JACOBI MATRICES FOR MEASURES MODIFIED BY A RATIONAL FACTOR. This paper decribes how, given the Jacobi matrix J for the measure d(t), it is possible to

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1 JACOBI MATRICES FOR MEASURES MODIFIED BY A RATIONAL FACTOR SYLVAN ELHAY AND JAROSLAV KAUTSKY y Abstrct. This pper decribes how, given the Jcobi mtrix J for the mesure d(t), it is possible to produce the Jcobi mtrix ^J for the mesure r(t)d(t) where r(t) is quotient of polynomils. The method uses new fctoring lgorithm to generte the Jcobi mtrices ssocited with the prtil frction decomposition of r(t) nd then pplies previouly developed summing technique to merge these Jcobi mtrices. The fctoring method performs best just where Gutschi's miniml solution method for this problem is wekest nd vice vers. This suggests hybrid strtegy which is believed to be the most powerful yet for solving this problem. The method is demonstrted on simple exmple nd some numericl tests illustrte its performnce chrcteristics. Key words. orthogonl polynomils, Jcobi mtrices, rtionl fctors modifying mesures, Guss qudrtures 1. Introduction. Let d(t) 0, mesure with nite, innite or semi-innite support [; b], be such tht the power moments (1) j = t j d(t) exist nd re nite for ll j = 0; 1; 2;. Let J be the (symmetric, tri-digonl) Jcobi mtrix, the elements of which re the coecients in the three-term recurrence reltion for the polynomils p j (t) j = 0; 1; 2; which re orthonorml with respect to d(t) on [; b]. Jcobi mtrices nd their ssocited orthogonl polynomils rise in connection with pproximte numericl integrtion, lest squres pproximtions for continuous nd discrete mesures, series expnsions nd continued frctions. The Jcobi mtrix for some d(t) cn, in principle, be computed from the power moments (1), but the build-up of roundo errors in this process is so severe tht in prctice this method is lmost useless. However, for certin of the clssicl mesures the Jcobi mtrices re explicitly known (see [2],[10] for some exmples) nd much eort hs gone towrds the investigtion nd development of methods which cn be viewed s trnsformtions from one or more known Jcobi mtrices, J 1 ; J 2 ;, corresponding to mesures d 1 (t); d 2 (t);, into the required Jcobi mtrix ^J for the mesure d^(t) which is relted to the d i (t). This is, of course, essentilly the sme s trnsforming one set or sets of orthogonl polynomils into set of others, but is done in numericl liner lgebr context. Thus, trnsformtion methods bsed on the modied moments (2) ^ j = p j (t)d^(t); j 0; (in which the p j re orthogonl with respect to the mesure d(t)), were proposed by Sck nd Donovn in [9]. The Lower Tringulr Lnczos (LTL) method of [8] (which is equivlent to the Modied Chebyshev method of [5]), is bsed on modied moments nd is of prticulr interest in this pper. Computer Science Deprtment, University of Adelide, South Austrli, y School of Informtion Science nd Technology, Flinders University, Bedford Prk, South Austrli, Pge 1, August 18, 1993

2 In [8], Kutsky nd Golub show, for r(t) 0 8t 2 [; b] polynomil, how to nd ^J for d^(t) = r(t)d(t) from J for d(t) (Polynomil Shift Implicit QR or PSIQR method). The present uthors used this pproch [7] to extend this work to the cse where the polynomil r cn chnge sign in [; b] nd use the methods so developed [4] to compute sequences of Kronrod-Ptterson type imbedded qudrture sequences for two clssicl mesures, e t on [0; 1] nd e t2 on [ 1; 1], for which Jcobi mtrices re explicitly known. More recently Elhy, Golub nd Kutsky show tht it is possible (see [3]), given the Jcobi mtrices J 1 nd J 2 for the mesures d 1 (t) nd d 2 (t), to produce the Jcobi mtrix J 3 corresponding to the mesure (3) d 3 (t) = 1 d 1 (t) + 2 d 2 (t); 1;2 constnts, if such mtrix J 3 exists (SUM method). Gutschi, in [5] nd [6], shows how to compute the modied moments of the mesures (4) nd d(t)=(t v); v =2 [; b]; (5) d(t)=((t x) 2 + y 2 ); x 2 R; y > 0; by mens of so-clled miniml solution to the three-term recurrence for the polynomils orthogonl with respect to d(t) (MS method). Once these modied moments re vilble it is possible to compute the corresponding Jcobi mtrix by mens of the LTL method. Note tht the denomintor in (5) is just (t v)(t v) with complex v = x+iy. Thus modictions such s the ones in (4) nd (5) llow for computtion of J corresponding to d(t) divided by either rel liner fctor or pir of complex conjugte fctors. Erlier work by Uvrov [11] uses the second solution nd Christoel-Drboux reltions to estblish determinntl expressions which relte the polynomils orthogonl with respect to some d(t) to those orthogonl with respect to r(t)d(t), r(t) rtionl frction with known fctors. However, there seems to be no esy wy to use them to determine the required Jcobi mtrix. The PSIQR nd SUM methods re bsed on orthogonl rottions nd re numericlly very stble. While the MS cn produce very ccurte Jcobi mtrices for mesures of the type (4) nd (5) it suers from the problem tht, for mny importnt cses, infesibly lrge initil mtrices my be required in order to produce quite moderte dimension result. In these cses the problem is compounded if we need to pply the MS method to (lredy divided) mesures such s (4) nd (5) repetedly, for exmple, to produce Jcobi mtrix tht corresponds to d(t) divided by product of fctors. Thus, there exist very stisfctory methods for Jcobi mtrices tht correspond to sums nd dierences of mesures, or products of mesures with polynomils, but there remins need for stisfctory methods to compute the Jcobi mtrix for mesure modied by quotient of polynomils. More precisely, we re interested in the following problem: Problem 1.1. Let the rtionl function r(t) = g n (t)=h m (t), g n nd h m polynomils of degree n nd m respectively, be such tht h m hs no zeros in [; b] nd r(t) 0, 8t 2 [; b]. Given nite J for d(t) nd ^J for d^(t) = r(t)d(t). Pge 2, August 18, 1993

