Key words. Companion matrix, quasiseparable structure, QR iteration, eigenvalue computation,

Transcription

1 A FAST IMPLICIT QR EIGENVALUE ALGORITHM FOR COMPANION MATRICES D. A. BINI, P. BOITO, Y. EIDELMAN, L. GEMIGNANI, AND I. GOHBERG Abstract. An implicit version of the QR eigenvalue algorithm given in [D. A. Bini, Y. Eidelman, I. Gohberg, L. Gemignani, SIAM J. Matrix Anal. Appl , no. 2, ] is presented for computing the eigenvalues of an n n companion matrix using On 2 flops and On memory storage. Numerical experiments and comparisons confirm the effectiveness and the stability of the proposed method. Key words. Companion matrix, quasiseparable structure, QR iteration, eigenvalue computation, complexity AMS subject classifications. 65F15, 65H17 1. Introduction. The paper concerns the efficient computation of the eigenvalues of companion matrices. A fast and numerically robust algorithm for this tas, devised in [4], is based on the use of the explicit QR iteration applied to an input companion matrix A C n n represented as a ran-one perturbation of a unitary matrix, namely, A U zw T, where U C n n is unitary and z, w C n. The computational improvement with respect to the customary methods is achieved by exploiting the quasiseparable Hessenberg structure in the QR iterates inherited from the ran properties of the input companion matrix A. More specifically, in [4] it is shown that each iterate A is a Hessenberg matrix expressed as the sum of a unitary matrix U plus a ran-one correction. The fast implementation of the explicit QR iteration taes in input a complete set of generators for the quasiseparable structure of A and returns as output a complete set of generators for the quasiseparable structure of A +1. The reduction of matrix operations to manipulating a set of On parameters enables one to perform each QR step in On floating point operations flops with On memory storage. In [4] it is also pointed out that in practice, due to rounding errors, some additional computations must be carried out in order to maintain both the quasiseparable structure of A and the unitary property of U. In the classical numerical linear algebra literature [12, 16, 2] it is generally claimed that the implicit QR method should be preferred to its explicit counterpart since it can be faster in the case of multiple shifts, more stable and, moreover, admits suitable variants for the case of real inputs. The multishift QR iteration proceeds as follows: A A q A Q R, QR factorization A +1 : Q H A Q, 1.1 where q z is a monic polynomial of degree one single-shift step or two double-shift step suitably chosen to accelerate the convergence. The bulge-chasing implicit QR techniques manage to perform the transformation A A +1 without explicitly forming the matrix q A. The implicit determi- Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, Pisa, Italy, {bini,gemignan}@dm.unipi.it, boito@mail.dm.unipi.it School of Mathematical Sciences, Raymond and Beverly Sacler Faculty of Exact Sciences, Tel-Aviv University, Ramat-Aviv, 69978, Israel, {eideyu, gohberg}@post.tau.ac.il This wor was partially supported by MIUR, grant number , and by GNCS-INDAM. 1

2 2 nation of A +1 from A was first described by Francis [11, 1] see also [12] and [16] and the references given therein. Let Q 1 be a Householder matrix chosen to annihilate the subdiagonal entries in the first column of q A. The transformation A Q H 1 A Q 1 corrupts the upper Hessenberg form of A by creating a bulge of size degq z at the top left corner of the matrix. It is shown that the computation of A +1 essentially consists of chasing the bulge down and to the right of the matrix until it disappears. The tas can be accomplished by a standard Hessenberg reduction employing a sequence Q 2,..., Q n 1 of Householder matrices. The resulting algorithm requires On 2 flops and it is provably bacward stable [14]. The first implicit fast and accurate QR eigenvalue algorithm for real companion matrices has been presented in [6]. The algorithm employs a factored representation of A as the product of a unitary Hessenberg by a quasiseparable upper triangular matrix. A similar representation was previously considered in [1] for the efficient eigenvalue computation of certain ran-one corrections of unitary matrices. The bulge-chasing procedure is performed in linear time by taing advantage of the special form of A. Specifically, the multiplication on the left by the elementary unitary matrix Q H j amounts to a suitable rearrangement of the Schur factorization of the unitary factor by performing a sequence of swapping operations. Moreover, the multiplication on the right by Q j involves the updating of the quasiseparable structure of the upper triangular factor via manipulations of its generators. At the end of this updating process an auxiliary compression step is still required to recover a minimal-order quasiseparable parametrization for the triangular factor. In this paper we modify the algorithm given in [4] to incorporate single-shift and double-shift implicit techniques, thus obtaining a fast adaptation of the implicit QR method for the case where the initial matrix A A C n n is in companion form. The novel scheme still follows from the representation of each iterate A as a ranone correction of a unitary matrix, namely, A U z w T. The resulting eigenvalue algorithm taes in input a condensed representation of U, the perturbation vectors z and w as well as the coefficients of the shift polynomial q z and returns as output the generators of A +1 computed by means of 1.1. Differently from the approach pursued in [3] and [4], here the use of a suitable factored representation of U maes it possible the updating of the decomposition during the bulge-chasing process in a stable way without any additional compression and/or re-orthogonalization step. A minimal quasiseparable representation of U is easily computed from its factored form and then used for finding the unitary matrices Q j involved in the bulge-chasing process. The proposed algorithm is therefore logically simple, efficient and numerically stable. It turns out that the QR iteration can be performed at the cost of On flops using On storage. Numerical experiments confirm that the algorithm is stable. Experimental comparisons are also included by showing that in the considered cases our algorithm outperforms the one in [6]. Algorithms for explicit and implicit QR iterations with Hermitian and small ran perturbations of Hermitian matrices may be found in [5, 7, 8, 9]. An implicit QR algorithm for companion matrices was also discussed in [15]. The paper is organized as follows. In Sect. 2 we recall the structural properties and introduce condensed representations for the matrices generated by the QR process applied to an input companion matrix. Fast algorithms to carry out both the single-shift and the double-shift implicit QR iteration applied to such matrices are described in Sect. 3 and Sect. 4, respectively. In Sect. 5 we address some complementary issues concerning deflation and stopping techniques while in Sect. 6 the results

