Matrices, Moments and Quadrature, cont d

Transcription

1 Jim Lmbers MAT 285 Summer Session Lecture 2 Notes Mtrices, Moments nd Qudrture, cont d We hve described how Jcobi mtrices cn be used to compute nodes nd weights for Gussin qudrture rules for generl positive, incresing mesure α(λ), which ensures tht the Jcobi mtrix J n is not only symmetric but lso positive definite Now, we consider the cse of the specific inner product with ssocited norm f, g f(λ)g(λ) dα(λ) u T f(a)g(a)u, f α f, f 1/2 (u T f(a) 2 u) 1/2 The underlying mesure α(λ) llows us to represent the qudrtic form u T f(a)u s Riemnn- Stieltjes integrl tht cn be pproximted vi Gussin qudrture Derivtion of the Lnczos Algorithm We now exmine the computtion of the required recursion coefficients α j xq j 1, q j 1, β j p j, p j 1/2, j 1 If we define the vectors then it follows tht Furthermore, x j q j 1 (A)u, r j p j (A)u, j 1, α j x T j Ax j, β j r j 2 r j p j (A)u (A α j I)q j 1 (A)u β j 1 q j 2 (A)u (A α j I)x j β j 1 x j 1 Putting ll of these reltions together yields the lgorithm r 0 u x 0 0 for j 1, 2,, n do β j 1 r j 1 2 x j r j 1 /β j 1 1

2 α j x T j Ax j r j (A α j I)x j β j 1 x j 1 end This is precisely the Lnczos lgorithm tht is often used to pproximte extreml eigenvlues of A, nd is closely relted to the conjugte grdient method for solving symmetric positive definite systems The vectors x 1,, x n re clled the Lnczos vectors The mtrix X n [ x 1 x n stisfies the reltions X T n X n I n, X T n AX n J n The second reltion follows from the bove formul for α j, s well s the reltion β j xq j 1, q j x T j 1Ax j, j 1 The Lnczos vectors llow us to express our pproximtion of the qudrtic form u T f(a)u, tht involves function of n N N mtrix, in terms of function of n n n mtrix We hve u T f(a)u (β 0 x 1 ) T f(a)(β 0 x 1 ) β 2 0(X n e 1 ) T f(a)(x n e 1 ) p 0, p 0 e T 1 X T n f(a)x n e 1 1, 1 e T 1 f(x T n AX n )e 1 (u T u)e T 1 f(j n )e 1 u 2 2[f(J n ) 11 It follows tht if the prticulr function f is conducive to computing the (1, 1) entry of tridigonl mtrix efficiently, then there is no need to compute the nodes nd weights for Gussin qudrture explicitly Now, suppose tht J n hs the spectrl decomposition J n U n Λ n U T n, where U n is n orthogonl mtrix whose columns re the eigenvectors of J n, nd Λ n is digonl mtrix tht contins the eigenvlues Then we hve u T f(a)u u 2 2e T 1 f(u n Λ n U T n )e 1 u 2 2e T 1 U n f(λ n )Un T e 1 n u 2 2 f(t j )u 2 1j n f(t j ) w j 2