3 Clerly, since d^(t) is non-negtive on the intervl [; b], there exist polynomils ^p j (t), j = 0; 1; 2;, ech of exct degree j, which re orthogonl on [; b] with respect to the inner product (f; g) = f(t)g(t)r(t)d(t): The importnce of stisfctory solution to Problem 1.1 is illustrted by the following ppliction. Suppose we know J for d(t) nd we require the Jcobi mtrix ^J for d^(t) = q(t)d(t), q some smooth function. If the intervl of orthogonlity is nite we cn pproximte q by polynomil g n nd use the PSIQR method to trnsform J into ^J for d^(t) = g n d(t). But on n innite intervl no polynomil cn usefully pproximte smooth function which does not grow beyond ll bounds t innity. Similrly, function with singulrity close to the support of d(t) my not be well pproximted by ny polynomil. This is just where rtionl pproximtions cn be used with good eect. Thus Jcobi mtrices hve importnt ppliction in the computtion of Gussin qudrture formule for mesures of the type d^(t) = r(t)d(t) where r is rtionl pproximtion to q. In the next section we derive new lgorithms which trnsform the known Jcobi mtrix J for the mesure d(t) into the Jcobi mtrix ^J corresponding to d(t) divided by liner or complex conjugte pir qudrtic fctor. We cll these, collectively, the Inverse Cholesky (IC) method. In x3 we indicte the IC lgorithms themselves nd briey review the MS method. The discussion in x4 leds to hybrid strtegy for hndling rtionl modiction which uses both the IC nd MS methods. In x5 we show simple exmple nd in x6 we summrize the results of some numericl tests. In x7 we mke some concluding remrks. Tbles of numericl results nd certin moments, which my be needed when using the IC method with the clssicl mesures, re displyed in x8. 2. The new methods. In x1 we mentioned tht the Jcobi mtrix for mesure modied by multipliction with polynomil is stbly nd eciently done with the PSIQR method. So our concern in the solution to Problem 1.1 is in the determintion of Jcobi mtrix tht corresponds to mesure d(t) modied by division with polynomil h m (t). We ssume tht the m (rel or complex conjugte pir) zeros of h m re known nd simple. We now estblish some nottion nd derive the reltions on which the IC method is bsed. Suppose tht the polynomils p(t) = (p 0 (t); p 1 (t); ; p n 1 (t)) T nd p n (t) re orthonorml on the intervl [; b] with respect to the non-negtive mesure d(t). Then there exists (symmetric, tridigonl) mtrix J = C C... A ; n the elements of which re the coecients in the three-term recurrence which the polynomils stisfy. In the following identity, s elsewhere in this pper, (6) tp(t) = Jp(t) + n p n (t)e n Pge 3, August 18, 1993

4 the vector e n is the lst column of n identity mtrix of pproprite dimension. Suppose now tht ^p(t) = (^p 0 (t); ^p 1 (t); ; ^p n 1 (t)) T is second set of polynomils orthonorml with respect to some mesure d^(t). The orthogonlity of the polynomils implies (multiply (6) on the right by p(t) T nd integrte) (7) p(t)p(t) T d(t) = I; tp(t)p(t) T d(t) = J nd there re corresponding reltions for the second set of orthonorml polynomils. Furthermore, there exist nonsingulr lower tringulr mtrix L, vector c nd non-zero constnt n, ll such tht (8) p(t) = L^p(t); p n (t) = c T ^p(t) + n ^p n (t): Thus, if ^J is the Jcobi mtrix in the identity corresponding to (6) for the polynomils ^p(t), ^p n (t) it is esy to show [4] tht (9) ^J = L 1 J L + e n b T ; where L 1 e n = n e n for some non-zero constnt n nd b = n n c. If we need the mtrix ^J of dimension n we cn get it by discrding the lst row nd column of the n + 1 dimension product L 1 JL, thus obviting the need to compute b. The methods we present in this pper determine the mtrix L directly from J nd some dditionl informtion. Our strting point is the theorem: Theorem 2.1. Let r(t) be ny nlytic function of t. Then (J ti)p(t)r(t) = (J ti)r(j)p(t) + n p n (t)(r(j) r(t)i)e n : If t is not n eigenvlue of J then (10) p(t)r(t) = r(j)p(t) + n p n (t)(j ti) 1 (r(j) r(t)i)e n : If v is such tht p n (v) = 0 then r(v) is n eigenvlue of r(j) nd p(v) is the corresponding eigenvector. Proof. From (6) we cn write (J ti)p(t) = n p n (t)e n : If r(t) is n nlytic function of t then J nd r(j) commute so we hve (J ti)(r(j) r(t)i)p(t) = n p n (t)(r(j) r(t)i)e n from which (10) immeditely follows. Multiplying (10) on the right by 1 p(t) nd integrting gives r(t) p(t)p(t) T d(t) = r(j) n p(t)p(t) T d(t) r(t) + p n (t)(j ti) 1 (r(j) r(t)i)e n p(t) T d(t) r(t) : Now if the polynomils p(t) re orthonorml with respect to the mesure d(t) nd the polynomils ^p(t) re orthonorml with respect to the divided mesure d^(t) = d(t) r(t) Pge 4, August 18, 1993

