::.i~:i,... ::.. ::i_:_:i..]'

Transcription

1 The Coditio of Vadermode-like Matrices Ivolvig Orthogoal Polyomials* Walter Gautschi Departmet of Computer Scieces Purdue Uiversity West Lafayette, Idiaa To my teacher, Alexader M. Ostrowski, i gratitude o his 90th birthday Submitted by Richard A. Brualdi ABSTRACT The coditio umber (relative to the Frobeius orm) of the X matrix P = [P;- 1 (x j)]?, j~ 1 is ivestigated, where p,( ) = p,( ; d"a) are orthogoal polyomials with respect to some weight distributio d "A, ad x j are pairwise distict real umbers. If the odes x j are the zeros of P, the coditio umber is either expressed, or estimated from below ad above, i terms of the Christoffel umbers for d"a, depedig o whether the p, are ormalized or ot. For arbitrary real x j ad ormalized p, a lower boud of the coditio umber is obtaied i terms of the Christoffel fuctio evaluated at the odes. Numerical results are give for miimizig the coditio umber as a fuctio of the odes for selected classical distributios d"a. 1. INTRODUCTION Let Pr( t) = Pr( t; d A.), r = 0, 1, 2,..., deote a sequece of orthogoal polyomials relative to some positive measure d"a( t) o the real lie. If i the Vadermode matrix the successive powers 1, t, t 2,.. are replaced by the successive orthogoal polyomials p 0 ( t ), p 1 ( t ), p 2 ( t ),..., there results the matrix [::i~:.i.,., ::.i~:i,... ::.. ::i_:_:i..]' p = P-l(xl) P-l(x2) P-l(x) (1.1) *Work supported i part by the Natioal Sciece Foudatio uder grat MCS LINEAR ALGEBRA AND ITS APPLICATIONS 52/53: (1983) 293 Elsevier Sciece Publishig Co., Ic., Vaderbilt Ave., New York, NY l00l /83/$3.00

2 294 WALTER GAUTSCHI which is osigular for pairwise distict odes x 1, x 2,.., x. We shall assume here that all odes are real. Our iterest is i the coditio of P. We fid it coveiet to cosider the coditio umber (1.2) with respect to the Frobeius orm IIAIIF = [tr(ara)]lf 2, or the closely related Turig coditio umber cod r( P) = - 1 cod p( P ). I Sectio 2 we discuss the case of orthoomwl polyomials {pr(-; da )} ad odes at the zeros ~~) of P Uormalized polyomials are cosidered i Sectio 3, ad arbitrary real odes i Sectio 4. I Sectio 5 we commet o the problem of miimizig the coditio umber i (1.2). 2. ORTHONORMAL POLYNOMIALS-NODES AT ZEROS OF P THEOREM 2.1. Let Pr( ; d A), r = 0, 1, 2,..., be the orthoomwl polyomials with respect to the (positive) measure da, ad xv = ~~>, v = 1,2,...,, the zeros ofp( ;da.). Let furthemwre Av=Ac;->, v=1,2,...,, deote the Christoffel umbers for da. The (2.1) REMARK. If ma(a), mh(a) deote, respectively, the arithmetic ad the harmoic mea of the (positive) umbers A 1, A 2,.,A, the result (2.1) may be restated i terms of the Turig coditio umber as (2.1') Lettig da. vary, for ay fixed positive iteger, over all positive measures which admit orthogoal polyomials of degree ~, it follows that codr(p), hece also cod p( P ), attais its miimum precisely whe A 1 = A 2 = = A. By a classical result [2] this is the case if ad oly if {Pr( ; da )} are the Chebyshev polyomials of the first kid.

3 VANDERMONDE-LIKE MATRICES 295 TABLE 1 THE CONDITION OF p FOR SOME CLASSICAL ORTHOGONAL POLYNOMIALS Chebyshev Legedre 2d kid Laguerre Hermite (0) 5.916(0) 2.076(2) 1.373(1) (1) 1.483(1) 1.005(6) 6.832(2) (1) 3.924(1) 7.770(13) 3.989(6) (1) 1.071(2) 1.924(30) 3.699(14) (2) 2.976(2) 6.607(63) 1.095(31) Proof of Theorem 2.1. Let P = P ad A = diag( A 1,..., A). From the discrete orthogoality property of orthoormal polyomials, if r=s} if r=~:=s ' r,s=0,1,...,-1, it follows that PA = Q is a orthogoal matrix. Therefore, so that IIPII~ = tr(ptp) = tr(a - 1 ), IIP- 1 11~ = tr( QAQT) = tr(a). The proof reveals that 1 j A v are the squares of the sigular values av of P, from which (2.1) follows also o accout of (2.2) The umerical behavior of the coditio umber i (2.1) is illustrated i Table 1 for some classical orthogoal polyomials. (The umbers i paretheses idicate decimal expoets.) 3. UNNORMALIZED POLYNOMIALS For uormalized orthogoal polyomials there seems to be o result comparable i simplicity to (2.1). However, we ca prove

4 296 WALTER GAUTSCHI 2 ~ THEOREM 3.1. Let dr= firpr-/t;d"a)d"a(t), r=1,2,..., ad Ll = max d r /mid r, where the maximum ad miimum are take over r = 1,2,...,. The, i the otatio of Theorem 2.1, if xv = ~~), v = 1, 2,...,, (3.1) Proof. Lettig P = P ad D = diag( d 1, d 2,..., d ), we ow fid that v pni 2 = Q is orthogoal. Therefore, tr(ptp) = tr(a - 112QTDQA ), tr( ( p-1) T p-1) = tr( v-112qaqtd-112). With (3.2) where ev is the vth coordiate vector, we thus have (3.3) Sice 11Qevll 2 = IIQTevll 2 = 1, the quatities rv, sv i (3.2) are Rayleigh quotiets of D ad A, respectively; hece, i particular, Furthermore midr ~ rv ~ maxdr. (3.4) r r L sv = tr( QAQT) = tr( A). v=l Therefore, (3.1) follows from (3.3) by replacig rv ad dv by the bouds i (3.4).

