Linear Systems With Coeæcient Matrices. Having Fields of Values. Ren-Cang Li. Department of Mathematics. University of California at Berkeley
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1 Linea Systems With Coeæcient Matices Having Fields of Values Not Containing The Oigin æ Ren-Cang Li Depatment of Mathematics Univesity of Califonia at Bekeley Bekeley, Califonia 9470 Mach 9, 994 Compute Science Division Technical Repot UCBèèCSD , Univesity of Califonia, Bekeley, CA 9470, Decembe, 994. This is a continuation of ë3ë addessing a poblem posed by Pof. Kahan. The poblem is the following: Given an n æ n ècomplexè matix A whose æeld of values FèAè does not contain the oigin, is it necessay to pivot when solving the linea system Ax b? It is well known FèAè is a compact convex set on the complex plane. Let's daw two pojecting lines ` and ` stating at the oigin and ëtangent" to the bounday of F èaè such that F èaè falls into the smalle section enclosed by ` and ` as shown in Figue. Let æ be the angle of the section. Clealy 0 æ é. Set, æ. It is poved in ë3ë that if is easonably æ This mateial is based in pat upon wok suppoted by Agonne National Laboatoy unde gant No and the Univesity of Tennessee though the Advanced Reseach Pojects Agency unde contact No. DAAL03-9-C-0047, by the National Science Foundation unde gant No. ASC , and by the National Science Infastuctue gants No. CDA and CDA Hee ëtangent" may not be the ight wod since the bounday of FèAè could not be smooth. So to be moe igous, we could say that ` and ` ae two suppot lines passing though the oigin.
2 y l θπ α α l Figue : A typical pictue. lage, no smalle than 0: èsayè, thee is no dange of instability in solving Ax b without pivoting. Howeve, if is athe small, æ èsayè, thee is a potentiality that elements might gow by a facto O æ. Recently, Ming 3 Gu ëë impoves this facto to O, which is optimal as fa as only ode æ is concened. This note adopts the idea developed in ëë whee the case A being eal is studied. We will give a bette bound which isasymptotically attainable. It is easy to veify that scala multiplications do not aæect element gowth in Gaussian elimination pocesses. Theefoe, without loss of any geneality, by otating the matix A by an angle æ as e iæ A we can assume that FèAè lies in the ight half plane and the angles between the y-axis and ` and between the negative diection of the y-axis and ` ae equal to. Set A H + is, whee H A + Aæ H æ S A, Aæ i S æ ae both Hemitian, and moeove H is positive deænite. When A is eal, FèAè is symmetic with espect to the x-axis, so eithe A itself o,a has the desied popety.
3 Doing Gaussian elimination on A, we get a decomposition A H + is LDM æ èè whee L and M ae unit lowe tiangula matices, D diagonal. Geneally, they ae all complex. The existence of the decomposition èè is guaanteed by the assumption we made on the FèAè èef. ë3ëè. Poposition Wite D diag èd d æææd n è. Then éd j é 0 j æææn whee éèæè denotes the eal pat of a complex numbe. Since H H æ is positive deænite, it has a unique Cholesky decomposition H GG æ, whee G is lowe tiangula. Now èè gives L, èh + isèl,æ DM æ L,æ è L, GG æ L,æ + il, SL,æ DM æ L,æ which yields èg æ L,æ è æ G æ L,æ + il, SL,æ DM æ L,æ : èè Let e j be the jth column of the n æ n identity matix. Compaing the jth diagonal enties of the two sides of èè leads to éd j kg æ L,æ e j k è3è since M æ L,æ is unit uppe tiangula. Theefoe 3 On the othe hand, kg æ L,æ D, k F nx j nx j kg æ L,æ D, e j k kg æ L,æ e j k jd j j n: è4è M, èh + isèm,æ M, LD è M, GG æ M,æ + im, SM,æ M, LD 3 D is not single-valued. But fo ou pupose it is good enough to pick any one of, def them and stick to it. D èd è,. 3
4 which yields èg æ M,æ è æ G æ M,æ + im, SM,æ M, LD, so éd j kg æ M,æ e j k and kg æ M,æ D, k F n: è5è It follows fom èè that LD ègg æ + isèm,æ D, èg + isg,æ èg æ M,æ D, D M D, L, ègg æ + isè D, L, GèG æ + ig, Sè: Thus kld k F p nkg + isg,æ k è6è kd Mk F p nkg æ + ig, Sk : è7è Notice that kg + isg,æ k kèg + isg,æ èèg æ, ig, Sèk kgg æ, is + is + SG,æ G, Sk kh + SH, Sk khk + ksh, Sk kg æ + ig, Sk kèg, isg,æ èèg æ + ig, Sk kgg æ + is, is + SG,æ G, Sk kh + SH, Sk khk + ksh, Sk : Togethe with è6è and è7è, we have 4 kjljjdjjm æ jk F kjld jjd Mjk F nkh + SH, Sk nèkhk + ksh, Sk è: è8è 4 By jxj, we mean its entywise absolute value, i.e. jxj def èjx ijjè. 4
5 To elate this bound to the angle æ, we obseve ksh, Sk kh èh, SH, èèh, SH, èh k kh k kh, SH, k kh k khk kh, SH, k : Lemma kh, SH, k max x60 x æ Sx x æ Hx tan æ : With those in mind, we get Theoem kjljjdjjm æ jk F nkhk "+ tan æ è : è9è Roughly speaking, the bound in ëë is the, one obtained by eplacing the numbe inside ëæë of è9è with + tan æ + 3 tan æ æ. In what follows, we ae going to pesent an example to shown that this inequality isat least asymptotically attainable in the sense that thee ae examples fo which the two sides of è9è ae abitaily close. Conside èef. ë3ëè è!è!è p A, p! è! p p, 0, + p p, p whee is positive. It is known the æeld of values of FèAè is a disk with cente and adius, i.e. p FèAè fz complex : jz, j g: So 0 6 FèAèiféèwhich will be assumed heeafteè. Fo this A, wehave è!è!è! A LDM æ 0, 0, 0 0 è!è!è! è jljjdjjm æ 0, 0 j, è! H A + Aæ, 0 0 +! 5
6 è S A, Aæ 0,i i i 0 è! SH, S 0 0 tan æ p, : +! Hence khk "+ kjljjdjjm æ jk F è, è + + tan æ è èkhk + ksh, Sk è Deæne a function fèè as follows: fèè def kjljjdjjm æ jk F è +,! è, è4 + è, è +è+ è è, è è è + è "+,, æ i, : khk h+, tan æ q è, è 4 + è, è +è+ è, è, è+ 3 4 è, è, 4 è, è3, 3 è, è4 + Oèè, è 5 è: It is easy to see fè0è p 0: , lim fèè fèè!, which shows that the inequalities è8è and è9è ae asymptotically attainable! And min 0 f èè 0: at p 3, 3 q 3,9+ p + 77 q 3,9+ p 77 3p 0: : To see how fast f èè appoaches pictoially, we efe the eade to Figue, whee the pictue on the left is the gaph of fèè and the one on the ight is that of, fèè. 6
7 f() 0.8 -f() Figue : The functions fèè and, fèè. Refeences ëë G. H. Golub and Ch. van Loan, Unsymmetic positive deænite linea systems, Linea Algeba and its Applications, 8è979è, 85í97. ëë M. Gu, Relating element gowth of Gaussian elimination to the æeld of values, 994. ë3ë R.-C. Li, Relations between the æeld of values of a matix and those of its Schu complements, pepint, 99. 7
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