Engineering Analysis ENG 3420 Fall 2009Lecture 19The inverse of a square Canonical base of an n-dimensional vector spaceThe response of a linear systemGauss-Seidel MethodJacobi IterationConvergenceDiagonal DominanceRelaxationNonlinear SystemsNewton-RaphsonEngineering 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 space100……000010……000001……000…………….000…….100000…….010000…….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 jthiteration values are found from the j-1thiteration 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 ithelement in the jthiteration, The method is ended when all elements have converged to a set tolerance.εa,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 λ is a weighting factor that is assigned a value between 0 and 2. 0<λ<1: underrelaxation λ=1: no relaxation 1<λ≤2: overrelaxationxinew=λxinew+1−λ()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-Raphsonmethod 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−
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