3 Thus we hve recovered the reltionship between the qudrture weights nd the eigenvectors of J n If we let n I[f f(λ) dα(λ), L G [f f(t j ) w j, then it follows from the fct tht L G [f is the exct integrl of polynomil tht interpoltes f t the nodes tht the error in this qudrture rule is I[f L G [f f (2n) (ξ) (2n)! n (λ t j ) 2 dα(λ), where ξ is n unknown point in (, b) The bove error formul cn most esily be derived by defining L G [f s the exct integrl of the Hermite interpolting polynomil of f, rther thn the Lgrnge interpolnt This is possible becuse for Gussin rule, the weights stisfy w j L j (λ) dα(λ) L j (λ) 2 dα(λ), where L j is the jth Lgrnge polynomil for the interpoltion points t 1,, t n Now, suppose tht the even-order derivtives of f re positive on (, b) Then, the error is positive, which mens tht the Gussin qudrture pproximtion is lower bound for I[f This is the cse, for exmple, if f(λ) 1 λ, or f(λ) e λt where t > 0 Prescribing Nodes It is desirble to lso obtin n upper bound for I[f To tht end, we consider modifying the Gussin rule to prescribe tht one of the nodes be λ, the smllest eigenvlue of A We ssume tht good pproximtion of cn be computed, which is typiclly possible using the Lnczos lgorithm, s it is well-suited to pproximting extreml eigenvlues of symmetric positive definite mtrix We wish to construct n ugmented Jcobi mtrix J n+1 tht hs s n eigenvlue; tht is, J n+1 I is singulr mtrix From the recurrence reltion for the orthonorml polynomils tht define the Jcobi mtrix, we obtin the system of equtions (J n+1 I )Q n+1 () 0, where the (unnormlized) eigenvector Q n+1 () is defined s Q n ws before Tht is, [Q n+1 () i q i 1 (), for i 1, 2,, n + 1 Decomposing this system of equtions into blocks yields [ Jn I e n e T n α n+1 [ Qn () q n () 3 [ 0 0

4 From the first n rows of this system, we obtin (J n I)Q n () q n ()e n, nd the lst row yields Now, if we solve the system we obtin which yields α n+1 q n 1 () q n () (J n I)δ() β 2 ne n, Q n () q n()δ(), α n+1 q n 1 () q n () β ( n q ) n()[δ() n q n () + [δ() n Thus we hve obtined Jcobi mtrix tht hs the prescribed node s n eigenvlue The error in this qudrture rule, which is known s Guss-Rdu rule, is I[f L GR [f f (2n+1) (ξ) (2n + 1)! (λ ) n (λ t j ) 2 dα(λ) Becuse the fctor λ is lwys positive on (, b), it follows tht this Guss-Rdu rule yields n upper bound on I[f if the derivtives of odd order re negtive, which is the cse for f(λ) 1 λ nd f(λ) e λt where t > 0 Similrly, if we prescribe the node λ b, then Guss-Rdu qudrture yield lower bound, which my be shrper thn the lower bound obtined from Gussin qudrture Now, suppose we wish to prescribe both λ nd λ b s nodes This yields wht is known s Guss-Lobtto qudrture rule To obtin such rule, we gin ugment the Jcobi mtrix J n to obtin new mtrix J n+1 tht hs both nd b s eigenvlues In contrst with Guss-Rdu rules, however, it is necessry to determine both nd α n+1 so tht this is the cse The recurrence reltion for orthonorml polynomils, nd the requirements tht both nd b re roots of q n+1 (x) yields the equtions 0 ( α n+1 )q n () q n 1 (), 0 (b α n+1 )q n (b) q n 1 (b), 4

5 or, in mtrix-vector form, [ qn () q n 1 () q n (b) q n 1 (b) [ αn+1 [ qn () bq n (b) The recurrence reltion lso yields the systems of equtions (J n I)Q n () q n ()e n, (J n bi)q n (b) q n (b)e n If we define δ() nd δ(b) to be the solutions of (J n I)δ() e n, (J n bi)δ(b) e n, it follows tht nd therefore Q n () q n ()δ(), Q n (b) q n (b)δ(b), q n 1 () q n ()[δ() n, q n 1 (b) q n (b)[δ(b) n It follows from [ 1 q n 1 () 1 q n() q n 1 (b) q n(b) [ αn+1 tht the recursion coefficients nd α n+1 stisfy [ [ 1 [δ()n αn+1 1 [δ(b) n [ b [ b The error in this Guss-Lobtto rule is I[f L GL [f f (2n+2) (ξ) (2n + 2)! n 1 (λ b)(λ ) (λ t j ) 2 Becuse nd b re the endpoints of the intervl of integrtion, nd (λ b)(λ ) < 0 on (, b), Guss-Lobtto qudrture yields n upper bound if the even-order derivtives of f re positive on (, b) The Cse of u v When using the bove qudrture rules to pproximte the biliner form u T f(a)v, where u v, cre must be exercised, becuse the underlying mesure α(λ) is no longer gurnteed to be positive nd incresing This cn led to negtive weights, which numericlly destbilizes the qudrture rule 5