5 then we hve, fter substituting with (8), (11) I = r(j)ll T + n (c T ^p(t) + n^p n (t))(j ti) 1 (r(j) r(t)i)e n^p(t) T d^(t)l T : The two cses of present interest re r(t) = t v nd r(t) = (t x) 2 + y 2. For the cse r(t) = t v reltion (11) reduces to (12) I = (J vi)ll T + n (c T ^p(t) + n ^p n (t))e n^p(t) T d^(t)l T ; = (J vi)ll T + n e n c T L T by orthogonlity. Note tht the mtrix L T is Cholesky-like fctor of (J vi) plus rnk-one correction to its lst row. For the second cse of interest, r(t) = (t x) 2 + y 2, we hve I = ((J xi) 2 + y 2 I)LL T + (13) n (c T ^p(t) + n ^p n (t))((j xi) + (t x)i)e n^p(t)d^(t)l T : Now the second term on the right of (13) reduces, by orthogonlity, to ( ) n (J 2xI)e n c T + e n c T t^p(t)^p(t) T d^(t) L T T + ^ n e n e n (14) = n n(j 2xI)e n c T + e n c T ^J o L T + ^ n e n e n T for some constnt ^ n. Bering in mind tht the mtrix ^J in (14) is tridigonl, it is esy to see tht (14) is mtrix, the rst n 2 rows of which vnish, so the L T mtrix for this cse is Cholesky-like fctor of (J xi) 2 + y 2 I plus rnk-two correction to its lst two rows. The methods we describe in this pper directly generte the mtrix L, in ech of the bove cses, from the top down. If the process is terminted before reching the row or rows which re ected by the low rnk corrections, then the prt of L produced to tht point will be correct nd the low rnk corrections need not be computed. 3. The lgorithms. In this section we derive the lgorithms for the Cholesky-like fctors of the inverse of symetric tridigonl nd pentdigonl mtrices. In this section we write L i:j;k:m to denote rows i through j of columns k through m of the mtrix L. (15) 3.1. The liner-fctor Inverse Cholesky lgorithm. Given symmetric tridigonl mtrix J, we require the fctor L which stises where the lower tringle L hs elements L = 0 I = JLL T + e n d T ; `n1 `n2 `n3 : : : `nn Pge 5, August 18, C ` : : : 0 `21 `22 0 : : : 0 `31 `32 `33 : : : A :

6 Noting tht e T i J = ( 0 : : : 0 i 1 i i 0 : : : 0 ) 1 < i < n we hve, considering only 1 j i n, ( ij ni d j ) = i 1 n X = i 1 `i 1;k`jk + i n X min i 1;j X X + i j 1 `i 1;k`jk + i `ik`jk + i n X `i+1;k`jk jx `ik`jk `i+1;k`jk + i`i+1;j`jj : Algorithm 3.1, which is bsed on this reltion, determines the L nd d of (15). Its input is symmetric tridigonl mtrix J nd `11 6= 0. Algorithm = 0 `21 = (1=`11 1`11 )= 1 for i = 2 : n 1 `i+1;1 = ( i 1`i 1;1 + i`i1 )= i end d 1 = ( n 1`n 1;1 + n`n1 ) for j = 2 : n s = j 2 L j 2;1:j 2 L T j;1:j 2+ j 1 L j 1;1:j 1 L T j;1:j 1+ j 1 L j;1:j 1 L T j;1:j 1 `jj = p s= j 1 for i = j : n 1 s = i 1 L i 1;minfi 1;jg L T j;minfi 1;jg+ i L i;1:j L T j;1:j+ i L i+1;1:j 1 L T j;1:j 1 `i+1;j = ( ij s)= i =`jj end d j = ( n 1 L n 1;1:minfn 1;jg L T j;1:minfn 1;jg + n L n;1:j L T j;1:j) end 3.2. The qudrtic-fctor Inverse Cholesky lgorithm. Here we require the lower tringle L which stises the reltion (16) I = J 2 LL T + e n d T + e n 1 f T : where, J being tridigonl, J 2 is pentdigonl nd denoted (for dierent 's nd 's) by J 2 = C C... A : n We ssume tht the principl 2 2 block of L is prescribed. We hve e T i J = ( 0 : : : 0 i 2 i 1 i i i 0 : : : 0 ) ; 2 < i < n 2; Pge 6, August 18, 1993