3 of extensive numerical experiments are given and compared. Finally, the conclusion and a discussion are the subjects of Sect Matrix structures under the QR iteration applied to a companion matrix: The classes H n and U n. In this section we analyze some matrix structures which play a fundamental role in the design of a fast adaptation of the shifted QR eigenvalue algorithm 1.1 applied to an input matrix A A C n n in companion form. For a given monic polynomial pz of degree n, pz p + p 1 z +... p n 1 z n 1 + z n n z ξ i, i1 3 the associated companion matrix A is defined by p 1 p 1 A p n 1 It is well nown that the set of eigenvalues of A coincides with the set of zeros ξ 1,..., ξ n of pz and this property provides the classical reduction between the computation of polynomial zeros and eigenvalues of companion matrices. Matrix methods based on the QR iteration 1.1 applied to a companion matrix are customary for polynomial root-finding: in fact, the MATLAB command roots relies on this approach. The general routines for Hessenberg matrices require On 2 flops and On 2 memory space per iteration. Fast adaptations of the QR eigenvalue algorithm applied to the companion matrix A can achieve better linear estimates both for the cost and for the storage. The computational improvement is due to the exploitation of certain additional matrix structures in the iterates A,, that follow from the companion form of A A. Specifically, it is worth noting that A A is an upper Hessenberg matrix which can be expressed as a ran-one perturbation of the unitary companion matrix U U associated with the polynomial z n 1, i.e., A p + 1 p 1. p n The Hessenberg shape is preserved under the QR iteration and, moreover, from 1.1 it follows that A +1 is a ran-one perturbation of a unitary matrix whenever A fulfills the same property. Therefore each matrix A,, generated by the QR iteration 1.1 applied to the initial companion matrix A A can be recognized as a member of a larger class of structured matrices. Definition 2.1. Let H n C n n be the class of n n upper Hessenberg matrices defined as ran-one perturbations of unitary matrices. That is, B H n if there exist MATLAB is a registered trademar of The Mathwors, Inc..

4 4 U C n n unitary and z, w C n such that B U zw T. 2.1 The vectors z z i n i1, w w i n i1 are called the perturbation vectors of the matrix B. The Hessenberg form of B implies some additional properties of the unitary matrix U. Definition 2.2. We define the class U n to be the set of n n unitary matrices U u i,j n i,j1 such that u i,j z i w j, 1 j i 2, 3 i n, 2.2 for suitable complex numbers z i and w j referred to as lower generators of the matrix U. Let B be a matrix from the class H n represented in the form 2.1 with the unitary matrix U and the vectors z z i n i1 and w w i n i1. Then the matrix U belongs to the class U n and the numbers z i, i 1,..., n and w i, i 1,..., n 2 are lower generators of the matrix U. In the reverse direction let U be a unitary matrix from the class U n with lower generators z i i 3,..., n and w i i 1,..., n 2. Tae arbitrary numbers z 1, z 2 and w n 1, w n and set z z i n i1, w w i n i1, the matrix B U zw T belongs to the class H n. In this paper we use essentially representations of the matrices from the class U n as a product U V F, where V and F are unitary matrices with zero entries above some superdiagonal and below some subdiagonal respectively. We consider in more details properties of such matrices. Denote by the direct sum of two matrices such that A B A B Lemma 2.3. Let W w ij n i,j1 be unitary matrix and let m be positive integer. The matrix W satisfies the conditions w ij for i > j + m if and only if W admits the factorization where. W W 1 W 2 W n m, 2.3 W i I i 1 W i I n i m, i 1,..., n m 2.4 with m + 1 m + 1 unitary matrices W i. The matrix W satisfies the conditions w ij for j > i + m if and only if W admits the factorization W W n m W n m 1 W 2 W with the unitary matrices W i i 1,..., n 2 of the form 2.3. Proof. Assume that w i,j, i > j + m. 2.6 The first column of the matrix W has the form w1 W :, 1,

5 where w 1 is an m + 1-dimensional column with the unit norm. We tae a unitary m+1 m+1 matrix W 1 such that W1 H f 1 1 T and then determine the matrix W 1 by the formula 2.4. We get W1 H W :, 1 e 1, where e 1 is the first vector from the standard basis in C n and since the matrix W1 W is unitary we conclude that W W 1 1 Ŵ 2 with a unitary n 1 n 1 matrix Ŵ2 which satisfies the condition 2.6. Next we apply the same procedure to the matrix ˆF 2 and so on and on the n m 1-th step we obtain the factorization Assume that the matrix W has the form 2.3 with the matrices W i i 1,..., n m of the form 2.4. We prove by induction in n that the condition 2.6 holds. For n m + 2 we have obviously W1 1 W W 1 W 2 1 W 2 and therefore w n,1 w m+2,1. Let for some n 2 the n 1 n 1 matrix where W W 1 W n m 1, W i I i 1 W i I n i m 1, i 1,..., n m 1 with m + 1 m + 1 matrices W i, satisfies the condition The matrix W defined via 2.3, 2.4 may be done in the form W In m 1 W W 1 W n m 1 W n m. 1 W n m 5 Hence it follows that W m + i + 1 : n 1, 1 : i W m + i + 1 : n, 1 : i, i 1,..., n m 1 which completes the proof of the first part of the lemma. Applying the first part of the lemma to the transposed matrix W T we prove the second part. Remar 2.4. Every matrix W i i 1,..., n 2 may be taen either as a m + 1 m + 1 Householder matrix or as a product of complex Givens rotations. For a matrix U U n we derive a condensed representation as a product of elementary unitary matrices. Definition 2.5. Let z z i n i1 be a vector and let V i, i 2,..., n be 2 2 unitary matrices. We say that the vector z is reduced by the matrices V i i 2,..., n if there exist complex numbers β i, i 2,..., n such that the relations β n z n, V H i zi β i+1 βi hold. Define the unitary matrix V C n n as the product, i n 1,..., 2, 2.7 V V n 1 V 2, 2.8

