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Analysis of a Fast Hankel Eigenvalue Algorithm Analysis of a Fast Hankel Eigenvalue Algorithm a b Franklin T Luk and Sanzheng Qiao a Department of Computer Science Rensselaer Polytechnic Institute Troy New York USA b Department of Computing and Software McMaster University Hamilton Ontario LS L Canada ABSTRACT This pap er analyzes the imp ortant steps of an O n log n algorithm for nding the eigenvalues of a complex Hankel matrix The three key steps are a Lanczostype tridiagonalization algorithm a fast FFTbased Hankel matrixvector pro duct pro cedure and a QR eigenvalue metho d based on complexorthogonal transformations In this pap er we present an error analysis of the three steps as well as results from numerical exp eriments Keywords Hankel matrix eigenvalue decomp osition Lanczos tridiagonalization Hankel matrixvector multipli cation complexorthogonal transformations error analysis INTRODUCTION The eigenvalue decomp osition of a structured matrix has imp ortant applications in signal pro cessing In this pap er nn we consider a complex Hankel matrix H C h h h h n n h h h h B C n n B C B C H B C A h h h h n n n n h h h h n n n n The authors prop osed a fast algorithm for nding the eigenvalues of H The key step is a fast Lanczos tridiago nalization algorithm which employs a fast Hankel matrixvector multiplication based on the Fast Fourier transform FFT Then the algorithm p erforms a QRlike pro cedure using the complexorthogonal transformations in the di agonalization to nd the eigenvalues In this pap er we present an error analysis and discuss the stability prop erties of the key steps of the algorithm This pap er is organized as follows In Section we discuss eigenvalue computation using complexorthogonal transformations and describ e the overall eigenvalue pro cedure and three key steps In Section we introduce notations and error analysis of some basic complex arithmetic op erations We analyze the orthogonality of the Lanczos vectors in Section and the accuracy of the Hankel matrixvector multiplication in Section In Section we derive the condition number of a complexorthogonal plane rotation and we provide a backward error analysis of the application of this transformation to a vector Numerical results are given in Section Supp orted by the Natural Sciences and Engineering Research Council of Canada under grant OGP COMPLEXORTHOGONAL TRANSFORMATIONS An eigenvalue decomp osition of a nondefective matrix H is given by H XDX T T We will pick the matrix X to b e complexorthogonal that is XX I So H XDX We apply a sp ecial Lanczos tridiagonalization to the Hankel matrix assuming that the Lanczos pro cess do es not prematurely terminate T H QJ Q where Q is complexorthogonal and J is complexsymmetric tridiagonal B C B C B C J B C B C A n n n T Let Q q q q Since Q Q I we see that n T q q j ij i We call the prop erty dened by as complexorthogonality corthogonality for short The dominant cost of the Lanczos metho d is matrixvector multiplication We prop ose an O n log n FFTbased n multiplication scheme so that we can tridiagonalize a Hankel matrix in O n log n op erations Given w C we n b want to compute the pro duct p H w Dene c C by T b c h h h h h h n n n n T n b Let w w w w Dene w C by n T b w w w w n n Let tv denote onedimensional FFT of a vector v itv onedimensional inverse FFT of v and comp onent b b wise multiplication of two vectors The desired pro duct is given by the rst n elements of the vector it t c t w The authors showed that Algorithm requires n logn O n ops real oatingp oint op erations To maintain symmetry and tridiagonality of J we apply complexorthogonal rotations in the QR iterations T J WDW where W is complexorthogonal and D is diagonal Thus we get with X QW The workhorse of our QRlike nn iteration is a complexorthogonal planerotation P C That is P is an identity matrix except for four strategic p ositions formed by the intersection of the ith and j th rows and columns c s p p ii ij G s c p p j i j j T where c s The matrix G is constructed to annihilate the second comp onent of a vector x x x We present our computational schemes in the following Overall Procedure Hankel Eigenvalue Computation Given H compute al l its eigenvalues Apply a Lanczos pro cedure Algorithm to reduce H to a tridiagonal form J Use Algorithm to compute the Hankel matrixvector pro ducts Apply a QRtype pro cedure Algorithm to nd the eigenvalues of J Algorithm Lanczos