A NONLINEAR OPTIMIZATION PROCEDURE FOR GENERALIZED GAUSSIAN QUADRATURES

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1 A NONLINEAR OPTIMIZATION PROCEDURE FOR GENERALIZED GAUSSIAN QUADRATURES JAMES BREMER, ZYDRUNAS GIMBUTAS, AND VLADIMIR ROKHLIN Abstrct We present new nonliner optimiztion procedure for the computtion of generlized Gussin qudrtures for brod clss of functions While some of the components of this lgorithm hve been previously published, we present simple nd robust scheme for the determintion of sprse solution to n underdetermined nonliner optimiztion problem which replces the continution scheme of the previously published works The performnce of the resulting procedure is illustrted with severl numericl exmples 1 Introduction Clssicl Gussin qudrture rules re optiml formuls for the evlution of integrls of the form (11) P (x)ω(x)dx where P (x) is polynomil nd ω : [, b] R + is n essentilly rbitrry nonnegtive weight function; they re optiml in sense tht n n-point Gussin qudrture rule integrtes polynomils of degree n 1 exctly with respect to the weight function ω They re extremely efficient s long s functions to be integrted re smooth or of the form f(x) = g(x)w(x) where g(x) is smooth nd w(x) is fixed nd known priori However, they perform poorly in number of instnces of gret importnce in computtionl mthemtics Of prticulr interest is the integrtion of functions f which dmit representtions of the form (12) f(x) = α i φ i (x), where the φ i re oscilltory, singulr, or both nd known priori but the coefficients α i re not Then ech of the functions φ i (x) cn exhibit different type of singulrity or different frequency of oscilltion, nd clssicl Gussin qudrture rules perform very poorly Mny such integrls rise in computtionl physics (see, for instnce, [4] nd [18]) It hs long been known tht Gussin qudrtures dmit generliztion to firly brod clss of systems of functions (see [11, 12, 15, 16, 13]) We dopt the terminology of [14, 20, 3], nd sy tht generlized Gussin qudrture for collection {φ 1,, φ 2n } of 2n functions nd nonnegtive weight function ω is n n-point qudrture rule (13) φ(x j )w j φ(x)ω(x) dx exctly integrting ech of the φ i with respect to the weight function ω The constructions found in [11, 12, 15, 16, 13] do not led redily to numericl lgorithms for the design of generlized Gussin qudrture rules In [14], numericl lgorithm is introduced for the construction of generlized Gussin qudrture rules for firly brod clss of functions The pproch is bsed on the observtion tht the nodes x j nd weights w j of generlized Gussin qudrture stisfy nonliner system of equtions The procedure of [14] is vrint of Newton s lgorithm coupled with continution scheme for the genertion of suitble initil point for the modified Newton itertions In [20] nd [3], the continution scheme of [14] is refined, improving Dte: June 30,

2 2 JAMES BREMER, ZYDRUNAS GIMBUTAS, AND VLADIMIR ROKHLIN the stbility of the lgorithm, nd new preprocessing step is dded in [20], gretly expnding its rnge of pplicbility The present pper introduces new numericl procedure for obtining generlized Gussin qudrture rules While we lso use Newton-type itertions to solve nonliner systems of equtions, our scheme differs from the lgorithm of [14, 20, 3] in tht the continution method is bndoned in fvor of simpler, more robust pproch The pper is structured s follows Section 2 contins mthemticl nd numericl preliminries In section 3, we describe certin stndrd numericl tools used by the lgorithm Section 4 describes the lgorithm in detil Section 5 contins severl exmples of qudrtures we hve obtined Finlly, in Section 6, we present conclusions nd discuss severl possible extensions of this work 2 Mthemticl nd numericl preliminries 21 Generlized Gussin nd Chebyshev qudrtures The qudrture formuls considered in this pper re of the form (21) φ(x j )w j where the x j nd ω j re rel numbers We will refer to the x j s the nodes nd the w j s the weights of the qudrture formul (21) They will be used to pproximte integrls of the form (22) φ(x)ω(x) dx where ω(x) is nonnegtive weight function Qudrtures re typiclly chosen so s to mke (21) exct for some set of functions, commonly polynomils of fixed order Clssicl Gussin qudrture rules consist of n nodes nd n weights which integrte polynomils of order 2n 1 exctly In [14], the notion of Gussin qudrture ws generlized s follows: DEFINITION 21 A qudrture formul will be referred to s Gussin with respect to set of 2n functions φ 1,, φ 2n : [, b] R nd weight function ω : [, b] R + if it consists of n nodes nd n weights nd integrtes exctly the functions φ i with respect to the weight function ω for ll i = 1,, 2n The weights nd nodes of Gussin qudrture will be referred to s Gussin weights nd nodes, respectively Although Chebyshev qudrtures re clssicl Gussin qudrtures on the intervl [ 1, 1] with respect to the weight function ω(x) = (1 x 2 ) 1/2, in prctice, Chebyshev nodes nd weights re often used to integrte functions on [ 1, 1] with respect to the weight function ω(x) = 1 This prctice leds to 2n-point qudrture which integrtes exctly polynomils of order 2n 1, nd motivtes the following definition: DEFINITION 22 A qudrture formul will be referred to s Generlized Chebyshev qudrture for set of 2n functions φ 1,, φ 2n : [, b] R nd weight function ω : [, b] R + if it consists of 2n nodes nd 2n weights nd integrtes exctly the functions φ i with respect to the weight function ω for ll i = 1,, 2n The weights nd nodes of Chebyshev qudrture will be referred to s Chebyshev weights nd nodes, respectively 22 Qudrture rules nd optimiztion Let m, n be integers with m 2n nd let x 1,, x n, w 1,, w n be qudrture rule (23) f(x)ω(x)dx f(x j )w j