6 6 where V i I i 1 V i I n i 1, 2 i n We say also that the vector z is reduced by the matrix V. From 2.8 by Lemma 2.3 it follows that V v i,j is a unitary lower Hessenberg matrix, i.e., v i,j, j > i Lemma 2.6. Let U C n n be a unitary matrix from the class U n with lower generators z i i 3,..., n, w j j 1,..., n 2 and let z 1, z 2, w n 1, w n be arbitrary numbers. Assume that the vector z z i n i1 is reduced by 2 2 unitary matrices V i, i 2,..., n 1, define the unitary matrix V via relations 2.8, 2.9. Then the unitary matrix F V H U f i,j has null entries below the second subdiagonal, i.e., f i,j, i > j Proof. Set w w 1,..., w n T, z z 1,..., z n T and define the matrix B C n n as in 2.1. It is clear that B is an upper Hessenberg matrix. Furthermore the relations imply that the vector g V H z g i has null components g i for i > 3. By Lemma 2.3 V is a lower Hessenberg matrix. Hence using the fact that B and V H are upper Hessenberg matrices we conclude that the matrix F V H U V H B + V H z w T satisfies the relations Lemma 2.7. Every matrix U from the class U n admits the decomposition U V F, 2.12 where V is a unitary lower Hessenberg matrix and F f ij is a unitary matrix satisfying the condition f i,j, i > j + 2. Moreover one can tae the matrix V in the form where V V n 1 V 2, 2.13 V i I i 1 V i I n i 1, 2 i n with 2 2 unitary matrices V i and the matrix F in the form where F F 1 F 2 F n 2, 2.15 F i I i 1 F i I n i 2, 1 i n with 3 3 unitary matrices F i Proof. Let z i i 3,..., n, w j j 1,..., n 2 be lower generators of the matrix U and let z 1, z 2, w n 1, w n be arbitrary numbers. Determine 2 2 unitary matrices V i such that the relations 2.7 hold, and define the unitary matrix V via

7 2.13, By Lemma 2.3 the matrix V is lower Hessenberg. Moreover the vector z z i n i1 is reduced by the matrix V and hence by Lemma 2.6 there exists a unitary matrix V of the form 2.13 such that V H U F is lower banded with bandwidth 2. Summing up, we obtain that any matrix B from the class H n can be represented as B U zw T V F zw T, where V and F are unitary matrices represented in the factored form as specified by 2.13, 2.14 and 2.15, 2.16, and z and w are the perturbation vectors. Hence, the matrix B H n is completely specified by the following parameters: 1. the unitary matrices V, 2,..., n 1 defining the matrix V ; 2. the unitary matrix F, 1,..., n 2 defining the matrix F ; 3. the perturbation vectors z and w. These parameters are also called the generating elements of the matrix B. The decomposition of U U n by means of the elementary matrices V and F is numerically robust but not easy to be manipulated under the QR iteration applied to B. In this respect a more suited condensed form of U is its quasiseparable parametrization introduced in [4]. The representation gives an explicit description of each entry of U as the product of certain vectors and matrices of small size. We will show that for every matrix from the class U n there exist vectors g j C 2, 1 j n, h j C 2, 1 j n, and matrices B j C 2 2, 2 j n, such that u i,j g T i B i,j h j, for j i, 2.17 where B i,j B i+1 B j for n j i + 1 and B i,j I 2 if i j. The elements g j, h j j 1,..., n, B j j 2,..., n are called upper generators of the matrix U. The next result shows that generators of U can be easily reconstructed from the two sequences of elementary matrices V and F defining the unitary factors V and F, respectively. The recurrence relations for the upper generators can also be used to compute some elements in the lower triangular part of U. In particular we provide the formulas for the subdiagonal elements of U involved in the QR iteration. Theorem 2.8. Let U u i,j be a unitary matrix from the class U n with the given factorization U V F, where the factors V and F defined by the formulas 2.13, 2.14 and 2.15, 2.14 using the given unitary 2 2 matrices V i i 2,..., n 1 and unitary 3 3 matrices F i i 1,..., n 2. The entries u i,j, max{1, i 2} j n, 1 i n, satisfy the following relations: u i,j g T i B i,j h j for j i ; u i,j σ j for 2 i j + 1 n, 2.19 where the vectors h and the matrices B are determined by the formulas h F 1: 2, 1, B +1 F 1: 2, 2: 3, 1 n 2, 2.2 and the vectors g and the numbers σ are computed recursively Γ 1 1, g T 1 1 ; 2.21

8 8 σ g T +1 Γ +1 V +1 Γ 1 F, 1,..., n 2, 2.22 σ n 1 Γ n 1 h n 1, g T n Γ n 1 B n with the auxiliary variables Γ C 1 2. Proof. Let the elements g i, h i i 1,..., N, B 2,..., n, σ 1,..., n 1 be given via Using the elements g, B we define the matrices G 1,..., N of sizes 2 via relations G colg T i B i, i1, 1,..., n It is easy to see that the relations 2.18, 2.19 are equivalent to G h U1 : + 1,, 1,..., n 1; U1 : n, n G n h n Next we set σ C V +1 V 2 F 1 F, 1,..., n 2. Using the formulas 2.13, 2.16 we get C 1 : + 2, 1 : + 2 C, I n 2 1,..., n Moreover using 2.12, 2.13, 2.15 we have U V N 1 V +2 C F +1 F N 2, 1,..., n 3; U C n 2. Furthermore from 2.13, 2.16 we get I+1 V n 1 V +2 and therefore I, F +1 F n 2 U1 : + 1, 1 : C 1 : + 1, 1 :, 1,..., n Next we prove by induction that C 1 : + 2, 1 : + 2 C 1 1 :, 1 : 1 G h G B +1 σ g T +1, 1,..., n 2. Γ For 1 using the formula C 1 V 2 F 1 and the formulas 2.13, 2.16 we have therefore 1 C 1 1 : 3, 1 : 3 F V 1. 2 From here using 2.21, g T 1 G 1 and 2.2, 2.22 with 1 we get C 1 1, 1 : 3 F 1 1, 1 : 3 g 1 h1 B 2 G1 h 1 G 1 B 2,

9 C 1 2 : 3, 1 : 3 V F 1 σ1 g T 2 Γ 2 which implies 2.28 with 1. Let for some with 1 n 3 the relation 2.28 holds. Using the equality G B G +1 one can rewrite 2.28 in the form g T +1 C 1 : + 1, 1 : G C 1 : + 2, 1 : Γ +1 Using the formula C +1 V +2 C F and the formulas 2.26, 2.13, 2.16 we obtain C +1 1 : + 3, + 1 : + 3 I+1 C 1 : + 1, 1 : G +1 Γ V +1 I. +2 F 1 +1 From here using the relation F +1 1 : 2, 1 : 3 h +1 B +2 and the formula 2.22 we get. C +1 1 : + 1, + 1 : + 3 G +1 h +1 G +1 B +2, C : + 3, + 1 : + 3 V +2 Γ+1 1 F +1 σ+1 g T +2 Γ +2 which completes the proof of Now combining the relations 2.27, 2.28 together we obtain the equalities 2.25 for 1,..., n 2. Moreover using the equality C n 2 U and the relation 2.28 with n 2 we have U1 : n, n 1 : n G n 2 B n 1 g T n 1 Γ n 1 Taing here B n, h n 1, h n, σ n 1, g T n as in 2.2, 2.23 we get Gn 1 h U1 : n, n 1 : n n 1 G n 1 B n h n g T n h n σ N 1 which completes the proof of the theorem. Remar 2.9. One can chec easily that the relations 2.17 are equivalent to the equalities. U, : N g T H, i 1,..., n 2.29 with 2 n + 1 matrices H defined via relations H rowb,i h i n i, 1,..., n In the next sections we provide a fast adaptation of the implicit QR iteration 1.1 applied for computing the complete set of eigenvalues of the input companion matrix A A H n. It taes in input the generating elements of the matrix A H n together with the coefficients of the shift polynomial q z and returns as output the generating elements of the matrix A +1 H n. The computation is carried out at the total cost of On flops using On memory space.