Tridiagonalization Given H this algorithm computes a tridiagonal matrix J T Initialize q such that q q v H q for j n T q v j j j r v q j j j j if j n q T r r j j j if break end j q r j j j v H q q j j j j end end n Algorithm Fast Hankel MatrixVector Multiplication Given w C compute p H w n b b Construct the vectors c and w as in and Calculate y C by b b y it t c t w T n The desired p C is given by the rst n elements of y p y y y y n n mm Algorithm ComplexSymmetric QR Step Given J C this algorithm implements one step of the QR method with the Wilkinson shift Initialize Q I Find the eigenvalue of J that is closer to J mmmm mm Set x J x J for k m T Find a complexorthogonal matrix P to annihilate x using x k T J P JP k k Q QP k if k m x J x J k k k k end if end for ERROR ANALYSIS FUNDAMENTALS Let u denote the unit roundo Since the ob jective of this pap er is to analyze the stability of our algorithm and not to derive precise error b ounds we carry out rstorder error analysis Sometimes we ignore mo derate constant co ecients The following results for the basic complex arithmetic op erations are from Higham f l x y x y j j u p f l xy xy j j p f l xy xy j j Using and similar to the real case rep eatedly applying we can show that for two nvectors x and y T T T f l x y x y y x x y where n u jy j jy j jxj jxj n n n n u We assume that nu So we can think of as approximately nu n LANCZOS TRIDIAGONALIZATION Analogous to the analysis of the conventional Lanczos tridiagonalization we shall show that column vectors q can j lose corthogonality Let T T b b b b b f l q v q v v j j j j j b for some v It follows immediately from that kv k kv k Thus n j T b b b b b q v jj kq k kv k j j n j j j and b b jb j kq k kv k j n j j If b b b b b q v v b q q r f l v b j j j j j j j for some v and q then and show that p b b b q k kq k kv k ukv b kqk j j j j b Denoting r v b q as the rst order error in r and using we get j j p p b b b b b q k jb j krk ukv b kq k u ukq k kv k j j j j j j j Now from we have p b b b b b b q f l r r r krk kr k j j j j j j that is p b b b b q r r krk kr k j j j j T b To check corthogonality we multiply q on the b oth sides of the ab ove equation j T T T T T b b b b b b b b b b q r r q q q r r q q q v b j j j j j j j j j j j T b b b v b from we get Since q is normalized and q j j j j T T b b b b q q q r r j j j j Next we derive the b ound for krk by using and p p b b b b q k kv b kq k kv k krk j j j j j It follows from and that kr r k krk krk p p b b b b ukq k u kv k kq k kv k j j j j p p p b b ukq k u kv k j j Finally and imply that T b b b b q q j jj kr r k kq k j j j j j p p p b b b kv k kq k u ukq k j j j n Using the approximation nu when nu and ignoring the mo derate constant co ecients we obtain n b b b n kq k kq k kv k j j j T b b u jq q j j j b j j j b b b b The ab ove inequality says that q can lose corthogonality when either j j is small or kq k is large recall that q j j j j b and denote computed results Hence the loss is due to either the cancellation in computing or the growth j j b b in kq k and not the accumulation of roundo errors In the real case kq k and is consistent with the j j b results in Paige whereas in the complex case the norm kq k can b e arbitrarily large This is an additional factor j that causes the loss of corthogonality HANKEL MATRIXVECTOR MULTIPLICATION The only computation in the fast Hankel matrixvector multiplication is b b y it t c t w Denoting the computed y as b b b y f l itt c t w j b we consider the computation of tw Assume that the weights in the FFT are precalculated and that the k j computed weights b satisfy k j j b k j k k where j j k j for all j and k Dep ending on the metho d that computes the weights we can pick cu cu log j or cuj where c denotes a constant dep endent on the metho d Following Higham we let b b b x tw and x f l tw We have p log n b n kxk kx xk log n where Approximately p n log n b kxk kx xk log n Similarly let b b b v tc and v f l tc Then p n log n b kv k kv v k log n b b Next we consider the comp onentwise multiplication z v x Denoting the computed z as b b b z f l v x we have approximately b kz z k u kzk Finally we consider b b b y itz and y f l itz Since F is the FFT matrix we have n F n F n n Thus similar to and we get p n n log b ky k ky y k log n As n F is unitary it follows that n p p b b b b zk n kzk ky k kitz k kF n kzk n n Consequently using and we get log n log n b b ky y k
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