7 nd denoting L s in the previous section, we hve, for 1 j i n, ( ij n;i 1 f j ni d j ) = i 2 minfi 2;jg X `i 2;k`jk + i 1 jx + i `ik`jk + i X + i j 1 jx minfi 1;jg X `i+1;k`jk `i+2;k`jk + i`i+2;j`jj : `i 1;k`jk Algorithm 3.2, which is bsed on this reltion, determines the L of (16). Its input is the symmetric pentdigonl mtrix J 2 nd the 2 2 principl submtrix of L. The lgorithm then nds the rest of L. The steps which compute d nd f re omitted for brevity. Algorithm 3.2. `3;1 = (1=`1;1 1`1;1 1`2;1 )= 1 `4;1 = ( 1`1;1 + 2`2;1 + 2`3;1 )= 2 for i = 4 : n 1 `i+1;1 = ( i 3`i 3;1 + i 2`i 2;1 + i 1`i 1;1 + i 1`i;1 )= i 1 end `3;2 = ( 1`2;1`1;1 + 1`22;1 + 1`2;1`3;1 + 1`22;2)=( 1`2;2 ) s = 1`2;1`1;1 + 2`22;1 + 2`2;1`3;1 + 2`2;1`4;1 + 2`22;2 + 2`2;2`3;2 `4;2 = (1 s)=( 2`2;2 ) for i = 3 : n 2 s = i 2 L i 2;1:minfi 2;2g L T 2;1:minfi 2;2g + i 1L i 1;1:minfi 1;2g L T 2;1:minfi 1;2g + i L i;1:2 L T 2;1:2 + i L i+1;1:2 L T 2;1:2 + i`i+2;1`t2;1 `i+2;2 = s=( i`2;2 ) end for j = 3 : n s = 0 if j > 3; s = s + j 3 L j 3;1:j 3 L T j;1:j 3 ; end if j > 4; s = s + j 4 L j 4;1:j 4 L T j;1:j 4 ; end s = s + j 2 L j 2;1:j L T j;1:j + j 2L j 1;1:j L T j;1:j + j 2L j;1:j 1 L T j;1:j 1 `j;j = p s= j 2 for i = j 1 : n 2 s = i 2 L i 2;1:minfi 2;jg L T j;1:minfi 2;jg+ i 1 L i 1;1:minfi 1;jg L T j;1:minfi 1;jg+ i L i;1:j L T j;1:j + i L i+1;1:j L T j;1:j + i L i+2;1:j 1 L T j;1:j 1 `i+2;j = ( ij s)=( i`j;j ) end end Strting vlues. The liner fctor IC method will be used for rel v. To strt it we need the vlue of (see (8)) where `11 = (~ 0 = 0 ; ) 1 2 d(t) ~ 0 = t v ; Pge 7, August 18, 1993

8 is vilble, from the tbles provided, for the clssicl mesures. The qudrtic fctor IC method will be used for complex conjugte pirs v, v where v = x + iy, x; y rel, y > 0. To strt it we need the principl 2 2 submtrix of L, `11 0 : `21 `22 The mesure d(t) hs here been modied by the fctor (t v)(t v) = (t x) 2 + y 2. Now denote t ~ j = j (t x) 2 d(t); j = 0; 1; 2: + y2 Using (8) nd orthognlity it quickly follows tht `2 11 = ~ 0 = 0 `21 = `11 (~ 1 =~ 0 1 )= 1 `2 22 = (~ 2 ~ 2 1 =~ 0)=( 0 2 1): For the clssicl mesures, the ~ j cn be found, gin in terms of the tbled quntities, by ~ 0 = ~ 1 = iy Z 1 b t v d(t) Z 1 b t v d(t) + ~ 2 = 0 + 2x~ 1 (x 2 + y 2 )~ 0 :! 1 t v d(t) 1 t v d(t)! x~ Modied moments from the miniml solution. For completeness, we stte here Gutschi's miniml solution (MS) method re-written to operte for normlized rther thn monic polynomils. Given J of dimension m, the zero-th moment, 0 of d(t), knot v outside the support intervl of d(t) nd n integer n = n(m; v) < m, the lgorithm produces the vector ^ = ( 0 ; 1 ; 2 ; : : : ; n 1 ) T. For rel v, the modied moments of the mesure (4) re the ^ i = i nd for complex v the modied moments of the mesure (5) re given by ^ i = Img( i) Img(v) : Algorithm 3.3. p m = 0 for i = m : 1 : 2 p i 1 = i 1 =(v i i p i ) end 0 = p 0 =( 1 p 1 (v 1 )) for i = 1 : n 1 i = i 1 p i end Pge 8, August 18, 1993

9 For the mesures in Tble 5 Gutschi proposes lower bound on n(m; v) which is lrge enough to ensure tht the ^ i re computed to suciently high ccurcy. As the point v pproches support of d(t) the vlue of n(m; v) increses nd lrger dimension J is needed for the sme ccurcy. 4. Use of the IC nd MS methods. Our discussion of just how to use the IC nd MS methods divides nturlly into two prts. The rst prt ddresses the question of how to use the methods repetedly to nd the J for mesure divided by generl polynomil of degree m > 2. The second prt ddresses the question of how to decide on which of the IC nd MS methods should be used for ech of the liner nd qudrtic fctors in the rtionl function denomintor. This issue rises becuse, s we show lter in the numericl tests, the IC nd MS methods behve very dierently from one nother ccording to whether the division fctors correspond to poles close to, or fr from, the support of the mesure d(t) Repeted divisions. In this section we compre () the strtegy of strting with known J, modifying it by division with fctor, modifying the result by division with the next fctor, nd so on, (the product strtegy), with (b) the strtegy of modifying the known J for division by ech of the fctors seprtely nd then using the summing method to produce the nl result (the summing strtegy). To strt, we compute the prtil frction decomposition of h m (t) = my (t v k ): In the cse where the zeros of h m re conjugte pirs, the prtil frction decomposition contins terms of the form (17) Ax + B (t x) 2 + y 2 ; y > 0: To compute the Jcobi mtrix for such term we use one of the lgorithms below for the denomintor rst nd then pply the PSIQR method to the result for the numertor The product strtegy. Denote by J s the Jcobi mtrix for the mesure (18) d(t) Q s (t v k) : In the product strtegy we strt from the given J 0 = J for d(t) nd sequentilly clculte J 1 ; J 2 ; : : :, nishing up with J m = ^J for d(t)=hm (t). Ech step in this sequence, in which we produce J s from J s 1, cn be performed, in principle, by either of the following lgorithms (for complex pirs of zeros the IC method cn be used to trnsform from J s 1 to J s+1 directly using only rel rithmetic, s indicted in x3, but the MS method requires complex oting point rithmetic). One stge using the MS method requires n lgorithm such s: Algorithm 4.1. Input: J s 1 of suciently lrge dimension j nd strting vlues s in x Pge 9, August 18, 1993