10 1 3. The Structured QR Iteration. In this section we present a fast modification of the implicit QR iteration 1.1 applied to an input matrix A H n expressed in terms of its generating elements, that is, given in the form A U zw T V F zw T V n 1 V 2 F 1 F n 2 zw T, where U V F is the factored representation of U U n and z and w are the perturbation vectors of A. For the sae of notational simplicity we omit the superscript which labels the QR steps. Let qz be a monic polynomial of degree l {1, 2} determined to speed up the convergence of the QR iteration. The cases l 1 and l 2 are referred to as the single-shift and the double-shift iteration, respectively. Moreover, let Q 1 G 1 I n l 1 be a unitary matrix suitably chosen to annihilate the subdiagonal entries in the first column of qa. The transformation A Q H 1 AQ 1 corrupts the upper Hessenberg form of A by creating a bulge of size l at the top left corner of the matrix. It can be shown that the computation of A 1 essentially consists of chasing the bulge down and to the right of the matrix until it disappears. The tas can be accomplished by a standard Hessenberg reduction employing a sequence Q 2,..., Q n 1 of unitary matrices. The cumulative unitary factor Q 1 such that A 1 Q 1H AQ 1 is given by the product Q 1 : Q 1 Q 2 Q n 1. The updating of the matrices A A 1 and U U 1 and of the vectors z z 1 and w w 1 can be carried out in n 1 steps according to the following rules: A 1 Q H 1 A; A A Q, A +1 Q H +1A, 1,..., n 1; A 1 : A n 1 Q n Here every A, 1,..., n 1 is an upper Hessenberg matrix. Moreover using the formula 2.1 and setting U 1 Q H 1 U; U U Q, U +1 Q H +1U, 1,..., n 1; U 1 : U n 1 Q n 1, 3.2 we get z 1 Q H 1 z; z +1 Q H +1z, 1,..., n 2; z 1 : z n w 1 w; w T +1 w T Q T, 1,..., N 1; w 1 : w n. 3.4 A U z w T, 1,..., n Here U are unitary matrices and z, w are vectors. Moreover the matrices Q H +1 are chosen to remove the bulge in the matrix A and therefore A is an upper Hessenberg matrix. Hence it follows that every matrix A belongs to the class H n. The structured QR iteration essentially consists of an efficient procedure for evaluating the unitary matrices Q j, 1 j n 1, combined with an efficient scheme for updating the factorized representation of U under the multiplication on the right and on the left by the matrices Q j. In the next two subsections we describe the structured QR iteration in the single-shift and in the double-shift case. Specifically, in Subsection 3.1 we present a detailed description of the fast single-shift iteration and give a formal proof of its correctness. Then in Subsection 3.2 we setch the fast double shift iteration putting in evidence the basic differences with the single-shift step. A proof of the correctness of the double-shift iteration proceeds in exactly the same way as for the single-shift case.

11 3.1. The Structured QR Iteration: The Single-Shift Case. The procedure FastQR ss for the fast computation of the generating elements of A 1 such that 11 A αi QR, QR factorization A 1 : Q H AQ 3.6 is outlined below. Given in input the generating elements V 2,..., n 1, F 1,..., n 2, z z i n i1, w w i n i1 of A together with the shift parameter α, the algorithm calculates the generating elements V 1 2,..., N 1, F 1 1,..., N 2 z 1 z 1 i n i1, w1 w 1 i n i1 of A1. The condensed representations of A 1 and U 1 are computed by the scheme , where Q j are Givens rotation matrices determined in the bulge-chasing process. Procedure FastQR ss 1. Using algorithm from Theorem 2.8 compute upper generators g i, h i i 1,..., n, B 2,..., n and subdiagonal entries σ 1,..., n 1 of the matrix U. 2. Set β n z n and for n 1,..., 3 compute β V H z 1, 1 : 2 β The third stage of the algorithm is to compute the Givens rotation matrices G, 1,..., n 1 and the updated perturbation vectors z 1, w 1 a Set z τ 1 z 1. Determine the complex Givens rotation matrix G 1 such that G1 H g T 1 h 1 z 1w 1 α 3.8 σ 1 z 2w 1 and compute z 1 1 z τ 2 σ τ 1 g τ 2 T G H 1 z 1 1 g T 1 h1 gt 1 B1 z 2 σ 1 g T b For 1,..., n 2 perform the following. Compute ρ ρ ρ ρ z 1 z τ +1 σ τ g τ +1 T h +1 z +2 w τ σ +1 w τ w +1 G. 3.1 Determine the complex Givens rotation matrix G +1 from the condition G H +1 Compute z 1 +1 z τ +2 σ τ +1 g τ +2 T ρ z τ +1 w1 ρ z +2 w 1 G H +1 G H +1 ρ z τ +1 z +2 g τ +1 T B +1 ρ g

12 12 c Compute and set w 1 n 1 w n 1 z 1 n w τ n 1 w n G n z τ n The fourth stage of the algorithm is to compute the generating elements V 1 2,..., n 1, F 1 1,..., n 2 of the matrix A 1. a Determine the complex Givens rotation matrix V τ 2 such that V τ 2 H τ 1 z 2 β with some number β 1 2 and compute the 3 3 matrix 1 F τ G H V τ 2 H 1 V 2 b For 1,..., n 3 perform the following. Compute the 3 3 matrix G H X V +2 β 3 V τ +1 1 F and the 4 4 matrix τ F Y 1 1 F +1 G I Determine the complex Givens rotation matrix Z from the condition Compute the matrix and the matrix X +1 1, 2 : 3Z. 3.2 W X +1 1 Z I2 D Z H 3.21 Y The unitary matrix W has the zero entry in the position 1, 3 and the unitary matrix D has the zero entry in the position 4, 1. Compute the factorization W 1 V τ +2 V with unitary 2 2 matrices V τ +2, V1 +1 and the factorization D F F τ with 3 3 unitary matrices F 1 c Compute F 1 n 2 F τ n 2, F τ +1. V 1 n 1 GH n 1V τ Gn 2 1 n 1, G n 1

