dvaced Sciece ad Techology Letters Vol.53 (ITS 4), pp.47-476 http://dx.doi.org/.457/astl.4.53.96 Estimatio of Bacward Perturbatio Bouds For Liear Least Squares Problem Xixiu Li School of Natural Scieces, Najig Uiversity of Posts ad Telecommuicatios, 3, Chia lixixiu4@63.com, lixx@jupt.edu.c bstract. Waldé, Karlso, ad Su foud a elegat explicit expressio of bacward error for the liear least squares problem. owever, it is difficult to compute this quatity as it ivolves the miimal sigular value of certai matrix. I this paper we preset a simple estimatio to this boud which ca be easily computed especially for large problems. Numerical results demostrate the validity of the estimatio. Keywords: liear least squares problem; bacward error; bacward stable; residual error Itroductio Cosider liear least squares (LS) problem m i x C m m x b with C, b C. () Let x be a approximate solutio of (), defie the bacward error sets m m E : E C, b ( E ) x m i b ( E ) x ad the smallest bacward x perturbatio ( x) m i E, () E The approximate solutio x ca be regarded as the exact solutio to a perturbed problem of (). small meas that x is the exact solutio to a earby problem of (), ad x ca be regarded as the computed result produced by a bacward stable algorithm. Cosequetly, F ca be used to test the stability of umerical algorithms. It is also used to moitor the covergece of iterative solutio methods ad to desig reliable stoppig criteria for these methods [6]. Waldé, Karlso, ad Su[] provided the followig explicit expressios. ISSN: 87-33 STL Copyright 4 SERSC
dvaced Sciece ad Techology Letters Vol.53 (ITS 4) Theorem..[] Let x ad be a real umber, r b x, ( r r / x ) m i,the /. r x r x (3) owever, from the view poit of practice, the mai aim of bacward perturbatio theory is to aalyze the accuracy of computed results, thus the bacward error boud should be computed efficietly, but (3) do't have the property. So how to estimate effectively is worth further researchig. May authors icludig Waldé, Karlso, ad Su have paid much attetio o the problem to derive explicit approximatio or fid upper ad lower bouds for [-6].The estimate derived by Karlso ad Waldé [3] i particular have bee studied by several authors, where / The purpose of this paper is to give a simple ad x r I r. cheaply computable estimate for.the paper is orgaized as follows. I Sectio, we propose the estimatio method for. Numerical examples are give to demostrate the validity of the estimatio i Sectio 3. Fially, some commets o the results are made i Sectio 4. Estimatio of Firstly, otice the followig facts:. Fact. r b x is the residual error of x to the liear system x=b, ad the residual error of x as the approximate solutio to () is where P r P ( b x ), (4). The reaso is as follows. Let x be the solutio to (), the ( ),. (5) x b I z z C Copyright 4 SERSC 47
dvaced Sciece ad Techology Letters Vol.53 (ITS 4) Let r=b-x, the r is a approximatio of r. Therefore, the residual error of x as the approximate solutio to () is r r r ( x x ). Based o ad (5), we have Cosequetly, r ( x x ) r P r. Fact. Let S be the set of the solutios to (), the x x ( I ) x ( ) ( )( ), (6) x x b x I z x z C / m i. (7) x S r x x r This fact ca be demostrated as follows. From (6), we have x x ( b x ) ( I )( z x ), therefore m i x x P r r, x S r ( x x ) x x, hece (7) is obtaied. ccordig to the applicatio bacgroud of LS problem, we assume m, ad ra()=p. Let m m U C U V be the sigular value decompositio of,where ad V C p p are uitary matrices, ad R is a positive diagoal matrix. The U r r / x U r r / x r C p, r C m p. Let Therefore x r r U r r, the, where, (8) r r r r r r G r r r r r r ad G have the same eigevalues. is uitary similar to. (9) Fact 3. The sufficiet ad ecessary coditio of is r ad Proof. Notice that the sufficiet ad ecessary coditio of is be a positive semi-defiite matrix. So if, we have ad rr is positive semi-defiite. ece r, r. r r rr, therefore r, r. Coversely, if. r, the G is a positive semi-defiite matrix. Therefore 47 Copyright 4 SERSC