10 () Use the MS method to produce 2(i + 2) modied moments. (b) Use the LTL method to directly get J s of dimension i. By comprison, one stge using the IC method would require Algorithm 4.2. Input: J s 1 of dimension i+1 nd strting vlues s in x () Use the IC method to produce L s of dimension i + 1. (b) Compute Ls 1 J s 1L s = J s e n b T s of dimension i + 1 (see (9)). (c) Discrd the lst row nd column of the product to get J s of dimension i. However, there re diculties with Algorithm 4.1. The estimtes of the size of j for given v k when d(t) is one of the clssicl (Jcobi, Lguerre or Hermite) mesures cn be quite lrge, if the point v k is close to the support of the intervl: for exmple to modify dimension i = 10 Lguerre Jcobi mtrix by the fctor (t 0:01) requires n initil Jcobi mtrix of dimension nerly j = 2000 when using stndrd double precision IEEE rithmetic. Thus, even for quite modest nl dimension ^J, very lrge initil mtrix my be needed. More problemtic is the fct tht the only estimtes for the size of the initil mtrix t ech stge re for the clssicl mesures nd there re not estimtes for the divided clssicl mesures. It is possible tht one could check the ccurcy of the process t ech stge by recomputing with vried estimtes of the strting size mtrix but clerly this process is unstisfctory. Indeed, it my even be necessry to bndon the clcultion nd strt gin from scrtch t some stge if the remining Jcobi mtrix is too smll to go to the next step. Using the Algorithm 4.2 requires n initil mtrix of only bout n + m but there re diculties s before: the strting vlues, which re computed from the zero-th moments of mesures such s (18) re not generlly vilble The summing strtegy. An lterntive strtegy uses the results of [3] mentioned in x1. If we denote the method for computing the Jcobi mtrix which corresponds to sum or dierence of mesures by SUM then the lgorithm we propose is: Algorithm 4.3. Given J of dimension n + 1 for d(t) () For ech k = 1 : m either (i) Use the MS method to produce 2(n + 2) modied moments for J ~ vk corresponding to d~(t) = d(t)=(t v k ) (or d~(t) = d(t)=(t v k )(t v k )). (ii) use the LTL method with these modied moments to get J ~ vk of dimension n for d~(t) or (i) Use the IC to produce L k for d~(t) = d(t)=(t v k ) (or d~(t) = d(t)=(t v k )(t v k )). (ii) Compute L 1 k JL k = J ~ vk e n b T v k of dimension n + 1 (or n + 2) (see (9)). (iii) Discrd the lst row (two rows) nd column (two columns) of the product to get J ~ vk of dimension n for d~(t). (b) Use method SUM to nd ^J from the seprte Jcobi mtrices. This scheme overcomes the disdvntges, mentioned bove, of the product strtegy since the strting vlues re known (for both the IC nd MS methods) for the clssicl mesures nd, in the cse of the MS method, ll the divisions use the sme initil mtrix nd its dimension cn be determined once t the outset from the estimtes given in [5]. If the numertor of ny term such s (17) chnges sign in the support of d(t), there my not exist Jcobi mtrix for it. However, by dding nd subtrcting Pge 10, August 18, 1993

