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A Fast Block Hankel Solver Based on an Inversion

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A Fast Block Hankel Solver Based on an Inversion A FAST BLOCK HANKEL SOLVER BASED ON AN INVERSION FORMULA FOR BLOCK LOEWNER MATRICES (1) Peter Kravanja Marc Van Barel 3 2 Abstract We prop ose a new O p n algorithm for solving complex np np linear systems that have blo ck Hankel structure where the blo cks are square matrices of size p p Via FFTs the blo ck Hankel system is transformed into a blo ck Lo ewner system An inversion formula enables us to calculate the inverse of the blo ck Lo ewner matrix explicitely The parameters that o ccur in this inversion formula are calculated by solving two rational inter p olation problems on the unit circle We have implemented our algorithm in Fortran Numerical examples are included Intro duction Let n and p b e p ositive integers Consider a sequence H H in 0 2n2 n1 pp C such that the np np blo ck Hankel matrix H H is regular k +l k l=0 n p 1 Let b C We consider the problem of computing x H b n1 I transforms the The np np blo ck exchange matrix E p k n1l k l=0 blo ck Hankel system H x b into the blo ck To eplitz system T x b where T HE and x E x For a given value of p the classical algorithms for solving a blo ck Hankel or a blo ck To eplitz system exploit the structure and 3 require less arithmetic op erations compared to O n for general linear systems 2 The socalled fast algorithms require O n arithmetic op erations while the 2 sup erfast ones need only O n log n op erations by using a divide and conquer strategy The ow of these metho ds is determined by the exact singularity of the square leading principal blo ck submatrices of H or T The fast metho ds compute the solutions corresp onding to successive nonsingular leading principal submatrices sections However in niteprecision arithmetic not only singular Received (1) Katholieke Universiteit Leuven Department of Computer Science Celestij nenlaan A B Heverlee Belgium Email napkravanjananetornlgov and marccskuleuvenacbe P Kravanja M Van Barel A fast block Hankel solver but also illconditioned sections should b e avoided The algorithms that have b een develop ed for this purp ose are called lo okahead algorithms They lo ok ahead from one wellconditioned section to the next one and jump over the illconditioned sections that lie in b etween For a more detailed overview concerning scalar Hankel and To eplitz systems we refer the interested reader to the intro duction of our pap er and the references cited therein For the blo ck To eplitz and the blo ck Hankel case the reader may lo ok at the intro duction and the references of Go o d lo okahead strategies are dicult to design Only recently a totally dierent approach was considered This approach transforms the blo ck Hankel or To eplitz matrix into a generalized Cauchy matrix Such a matrix can b e factorized using pivoting without destroying the structure Hence the Cauchy system can b e solved without using lo okahead and the solution can b e transformed back to the solution of the original system We refer the interested reader to and the references cited therein In we presented such a transformation approach based on an inversion formula for Lo ewner matrices In this pap er we will generalize this approach to the blo ck case The outline of this pap er is as follows In Section we will show how the blo ck Hankel system H x b can b e transformed into a blo ck Lo ewner 0 0 2 1 system Lx b in O p n log n ops An explicit formula for L enables us 0 1 0 to calculate x as L b This inversion formula for blo ck Lo ewner matrices is discussed in Section It involves certain p p matrices that can b e computed by solving two rational interp olation problems on the unit circle In Section 3 2 we present an O p n algorithm to solve these interp olation problems We conclude with numerical examples in Section Transformation into a blo ck Lo ewner system Let y y z z b e n mutually distinct complex numb ers and 1 n 1 n dene y y y and z z z Let Ly z b e the class of np np 1 n 1 n blo ck matrices i o n h n pp C D k l j C C D D C Ly z 1 n 1 n y z k l k l=1 The elements of Ly z are called blo ck Lo ewner matrices They b ear the name of Karl Lo ewner who studied them for p in the context of rational interp olation and monotone matrix functions pp pp The set Ly z is a C mo dule and a submo dule of the C mo dule of all the complex np np blo ck matrices having p p blo