3 A NONLINEAR OPTIMIZATION PROCEDURE FOR GENERALIZED GAUSSIAN QUADRATURES 3 which is exct for the functions φ 1,, φ m Obviously, the weights w j nd nodes x j of such qudrture stisfy the (generlly underdetermined) nonliner system of equtions (24) where F 1 (x 1,, x n,w 1,, w n ) = b 1 F 2 (x 1,, x n,w 1,, w n ) = b 2 F m (x 1,, x n,w 1,, w n ) = b m, (25) F i (x 1,, x n, w 1,, w n ) = nd (26) b i = φ i (x j )w j, φ i (x)ω(x)dx When m = 2n, the nodes x j nd weights w j in (24) form generlized Gussin qudrture for the functions φ 1,, φ 2n In [11] nd [13], the existence of unique solution of (24) is proven under certin conditions on the system of functions φ 1,, φ 2n It is observed in [14, 20, 3] tht solutions, or t lest pproximte solutions, of (24) exist for mny systems of 2n functions not stisfying those conditions In this pper, we will encounter systems of the form (24) where m < 2n Under these conditions, the system does not dmit unique solution Non-uniqueness is not, however, n obstruction to the determintion of qudrture rule for the functions φ 1,, φ m since such rule is given by ny solution of the system (24) Indeed, even in the cse when there is no exct solution of (24), it is often possible to construct n pproximte qudrture for the functions φ 1,, φ m We close this section with the following definition: DEFINITION 23 Suppose φ 1,, φ m : [, b] R re squre integrble with respect to the nonnegtive integrble weight function ω We sy tht qudrture rule x 1,, x n, w 1,, w n integrtes φ 1,, φ m with precision ɛ > 0 if m (27) F i (x 1,, x n, w 1,, w n ) b i 2 ɛ 2, where (28) F i (x 1,, x n, w 1,, w n ) = nd (29) b i = for ll i = 1,, m φ i (x j )w j φ j (x)ω(x)dx 23 Qudrture nd interpoltion It is well known tht when Chebyschev nodes re used for polynomil interpoltion on the intervl [ 1, 1], the procedure is numericlly stble nd the convergence properties re close to optiml (see [19] nd [6]) In this subsection, we prove tht the nodes of ny Gussin qudrture nd mny generlized Gussin qudrtures led to stble interpoltion formuls The principl nlytic tool of this subsection is the following obvious theorem (see, for exmple, [20]) THEOREM 21 Suppose tht the weight function ω : [, b] R is nonnegtive nd the functions φ 1,, φ n : [, b] R re orthonorml with respect to ω Suppose further tht the n-point qudrture

4 4 JAMES BREMER, ZYDRUNAS GIMBUTAS, AND VLADIMIR ROKHLIN rule x 1,, x n, w 1,, w n is exct for ll products of the form φ i (x)φ j (x) nd w i > 0 for ll 1 i n Then the n n mtrix A with entries (210) A ij = w j φ i (x j ) is orthogonl Now let f be function of the form (211) f(x) = α i φ i (x) We would like to construct n interpoltion formul on the intervl [,b] for functions of this form; tht is, given the vlues f 1,, f n of function of the form (211) t collection of points x 1,, x n, we would like formul for determining the coefficients α 1,, α n Let α be the vector α = (α 1,, α n ), let F be the vector F = (f 1, f 2,, f n ), nd finlly, let B be the n n mtrix with entries (212) B ij = φ i (x j ) Then (213) F = Bα, nd ssuming tht B is invertible, it follows tht (214) α = B 1 F If the condition number of B is not too high, then (214) is numericlly stble formul for computing the coefficients α 1,, α n The following observtion is the principl observtion of this subsection Observtion 21 Under the conditions of Theorem 21, the mtrix B in the interpoltion formul (214) is of the form (215) B = DQ, where D is digonl mtrix with entries (216) D ii = 1 wi nd Q is orthogonl Thus (217) α = Q D 1 F In other words, the coefficients α cn be obtined by pplying digonl mtrix followed by n orthogonl mtrix 24 The dmped Guss-Newton method The dmped Guss-Newton method is well-known itertive technique for the solution of nonliner lest-squres problems It converges under very generl conditions, nd does not require tht the Jcobin of the system be nonsingulr Here we give only elementry detils; more thorough tretment cn be found in [7] Suppose tht R : R n R m is continuously differentible function of the form (218) R(x) = nd let J(x) be the Jcobin (219) J(x) = r 1 r 1 (x) r 2 (x) r m (x) x 1 (x) r m x 1 (x) r 1 x n (x) r m x n (x)

5 A NONLINEAR OPTIMIZATION PROCEDURE FOR GENERALIZED GAUSSIAN QUADRATURES 5 of R t the point x The dmped Guss-Newton method is numericl method for minimizing the function (220) f(x) = 1 m r j (x) 2 2 It belongs to brod clss of numericl optimiztion methods which proceed from n initil guess x 0 by forming sequence x 1, x 2, of itertes vi the formul (221) x k+1 = x k + α k d k where d k is referred to s the serch direction t itertion k nd α k is crefully chosen step size The primry purpose of this section is Theorem 22 below, which gives conditions under which n itertion of the form (221) converges We strt with the following definition: DEFINITION 24 Let f : R n R be continuously differentible function nd consider ny itertion of the form (221) We sy tht the step length α k nd step direction d k stisfy the Wolfe conditions t the point x k if (222) nd (223) for some constnts λ (0, 1) nd β (λ, 1) The following theorem cn be found in [7]: f(x k + α k d k ) f(x k ) + λα k f(x k ) d k, f(x k + α k d k ) d k β f(x k ) d k THEOREM 22 Suppose tht f : R n R is continuously differentible function bounded from below, nd ssume tht there exists γ 0 such tht (224) f(x) f(y) 2 γ x y 2 for every x nd y in R n Consider ny itertion of the form (221) such tht for ech k = 0, 1, the pir (d k, α k ) stisfies the Wolfe conditions If, in ddition, one of the conditions (225) or (226) holds for ech k = 0, 1,, then either f(x k ) d k < 0 f(x k ) = 0 nd d k = 0 (227) f(x k ) = 0 for some k 0, or (228) lim k f(x k ) d k d k 2 = 0 Remrk 21 Theorem 22 sttes tht either the sequence x k converges to criticl point for the function f or the direction d k become orthogonl to the grdient of f In prctice, it is esy to void the lter condition, thus ensuring the convergence of f(x k ) to 0 Remrk 22 Under mild conditions on the function f, given sequence of {d k } stisfying condition (225), there exists sequence of α k stisfying the Wolfe conditions (see, for exmple, [7] Theorem 632) In the cse of the dmped Guss-Newton method, the serch direction d k is chosen s solution to the lest-squres problem (229) min d k J(x k )d k + R(x k ) 2, which is n ffine pproximtion to the nonliner lest-squres problem Since min x R(x) 2 (230) f(x k ) = J (x k )R(x k ),

