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Unformatted text preview: Quiz-2 Review November 13, 2006 Singular Value Decomposition: SVD Any rectangular m n matrix A can be decomposed into a product of 3 matrices as shown in Equation 1, where m n is a diagonal matrix containing singular values i of matrix A such that i = i , where i are the eigenvalues of matrix AA T . U and V are orthonormal matrices i.e. V T = V 1 , U T = U 1 . The columns of U and V are called left sin- gular and right singular vectors of A respectively. This decomposition is called Singular Value Decomposition (SVD) of matrix A. A m n = U m m m n V n T n (1) . In matlab SVD of any matrix A can be performed using the command svd . The condition number of matrix is defined as cond ( A ) = max / min . SVD can be used to decouple a noisy signal into useful component and noise. Another useful application of SVD is in the solution of linear system of equations. The solution of a system of equation A x = b is given in Equation 2, where we replace 1 with if i 0. If there are i more equations than unknowns then the equation gives a least squares solution of the overdetermined set of linear equations. x = V 1 1 1 2 . . . 1 n U T b (2) 1 Cite as: Sandeep Sharma, course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Ordinary Differential Equations: ODE and Differential Al- gebraic Equations: DAE The general form of ODE is given in Equation 3, where X is a vector of length n and F is a set of n functions. dX = F ( X ) (3) dt If we have an n th order ordinary differential equation, it can be con- verted into n ordinary differential equations. To do that we define n 1 new variables y i s, where each y i = d i y To solve a set of ODEs dt i . we require the boundary conditions to be specified at t = t . If all the Boundary conditions are not specified at at the same boundary then we can either use shooting method or solve the system as a boundary value problem. The solution of Equation 3 can be obtained by performing numerical integration as shown in Equation 4. t X ( t ) = X ( t ) + F ( X ( t )) dt (4) t If we are given the value of function F ( X ( t )) at discrete points t 1 , t 2 ... t 3 , we can use Trapezoid rule or Simpsons rule to integrate the function. The local errors (error incurred per step) of Trapezoid rule and Simpsons rule are O ( t 3 ) and O ( t 5 ) respectively. We can in- crease the accuracy of the integration by using a more accurate method or decreasing t . Using too small of a time step leads to rounding off errors. To overcome this problem the concept of Richardson Extrapo- lation is used. When applied to trapezoid rule we get the result shown in Equation 5. Notice that the error properties...
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