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 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 algorithm Calculate the square function. –h(x) = [  f(x)) 2 ] Calculate a scalar value –z 0 = h(x) Calculate the gradient –dx = - dh/dx

Minimization algorithm Multiple loops to convergence x new = x old + dx ; z 1 = h(x new ); dif = z 1 - z 0 ; if dif > 0 dx = dx/2 x new = x old + dx else end loop endif

Program 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 Problem f 1 (x,y) = x 2 + y f 2 (x,y) = x 2 - y

Example of the 2-D Problem The gradient function: h(x,y) =[( x 2 + y 2 - 1) 2 +( x 2 - y) 2 ] The derivative of the function: dh=[ -(4(x 2 + y 2 -1)x + 4( x 2 - y)x) -(4(x 2 +y 2 - 1)y - 2( x 2 - y))]

Example of the 3-D Problem f 1 (x,y,z) = x 2 + 2y 2 + 4z f 2 (x,y,z) = 2x 2 + y 3 + 6z f 3 (x,y,z) = xyz + 1

Example of the 3-D Problem The gradient function: h(x,y,z) = [ (x 2 + 2y 2 + 4z 2 - 7) 2 + (2x 2 + y 3 + 6z ) 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 6 LU Decomposition of Matrices

LU 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 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 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.

Decomposition

LU 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 Decomposition The matrices are represented by

Equation 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 Solving First step - forward substitution

Equation Solving Second step - back substitution

LU Decomposition (Crout’s reduction) Matrix decomposition

LU Decomposition (Doolittle’s method) Matrix decomposition

Cholesky’s method Matrix is decomposed into: where, l ii = u ii

LU Decomposition (Crout’s reduction) Matrix decomposition

Crout’s Reduction The method alternates from solving from the lower triangular to the upper triangular

Crout’s Reduction

General formulation of Crout’s These are the general equations for the component of the two matrices

Example The matrix is broken into a lower and upper triangular matrices.

LU Decomposition (Doolittle’s method) Matrix decomposition

Doolitte’s method The method alternates from solving from the upper triangular to the lower triangular

General formulation of Doolittle’s The problem is reverse of the Crout’s reduction, starting with the upper triangular matrix and going to the lower triangular matrix.

LU Programs There are two programs –LU_factor –LU_factor - the program does a Doolittle decomposition of a matrix and returns the L and U matrices –LU_solve –LU_solve uses an L and U matrix combination to solve the system of equations.

Example The matrix is broken into a lower and upper triangular matrices.

Summary Nonlinear scalar gradient method uses a simple step to find the crossing terms. Setup of the LU decomposition techniques.

Homework Check the Homework webpage