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# NUMERICALMATRIXANALYSIS MATH543

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This 18 page Class Notes was uploaded by Camryn Rogahn on Tuesday October 20, 2015. The Class Notes belongs to MATH543 at San Diego State University taught by P.Blomgren in Fall. Since its upload, it has received 24 views. For similar materials see /class/225266/math543-san-diego-state-university in Mathematics (M) at San Diego State University.

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Date Created: 10/20/15

Peter Blom in V9 M Peter Blomgren blomgren peter gmail com Department of Mathematics and Statistics San Diego CA 9218277720 httpterminussdsuedu Spring 2010 5 Mayan a nu Pml ms 0 Solution 0 Recap 9 Least Squares Problems 0 Problem Language 0 Problem Setup the Vandermonde Matrix 0 Formal Statement 9 LSQ The Solution 0 Pseudoilnverse 0 The MoorePenrose Matrix lnverse 0 3 5 Algorithms for the LSQ Problem Peter Blom Previously Computing the QR factorization 3 ways Gram Schmidt Orthogonalization Modified vs Classical Householder Triangularization Modified Gram Schmidt Householder Numerically stable Even better stability Useful for iterative methods Not as useful for iterative methods Triangular Orthogonalizationquot Orthogonal Triangularizationquot ARlerHRnQ QnHAQZQlAR Work N 2mn2 flops Work N 2mn2 7 27 73 flops Note No Q at this lower cost quot 5quot Peter Blomgren Least Squares Problems 1 Least Squares Problems Q quot ol Iluo S Iitiun Least Squares Least squares datamodel fitting is used everywhere social sciences engineering statistics mathematics In our language the problem is expressed as an overdetermined system Aiamp AECm mgtm Since A is tall and skinny we have more equations than unknowns The least squares solution is defined by EiAi is arg min SEEC Least Squares Some Language The quantity F b 7 A is known as the residual and since our problem is overdetermined we cannot in general hope to find an 52 such that F0 0 Minimizing some norm of F6 is a close second best The choice of the 2 norm leads to a problem that is easy to work with and it is usually the correct choice for statistical reasons computing the least squares solution yields the Maximum Likelihood estimate under certain conditions independent identically distributed variables etc Peter Blomgren Least Squares Problems Recap Least Squares Prohle s be Solution Example Polynomial Data Fitting Figure Illustrating the leastisquares polynomial fit of degrees 1 2 3 6 12 and 18 o d t containing 38 points The top panel of each figure shows the dataset and the fitted polynomial the bottom panel shows the residual as a function of thm polynomial degree r Least Squares Problems Problem Set the Vandermonde Matrix quot y I r v LSQ Iluo Solution Least Squares Problem Set Up So How do we achieve this miracle of data fitting7l We flip back to lecture 2 and express our system using the Vandermonde matrix c b 1 X1 x12 de 0 0 1 2 d C1 b1 xz x2 x2 A 7 c 62 7 b b2 7 1 X x2 Xd 39 39 where the fitting polynomial is described using the coefficients E px Co clx czx2 cdxdi Given the locations of the points i and a particular set of coefficients E the matrixvector product Iquot AE evaluates the polynomial in those points ie T px17 p097 i i i pXm Peter Blon ren quares Problems Recap Least Squares Problems Problem See 19 Vandermonde Matrix r LSQ The Sollitiul Least Squares Thinking About Projectors We can think of the least squares problem in as the problem of finding the vector in rangeA which is closest to b df Since we are measuring using the 2 norm closest e closest In the sense of Euclidean distance We look to minimize the residual F b7 A52 The minimum residual must be orthogonal to rangeA Peter Blomgren 39 Least Squares Problems Recap Least Squares Problems LSQ Tho Solmion 22mm Banana Thetawem may 9mm Let A 6 CW m 2 n and l E Cm he given A vector 6 C minimizes the residual norm Hsz Mb 7 Ail 2 thereby solving the least squares problem if and only if L rangeA that is AF 0 gt AAquotlt Al gt Ai PE Where the