Engineering Analysis ENG 3420 Fall 2009Lecture 19The inverse of a squareCanonical base of an n-dimensional vector spaceThe response of a linear systemGauss-Seidel MethodJacobi IterationConvergenceDiagonal DominanceSlide 10RelaxationNonlinear SystemsNewton-RaphsonSlide 14Engineering Analysis ENG 3420 Fall 2009Dan C. MarinescuOffice: HEC 439 BOffice hours: Tu-Th 11:00-12:0022Lecture 19Lecture 19Last time: •Midterm: solutions and discussions Today:TodayThe inverse of a matrixIterative methods for solving sytems of linear equationsGauss-SiedelJacobiNext TimeRelaxationNon-linear systemsThe inverse of a square If [A] is a square matrix, there is another matrix [A]-1, called the inverse of [A], for which [A][A]-1=[A]-1[A]=[I] The inverse can be computed in a column by column fashion by generating solutions with unit vectors as the right-hand-side constants:A x1 100A x2 010A x3 001A  1 x1x2x3 Canonical base of an n-dimensional vector space 100……000 010……000 001……000 ……………. 000…….100 000…….010 000…….001The response of a linear systemThe response of a linear system to some stimuli can be found using the matrix inverse.Interactions response  stimuli sArIAAsAArAsAr1111Gauss-Seidel MethodThe Gauss-Seidel method is the most commonly used iterative method for solving linear algebraic equations [A]{x}={b}.The method solves each equation in a system for a particular variable, and then uses that value in later equations to solve later variables. For a 3x3 system with nonzero elements along the diagonal, for example, the jth iteration values are found from the j-1th iteration using:x1jb1 a12x2j 1 a13x3j 1a11x2jb2 a21x1j a23x3j 1a22x3jb3 a31x1j a32x2ja33Jacobi IterationThe Jacobi iteration is similar to the Gauss-Seidel method, except the j-1th information is used to update all variables in the jth iteration:a) Gauss-Seidelb) JacobiConvergenceThe convergence of an iterative method can be calculated by determining the relative percent change of each element in {x}. For example, for the ith element in the jth iteration, The method is ended when all elements have converged to a set tolerance.a,ixij xij 1xij100%Diagonal DominanceThe Gauss-Seidel method may diverge, but if the system is diagonally dominant, it will definitely converge.Diagonal dominance means:aii aijj1jinRelaxationTo enhance convergence, an iterative program can introduce relaxation where the value at a particular iteration is made up of a combination of the old value and the newly calculated value:where  is a weighting factor that is assigned a value between 0 and 2.0<<1: underrelaxation=1: no relaxation1<≤2: overrelaxationxinewxinew 1 xioldNonlinear SystemsNonlinear systems can also be solved using the same strategy as the Gauss-Seidel method - solve each system for one of the unknowns and update each unknown using information from the previous iteration.This is called successive substitution.Newton-RaphsonNonlinear systems may also be solved using the Newton-Raphson method for multiple variables.For a two-variable system, the Taylor series approximation and resulting Newton-Raphson equations are:f1,i1f1,i x1,i1 x1,i f1,ix1 x2,i1 x2,i f1,ix2x1,i1x1,if1,if2,ix2 f2,if1,ix2f1,ix1f2,ix2f1,ix2f2,ix1f2,i1f2,i x1,i1 x1,i f2,ix1 x2,i1 x2,i f2,ix2x2,i1x2,if2,if1,ix1