11 constnt to the numertor we cn rewrite (17) s two seprte terms ech of which does not chnge sign on the support. We cn then sum ll the terms which re positive on the support nd then subtrct those which re negtive. Since the nl Jcobi mtrix corresponds to mesure which is positive on the support (by ssumption), the result must exist nd be computble this wy Distribution of the poles. Consider the cse of modifying J for d(t) to ^J for d^(t) = d(t)=(t v), v rel or complex, outside the support, [; b], of d(t). The closer is v to [; b], the lrger j, the dimension of J, needs to be for the MS method to mintin ccurcy. The estimtes given in [5] for the size j when d(t) is one of the Jcobi, Lguerre or Hermite mesures, re reported there to be relistic in the sense tht, using J of dimension 5 fewer thn given by the estimte rrely gve sucient ccurcy nd the given estimte ws only occisionlly required to be incresed by bout 5 or 10. Thus close to the support, the MS method cn be quite unstisfctory. By contrst, the IC method, s the numericl tests reported in x6 show, returns poorer results s the point v moves wy from the support but is extremely ccurte close to the support. Our numericl evidence lso suggests tht this eect is more pronounced for the mesures with innite or semi-ininte support. The obvious solution then, is to use Algorithm 4.3, pplying the MS method on points fr from [; b] nd the IC methods on those close to [; b]. Both methods cn be pplied to points which re neither close nor fr nd the two results used for conrmtion. Thus, Algorithm 4.3 provides stble nd ecient method for solution to Problem A simple exmple. The function ln(1 + 4t) hs power series ln(1 + 4t) = 4 t 8 t 2 + nd Pde pproximtion (19) 64 t t t t6 3 r(t) = g 3 (t)=h 3 (t) = 60 t t t t t t t7 7 + O(t 8 ); which, on the subintervl [0; 20], hs error bounded, in bsolute vlue, by bout 0:228. The denomintor h 3 (t) hs zeros f 1=2; 1 4 (5 p 15)g f 0:5; 0:28175; 2:21825g nd the numertor g 3 (t) hs zeros f0; :3; 1g so both g 3 (t) nd h 3 (t) do not chnge sign on the support [0; 1] of the Lguerre mesure. The prtil frction decomposition of 1=h 3 (t) is A 1 + A 2 + A 3 t v 1 t v 2 t v 3 = 1=18 t + 1= =(15 3 p 15) 6 (t + (5 p 15)=4) =( p 15) (t + (5 + p 15)=4) : We demonstrte the technique by nding the dimension 7 Jcobi mtrix which corresponds to the mesure d(t) = e t r(t) on the intervl [0; 1] from the (known, see [2]) Jcobi mtrix for the mesure d(t) = e t on the sme intervl. Since Pge 11, August 18, 1993

12 the processes we employ require us to discrd one or two rows nd columns of the computed mtrices, we ssume tht the strting mtrix is suciently lrge to give result of the required dimensions. All the clcultions were done in IEEE stndrd double precision rithmetic (mchine epsilon 2: ). Let us denote the following power moments by Z 1 t ~ jvi = j e (20) t dt; i = 1; 2; 3; j = 0; 1; 2; : t v i 0 To strt with, we use Tble 6 to compute ~ 0vi for ech of the zeros v i. These numbers (computed to 50D with Mple V nd correctly rounded) re given in Tble 1. Of v 1 9: v 2 1: : v 3 Tble 1 Vlues of ~ for the three zeros of h 0v i 3(t). course, 0 = 1. We next use Algorithm 4.3 (choosing the IC method option) to compute the three Jcobi mtrices J ~ vi, i = 1; 2; 3, ech of which corresponds to the mesure e t =(t v i ). We now pply the method SUM, referred to in x1, to J ~ v2 nd J ~ v3 to compute the Jcobi mtrix, J temp1, corresponding to the sum of the two positive terms e t A 2 =(t v 2 ) nd e t A 3 =(t v 3 ). The constnts i, i = 1; 2 of (3) re here chosen s A i ~ 0vi =(~ 0v2 + ~ 0v3 ). Next, we use the dierencing version of SUM to compute the Jcobi mtrix, J temp2 corresponding to the mesure e t A2 t v 2 + A 3 t v 3 + e t A 1 t v 1 : (recll tht A 1 < 0) from J temp1 nd J v1. Finlly, we pply the PSIQR method (see x1) to J temp2 with the polynomil g 3 (t) to get ^J corresponding to r(t)d(t). After the PSIQR process we discrd the lst 2 rows nd columns of the mtrix ^J. The resulting mtrix hs digonl j nd subdigonl j components shown in Tble 2 nd the 7-point Guss qudrture formul (21) Q(f) = nx j=1 w j f(t j ) found from it is in Tble 3. Of course, this Jcobi mtrix determines ll the Guss qudrtures (for the modied mesure r(t)e t ) with seven or fewer points. It is worth noting tht we could hve run the processes in dierent order: strt with the Jcobi mtrix for e t, pply the PSIQR method with polynomil g 3 (t) nd pply the IC method to this mtrix once for ech of the poles of r(t). The three resulting mtrices could then be merged using the SUM method. Although it is not relible test of the ccurcy of qudrture fromul, we pplied Q(f) to the powers of t to conrm tht it is indeed the required formul. We expect Q(f) to compute the integrls (22) I pprx (t j ) = Z 1 t j e t r(t)dt 0 Pge 12, August 18, 1993

13 j j j 1 1: : : : : : : : : : : : : Tble 2 The Jcobi mtrix for r(t)e t. j t j w j 1 3: : : : : : : : : : : : : : Tble 3 The 7-point Guss qudrture formul for r(t)e t. to within modest multiple of mchine ccurcy for the rnge j = 0; 1; 2; : : :; 13 nd we expect I pprx (t j ) Q(t j ) to be quite dierent from (mchine) zero for j = 14. Further, we expect tht Q(f) will pproximte (23) I exct (t j ) = Z 1 0 t j e t ln (1 + 4t)dt to roughly the ccurcy tht I pprx (t j ) pproximtes I exct (t j ). In fct, I pprx (t j ) pproximtes I exct (t j ) with reltive error tht vries lmost linerly between nd s j rnges from 0 to 14. Once ~ 0vi is known, the numbers ~ jvi, j > 0 cn esily be computed from the recursion (24) ~ jv = v~ j 1;v + j 1 ; j > 0: From these nd the prtil frction decomposition, we cn nd the I pprx (t j ) of (22). Furthermore, the numbers j = I exct (t j ) of (23) cn be found from the recursion j = j j 1 + ~ j; 1 ; 0 = ~ 4 0; 1 4 once (24) hs been used with v = 1=4 to compute the ~ j; 1. The bsolute vlues of the reltive errors for Q(t j ) pproximting the numbers in (22) nd (23) re 4 summrized in Tble 4 nd follow our expecttions. 6. Numericl Tests. In this section we report the results of some tests which compre the performnce of the IC method nd the MS method nd which identify two signicnt dierences in the wy these methods behve. Our tests were pplied to ll the clssicl mesures shown in Tble 5. In ll cses the clcultions were done Pge 13, August 18, 1993

