a positive scalar value and use a gradient to find the zero of multiple functions.
Minimization algorithmMinimization algorithm
• Calculate the square function.– h(x) = [f(x))2]
• Calculate a scalar value – z0 = h(x)
• Calculate the gradient– dx = - dh/dx
Minimization algorithmMinimization algorithm
• Multiple loops to convergence
xnew = xold + dx ; z1 = h(xnew ); dif = z1 - z0;
if dif > 0dx = dx/2
xnew = xold + dx
elseend loop
endif
Program FFMINProgram FFMIN
• The program is adapted from the book to do a minimization of scalar and uses a gradient technique to find the roots.
Example of the 2-D ProblemExample of the 2-D Problem
f1(x,y) = x2 + y2 - 1
f2(x,y) = x2 - y
Example of the 2-D ProblemExample of the 2-D Problem
The gradient function:
h(x,y) =[( x2 + y2 - 1)2 +( x2 - y) 2]
The derivative of the function:
dh=[-(4(x2 + y2-1)x + 4( x2 - y)x)
-(4(x2 +y2 - 1)y - 2( x2 - y))]
Example of the 3-D ProblemExample of the 3-D Problem
f1(x,y,z) = x2 + 2y2 + 4z2 - 7
f2(x,y,z) = 2x2 + y3 + 6z2 - 10
f3(x,y,z) = xyz + 1
Example of the 3-D ProblemExample of the 3-D Problem
The gradient function:
h(x,y,z) = [ (x2 + 2y2 + 4z2 - 7)2 +
(2x2 + y3 + 6z2 - 10)2 +
(xyz + 1) 2]
End of material on Exam 1
Exam 1
Chapter 1 through 5
Monday July 3, 2002
open book and open notes
Chapter 6Chapter 6
LU Decomposition of Matrices
LU DecompositionLU Decomposition
A modification of the elimination method, called the LU decomposition. The technique will rewrite the matrix as the product of two matrices.
A = LU
LU DecompositionLU Decomposition
The technique breaks the matrix into a product of two matrices, L and U, L is a lower triangular matrix and U is an upper triangular matrix.
LU DecompositionLU Decomposition
– Crout’s reduction (U has ones on the diagonal)– Doolittle’s method( L has ones on the diagonal)– Cholesky’s method ( The diagonal terms are the
same value for the L and U matrices)
There are variation of the technique using different methods.
DecompositionDecomposition2 1 1 2 0 0 1 0.5 0.5
0 4 2 0 4 0 0 1 0.5
6 3 0 6 0 3 0 0 1
1 0 0 2 1 1
0 1 0 0 4 2
3 0 1 0 0 3
1 0 0 2 1 1
0 2 0 0 2 1
3 0 1 0 0 3
LU Decomposition SolvingLU Decomposition Solving
Using the LU decomposition
[A]{x} = [L][U]{x} = [L]{[U]{x}} = {b}
Solve
[L]{y} = {b}
and then solve
[U]{x} = {y}
LU DecompositionLU Decomposition
The matrices are represented by
11 12 13 14 11 11 12 13 14
21 22 23 24 21 22 22 23 24
31 32 33 34 31 32 33 33 34
41 42 43 44 41 42 43 44 44
a a a a 0 0 0
a a a a 0 0 0
a a a a 0 0 0
a a a a 0 0 0
Equation SolvingEquation Solving
What is the advantage of breaking up one linear set into two successive ones?
– The advantage is that the solution of triangular set of equations is trivial to solve.