13 Theorem 3.1. Let A U zw T be a matrix from the class H n with generating elements V 2,..., n 1, F 1,..., n 2, z z i n i1, w w i n i1 such that the vector z is reduced by the matrix V V n 1 V 2, V i I i 1 V i I n i 1, 2 i n 1. Then the output data V 1 2,..., N 1, F 1 1,..., N 2, z 1 z 1 i n i1, w1 w 1 i n i1 returned by FastQR ss are the generating elements of the matrix A 1 satisfying 3.6 and, moreover, z 1 is reduced by the matrix V 1 V 1 n 1 V 1 2, V 1 i I i 1 V 1 i I n i 1, 2 i n Proof. Using Stage 1 of the algorithm we obtain upper generators and subdiagonal entries of the matrix U. On Stage 2 of the algorithm we use the fact that the vector z is reduced by the matrix V and compute the parameters β, N,..., 3 using the formulas 2.7. To justify Stage 3 of the algorithm we prove by induction that the formulas 3.8, 3.11 determine the unitary matrices G, 1,..., n 1 such that the unitary matrices Q I 1 G I n 1 mae the transformations and moreover for 1,..., n 1 the relations U + 1, : n hold and the vectors z, w have the form σ τ g τ +1 T H z z 1 1,..., z1, zτ +1, z +2,..., z n T, 3.28 w w 1 1,..., w1 1, wτ, z +1,..., z n T where the matrices U and the vectors z, w are determined in , hold. On the first step we deal with the matrix A 1 Q H 1 A U 1 z 1 w T 1 with U 1 Q H 1 U, z 1 Q H 1 z, w 1 w. We have g T A αi1 : 2, 1 1 h 1 z 1 w 1 α σ 1 z 2 w 1 and determine the complex Givens rotation matrix G 1 from the condition 3.8. Next using the formula 2.29 with 1, 2, the formula 2.19 with 1 and the relation H 1 h 1 B 2 H 2 we get g T U1 : 2, 1 : N 1 h 1 g T 1 B 2 H 2 σ 1 g T. 2 H 2 Using the formula 3.9 we obtain 3.27, 3.28, 3.29 with 1. Let for some with 1 n 2 the statement of induction holds. From the formulas 3.2 it follows that U + 2, : n U + 2, : n.

14 14 Furthermore using the formulas 2.29, 2.19 we obtain U + 2, + 1 : N σ +1 g T +2 H +2 and from the formulas 3.5, A + 2, and 3.28, 3.29 it follows that U + 2, z +2 w τ. Moreover using the formula 3.27 and the equality H +1 h +1 B +2 H +2 we obtain U +1 : +2, : N We have σ τ g τ +1 T h +1 g τ +1 T B +2 H +2 z +2 w τ σ +1 g T +2 H A A Q U Q z w T Q U z w T +1. Postmultiplication of a matrix C by the matrix Q means only postmultiplication of the columns, + 1 of C by the matrix G. Hence applying the formula 3.1 we obtain and U + 1 : + 2, : n ρ ρ g τ +2 T B +2 H +2 ρ ρ g T +2 H +2 w +1 w 1 1,..., w1, wτ +1, w +2,..., w n T. Now the elements in the, + 1,, + 2 positions in the matrix A are ρ z τ +1 w1, ρ z +2 w 1 and we determine the matrix of Givens rotation G+1 H from the condition Next consider the matrix A +1 Q H +1A Q H +1U Q H +1z w T +1 U +1 z +1 w T +1. Premultiplication of a matrix C by the matrix Q H +1 means only premultiplication of the rows + 1, + 2 of C by the matrix G+1 H. Hence using the formulas 3.13, 3.12 we obtain U , + 1 : N σ τ +1 g τ +2 T H +2 and z +1 z 1 1,..., z1, z1 +1, zτ +2, z +3,..., z n T. Thus we have determined the Givens rotation matrices G, 1,..., n 1 and the vectors z n 1, w n 1. To finish the proof of Stage 3 notice that the last coordinates of the vectors z n 1 z 1 and w n w 1 are given by the formulas 3.15 and Now we will justify Stage 4 of the algorithm. By the definition of generating elements we have U V n 1 V 2 F 1 F n 2, 3.31

15 15 where V i I i 1 V i I n i 1, F i I i 1 F i I n i 2. We should prove that the matrix U 1 admits the factorization where U 1 V 1 n 1 V 1 2 F 1 1 F 1 n 2, 3.32 V 1 i I i 1 V 1 i I n i 1, F 1 i I i 1 F 1 i I n i 2 with the matrices V 1 i, F 1 i determined via 3.16, 3.26, and moreover the vector z 1 is reduced by the matrix V 1. To do this we prove by induction that every unitary matrix U from 3.2 admits the factorization with where U ˆV ˆF, 1,..., n ˆV V n 1 V +2 V τ +1 V 1 V 1 2, ˆF F 1 1 F 1 1 F τ F +1 F n 2, V τ +1 I V τ +1 I n 2, F τ I 1 F τ I n 2 and the matrices V i, F i and V 1 i, F 1 i are defined above. Moreover we will prove that the vector z defined in 3.3 is reduced by the matrix ˆV, i.e., β n z n, V H i zi β i+1 βi, i n 1,..., + 2; 3.34 V τ +1 β H z τ β 1 +1, 3.35 V 1 i H z 1 i β 1 i+1 β 1 i, i,..., From 3.31 it follows that the matrix U 1 Q H 1 U has the form U 1 V n 1 Ṽ3Q H 1 V 2 F 1 F 2 F n 2. Taing the matrix V τ 2 from 3.16 and the matrix F τ 1 from 3.17 we obtain U 1 ˆV 1 ˆF 1 with ˆV 1 V n 1 V 3 V τ 2, ˆF 1 F τ 1 F 2 F n 2. Furthermore using the relations 2.7 with i n,..., 3 and 3.16 we conclude that the vector z 1 is reduced by the matrix ˆV 1 V n 1 V 3 V τ 2.