cks Since addition of a constant matrix to all the n p p matrices C D leads to the same k l blo ck Lo ewner matrix its dimension is n The set of all the complex based on an inversion formula for block Loewner matrices np np blo ck Hankel matrices having p p blo cks also forms a submo dule of dimension n Blo ck Hankel and blo ck Lo ewner matrices are even more closely related In the scalar case p a theorem of Fiedler asserts that every Hankel matrix can b e transformed into a Lo ewner matrix and vice versa In this section we will formulate this theorem for the blo ck case and discuss 2 our O p n log n implementation of the transformation First we have to deal with some preliminaries concerning Vandermonde matrices Let t t b e n complex numb ers and dene t t t The 1 n 1 n Vandermonde matrix with no des t t is given by 1 n n1 t t 1 1 V t V t t 1 n n1 t t n n Let f z b e the monic p olynomial of degree n that has zeros t t t 1 n f z z t z t t 1 n and dene Y f z z t j n tj k k 6=j Note that f z is a monic p olynomial of degree n for j n Dene tj the n n matrix W t by the equation f z t1 f z t2 z W t n1 z f z tn This means that the j th row of W t contains the co ecients of f z when tj n1 written in terms of the standard monomial basis z z Then T W t V t diag f t f t D t t1 1 tn n The Vandermonde matrix V t is regular if and only if its no des t t are 1 n mutually distinct In that case implies that W t is regular and also that 1 T 1 V t W t D t Let V y z b e the n n Vandermonde matrix with no des y y and 1 n z z and similarly for W y z 1 n Recall that the Kronecker pro duct b etween two matrices A a k l k l=1 C and B C is dened as the blo ck matrix A B a B C k l k l=1 P Kravanja M Van Barel A fast block Hankel solver T Theorem The matrix L W y I H W z I is a blo ck Lo ewner p p matrix in Ly z whose p p matrices C C D D are given by up 1 n 1 n pp to an arbitrary additive matrix C C 1 H 0 H 1 C n W y z I p D 1 H 2n2 D n Proof The scalar case p has b een proven by Fiedler Theorem The generalization to the blo ck case is straightforward The Kronecker pro duct A B is regular if A and B are b oth regular This implies that W y I and W z I are regular and thus L is regular p p A judicious choice of the p oints y and z enables us to transform the blo ck 0 0 2 Hankel system H x b into a blo ck Lo ewner system Lx b in O p n log n k 1 ops Let exp in and supp ose from now on that y for k n1 k n That is let y Let exp in and supp ose n1 from now on that z y for k n That is let z k k Note that 2 2n2 3 2n1 y and z Let b e the n n Fourier matrix n n1 p V n n H amount to an inverse discrete Matrixvector pro ducts involving n n Fourier transform DFT and can b e evaluated via the celebrated inverse fast Fourier transform FFT in O n log n ops Finally let D and D b e n n the n n diagonal matrices n1 n1 D diag and D diag n n Theorem The solution to the blo ck Hankel system H x b is given by p n1 H 0 n D D I x x n n p n p 0 1 0 0 where x L b with b n D I b n p n 1 T 0 Proof Theorem implies that x H b is given by x W z I x p 0 1 0 0 where x L b with b W y I b In the pro of of Theorem it p is shown that p p T n1 H n D and W z n D D W y n n n n n based on an inversion formula for block Loewner matrices T T As W z I W z I the result follows p p It follows that 0 x x r r 0 x x p p+r p+r n1 H n D D n n n 0 x x (n1)p+r (n1)p+r and 0 b b r r 0 b b p p+r p+r n D n n 0 b b (n1)p+r (n1)p+r 0 0 for r p Thus the transformations b b and x x can b e done in O pn log n oating p oint op erations Let P b e the p ermutation matrix dened by y 1 y n P z 1 2n1 z n let b e the n n Fourier matrix 2n 2n1 p V 2n n 2n1 and let D diag 2n Theorem The p p matrices C C D D of the blo ck Lo ewner 1 n 1 n pp matrix L are given by up to an arbitrary additive matrix C C 1 H 0 H 1 p C n T n P D I 2n 2n p D 1 H 2n2 D n p T Proof In Theorem it is shown that W y z n P D 2n 2n The result then follows immediately from Theorem P Kravanja M Van Barel A fast block Hankel solver The previous theorem implies that C 1 k l H 0 k l H 1 k l p C n k l T n P D 2n 2n D 1 k l H 2n2 k l k l D n k l 2 for k l p Thus H can b e transformed into L in O p n log n ops Note As we already mentioned in the pro of of Theorem in Theorem it is shown that p p T n1 H W y n D and W z n D D n n n n n p p T n and W z n are unitary The Kronecker pro duct of two Thus W y unitary matrices is a unitary matrix As T p p L n W y I H W z I p p n n 1 1 1 it follows that kLk nkH k and kL
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