5 V ANDERMONDE-LIKE MATRICES ARBITRARY REAL NODES We ow cosider arbitrary real odes x", but assume ormalized orthogoal polyomials Pr( ; d ~ ). We recall the defiitio of the Christoffel fuctio (see, e.g. (1]): ~(x 0 ) = mi i p 2 (t) d~(t ), pep-1 IR p(x 0 ) = 1 x 0 E IR, (4.1) or, equivaletly, -1 [~(x)] - 1 = L PHx), k=o X E IR. (4.2) THEOREM 4.1. Let x 1, x 2,., x be pairwise distict real umbers ad { Pr( ; d ~ )} the orthoormal polyomials with respect to the (positive) measure d ~. The (4.3) Proof Let v = 1, 2,...,, be the fudametal Lagrage iterpolatio polyomials for the odes x 1, x 2,,x, ad let The, as is easily see, lv(t)= L a"jlpjl-1(t). JL=1

6 298 WALTER GAUTSCHI Cosequetly, 1 L z;(t)da.(t)= 1 L:L:a.,JLpJL_l(t)L:a.,KpK_l(t)dA.(t) IRv=l IR v J.L K ad therefore I!P- 1 11~= 1 I: z;(t)da.(t). IRv=l (4.4) Sice l, E IP-l ad Z.,(x.,) = 1, it follows from (4.1) that O the other had, usig ( 4.2), IIP- 1 11~ ~ L A( x.,) (4.5) v=l The assertio ( 4.3) ow follows immediately from ( 4.5), ( 4.6). (4.6) v = 1,2,...,, as We remark that (4.3) holds with equality if x, = ~~>, follows from Theorem 2.1 ad the fact that A.(~~))= A.<;>, v = 1,2,...,. We also remark that Theorem 4.1 remais valid, with essetially the same proof, if the odes are complex ad A( ) is defied as i (4.1), with p 2 (t) replaced by lp(t ) MINIMIZING THE CONDITION NUMBER A iterestig problem is to determie the optimally coditioed matrix P for ay fixed measure da., i.e. to fid the odes x 1, x 2,,x which

7 VANDERMONDE-LIKE MATRICES 299 miimize the coditio umber cod F( P) over all pairwise distict real odes. We report here o attempts to solve this problem umerically. Recall from (2.2) that where m A (a 2 ), m H( a 2 ) are, respectively, the arithmetic ad the harmoic mea of the squares of the sigular values av of P. It follows that cod p( P) ~, so that the smallest possible coditio umber (attaied for the Chebyshev measure ad Chebyshev odes; cf. Remark to Theorem 2.1) is equal to. Assumig ormalized polyomials p/ ; da ), the coditio umber cod p( P ), or rather its square, ca be writte explicitly as the product of the two expressios i (4.4) ad (4.6). Both expressios, icludig their gradiets, ca be computed fairly easily, the itegral i (4.4) ad similar itegrals ivolved i the gradiet beig evaluated (exactly) by the -poit Gauss Christoffel quadrature rule associated with the measure d A. Usig this computatio i cojuctio with a miimizatio algorithm, for which we selected the procedures i [3], we were able to obtai the results show i Tables 2 ad 3. Although oly local extrema ca be foud i this maer, the closeess of the miimum to the absolute miimum i some of the examples suggests that the results are ideed optimal to withi the precisio give. I Table 2 we show the "optimal" odes ad the miimum coditio umber for Legedre polyomials ( d A( t) = dt o [ - 1, 1 ]). Table 3 displays oly the optimal coditio umber (without odes) for some of the other TABLE 2 OPTIMALLY CONDITIONED MATRIX p FOR LEGENDRE POLYNOMIALS XV codf(p) XV codf(p) 2 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

8 300 WALTER GAUTSCHI TABLE3 OPTIMAL CONDITION OF p FOR SOME CLASSICAL ORTHOGONAL POLYNOMIALS Chebyshev 2d kid Laguerre Hermite (1) (1) (3) ( 1) (1) < (6) (3) classical polyomials. I the case = 20 of Laguerre polyomials the miimizatio algorithm could ot be made to coverge withi a reasoable amout of time. Iterestigly, some of the odes i the Laguerre case tur out to be egative. For = 2 it ca be show by direct computatio that the optimal coditio always equals codp(p 2 ) = 2, ad that the optimal odes are the zeros ~ 1, ~ 2 of p 2 ( ; d A), provided the measure d A is "symmetric" i the sese frtda(t)= frt 3 da(t)=o. I the Laguerre case, the optimal odes are x 1 =0,x 2 =2. REFERENCES 1 P. G. Nevai, Orthogoal polyomials, Mem. Amer. Math. Soc. 18, No. 213, Amer. Math. Soc., Providece, R.I., C. Posse, Surles quadratures, Nouv. A. Math. (2) 14:49-62 (1875). 3 D. F. Shao ad K. H. Phua, Remark o Algorithm 500, ACM Tras. Math. Software 6: (1980). Received 21 August 1981; revised 5 October 1981