6 6 JAMES BREMER, ZYDRUNAS GIMBUTAS, AND VLADIMIR ROKHLIN we hve (231) f(x k ), d k = R(x k ), J(x k )d k The serch direction d k is chosen so tht J(x k )d k is the projection of R(x k ) onto the column spce of J(x k ) It follows tht (232) f(x k ), d k = R(x k ), J(x k )d k 0 If we choose d k to be 0 in the event tht f(x k ), d k = 0, then we obtin the following theorem: THEOREM 23 Suppose tht R : R n R m is continuously differentible function of the form (218), f : R n R is given by (220), nd the Jcobin, J(x), of R is given by (219) Further suppose tht J(x) 2 is bounded on R n nd there exists constnt γ > 0 such tht J(x) J(y) 2 γ x y 2 for ll x nd y in R n If x k is defined by the itertion x k+1 = x k + α k d k where, for ech k = 1, 2,, d k is solution of the lest-squres problem J(x k )d k + R(x k ) 2 = min v J(x k )v + R(x k ) 2, nd the sequence {α k } is chosen so tht the pirs (d k, α k ) stisfy the Wolfe conditions for k = 0, 1,, then either or f(x k ) = 0 for some k 0 f(x k ) d k lim = 0 k d k 2 25 Singulr vlue decomposition The SVD is ubiquitous tool in numericl nlysis (see, for instnce, [8]) Here we discuss it in the cse of rel mtrices LEMMA 21 (SVD) For ny n m rel mtrix A, there exist, for some integer k, n n k rel mtrix U with orthonorml columns, n m k rel mtrix V with orthonorml columns, nd k k rel digonl mtrix Σ with positive digonl entries σ 1 σ 2 σ k > 0, such tht (233) A = U Σ V The digonl entries of Σ re clled singulr vlues, the columns of the mtrix U re clled the left singulr vectors, nd the columns of the mtrix V re clled right singulr vectors One of the most common pplictions of the SVD is the pproximtion of mtrices; if we let Σ p denote the digonl mtrix with entries { σ i if i p D ii = 0 otherwise, then (234) A UΣ p V 2 = σ p+1 The mtrix UΣ p V is, in fct, the optiml rnk p pproximtion of the mtrix A (see, for instnce, [8]); tht is, (235) min A k A A k 2 = σ p+1 where A k rnges over the set of ll n m mtrices of rnk k

7 A NONLINEAR OPTIMIZATION PROCEDURE FOR GENERALIZED GAUSSIAN QUADRATURES 7 26 QR decompositions The singulr vlue decomposition provides the mens to construct n optiml rnk k pproximtion to given mtrix; however, the SVD cn be expensive to form, nd other less computtionlly expensive mtrix fctoriztions cn be used to compress mtrices in lieu of the SVD By slight buse of terminology, we will refer to the fctoriztion in the following obvious lemm s QR decomposition LEMMA 22 For ny n m rel mtrix A, there exist n integer k, n n k rel mtrix Q with orthonorml columns, n m m permuttion mtrix Π, nd k m rel mtrix R with the block form R = ( T M ), where T is k k upper tringulr mtrix with positive digonl entries nd M is generl k (m k) mtrix, such tht (236) AΠ = QR Moreover, there is one-to-one correspondence between m m permuttion mtrices Π nd decompositions of the form (236) The following theorem cn be found (in stronger form) in [10] THEOREM 24 Suppose tht A is rel m n mtrix with singulr vlues σ 1 σ 2 σ r > 0 For ny integer p, 0 < p r, there exists n m m permuttion mtrix Π such tht the QR decomposition uniquely determined by Π is of the form ( ) R11 R (237) AΠ = Q 12, 0 R 22 where R 11 is p p upper tringulr mtrix with positive digonl entries, nd (238) R (n k)σ p+1 Remrk 23 Theorem 24 implies tht given ny rel m n mtrix A with singulr vlues σ 1 σ 2 σ r > 0 nd ny integer 0 < k < r, there is permuttion mtrix Π such tht the well-known Grm-Schmidt orthogonliztion procedure (see, for exmple, [8]) pplied to the mtrix AΠ results in the pproximte mtrix fctoriztion (239) A QS 2 cσ k+1, where Q is n m k mtrix with orthonorml columns nd S is k n mtrix S Note tht the error bound (239) is close to the optiml error for rnk k pproximtions to A Remrk 24 When properly implemented, the modified Grm-Schmidt lgorithm with reorthogonliztion is relible method for obtining QR decomposition (see, for instnce, [2]) While there re no gurnteed bounds of the form (239) for this lgorithm, it does well in prctice In [10], robust, provbly stble lgorithms re introduced for producing QR decompositions tht stisfy bounds of the form (239) 27 Anlogs of mtrix fctoriztions for finite sequences of functions Here we introduce nlogs of the mtrix fctoriztions of the preceding two subsections for finite sequences of functions We begin with the following obvious generliztion of Lemm 22 LEMMA 23 For ny finite sequence φ 1,, φ m : [, b] R of squre integrble functions, there exists n integer k m, permuttion Π S m, collection of orthonorml functions q 1,, q k : [, b] R, nd n k m rel mtrix R = (r ij ) with the block form ( T M ) where T is k k upper tringulr mtrix with positive digonl entries nd M is generl k (m k) mtrix, such tht (240) φ Π(j) (x) = mx(j,k) r ij q i (x)

8 8 JAMES BREMER, ZYDRUNAS GIMBUTAS, AND VLADIMIR ROKHLIN for ll j = 1,, m Moreover, there is one-to-one correspondence between decompositions of this form nd permuttions Π S m By nlogy with the finite dimensionl cse, we will refer to decompositions of this type s QR decompositions The functions q i (x) will be referred to s QR functions nd we will cll the digonl entries r ii of R the normlizing fctors of the decomposition As in the cse of mtrices, common ppliction of expnsions of the form (240) is the compression of the functions φ j ; tht is, often permuttions Π S m cn be found for which the truncted series p (241) ij q i (x), with p < k, provide good pproximtions to the functions φ Π(j) Now, we restte result found in [3], generlizing the SVD to the cse of finite sequence of functions THEOREM 25 Suppose tht the functions φ 1,, φ m : [, b] R re squre integrble Then there exist n integer k, finite orthonorml sequence of functions u 1,, u k : [, b] R, n m k mtrix V = (v ij ) with orthonorml columns, nd sequence s 1 s 2 s k > 0 R such tht (242) φ j (x) = k u i (x)s i v ji for ll x [, b] nd ll j = 1,, m The sequence s 1,, s k is uniquely determined by k By nlogy with the finite-dimensionl cse, we will refer to this decomposition s the SVD of finite sequence of functions We cll the functions u i the singulr functions, the columns of V the singulr vectors, nd the vlues s i the singulr vlues The SVD is clerly useful tool for the compression of the sequence φ 1,, φ m : if we let φ j (x) denote the p-term trunction (243) φj (x) = of the sum (242), then p u i (x)s i v ji (244) φ j (x) φ j (x) s p+1 for j = 1,, m However it is obtined, using n pproximte representtion p (245) φ j (x) α i q i (x), integrls of the form (246) cn be pproximted s (247) φ j (x)ω(x)dx = φ j (x)ω(x)dx ( p ) α i q i (x) ω(x) dx p α i q i (x)ω(x)dx; thus, qudrture which is exct for the integrls (248) q i (x)ω(x)dx