16 16 Assume that for some with 1 n 3 the induction statement holds. Hence the matrix U +1 Q H +1 U Q has the form with U +1 T +1 S +1 T +1 V n 1 Ṽ+3Q H +1V +2 V τ 1 +1 V V 1 2, We have and S +1 F 1 1 F 1 τ 1 F F +1Q F +2 F n 2. Q H +1V +2 V τ +1 I X +1 I n 1 F τ F +1Q I 1 Y I n 3 with the 3 3 unitary matrix X +1 from 3.18 and the 4 4 unitary matrix Y from Next we determine a 2 2 unitary matrix Z from the condition 3.2 and compute the matrix W by the formula The unitary 3 3 matrix W has the zero entry in the position 1, 3. Hence this matrix can be factorized in the form 3.23 with unitary 2 2 matrices V τ +2, V1 +1. Set Z I +1 Z I n 3 and furthermore ˆV +1 T +1 Z, ˆF +1 Z H S +1. We have where U +1 ˆV +1 ˆF +1, 3.37 ˆV +1 V n 1 V +3 Q H +1V +2 V τ +1 Z V 1 V 1 2 V n 1 V +3 V τ +2 V V Ṽ 1 2 which yields the desired representation for the matrix ˆV +1. Let us chec that the vector z +1 is reduced by the matrix V +1. Indeed using the relations 3.18, 3.21, 3.23 we have 1 Z H 1 V +1 H 1 1 V τ τ V +1 H 1 1 V+2 H +2 H G+1 1 Moreover using the relations 3.12, 3.35 and the relation 3.34 with i + 2 we.

17 17 get 1 1 Z H Z H V τ +1 H 1 1 Z H V τ +1 H 1 1 V τ +1 H 1 zτ +1 β +2 V H +2 1 V+2 H 1 Z H G+1 1 zτ +1 z +2 β +3 β +1 z 1 +1 z τ +2 β +3 β +1. Hence it follows that V τ +2 β H z τ β 1 +2, V 1 +1 H z 1 +1 β 1 +2 β 1 +1 which together with the relations 3.34 with i N 1,..., + 3 and 3.36 implies that the vector z +1 is reduced by the matrix V +1. Thus applying Lemma 2.6 to the matrix U +1 we get Furthermore we get Here we have ˆF , ˆF +1 F 1 1 F 1 1 ZH F τ F +1Q F +2 F n 2. Z H F τ F +1Q with the 4 4 matrix D from This implies ˆF +1 F 1 1 F 1 1 I 1 D I N 3 I 1 D I n 3 F +2 F N 2. We have D 4, 1 ˆF +1 +3, and using 3.38 we conclude that the 4 4 unitary matrix D has the zero entry in the 4, 1 position. Hence the matrix D admits the factorization 3.24 with 3 3 unitary matrices F 1, F τ +1. Hence it follows that ˆF +1 F 1 1 F F F τ +1 F +2 F N 2. which completes the proof of Thus the unitary matrix U n 2 admits the factorization U n 2 ˆV n 2 ˆF N 2 and therefore the matrix U 1 Q H n 1U n 2 Q n 2 Q n 1 has the form U 1 Q H n 1V τ 1 n 1 V n 2 V 1 2 F 1 1 F 1 τ n 3 F n 2 Q n 2Q n 1.

18 18 From here using 3.25, 3.26 we obtain the desired factorization 3.32 for the unitary matrix U 1. Next using 3.12 with n 1 and 3.25 we get V 1 n 1 H z 1 n 1 β 1 n V 1 n 1 H G n 1 Gn 1 H 1 p n 1 p 1 N β 1 n 1 which together with 3.36 with n 2 means that the vector z 1 is reduced by the matrix V 1 V 1 n 1 V 1 2. Remar 3.2. Notice that in the subsequent iteration the parameter β 3 may be updated by the formula 3.35 with 2, i.e., 3 τ z 3 V τ β 4 β 1 3. In this case the Stage 2 of FastQR ss can be sipped The Structured QR Iteration: The Double Shift Case. The double shift technique follows again the scheme In this case, however, the matrices Q are of the type Q I 1 G I n 2, where G is a 3 3 unitary matrix and 1 n 2. The goal is to reduce the matrix qa A α 1 I n A α 2 I n in upper Hessenberg form, where α 1 and α 2 are the eigenvalues of the 2 2 trailing bloc in the matrix A. At the beginning, Q 1 is chosen in such a way that it reduces the first column of qa. The lower triangular part of the matrix A 1 obtained after 1 steps of is given by trila 1, a 1 1 γ 1 1 a γ 1 2 a 1 1 δ 1 â θ 1 ˆγ â +1 ξ 1 η ˆγ +1 a γ n 1 a n In general, the matrix G is chosen so that G H δ 1 â θ 1 ˆγ â +1 ξ 1 η ˆγ +1 β 1 1 a 1 ˆγ 1 a 1 +1 γ 1 +1 Observe that, while in classical implicit QR with double shift the matrix G is only required to reduce the first column of the bulge, in this case the whole bulge should be eliminated in order to ensure that A attains again an upper Hessenberg form. The bulge chasing step to be applied to the matrix U can be described as follows..

19 19 Set R U + 1 : + 4, : + 3, we have and next and R σ t ρ +1 g t +1 T h +2 g t +1 T B +3 h +3 z t +2 wt z +3 w t z +4 w t σ 1 z t +2 w1 σ τ +1 g τ +2 T h +2 g τ +2 T B +3 h +3 z +3 w τ +1 σ +2 g +3 T h +3 z +4 w τ +1 z +4 w +2 σ +4 G H +1 1 R G 1 σ t +1 ρ +2 g t +2 T h +3 z τ +3 w1 z +4 w 1 z τ +3 wt +1 σ τ +2 g τ +3 T h +3 z +4 w t +1 z +4 w τ +2 σ +4 The updated vectors z +1 and w +1 are defined so that w 1 w t w t +1 G H w τ w τ +1 w z 1 z t +1 z τ +2 G+1 H z t z τ +1 z +2 Once determined through the bulge chasing process, the matrices Q j are used to compute the product representation for the updated U. Let us describe the general update step, using for the product representation the same notation as in the previous section. In order to exploit the bulge chasing properties of the matrices Q j, consider an update step where the current unitary term U is multiplied by Q on the right and by Q H +1 on the left. We have: Q H +1 U Q Q H +1 V n 1 V +3 V τ +2 V t +1 V 1 2 F 1 1 F t.. F τ +1 F +2F n 2 Q V n 1 QH +1 V +3 V τ +2 V t +1 V 1 2 F 1 1 F t F τ +1 F +2 Q F n 2. Let us examine more closely the product P +1 Q H +1 V +3 V τ +2 V t +1. We have: G H +1 G H +1 1 I2 1 V +3 1 V τ +2 1 V t +1, I 2