orthogonal projector P E Cm maps Cm onto rangeA The n x n system AA Z Ab the normal equations is non singular if and only ifA has full rank ltgt The solution 52 is unique if and only ifA has full rank Peter Blomgren Language The Pseudo Inverse Hence if A has full rank the least squaressolution is is uniquely determined by as AA 1A E The matrix Al 2 AA 1A is known as the pseudo inverse of A With this notation and language the least squares problem comes down to computing one or both of We will look at 3 algorithms for accomplishing this 5351 Peter Blomgren Least Squares Problems The MoorePenrose Matrix Inverse Pseudo Inverse Given B 6 CW the Moore Penrose generalized matrix inverse is a unique pseudo inverse Bi satisfying i BBTB B ii BTBBT Bi iii BBT 88 iv BTB BTB The MoorePenrose inverse is often referred to in the literature so it is a good thing to know what it is Peter Blomgren Least Squares Problems Leasrsmrares Pro 7 r LSQ The Solution iurms for are LSQ Problem Take1 The Normal Equations Ian l flops The classical straight forward boneheaded way to solve the least squares problem is to solve the normal equations AAS Ab The preferred way of doing this is by computing the Cholesky factorization details to follow at a later date AA CE RR where R is an upper triangular matrix Hence the equivalent system RR lt AE Al RR 1A can be solved by a forward and a backward substitution sweep m Peter Blomgren Least Squares Problems est vi ir new he Solution idnns for die LSQ Problem Take2 The SVD 2mn2 lln3 flops If when we can compute the reduced SVD A Bi V we can use 7 to express the projector P and end up with the linear system of equations Di w 70 and we get is by is Vf lU S Here the pseudo inverse is AT Vi lUf 535 Peter Blomgren Least Squares Problems Leasrsmrares Pro 7 r LSQ The Solution iurms for are LSQ Problem Take3 The QR Factorization With the reduced QR factorization the game unfolds like this GivenAAA we can project I to the range of A using P 6262 then the system an ears has a unique solution given by is R 1Qb Al R 1Q Note that we do not need Q explicitly only the action QE which we can get cheaply from the Q less version of Householder triangularization Peter Blomgren Least Squares Problems Say we computed P using the Householder Q less QR factorization but forgot to compute Qb is everything lost No we can still compute is using the following sequence 2 R lR AE F 57A R lR AF i i The first step solves the semi normal equationsquot RR lt AE The remaining three steps takes one step of iterative refinement to reduce roundofF error Peter Blomgren Least Squares Problems 1 Least 51 wres PrulIIen Iquot 5h 4 L5 The Solution 35 Algorithms for the LSQ Problem Algorithms for Least Squares Comments Method Work flops Comment 3 39 39 Normal Equations N mng n7 Fastest sensitive to rdoundofT er rors Not recommen e Your everyday choice Can break 3 QReFaCtorization N Zmn2 7 2L I I when A Is close to rankedeflclent 3 The Big HammerTM more stable SVD N 2mn2 11n3 than the QR approach but requires more computational wor Additional Comment If rn gt n then the work for QR and SVD are both dominated by the first term Zmng and the computational cost of the SVD is not excessive However when rn z n the cost of the SVD is roughly 10 times that of the QRefaCtorization p Pralinequot Iquot 5h 4 The Solution 35 Algorithms for die LSQ Problem Looking Forward We can now compute and use one of the big important tools of numerical linear algebra the QR factorization Next we finally formalize the discussion on numerical stability and then we take another look at some of our algorithms in the light of stability considerations LsQ Problem HW4 Due Friday March 5 2010 HW4 Exploratory Implement modified Gram Schmidt QR factorization Work through experiment 1 and 2 in lecture 9 of Trefethen amp Bau Make sure your versions of classical and modified GS can reproduce figure 91 Do exercises 91ab and 92ab For additional non mandatory fun do exercises 91c and 92c Peter Blomgrel Least Squares Problems

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