6 To construct Gussin, Guss-Rdu or Guss-Lobtto qudrture rules, the unsymmetric Lnczos lgorithm must be used, with left initil vector u nd right initil vector v This lgorithm yields two biorthogonl sequences of Lnczos vectors such tht x j q j 1 (A)v, y j q j 1 (A)u, j 1, 2,, n, y T i x j δ ij However, becuse of the symmetry of A, we hve q j (λ) ±q j (λ), j 0, 1,, n 1, so ech underlying sequence of polymomils is still orthogonl with respect to the mesure α(λ) These sequences stisfy 3-term recurrence reltions β j q j (x) p j (x) (x α j )q j 1 (x) γ j 1 q j 2 (x), γ j q j (x) p j (x) (x α j ) q j 1 (x) β j 1 q j 2 (x), for j 1, with q 1 (x) q 1 (x) 0, p 0 (x) p 0 (x) 1, nd β 0 γ 0 p 0, p 0 It follows tht α j q j 1, xq j 1 y T j Ax j, γ j β j p j, p j r T j r j, where r j p j (A)u The fctoriztion of γ j β j into γ j nd β j is rbitrry, but is normlly chosen so tht γ j ±β j If, for ny index j 0, 1,, n 1, γ j β j, then the Jcobi mtrix J n is no longer symmetric In this cse, the eigenvlues of J n re still the Gussin qudrture nodes They re still rel nd lie within the intervl (, b), becuse they re the roots of member of sequence of orthogonl polynomils However, the weights re now the products of the first components of the left nd right eigenvectors, which re no longer gurnteed to be the sme becuse the Jcobi mtrix is not symmetric It follows tht the weights re no longer gurnteed to be positive, s they re in the cse of symmetric Jcobi mtrix As long s the unsymmetric Lnczos lgorithm does not experience serious brekdown, in which γ j β j 0, n-node Gussin qudrture is exct for polynomils of degree up to 2n 1, s in the cse of qudrtic forms However, becuse the weights cn be negtive, typiclly lterntive pproches re used to pproximte biliner forms using these qudrture rules Some such pproches re: Using perturbtion of qudrtic form The biliner form is rewritten s u T f(a)v 1 δ [ut f(a)(u + δv) u T f(a)u, where the prmeter δ is chosen sufficiently smll so tht the mesure of the perturbed qudrtic form is still positive nd incresing 6

7 Using difference of qudrtic forms: u T f(a)v 1 4 [(u + v)t f(a)(u + v) (u v) T f(a)(u v), nd the symmetric Lnczos lgorithm cn be used to pproximte both qudrtic forms A block pproch: the 2 2 mtrix integrl [ T [ u v f(a) u v is insted pproximted The block Lnczos lgorithm, due to Golub nd Underwood, is pplied to A with the initil block [ u v The result is 2n 2n block tridigonl mtrix with 2 2 blocks plying the role of recursion coefficients The eigenvlues of this mtrix serve s qudrture nodes, nd the first two components of ech eigenvector, tken in outer products with themselves, ply the role of qudrture weights One importnt drwbck of ll of these pproches is tht it is not possible to obtin upper or lower bounds for u T f(a)v, wheres this is possible when it is pproximted by single qudrture rule Guss-Rdu nd Guss-Lobtto rules cn lso be constructed in similr mnner s for qudrtic forms Detils cn be found in the pper of Golub nd Meurnt 7