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Unformatted text preview: Quiz2 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 Σ m × n is a diagonal matrix containing singular values σ i of matrix A such that σ i = √ λ i , where λ 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 Σ 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 ) = σ max /σ 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 σ i ≈ 0. If there are σ i more equations than unknowns then the equation gives a least squares solution of the overdetermined set of linear equations. ⎞ ⎛ x = V · ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 σ 1 ··· ··· 1 σ 2 ··· ··· . . . 1 ··· σ 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 Simpson’s rule to integrate the function. The local errors (error incurred per step) of Trapezoid rule and Simpson’s rule are O (Δ t 3 ) and O (Δ t 5 ) respectively. We can in crease the accuracy of the integration by using a more accurate method or decreasing Δ 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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 Fall '04
 ClarkColton
 Numerical Analysis, Optimization, Boundary value problem, Singular value decomposition, Gradient descent, Numerical ordinary differential equations

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