Equation SolvingEquation Solving
• First step - forward substitution
N,2,i ,yb1
y
by
1i
1jjiji
iii
11
11
Equation SolvingEquation Solving
• Second step - back substitution
,11,N i ,xy1
x
yx
N
1ijjiji
iii
NN
NN
LU Decomposition (Crout’s reduction)LU Decomposition (Crout’s reduction)
Matrix decomposition
11 12 13 14 11 12 13 14
21 22 23 24 21 22 23 24
31 32 33 34 31 32 33 34
41 42 43 44 41 42 43 44
a a a a 0 0 0 1
a a a a 0 0 0 1
a a a a 0 0 0 1
a a a a 0 0 0 1
l u u u
l l u u
l l l u
l l l l
LU Decomposition (Doolittle’s method)LU Decomposition (Doolittle’s method)
Matrix decomposition
11 12 13 14 11 12 13 14
21 22 23 24 21 22 23 24
31 32 33 34 31 32 33 34
41 42 43 44 41 42 43 44
a a a a 1 0 0 0
a a a a 1 0 0 0
a a a a 1 0 0 0
a a a a 1 0 0 0
u u u u
l u u u
l l u u
l l l u
Cholesky’s methodCholesky’s method
Matrix is decomposed into:
where, lii = uii
11 12 13 14 11 11 12 13 14
21 22 23 24 21 22 22 23 24
31 32 33 34 31 32 33 33 34
41 42 43 44 41 42 43 44 44
a a a a 0 0 0
a a a a 0 0 0
a a a a 0 0 0
a a a a 0 0 0
l u u u u
l l u u u
l l l u u
l l l l u
LU Decomposition (Crout’s reduction)LU Decomposition (Crout’s reduction)
Matrix decomposition
11 12 13 14 11 12 13 14
21 22 23 24 21 22 23 24
31 32 33 34 31 32 33 34
41 42 43 44 41 42 43 44
a a a a 0 0 0 1
a a a a 0 0 0 1
a a a a 0 0 0 1
a a a a 0 0 0 1
l u u u
l l u u
l l l u
l l l l
Crout’s ReductionCrout’s Reduction
11 11
21 21
31 31
41 41
a
a
a
a
l
l
l
l
The method alternates from solving from the lower triangular to the upper triangular
1211 12 12 12
11
1311 13 13 13
11
1411 14 14 14
11
aa
aa
aa
l u ul
l u ul
l u ul
Crout’s ReductionCrout’s Reduction
21 12 22 22 22 22 21 12
31 12 32 32 32 32 31 12
41 12 42 42 42 42 41 12
1 a a
1 a a
1 a a
l u l l l u
l u l l l u
l u l l l u
23 21 1321 13 22 23 23 23
22
24 21 1421 14 22 24 24 24
22
aa
aa
l ul u l u u
l
l ul u l u u
l
General formulation of Crout’sGeneral formulation of Crout’s
n,2,j j,i l
ulau
N,1,i i,j ulal
ii
1i
1kkjikij
ij
1j
1kkjikijij
These are the general equations for the component of the two matrices
ExampleExample
100
110
3/23/11
13/42
03/71
003
122
321
213
The matrix is broken into a lower and upper triangular matrices.
LU Decomposition (Doolittle’s method)LU Decomposition (Doolittle’s method)
Matrix decomposition
11 12 13 14 11 12 13 14
21 22 23 24 21 22 23 24
31 32 33 34 31 32 33 34
41 42 43 44 41 42 43 44
a a a a 1 0 0 0
a a a a 1 0 0 0
a a a a 1 0 0 0
a a a a 1 0 0 0
u u u u
l u u u
l l u u
l l l u
Doolitte’s methodDoolitte’s method
1414
1313
1212
1111
au
au
au
au
11
4141411141
11
3131311131
11
2121211121
u
alaul
u
alaul
u
alaul
The method alternates from solving from the upper triangular to the lower triangular
General formulation of Doolittle’sGeneral formulation of Doolittle’s
n,2,i i,j u
ulal
N,1,j j,i ulau
ii
1j
1kkjikij
ij
1i
1kkjikijij
The problem is reverse of the Crout’s reduction, starting with the upper triangular matrix and going to the lower triangular matrix.
LU ProgramsLU Programs
• There are two programs– LU_factorLU_factor - the program does a Doolittle
decomposition of a matrix and returns the L and U matrices
– LU_solveLU_solve uses an L and U matrix combination to solve the system of equations.
ExampleExample
• The matrix is broken into a lower and upper triangular matrices.
3 1 2 1 0 0 3 1 2
1 2 3 1 3 1 0 0 7 3 7 3
2 2 1 2 3 4 7 1 0 0 1
SummarySummary
• Nonlinear scalar gradient method uses a simple step to find the crossing terms.
• Setup of the LU decomposition techniques.
HomeworkHomework
• Check the Homework webpage