14 Error of Q(t j ) for j I exct (t j ) I pprx (t j ) 0 1: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Tble 4 Reltive errors of Q(t j ) in IEEE stndrd double precision rithmetic (mchine epsilon 2: ). The tbles in this section show the ccurcy of computed Jcobi mtrices when compred with the explicitly known results. For ech rel knot v, or complex knot x + iy, the entry heded N in the tble shows the number N = n(m; v) of recurrence coecients which the MS method requires. Where this number ws considered too lrge to be fesible we used 2000 coecients nd disply the required N in prentheses. The remining columns show the ccurcy of the digonl j nd subdigonl j elements produced by the IC nd MS methods. Thus for the columns heded we show ( min j j exct log 10 pprox j exct j nd, except where they re zero by virtue of the symmetry of the mesure d(t), similr quntities for the digonls j. Recll tht the IC method produces mtrix L from which we compute the required Jcobi mtrix using (9). We computed the 2-norm condition numbers for the L mtrices tht re generted by the IC process for these tests nd found tht they remin modest for ll the cses shown. For n = 10 ll but 5 re smller thn 80, nd the blnce re smller thn 700. For n = 50 they re ll smller thn The tests themselves exploit the fct tht the PSIQR nd either the IC or MS re inverses of ech other if the polynomil r(t) is rel liner fctor (r(t) = t v) or the product of complex conjugte pir of fctors (r(t) = ((t x) 2 + y 2 ), y > 0). Since the PSIQR method produces mtrices which re ccurte to bout mchine epsilon, the dierence between the strting mtrix nd the result, fter pplying both the PSIQR nd either IC or MS, is good mesure of the error. This error is just wht we disply in the tbles of this section. The tbles illustrte tht the MS method performnce improves s the point v moves wy from the support of d(t) nd lso shows tht the MS method is better overll on mesures which hve nite support. By contrst, the IC method Pge 14, August 18, 1993 ) :

15 performnce improves s the point v pproches the support intervl nd is better overll for the mesures with innite support. These phenomen re clerly shown s the point v pproches the support intervl from the left, in the cse of liner fctors (Tbles 7 nd 8) nd s it pproches the origin long the line y = x in the cse of complex pir fctors or for the Hermite mesure (Tbles 9, 10 nd 11). Tble 12 shows the errors for the methods when the point x + iy moves prllel but close to the x-xis from x = 5 to x = 5 nd illustrtes further the behviour of the two methods close to nd fr from the support. Most of the tbles in this section show the results of the tests pplied to produce mtrices of dimension n = 10. However, these tbles re representtive of lrger cses s the exmples in Tbles 13 nd 14, with n = 50, show. The loss of ccurcy seems to be quite grdul s n increses. We used the non-symmetric Jcobi mesure (with mesure prmeters = 1=2, = 1=7) s n exmple of nite support mesure to illustrte the performnce of the methods on both the j nd the j components of the mtrix. The Legendre, Gegenbuer nd Chebyshev mesures give similr results. The Hermite nd Lguerre mesures used here hve the mesure prmeter = 1=3. The clcultion seems, however, to be reltively insensitive to this prmeter nd other vlues of tht we tested gve very similr results. These dierences in performnce suggest hybrid method for the solution of Problem 1.1. The decision bout which of the IC or MS methods to choose in Algorithm 4.3 should be driven by the phenomen the results here indicte: the closeness of the point in question to the support intervl nd whether or not the suport intervl is innite or semi-innite. Where either method is suitble they could both be used nd the results compred for veriction. 7. Conclusions. We hve derived new method, clled the IC method, for the computtion of the Jcobi mtrix corresponding to mesure modied by rtionl function. Our method is bsed on new fctoring lgorithm which cn be used to produce the Jcobi mtrix corresponding to known mesure divided by liner or qudrtic fctor. We use the fctoring lgorithm once for ech pole or conjugte pir of poles nd then pply the uthors' SUM method to nd the Jcobi mtrix corresponding to the given mesure divided by the denomintor of the rtionl function. We then pply the PSIQR method to this for the numertor. The IC method requires certin strting vlues which re expressed, for the clssicl mesures, in terms of well known specil functions. We computed these to very high ccurcy without diculty by use of the symbolic mnipultors Mple nd Mthemtic. Subroutine pproximtions for these specil functions re lso redily vilble from softwre librries such s Netlib. We hve shown tht the product strtegy of x4 hs serious problems nd tht our new summing strtegy overcomes these problems. We hve lso shown tht the IC method performs best just where the previously known MS is wekest nd vice vers. This hs led to hybrid strtegy which we believe is the most powerful yet for this problem. The new method ws demonstrted on simple exmple nd some tbles, which re representtive of the numericl tests we rn, show the pttern of performnce. Problem 1.1 remins dicult nd stble method which does not require the poles of the rtionl function to be known is still desirble. 8. Tbles. Pge 15, August 18, 1993