20 2 that is, multiplying on the left by Q H +1 spoils the lower Hessenberg structure of V +3 V τ +2 V t +1. Analogously, multiplying on the right by Q spoils the structure of the product F t F t F τ +1 F +2: I 2 1 F τ +1 G 1 I 2 I2 F +2 G In order to restore these structures, define a 3 3 unitary matrix Z +2 such that P +1 Z H +2 is again lower Hessenberg, where Z +2 I +1 Z +2 I n 4, and consider the updated U in the form Q H +1 U Q V n 1 Q H +1 V +3 V τ +2 V t +1 ZH +2 V 1 2 I 2. F 1 1 Z +2 F t F τ +1 F +2 Q F n 2. Since P +1 Z+2 H is lower Hessenberg, it can be re-factorized as where P +1 Z H +2 V τ +3 V t +2 V 1 +1, V τ +3 I +2 V τ +3 I n 4, V t +2 I +1 V t +2 I n 3, V 1 +1 I V 1 +1 I n 2, for suitable 2 2 unitary matrices V τ +3, Vt +2 and V1 +1. Because of the way in which the matrices Q j have been chosen, the matrix Z +2 must also have the property of restoring the structure of the product F t F τ +1 F +2. It is therefore possible to compute 3 3 unitary matrices F 1, F t τ +1 and F +2 such that where Z +2 F t F τ +1 F +2 Q F 1 F t +1 F τ +2, F 1 I 1 F 1 I n 2, F t +1 I F t +1 I n 3, F τ +2 I +1 F τ +2 I n Convergence Criteria and Deflation Techniques. Deflation is an important concept in the practical implementation of the QR iteration. Deflation amounts to setting a small subdiagonal element of the Hessenberg matrix to zero. This is called

21 deflation because it splits the Hessenberg matrix into two smaller subproblems which may be independently refined further. For the sae of clarity, suppose that after iterations of the QR algorithm the subdiagonal entry β n s of A becomes small enough to be considered negligible. A customary criterion requires that such an entry be small compared with its diagonal neighbors β n s u a n s,n s + a n s+1,n s+1. If this condition is satisfied, then β n s is set to zero which maes A a bloc upper triangular matrix: A A 1 A 2 where A 1 C n s n s and A 2 C s s. Now A 1 and A 2 can be reduced into upper triangular form separately. The process is called deflation and continuing in this fashion, by operating on smaller and smaller matrices, we may approximate all the eigenvalues of A. Specifically, if s 1 or s 2 then the computed eigenvalues of A 2 may be taen to be eigenvalues of the initial matrix A. The matrix U can be partitioned accordingly with A : where u u 1 u 2 U, w U 1 H u 2 w 1 w 1 w 2, U 2,, u 1, w 1 C n s, u 2, w 2 C s. Observe that U 1 and U 2 are not generally unitary matrices and, therefore, the matrices A 1 and A 2 are specified in terms of the whole matrix U as follows: A 1 I n s U In s + u H 1 w 1, 21 and A 2 I s U Is + u H 2 w 2. These representations enable the fast QR iteration to be applied to the input matrices A 1 and A 2. Observe that, even though the whole matrix U is required for representation purposes, woring only on A 1 or A 2 means that only part of the generators are actually involved in the computation. Remar 4.1. The implicit QR method for eigenvalue computation relies heavily on the implicit Q theorem see e.g. [12], which applies to irreducible Hessenberg matrices. As a consequence, deflation is a crucial step in implicit QR and it is necessary to perform it whenever possible.

22 22 A more subtle form of deflation should also be applied when two consecutive subdiagonal entries of A become small typically, small enough for their product to be less than or equal to the machine epsilon. In this case, the next bulge chasing process should only be applied to the submatrix A r+1 : n, r+1 : n, where r is the index such that the small subdiagonal entries are A r + 1, r and A r + 2, r + 1. Indeed, it turns out that beginning the bulge chasing process from the r + 1-st row introduces negligible perturbations of the Hessenberg structure in the entries of indices r + 2, r and r + 3, r in the double shift case. This is an idea that dates bac to Francis and is explained in detail in Wilinson s classical boo [17], Chapter 8 Section 38. It is however often overlooed in textboos, so we recall it briefly here. Let A be an upper Hessenberg matrix having small elements ɛ 1 and ɛ 2 in positions r + 1, r and r + 2, r + 1, for a certain index r, and consider the following partition: A ɛ 1 ɛ 2 X Y E W. 4.1 With this notation we have: Proposition 4.2. Let µ be any eigenvalue of W. Then µ is also an eigenvalue of a matrix A, of the same size as A, which differs from A only by an element ɛ 1 ɛ 2 /Ar + 1, r + 1 µ in position r + 2, r. As a consequence of Proposition 4.2, if the product ɛ 1 ɛ 2 is small enough then the next QR iteration can be applied to the submatrix W, ignoring the fact that E is not numerically zero and performing in fact a sort of deflation. In practical implementations, the search for a suitable partition is generally performed from bottom up, so that one always wors on the smallest possible trailing submatrix. More precisely, before starting each QR iteration on the current iterate A, the following procedure is performed: - loo for the smallest index s such that A n s + 1, n s is negligible; - if s 1 or s 2 for the double shift case, then one or two eigenvalues have been found; exit procedure; - else loo for the largest index r, with n s r n 2 or n s r n 3 for the double shift case, such that beginning the bulge chasing process from the r + 1-st row introduces negligible perturbations of the Hessenberg structure in the entries of indices r + 2, r and r + 3, r for the double shift case; - perform bulge chasing on the trailing submatrix A r + 1 : n, r + 1 : n. See also [13] for practical criteria and suggestions in the classical non-structured case. 5. Numerical Experiments. The algorithms for eigenvalue computation described in the previous sections have been implemented in Fortran 95, for the single and double shift versions. These programs deal with the particular case of the companion matrix A associated with a given polynomial P x n c x and have been used to test the efficiency and stability of the proposed algorithms on several examples. The software is available upon request to the authors. The following definitions of errors are used in tests for stability: Absolute forward error a.f.e.: this is the absolute distance between the