9 A NONLINEAR OPTIMIZATION PROCEDURE FOR GENERALIZED GAUSSIAN QUADRATURES 9 for i = 1,, p, cn be used s n pproximte qudrture for the integrls (249) for j = 1,, m φ j (x)ω(x)dx 28 Underdetermined systems The purpose of this short subsection is the sttement of two lemms on the existence nd behvior of sprse solutions of underdetermined systems We begin with the following result on the existence of well-behved solutions to underdetermined lest-squres problems, whose proof is n elementry exercise in liner lgebr LEMMA 24 Suppose tht A is n n m mtrix, n < m, with left singulr vectors u 1,, u k nd singulr vlues σ 1 σ k > 0 For 0 < p k, let U p be the subspce of R n spnned by u 1,, u p, nd let Proj p : R n U p denote the projection opertor R n U p Then, given ny vector b R n nd ny integer 0 < p k, there exists vector x R m with no more thn p nonzero entries such tht (250) Ax b 2 = Proj p b b 2 nd (251) x 2 σ 1 σ p b 2 Let A be n n m mtrix with n < m nd consider the lest-squres problem (252) min x Ax b 2, where b is given vector in R n The problem is underdetermined nd therefore the condition number of A is infinite However, Lemm 24 implies tht if b is in the spn of smll number of singulr vectors of A, then well-behved pproximte solution x to the lest-squres problem (252) cn still be found The second lemm is stronger, but more specilized result, whose proof is n esy corollry of the following theorem, which ppers s Theorem 2 in [17] THEOREM 26 Suppose tht S is n rbitrry set, n is positive integer, f 1,, f n re bounded complex-vlued functions on S, nd ɛ is positive rel number such tht (253) ɛ 1 Then, there exist n points x 1,, x n in S nd n functions g 1,, g n on S such tht (254) g k (x) 1 + ɛ for ll x in S nd k = 1, 2,, n, nd (255) f(x) = f(x k )g k (x) k=1 for ll x in S nd ny function f defined on S vi the formul (256) f(x) = c k f k (x) Moreover, if the set S is finite, then g 1,, g n cn be chosen so tht (254) holds with ɛ = 0 LEMMA 25 If k=1 (257) Ax = b, where A is n m n mtrix of rnk m, then there exists vector x R n with no more then m nonzero entries such tht A x = b, nd x 1 m x 1

10 10 JAMES BREMER, ZYDRUNAS GIMBUTAS, AND VLADIMIR ROKHLIN Proof: By Theorem 26, there exists n m n mtrix G whose entries re bounded by 1 nd n m m mtrix of à consisting of m columns A i 1,, A im of A such tht (258) A = ÃG Since Ax = b, it follows tht (259) ÃGx = b If we let y = Gx R m nd define x by the formul { y k if j = i k, (260) x j = 0 otherwise, then x hs t most m nonzero entries, A x = b, nd m (261) x 1 = x i (262) (263) (264) s desired m = g ij x j m x j m x 1, Remrk 25 For sets S which re finite, the proof of Theorem 26 given in [17] is constructive, but the procedure is computtionlly infesible To wit, in the specil cse of Lemm 25, the scheme of [17] mounts to the choice of submtrix à consisting of set of columns A i 1,, A im of A which mximize the determinnt (265) det ( ) A i1 A i2 A im over the collection of ll sets of m columns of the mtrix A The existence of the mtrix G follows since det(ã) 0 by construction, nd the bound on the entries of G follows from Crmer s rule Remrk 26 In prctice, the modified Grm-Schmidt procedure with reorthogonliztion cn be used to find set of m columns A i1,, A im of the m n rnk m mtrix A such tht (266) det ( ) A i1 A i2 A im is comprble to the supremum (267) sup j 1,,j m det ( A j1 A j2 A jm ) over ll collections of m columns of A Indeed, the modified Grm-Schmidt procedure is nothing more thn greedy lgorithm for finding n pproximte solution to the optimiztion problem (268) rgmx j 1,,j m det ( A j1 A j2 A jm ) An obvious modifiction of the rgument in [17] (which is sketched bove in Remrk 25) shows tht once such set of columns i 1,, i m hs been found, the mtrix A cn fctored s A = ÃG, where à is n m m submtrix of A nd G is n m n mtrix whose entries re bounded in bsolute vlue, but not necessrily by 1

11 A NONLINEAR OPTIMIZATION PROCEDURE FOR GENERALIZED GAUSSIAN QUADRATURES 11 Remrk 27 Strong rnk reveling QR fctoriztions (see [10] nd [2], for instnce) identify spnning set A i1,, A im of columns of n m n mtrix A of rnk m for which the rtio of det ( ) A i1 A i2 A im to sup det ( ) A j1 A j2 A jm j 1,,j m is gurnteed to stisfy lower bound, thus ensuring tht stble fctoriztion of the form A = ÃG, where à is n m m submtrix of A, cn be found 29 The Shermn-Morrison-Woodbury formul The Shermn-Morrison-Woodbury formul gives n expression for the rnk-k updte (A + UV t ) 1 of the inverse of mtrix A The following Lemm cn be found, for instnce, in [8]: LEMMA 26 Suppose tht A is n invertible n n mtrix, U is n n k mtrix, nd V is n k n mtrix If the rnk-k updte (269) (A + UV t ) of the mtrix A is invertible, then its inverse is (A + UV t ) 1 = A 1 A 1 U(I + V t A 1 U) 1 V t A 1 In this pper, given n m n rel mtrix A, we will utilize the Shermn-Morrison-Woodbury formul to perform specific type of updte to the inverse of the m m mtrix AA t In prticulr, we wish to updte the inverse of AA t in order to form the inverse of the mtrix BB t, where B is obtined from A by deleting its j th column Tht is, B = A uv t, where u is the m 1 vector which is the j th column of the mtrix A nd v is the n 1 vector with entries { 1 if i = j (270) v i = 0 otherwise A simple clcultion shows tht ( A uv t ) ( A uv t) t = AA t uv t A t Au t v + uv t vu t (271) (272) = AA t uu t In other words, if we form the mtrix B by deleting the j th column of the mtrix A, then BB t cn be formed from AA t by rnk-1 updte The Shermn-Morrison-Woodbury formul then implies tht the inverse of BB t cn be computed from (AA t ) 1 vi rnk-1 updte 3 Numericl pprtus 31 Nested Legendre discretiztions of sequences of functions In this pper, we re confronted with sequences φ 1,, φ m of functions defined on the intervl [, b] for which we wish to construct generlized Gussin qudrture rule The collection of functions φ j hs the following properties: Ech function φ j is integrble on [, b] nd nlytic except t smll number of points, The totl number functions is quite lrge (eg, m = 10, 000), The rnk of the set φ 1,, φ m is low (eg, 40) to high precision