16 8.1. Zero-th moments of clssicl mesures nd divided clssicl mesures. The following re sometimes referred to s the clssicl mesures: Nme d(t) Intervl Constrints Legendre 1 [ 1; 1] Chebyshev (1 t 2 ) 1=2 [ 1; 1] Gegenbuer (1 t 2 ) [ 1; 1] > 1 Jcobi (1 t) (1 + t) [ 1; 1] ; > 1 Lguerre t e t [0; 1] > 1 Hermite jtj e t2 [ 1; 1] > 1 Tble 5 Clssicl mesures. The tble below shows the zero-th moments, 0, of these mesures nd the zero-th moments of these when divided by liner fctor, ~ 0 = 1 t v d(t); v = x + iy 62 [; b]: In the cse of the Hermite mesure we require y > 0, but v my be rel for the other mesures. Nme 0 ~ 0 Legendre 2 ln((v 1)=(v + 1)) Chebyshev = p v 2 1 Gegenbuer Jcobi (+1) 2 (2+2) (+1) (+1) (++2) (1 v) B(1 + ; 1 + ) 2F 1 (1; + 1; 2 + 2; 2 1 v ) (1 v) B(1 + ; 1 + ) 2F 1 (1; + 1; + + 2; 2 1 v ) Lguerre ( = 0) 1 e v E 1 ( v) Hermite ( = 0) p Tble 6 Moments of clssicl mesures. ie v 2 erfc( iz) Here the Gmm, Bet, nd Hypergeometric functions re dened by (z) = Z 1 t z 1 e t dt; Re(z) > 0; 0 B(z; u) = (z) (u)= (z + u); Pge 16, August 18, 1993

17 2F 1 (; b; c; z) = (c) () (b) 1X j=0 ( + j) (b + j) z j (c + j) j! : Further, E 1 (v) = Z 1 e vt dt; Re(v) > 0 1 t is the Exponentil Integrl nd erfc(z) is the Complementry Error Function Z erfc(z) = p 2 1 e t2 dt: For detils, see [1]. Built-in functions re vilble in the symbolic mnipultors Mple nd Mthemtic for ll the entries in the tbles bove nd we hve used them to compute high precision zero-th moments, for rel nd complex v without diculty Tbles for the results of numericl tests. Liner fctor -log(mximum errors) IC MS v N Tble 7 Errors for Jcobi mesure = 1=3; = 1=7, n = 10 z Liner fctor -log(mximum errors) IC MS v N (1117) (9033) (84032) Tble 8 Errors for Lguerre mesure = 1=3, n = 10 Pge 17, August 18, 1993

18 Qudrtic fctor v = x + iy -log(mximum errors) IC MS x y N (18049) Tble 9 Errors for Jcobi mesure = 1=3; = 1=7; n = 10 Qudrtic fctor v = x + iy -log(mximum errors) IC MS x y N (2442) (4591) (41270) Tble 10 Errors for Lguerre mesure = 0; n = 10 Qudrtic fctor v = x + iy -log(mximum errors) IC MS x y N (4749) (17591) ( ) Tble 11 Errors for Hermite mesure = 1=3; n = 10 Pge 18, August 18, 1993

19 Qudrtic fctor v = x + iy -log(mximum errors) IC MS x y N Tble 12 Errors for Lguerre mesure, v moving prllel to x xis. = 0; n = 10 Qudrtic fctor v = x + iy -log(mximum errors) IC MS x y N Tble 13 Errors for Legendre mesure n = 50 Qudrtic fctor v = x + iy -log(mximum errors) IC MS x y N (2980) (5319) (43398) Tble 14 Errors for Lguerre mesure n = 50 Pge 19, August 18, 1993

20 REFERENCES [1] M. Abrmowitz nd I.A. Stegun. Hndbook of mthemticl functions. Dover Publictions, New York, [2] T.S. Chihr. An Introduction to orthogonl polynomils. Gordon nd Brech, New York, [3] S. Elhy, G.H. Golub, nd J. Kutsky. Jcobi mtrices for sums of weight functions. BIT, 32:143{166, [4] S. Elhy nd J. Kutsky. Generlized Kronrod Ptterson type imbedded qudrtures. Aplikce Mtemtiky, 37(2):81{103, [5] W. Gutschi. Miniml solutions of three-term recurrence reltions nd orthogonl polynomils. Mth. Comp., 36(154):547{554, [6] W. Gutschi. On generting orthogonl polynomils. SIAM J. Sci. Sttist. Comput., 3:289{ 317, [7] J. Kutsky nd S. Elhy. Guss qudrtures nd Jcobi mtrices for weight functions not of one sign. Mth. Comp., 43(168):543{550, [8] J. Kutsky nd G.H. Golub. On the clcultion of Jcobi mtrices. Liner Algebr Appl., 52/53:439{455, [9] R.A. Sck nd A.F. Donovn. An lgorithm for Gussin qudrture given modied moments. Numer. Mth., 18:465{478, 1971/72. [10] G. Szego. Orthogonl polynomils. AMS Colloquium Publictions 23. Americm Mthemticl Society, Providence, Rhode, RI, 4th edition, [11] V.B. Uvrov. The connection between systems of polynomils orthogonl with respect to dierent distribution functions. USSR Computtionl Mth. nd Mth. Phys., 9(6):25{36, Pge 20, August 18, 1993