23 LAPACK FASTQR LAPACK FASTQR time.5.4 logtime deg deg Fig Comparison of running times Example 5.1. eigenvalues computed by the structured method which will be called here FASTQR for reference purposes and the eigenvalues computed by LAPACK routines ZGEEV and DGEEV, which are assumed to be correct. The absolute forward error is computed as the -norm of the difference between the two output vectors obtained from FASTQR and LAPACK and sorted by decreasing absolute value. An estimate for this error, based on theoretical arguments, is given by the quantity δ ɛ A 2 max{condeiga}, where ɛ is the machine epsilon and, using Matlab terminology, condeiga is the vector of eigenvalue condition numbers for the matrix A. Matrix relative bacward error: it is defined as m.b.e. Q aa Q a V f F f p f q T f A, where A A is the initial matrix in the QR iteration process, Q a is the accumulated unitary similarity transformation and V f, F f, p f, q f are generators for the product structure of the final iterate A f. Coefficient relative bacward error: it is a vector whose entries are defined as c.b.e c c c if c, where { c } 1,...n are the coefficients, computed in high precision, of the polynomial whose roots are the eigenvalues of A given by our structured method. We start with some examples of the behavior of the single shift version of FASTQR applied to complex polynomials. Most examples in this section are taen from [4] and [6]. Example 5.1. Given a positive integer n, define P x n a + ib x, where a and b are uniformly distributed random numbers in [ 1, 1].

24 y 2.2*x logtime logdeg Fig Experimental estimate of time growth Example 5.1. Random polynomials as in Example 5.1 are particularly useful for tests on computational speed and behavior on polynomials of high degree. Figure 5.1 shows, on the left, running times for FASTQR and LAPACK on such random polynomials, for degrees ranging between 5 and 35. On the right we report the log-scale plot of the same data, where it can be seen that FASTQR becomes faster than LAPACK for polynomials of degree about 1. Figure 5.2 is a log-log plot of the running times of FASTQR on polynomials of degrees ranging between 5 and 2. A linear fit of this plot shows that the growth of running time is indeed quadratic in n, since the slope of the line is about 2.2. The number of required QR iteration is generally less than 3 per eigenvalue. Polynomials in Example 5.1 are generally rather well-conditioned and the behavior of FASTQR is satisfactory both for bacward and forward stability more details are given in Example 5.4 for the double shift version. In the next example, coefficients are still random but chosen with an unbalanced distribution: Example 5.2. Given a positive integer n, define P x x n + n 1 j a jx j, with a j u j 1 vj, where u j is a uniformly distributed random number in [ 1, 1] and v j is a uniformly distributed random number in [ 5, 5]. The behavior of FASTQR on such polynomials is shown in the following table: n m.b.e. a.f.e. δ Bacward errors are small, which ensures bacward stability. The growth of the

25 LAPACK SSS QR FASTQR LAPACK SSS QR FASTQR time.8.6 time deg deg Fig Comparison of running times Example 5.4. forward error mirrors closely the behavior of the quantity δ, therefore the output of our method is consistent with theoretical expectations. Example 5.3. Given a positive even integer n, define P x n/2 1 n/2 x 2+.5 n 1 i sin 2+.5 n 1, where i 2 1. This is the monic polynomial with zeros equally spaced on the curve x t + i sinπt with 1 t 1. Results for this class of test polynomials are shown in the following table: n a.f.e. δ Next, let us consider some examples with real polynomials, to which we apply the double shift version of FASTQR. Example 5.4. Given a positive integer n, define P x n c x, where c is a random number uniformly distributed in [ 1, 1]. As in Example 5.1, we use these polynomials to compare running times and chec that the growth of the running time is indeed quadratic. In this case we are also able to introduce a comparison with the structured implicit QR method proposed in [6], denoted as SSS-QR, a Fortran 95 implementation of which is available on-line for the case of real polynomials. Figure 5.3 shows on the left the running times for FASTQR, LAPACK and SSS-QR for polynomials of degrees ranging between 5 and 4 and on the right the corresponding logarithmic plot. FASTQR becomes faster than LAPACK for polynomials of degrees between 15 and 2. Figure 5.4 shows a log-log plot of the running times of FASTQR for polynomials of degrees ranging between 5 and 2, with a linear fit; the slope here is about 2.2. The number of double shift iterations is generally less than 1.5 per eigenvalue. The logarithms of the absolute forward errors computed for these same polynomials are plotted in Figure 5.5. It is interesting to notice that, while for degrees up to 1 there

26 y 2.2*x logtime FASTQR linear fit logdeg Fig Experimental estimate of time growth Example 5.4. is a growth of the forward errors which is however expected, for degrees higher than 1 the errors stay roughly the same. This is also the case for experiments made for degrees 3 and 5. This suggests that FASTQR reliably computes eigenvalues even for very high degrees. Example 5.5 Roots of 1. Given a positive integer n, define P x x n 1. We use these polynomials to compare the running times of FASTQR to the running times of LAPACK and SSS-QR; the regular and logarithmic plot for polynomials of degrees ranging from 5 to 4 are found in Figure 5.6. Here FASTQR becomes faster than LAPACK for n 5. The computation of the roots of unity with FASTQR is generally very accurate forward errors are about 1 15 for the polynomials used in this experiment. Example 5.6 Multiple roots. Given a positive integer n, define P x 2 c x x 1 n, where c is a random number in [ 1, 1]. In this case, the computation of roots with multiplicity 1 is very accurate, whereas the multiple root 1 suffers from ill conditioning. The following table shows forward and bacward errors for several values of n. For the coefficient bacward error we give an interval containing all entries of the error vector. n a.f.e. c.b.e. δ , , , , Example 5.7. Define P x x 2 + x x + 1. We obtain here a.f.e ; the entries in the coefficient bacward error

27 log1forward error deg Fig Absolute forward errors Example LAPACK SSS QR FASTQR 1 LAPACK SSS QR FASTQR 3 time time deg deg Fig Comparison of running times Example 5.5. are in the range, , which is roughly the same result given in [6] and the same as one would obtain using the Matlab function roots. Example 5.8. Define P x as the monic polynomial whose roots are, in Matlab notation, [ 2.1 :.2 : 1.7]. For this example we have a forward error of , which is consistent with the estimate given by δ The entries in the coefficient bacward error

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