12 12 JAMES BREMER, ZYDRUNAS GIMBUTAS, AND VLADIMIR ROKHLIN In order to construct n efficient qudrture rule, we will tke dvntge of rnk deficiency to compress the sequence φ 1,, φ m This mens, more specificlly, tht we first form set u 1,, u k of orthonorml functions on [, b] such tht ech φ i cn be pproximted by sum of the form (31) φ i (x) nd ech u j is defined by sum (32) u j (x) = k α ij u j (x), m β ij φ i (x) We then build generlized Gussin qudrture rule for the functions u 1,, u k In the course of constructing tht qudrture, it will be necessry to evlute the functions u j t rbitrry points on the intervl [, b] If the sums (32) involve lrge number of terms, or if the evlution of the φ i is expensive, then it is imprcticl to use formul (32) to evlute the u j It will therefore be necessry to represent the function u j in mnner which llows for their rpid evlution An obvious lterntive to evluting sums of the form (32) directly is to represent the functions u j vi polynomil interpoltion Let x 1,, x n be mesh of interpoltion points on the intervl [, b] nd suppose tht the Lgrnge polynomils interpolting u 1,, u k t these mesh points represent the functions u j to given precision Then, clerly, the Lgrnge polynomil interpolting the function (33) ψ i = α ij u j (x) t x 1,, x n pproximtes φ i with controllble precision As ws discussed in Subsection 23, if Gussin nodes re used s interpolting points, then the resulting procedure is numericlly stble However, when the functions φ j re not smooth, polynomil interpoltion becomes inefficient nd cn fil completely for sufficiently singulr functions In such cses, where the functions to be interpolted re singulr, it is customry to use n dptive interpoltion scheme insted Tht is, the intervl [, b] is subdivided into collection of subintervls such tht ech function φ j is ccurtely interpolted by low order polynomil on ech subintervl In this subsection, we describe numericl procedure for the pproximtion of sequence of functions vi nested Guss-Legendre polynomil interpoltion The procedure is introduced in [20], but we reproduce it here since it is n integrl prt of the lgorithm The input to this procedure is collection φ 1,, φ m of functions integrble on [, b], precision ɛ > 0, nd resonbly lrge integer k which controls the number of interpoltion nodes used on ech subintervl (for the computtions in this pper, we used k = 30) The lgorithm proceeds in three stges Stge 1 In the first stge, the following procedure is used to discretize ech of the φ i seprtely Tht is, the following sequence of steps is repeted for i = 1,, m: 1 Construct the 2k Legendre nodes x 1,, x 2k on the intervl [, b] 2 Let P (x) denote the Lgrnge polynomil of order 2k 1 interpolting φ i t the mesh points x 1,, x 2k Construct the coefficients α 1,, α 2k of P (x) in the expnsion P (x) = 2k 1 j=0 where L j (x) is the j th order Legendre polynomil 3 If the inequlity α j L j (x) (34) 2k j=k+1 α j 2 < ɛ

13 A NONLINEAR OPTIMIZATION PROCEDURE FOR GENERALIZED GAUSSIAN QUADRATURES 13 is stisfied, then we conclude tht the order k Legendre expnsion for φ i on the intervl [, b] is sufficient Otherwise, we split the intervl [, b] into two subintervls, [, (+b)/2] nd [(+b)/2, b)], nd repet the procedure recursively for ech of the subintervls Stge 2 Store the endpoints of ech subintervl constructed in Stge 1 in n rry Sort the rry nd eliminte multiple elements The resulting rry of points on the intervl [, b] is the rry of endpoints of the subintervls of the finl subdivision Stge 3 Construct the k point Legendre discretiztion on ech the subintervls obtined in Stge 2 for ech of the functions φ i (x) Remrk 31 The scheme of this subsection is resonbly relible mechnism for the discretiztion of sets of functions with singulrities The stopping condition (34) is nlogous to tht usully used to terminte n dptive qudrture procedure Just s ny such qudrture procedure cn fil for crefully designed counterexmples, so too cn the procedure of this section fil under certin circumstnces The problem, however, is not encountered in the exmples of this pper nd whenever the uthors hve encountered it in prctice, it hs been esy to rectify 32 Compression of finite sequence of functions This subsection describes numericl procedure for compressing finite sequence of functions by pproximting either the singulr vlue decomposition or QR decomposition of the sequence We begin with two definitions, the second of which is dpted from [20] nd [3] In wht follows, P C([, b]) refers to the vector spce of piecewise continuous functions on the intervl [, b] DEFINITION 31 A k-point liner interpoltion scheme on the intervl [, b] is collection of linerly independent functions φ 1, φ k in P C([, b]) nd set of distinct nodes x 1,, x k in [, b] together with liner mpping T : P C([, b]) spn{φ 1,, φ k } such tht (35) T f(x j ) = f(x j ) for ll j = 1,, k We cll the x 1,, x k interpoltion nodes, the mpping T the interpoltion mpping, nd the φ j interpolting functions We lso sy tht the coefficients α 1,, α k of T f(x) with respect to the bsis {φ 1,, φ k } re the interpoltion coefficients for the function f Associted with every k-point liner interpoltion scheme is n invertible liner trnsformtion R k R k which tkes the vlues f(x 1 ),, f(x k ) of function f t the interpoltion nodes to the k interpoltion coefficients for the function The imge T f of f under the interpoltion mpping is, of course, determined by these interpoltion coefficients We will often refer to T f s the function defined by the interpoltion scheme nd the vlues f(x 1 ),, f(x k ) Remrk 32 It is generlly possible to extend the definition of interpoltion scheme to encompss interpolting functions φ not in P C([, b]), s well s interpoltion mppings T defined on lrger spces We must keep in mind, however, tht eqution (35) requires tht both T f(x) nd f(x) be defined pointwise DEFINITION 32 We sy tht the the combintion of k-point liner interpoltion scheme on [, b] with nodes y 1,, y k [, b] nd interpolting functions φ 1,, φ k, nd k-point qudrture rule with nodes x 1,, x k [, b] nd weights w 1,, w k preserves inner products if The nodes x 1,, x k of the qudrture rule coincide with the nodes y 1,, y k of the interpoltion scheme, nd The qudrture rule is exct for ll products φ i (x)φ j (x) of the interpolting functions Remrk 33 The second condition in definition 32 together with the ssumption of the linerity of the interpoltion scheme ensures tht for ny two piecewise continuous functions f(x) nd g(x) the qudrture rule is exct for the integrl (36) T f(x)t g(x)dx The following re exmples of qudrture nd interpoltion schemes which preserve inner products:

14 14 JAMES BREMER, ZYDRUNAS GIMBUTAS, AND VLADIMIR ROKHLIN Exmple 31 The combintion of clssicl Gussin qudrture nd Lgrnge interpoltion t the sme Gussin nodes preserves inner products, since polynomils interpoltion on n nodes produces n interpolting polynomil of order n 1 nd the product of ny two such polynomils is exctly integrted by n n point Gussin qudrture Exmple 32 Nested Gussin interpoltion like tht of the preceding section coupled with corresponding nested Gussin qudrture formuls lso preserve inner products, since on ech subintervl Gussin interpoltion is coupled with Gussin qudrture formul Note tht the interpolting functions in this exmple re not continuous, but rther piecewise continuous The following theorem, which is reformultion of Theorem 45 in [20], will be used in pproximting the singulr vlue decomposition (or QR decomposition) of sequence of functions THEOREM 31 Suppose tht the combintion of qudrture rule with nodes x 1,, x n nd weights w 1,, w n, nd liner interpoltion scheme with interpolting functions ψ 1,, ψ n nd interpoltion mpping T preserves inner products on [, b] Suppose further tht φ 1,, φ m is collection of piecewise continuous functions on [, b], U = (u ij ) is n n k mtrix with orthonorml columns, nd R = (r ij ) is k m mtrix such tht (37) φ j (x i ) k w i = u il r lj for ll j = 1,, m nd i = 1,, n If the functions u j (x) re defined for ll j = 1,, k vi the interpoltion scheme nd the vlues l=1 (38) u j (x i ) = u ij wi, then: 1 The functions u j (x) re orthonorml; tht is, u i (x)u j (x)dx = δ ij for ll i, j = 1,, k 2 The sequence of functions φ 1,, φm defined by the formul k φ j (x) = u i (x)r ij is identicl to the sequence of functions produced by smpling the functions φ 1,, φ m t the points {x i } nd then interpolting with the interpoltion scheme on [, b] The lgorithm described below uses s input sequence of functions φ 1,, φ m : [, b] R nd qudrture nd interpoltion scheme which preserves inner products The nodes nd weights of the qudrture will be denoted by x 1,, x n nd w 1,, w n, respectively The output of the lgorithm depends on whether the SVD or QR decomposition is used for compression In either cse, the output is sequence u 1,, u k of orthonorml vectors nd sequence σ 1,, σ k of positive rel numbers When the SVD is used, the orthonorml vectors u 1,, u k pproximte the singulr vectors for the sequence φ 1,, φ m nd the σ 1,, σ k pproximte the corresponding singulr vlues In the cse of the QR decomposition, the orthonorml vectors u 1,, u k pproximte collection of QR vectors for φ 1,, φ m nd the σ j re the corresponding normlizing fctors Description of the lgorithm 1 Construct the n m mtrix A with entries A ij = φ j (x i ) w i 2 Compute either the SVD of the mtrix A or QR decomposition for the mtrix A In the first cse, produce the fctoriztion A = UΣV,

15 A NONLINEAR OPTIMIZATION PROCEDURE FOR GENERALIZED GAUSSIAN QUADRATURES 15 where U = (u ij ) is n n k mtrix with orthonorml columns, V = (v ij ) is n m k mtrix with orthonorml columns, nd Σ is digonl mtrix whose j th digonl entry is σ j In the second cse, produce the fctoriztion AΠ = UR, where U = (u ij ) is n n k mtrix with orthonorml columns nd R is n k m trpezoidl mtrix with digonl entries σ 1,, σ k 3 Construct the n k vlues u j (x i ) defined by the formul (39) u j (x i ) = u ij wi 4 For ny desired point x [, b], evlute the functions u i : [, b] R using the interpoltion scheme on [, b] Remrk 34 Theorem 31 ensures tht the ccurcy of the pproximtion produced by the lgorithm of this section depends primrily on the ccurcy of the underlying interpoltion scheme 33 Construction of Chebyshev qudrtures In this subsection, we describe numericl lgorithm for the construction of Chebyshev qudrture for finite sequence of functions Given pre-existing n-point qudrture formul x 1, x 2,, x n, w 1, w 1,, w n, where n > k, which exctly integrtes the u 1,, u k, it is strightforwrd to construct Chebyshev qudrture for u 1,, u k Becuse the pre-existing qudrture rule exctly integrtes the input functions u 1,, u k, the mtrix eqution (310) where r i is defined by u 1 (x 1 ) u 1 (x 2 ) u 1 (x n ) u 2 (x 1 ) u 2 (x 2 ) u 2 (x n ) u k (x 1 ) u k (x 2 ) u k (x n ) r i = u j (x)dx w 1 w 2 w n = for ll i = 1,, k, is stisfied By Lemm 25, there exist i 1,, i n nd w 1, w k such tht (311) nd (312) u 1 (x i1 ) u 1 (x i2 ) u 1 (x in ) u 2 (x i1 ) u 2 (x i2 ) u 2 (x in ) u k (x i1 ) u k (x i2 ) u k (x in ) k w j k w j w 1 w 2 w k 0 0 = In other words, there is by Lemm 25 k-point qudrture formul x i1,, x ik, w 1,, w k for the k functions u 1,, u k Moreover, the inequlity (312) ensures tht the k-point qudrture formul is numericlly stble ssuming the weights of the originl qudrture re resonbly smll The following lgorithm is computtionl procedure for constructing such qudrture vi the modified Grm-Schmidt lgorithm with double orthogonliztion, or one of its vrints This procedure does not relize the theoreticl bound (25) gurnteed by Lemm 25 In prctice, however, it still leds to the construction of stble Chebyshev qudrture formuls for the input functions; see Subsection 28 for more discussion of the numericl stbility of the modified Grm- Schmidt lgorithm nd its vrints r 1 r 2 r k r 1 r 2 r k,