dvaced Sciece ad Techology Letters Vol.53 (ITS 4) Fact 4. r if ad oly if x=b is cosistet. This fact ca be demostrated as follows.if we split R ( U ) R ( ), R ( U ) R ( ), with m p U C, where p=ra(), the U [ U, U ].From (8), we have x=b is cosistet. r U b b R ( ) r U ( b x ) U b. So Sice P r r, hece x=b is cosistet ad P r. We ote that the method of least squares is maily used i the case where x=b is icosistet, so the geeral case of (3) is r / x Theorem. Let,b, r, r, r, r, U,,, be defied as above. The / () r / x : () Proof. Let Q r r e be the QR factorizatio of r, where Q is a uitary matrix, ad e deotes the first colum of I, we have m p I p I p U r r U r r r r e T T Q Q r e r r e e For a ermite matrix M ( m ), it is well ow that ( M ) m i{ m }. ij m i ii ece Substitutig () ito (), we get ( ) x r r r r / x. From (8) r U r, hece / T. i r () (3) r U U r r (4) ccordig to (3) ad (4), the approximate estimatio () is obtaied. 3 Numerical examples I Sectio we have derived a estimate for the bacward error. I this sectio, we preset two umerical examples to demostrate the validity of our estimatio for large problems. Copyright 4 SERSC 473
dvaced Sciece ad Techology Letters Vol.53 (ITS 4) Example 3.. The coefficiet matrix is"well85.mtx", a 85 7 matrix from the Matrix Maret [7] with 8758 o-zeros etries, ( )., the right had side b is"well85-rhs.mtx". Example 3.. The coefficiet matrix is" illc33.mtx", a 33 3 matrix from the 4 Matrix Maret [7] with 473 o-zeros etries, ( ).9, the right had side b is" illc33-rhs.mtx". pplyig the iterative method LSQR proposed i [6] to the least square problems () geerates a sequece of approximate solutios { x }. Fig.. bsolute errors of ad ( ) for LSQR iterate x i Example3.. x Fig.. Ru time i computig ( x ) ad their estimatios i Example3.. 474 Copyright 4 SERSC
dvaced Sciece ad Techology Letters Vol.53 (ITS 4) Fig. 3. The compariso betwee ( x ) ad, ( x ) for LSQR iterate x i Example 3.. Fig.4. Ru time i computig ( x ) ad therir estimatios i Example3.. The estimatio are compared with ad ( x ) i Fig. ad Fig.3, which show that for every purported solutio of LS (), is a respectively good ad efficiet Copyright 4 SERSC 475
dvaced Sciece ad Techology Letters Vol.53 (ITS 4) estimate of ( x ). Oly whe x is a good approximatio to the exact solutio of (), the asymptotic property of is displayed. Ruig time i computig ad ( x ), ( x ) are give i Fig. ad Fig.4, which show that computatioal complexity of is much smaller tha ad. Therefore, is a practical ad cheaply computable estimate of the bacward error for the LS problem. ( x ) 4 Coclusio I this paper, we propose a efficiet estimatio for the bacward error of least square problem. Numerical experimets show that i terms of the calculatio accuracy ad computatioal complexity, our result is good. cowledgemets. This wor is supported by the Natural Sciece Foudatio of Chia uder Grat No. 4. Refereces. B. Waldé, R. Karlso, ad J.G. Su, Optimal bacward perturbatio bouds for the least squares problem, Numer. Li. lg. ppl., (3) (995), 7-86.. M. Gu, Bacward perturbatio bouds for liear least squares problems, SIM J. Mat. al. ppl. (999), 363-37. 3. R. Karlso, B. Waldé, Estimatio of optimal bacward pertutbatio bouds for liear least squares problem, BIT, 37(997), 86-869. 4. Zheg Su, Bouds for a estimate of the optimal bacward error for liear least squares problems, Iteratioal Joural of Computer Mathematics, 88(), 743-75. 5. S.Gratto, P.Jirae, T.P, David, Simple bacward error bouds for liear least squares problems, Li. lg. ppl.,439(3),78-89. 6. P.Jirae, T.P, David, Estimatig the bacward error i LSQR, SIM J. Matrix al. ppl., 3(4), 55-74. 7. R. F. Boisvert,et.al. The Matrix Maret: a web repository for test matrix data. http://math.ist.gov/matrixmaret/. 476 Copyright 4 SERSC