Sinwook Lee Digital Media Lab. HYU

 Linear Equations  To gain the appropriate solution, 1..n of equations are necessary.  The num of equations < n  underdetermined there may be many solutions  The num of equations > n  overdeterminded there may be no solutions

Letting A = (a ij ), x = (x i ), b = (b i ), as Ax = b A is nonsingular

 “LUP decomposition” is introduced for a fast and stable method  The process with LUP decomposition  The process to fine LUP matrices effeciently

 The idea behind LUP is to find three n x n matrices L, U, and P such that PA = LU where  L is a unit lower-triangular matrix  U is an upper-triangular matrix  P is a permutation matrix

 LUP decomposition satisfied for Ax = b, PA = LU  PAx = Pb LUx = Pb, Let us define y = Ux, where x is the desired solution vector Ly = Pb  In the Ly = Pb, we solve the unknown vector y by a method called “forward substitution”  In the y = Ux, we solve the unknown vector x by a method called “backward substitution”

….. Ly = Pb

Ux = y

 PA = LU  We start with the case  A is an nxn nonsingular matrix  P is absent(or, equivalently, P = I n )   A = LU  We perform LU decomposition is called Gaussian elimination

vA’

Matrix Algebra

 In the Ax = b, We must pivot on off-diagonal elements of A to avoid dividing by 0 as well as very small value  In case that a 11 ~ a 1n are 0, we can exchange first row and k’th row