16 16 JAMES BREMER, ZYDRUNAS GIMBUTAS, AND VLADIMIR ROKHLIN The lgorithm uses s input sequence u 1,, u k of functions nd pre-existing qudrture x 1,, x n, w 1,, w n, with n > k, which exctly integrtes the functions u 1,, u k, nd produces s output k-point qudrture formul consisting of k nodes x 1,, x k {x 1,, x n } nd k weights w 1,, w k Description of the lgorithm 1 For the vector r R k whose i th entry is the sum r i = u i (x j )w j, which is the vlue of the integrl 2 Form the k n mtrix B with entries u i (x)dx B ij = u i (x j ) w j 3 Use the pivoted Grm-Schmidt lgorithm with double reorthogonliztion to select set B i1,, B ik of spnning columns for the mtrix B nd form the fctoriztion ( ) R11 R B = Q 12, 0 R 22 where R 11 is k k upper tringulr mtrix, Q is k k orthogonl mtrix, ( Bi1 B i2 B ik ) = QR11, nd R 22 2 is smll 4 Use bck substitution to construct solution z R k to the k k system of equtions R 11 z = Q r, which is, of course, lest squres solution of the system ( Bi1 B i2 B ik ) z = r 5 Form the new k-point qudrture x 1,, x k, w 1,, w k by letting for ll j = 1,, k x j = x ij nd w j = z j wj Remrk 35 The columns of the mtrix B re scled by the squre roots of the qudrture weights so tht the l 2 norms of columns of B computed in Step 3 by the Grm-Schmidt orthonormliztion procedure re proportionl to the qudrture weight for the corresponding column 4 Numericl lgorithm This section describes numericl lgorithm for the construction of the nodes nd weights of n pproximte qudrture rule for sequence of functions The lgorithm s input is sequence of functions (41) φ 1,, φ m : [, b] R nd two ccurcies, ɛ disc nd ɛ qud The first, ɛ disc, controls the ccurcy of the scheme used to discretize nd compress the input functions φ 1,, φ m, nd the second, ɛ qud, is the desired ccurcy for the qudrture rule The lgorithm s output is qudrture rule consisting of set of nodes x 1,, x l nd set of weights w 1,, w l such tht l (42) φ i (x)dx φ i (x j )w j for ll i = 1,, m The integer l depends on the numericl rnk of the input set φ 1,, φ m nd both ɛ disc nd ɛ qud The lgorithm proceeds in three stges In the first stge, the numericl techniques of the preceding section re used to discretize nd compress the functions φ 1,, φ m In the second,

17 A NONLINEAR OPTIMIZATION PROCEDURE FOR GENERALIZED GAUSSIAN QUADRATURES 17 suboptiml qudrture rule is obtined for the compressed collection of functions Finlly, in the third phse, n optimiztion procedure is used to reduce the number of points needed by the qudrture Stge 1: Discretiztion nd compression In this stge the following sequence of steps is performed to discretize nd compress the input functions 1 Use the technique of Section 31 to discretize the functions φ 1,, φ m, so tht they re ll represented to the precision ɛ disc Let φ 1 (x),, φm (x) denote the resulting discretiztions Also, let x 1,, x n denote the discretiztion nodes nd w 1,, w n the weights of the ssocited qudrture 2 Apply the procedure of Subsection 32 to compress the sequence φ 1,, φ n vi either QR decomposition or the singulr vlue decomposition Denote the resulting orthonorml functions by u 1,, u p nd the ssocited singulr vlues or normliztion fctors by λ 1 λ 2 λ p > 0 3 Discrd ny of the functions u 1,, u p corresponding to singulr vlue (or normliztion fctor) λ j ɛ qud 4 Denote the functions obtined in this mnner by u 1,, u k nd the ssocited singulr vlues (normliztion fctors) by λ 1 λ 2 λ k > 0 Remrk 41 It is importnt tht ɛ qud be somewht lrger thn ɛ disc (for the exmples of this pper, ɛ qud /ɛ disc 100) Becuse the functions φ 1,, φ m re discretized to precision ɛ disc, singulr vectors u j corresponding to singulr vlues λ j comprble to ɛ disc contin little informtion bout the functions φ 1,, φ m This mens tht constructing qudrture formul which fithfully integrtes such singulr functions is pointless, nd moreover, since the u j corresponding to singulr vlues λ j comprble to ɛ svd re not necessrily smooth, even when the the functions φ 1,, φ m re very smooth, it cn cuse difficulties with the the lgorithm described in Step 3 below Stge 2: Construction of k-point qudrture rule We now pply the procedure of Subsection 33 to construct k-point qudrture formul for φ 1,, φ m We use s inputs to tht procedure the functions u 1,, u k nd the nested Gussin qudrture x 1,, x n, w 1,, w n constructed in Stge 1 Note tht since the u 1,, u k re defined vi the nested Gussin interpoltion scheme, the qudrture x 1,, x n, w 1,, w n is exct for the functions u 1,, u k Denote the resulting k qudrture nodes by x 1,, x k nd the k qudrture weights by w 1,, w k The weights of the nested Gussin qudrture scheme stisfy the bound (43) w j (b ), since w j > 0 for ll j = 1,, n nd the qudrture rule is exct for the function f(x) = 1 It follows from the discussion in Subsection 33 tht the new k-point qudrture formul is numericlly stble Becuse the discretiztions φ 1,, φm re well pproximted by functions in the spn of the u 1,, u k, qudrture rule exct for the functions u 1,, u k will pproximtely integrte the functions φ 1,, φm (nd, of course, it follows tht such qudrture rule will lso serve for the functions φ j (x), ssuming the discretiztions constructed in Step 1 re sufficiently ccurte) So the new k-point qudrture rule x 1,, x k, w 1,, w k is n pproximte qudrture for the input functions φ 1,, φ m Stge 3: Point-by-point reduction of the qudrture rule In this stge, beginning with n n-point qudrture rule x 1,, x n,w 1,, w n, we repetedly pply the following sequence of steps, which ttempt to reduce n n-point qudrture rule for u 1,, u k to n (n 1)-point qudrture rule for u 1,, u k To perform these clcultions, we use the observtion of Subsection 22; nmely, tht the nodes nd weights of qudrture rule exct for sequence of functions stisfy system of nonliner equtions Given one of the n qudrture nodes x j, we cn use this fct to compute n (n 1) point qudrture for the functions u 1,, u k vi the dmped Guss-Newton lgorithm described in

18 18 JAMES BREMER, ZYDRUNAS GIMBUTAS, AND VLADIMIR ROKHLIN Subsection 24: we simply form the system of k nonliner equtions in n 1 unknowns described in Subsection 22 nd use s n initil guess for the Guss-Newton itertions the current qudrture nodes nd weights, excluding the chosen point x j nd its corresponding weight w j The only problem with this pproch is the selection of the qudrture node x j to eliminte There re n possible nonliner systems, ech corresponding to one of the n points which cn be omitted It is computtionlly expensive to serch for sufficiently ccurte (n 1)-point qudrture rule by solving ech of the n resulting nonliner systems of equtions vi the dmped Guss-Newton method Tht difficulty is surmounted vi the procedure described in Step 1, wherein the direction of the step for the first itertion of the dmped Guss-Newton method is computed for ech of the n possible nonliner systems The procedure is expedited by using the Shermn-Morrison-Woodbury updte formul, which leds to computtionlly fesible scheme, nd the results re used to determine the order in which to remove qudrture nodes Step 1: Rnk the remining nodes 1 For ech i = 1,, k, compute the integrls u i (x)dx using the originl nested Legendre qudrture rule formed in Step 1 Form the vector r 1 r 2 r = r k 2 Form the Jcobin mtrix u 1(x 1 )w 1 u 1(x 2 )w 2 u 1(x n )w n u 1 (x 1 ) u 1 (x 2 ) u 1 (x n ) u 2(x 1 )w 1 u 2(x 2 )w 2 u 2(x n )w n u 2 (x 1 ) u 2 (x 2 ) u 2 (x n ) J = u k (x 1)w 1 u k (x 2)w 2 u k (x n)w n u k (x 1 ) u k (x 2 ) u n (x n ) for the nonliner system (24) in Subsection (22), nd compute the inverse A = (JJ t ) 1 of the product JJ t 3 For ech node x k, first use the Shermn-Morrison-Woodbury formul (see Subsection 29) to form the mtrix A k = (J k J k t ) 1 vi two rnk-1 updtes to A, where the mtrix J k is obtined from J by deleting its k th nd (k + n) th columns Tht is, J k is obtined from J by deleting the contributions from the node x k nd its corresponding weight w k Next, compute the dmped Guss-Newton step direction x k for the nonliner system obtined by omitting the point x k ; ie, find solution x k to the lest squres problem rgmin J k x r 2, x where r is the vector of definite integrls computed bove The solution to the minimiztion problem is found by solving the norml equtions x k = (J k J t k) 1 J t kr = A k J t kr 4 For ech k = 1,, n, let η k denote the l 2 norm of the solution vector x k We will refer to the vlue η k s the significnce of the node x k 5 Renumber the nodes nd weights of the qudrture so tht {x 1,, x n } re rrnged in order of incresing η j

19 A NONLINEAR OPTIMIZATION PROCEDURE FOR GENERALIZED GAUSSIAN QUADRATURES 19 x i E E E E E E E E E E E E E E E E E E E E E E E E E E+00 w i E E E E E E E E E E E E E E E E E E E E E E E E E E-01 x i E E E E E E E E E E E E E E E E E E E E E E E E E E+00 w i E E E E E E E E E E E E E E E E E E E E E E E E E E-01 Tble 1 Qudrture formuls for the functions of the form (52) with α [ 6, 10] nd b = 20 The 26-point rule on the left ws generted with the QR vrint of the lgorithm, nd the 26-point rule on the right with the SVD vrint Both chieve full double precision ccurcy Step 2: First pss through the nodes For ech j = 1,, n, perform the following sequence of steps: 1 Form n initil guess x 1,, ˆx j,, x n nd w 1,, ŵ j,, w n for the dmped Guss-Newton method, which excludes the node x j nd its corresponding weight w j 2 Perform smll number (eg, 4) of dmped Guss-Newton itertions to form (n 1)-point qudrture rule x 1,, x n 1, w 1,, w n 1 for the functions u 1,, u k 3 Mesure the pproximtion error 2 k n 1 ɛ j = u i ( x j ) w j r i for this new qudrture rule 4 If the error ɛ j for the qudrture rule is sufficiently smll (ie, ɛ j ɛ qud ), then we ccept this (n 1)-point qudrture rule nd go to Step 4 If this sequence of steps completes without finding qudrture rule with cceptble ccurcy, then we renumber the points x 1,, x n nd weights w 1,, w n so tht nd move onto Step 3 Step 3: Second pss through the nodes ɛ 1 ɛ 2 ɛ n, We rrive t this stge only if we were unble to find stisfctory (n 1)-point qudrture rule by tking smll number of dmped Guss-Newton steps For ech j = 1,, n we perform the following sequence of steps: 1 Form n initil guess x 1,, ˆx j,, x n nd w 1,, ŵ j,, w n for the dmped Guss-Newton method, which excludes the node x j nd its corresponding weight w j 2 Use the dmped Guss-Newton lgorithm to form n (n 1)-point qudrture rule x 1,, x n 1, w 1,, w n 1 for the functions u 1,, u k In this step, the limit m on the number of itertions should be lrge (for the exmples of this pper, m = 30)

20 20 JAMES BREMER, ZYDRUNAS GIMBUTAS, AND VLADIMIR ROKHLIN x i E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00 w i E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-02 x i E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00 w i E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-02 Tble 2 Qudrture formuls for the functions of the form (52) with α [ 6, 10] nd b = 50 The 33-point rule on the left ws generted with the QR vrint of the lgorithm, nd the 33-point rule on the right with the SVD vrint Both chieve full double precision ccurcy 3 Mesure the pproximtion error k n 1 ɛ j = u i ( x j ) w j u i (x)dx for this new qudrture rule 4 If the error ɛ j for the qudrture rule is sufficiently smll (ie, ɛ j ɛ qud ), then we ccept this (n 1)-point qudrture rule nd go to Step 4 Step 4: Form the (n 1)-point qudrture If n (n 1) point qudrture rule with sufficient precision hs been found, then we ccept this rule nd repet the procedure of this stge for the newly formed (n 1) point qudrture, beginning with Step 1 Otherwise, we ccept the n-point qudrture rule which ws the input to this stge nd the lgorithm termintes Remrk 42 The process termintes when n n-point qudrture rule cnnot be reduced to n (n 1)-point qudrture rule without n uncceptble loss of precision Remrk 43 We would like to reiterte tht the qudrture rules obtined by this lgorithm re not exct for the functions φ 1,, φ m, but rther depend on the precision of the pproximtion used nd the pointwise properties of the φ j 5 Numericl Exmples We hve implemented the lgorithm of this pper for the computtion of efficient qudrture rules nd tested it on number of exmples Below we present severl of these exmples in order to demonstrte the performnce of the lgorithm It ws implemented in Fortrn 77, compiled with the Lhey-Fujitsu FORTRAN 95, nd ll timings were mesured on 20 GHz Intel Core 2 Duo processor with 2GB of RAM (no prlleliztion ws utilized) Where possible, computtions were performed in double precision (FORTRAN REAL*8) rithmetic; however, in order to chieve double precision ccurcy for qudrture rules, it is necessry to perform the computtions in 2