Math 270A: Numerical Linear Algebra

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1 Mth 70A: Numericl Liner Algebr Instructor: Michel Holst Fll Qurter 014 Homework Assignment #3 Due Give to TA t lest few dys before finl if you wnt feedbck. Exercise 3.1. (The Bsic Liner Method for Liner Systems 1. Recll tht the trnspose A T of n n n mtrix A is defined s A T ij = A ji. Another chrcteriztion of the trnspose mtrix A T is tht it is the unique djoint opertor stisfying where (, is the usul Eucliden inner-product, (Au, v = (u, A T v, u, v R n, (u, v = n u i v i. Show tht these two chrcteriztions re equivlent.. Given n SPD (symmetric positive definite n n mtrix A, show tht it defines new inner-product (u, v A = (Au, v = i=1 n (Au i v i, u, v R n, i=1 clled the A-inner-product. I.e., show tht (u, v A is true inner-product, in tht it stisfies the three inner-product properties. 3. The A-djoint of mtrix M, denoted M, is defined s the djoint in the A-inner-product; i.e., the unique mtrix stisfying: (AMu, v = (Au, M v, u, v R n. Show tht tht n equivlent definition of M is M = A 1 M T A. 4. Now tht we hve new inner-product (u, v A, give definition of the norm it induces, which we will denote u A. Give n rgument tht it stisfies the three norm properties. 5. Use the vector norm u A to define the opertor (mtrix norm M A which it induces for ny n n mtrix M. Give n rgument tht this definition gives us ccess to the following inequlity: Mu A M A u A, u R n. 6. Use the A-norm to define the A-condition number of ny squre n n mtrix M, which we denote s κ A (M. 7. Prove the following: If M R n n is A-self-djoint for n SPD A R n n, then ρ(m = M A, κ A (M = λ mx(m λ min (M.

2 8. Let A, B R n n be SPD, nd let α R with α 0. Our gol is to solve the liner system Au = f for the unknown u R n, given f R n, nd given B A 1. Derive the following identity nd rgue tht it hs unique solution u R n tht grees with the solution to Au = f: u = (I αbau + αbf. 9. Derive the prmeterized form of the Bsic Liner Method for the problem Au = f: u n+1 = (I αbau n + αbf. 10. Use the Bnch Fixed Point Theorem to prove tht the BLM converges if α is restricted to subset of R, nd give tht subset. 11. Prove tht when the optiml vlue is chosen for α, the following error bound holds for the Bsic Liner Method: e n+1 ( n+1 1 e 0, 1 + κ A (BA where e n+1 = u u n+1, nd where κ A (BA denotes the A-condition number of BA. 1. Drive more generl error bound for the Bsic Liner Method, similr to the result bove, but one tht holds for ny vlue of α. (Hint: The norm of the error propgtor E = I BA will pper more nturlly thn the condition number of BA. 13. Assume we hve the optiml vlue for α. Assuming tht we would like to chieve the following ccurcy in our itertion fter some number of steps n: e n+1 A e 0 A < ɛ, use the pproximtion: ln ( ( ( 1/ = ln = ( 1/ [ ( ( ( ] <, to show tht we cn chieve this error tolernce with the Bsic Liner Method if n stisfies: n = O (κ A (BA ln ɛ. 14. Derive similr itertion bound result tht holds for ny α. 15. Mny types of mtrices hve O(1 non-zeros per row (the finite difference nd finite element discretiztions of ordinry nd prtil differentil equtions we looked t briefly in 70A, nd will look t more in 70C, lwys generte such mtrices. Assume tht the cost to store nd pply B is O(n. Prove tht the cost of one itertion of BLM will be O(n, where n is the dimension of the problem, i.e., A is n n n mtrix. 16. Wht is the overll complexity (in terms of n nd κ A (BA to solve the problem with the BLM to given tolernce ɛ? (Assume you hve the optiml α. If κ A (BA cn be bounded by constnt, independent of the problem size n, wht is the complexity? Is this then n optiml method?

3 Exercise 3.. (Derivtion of the Conjugte Grdient Method. In this problem, we will derive the conjugte grdient (CG method, n extremely importnt itertive method for solving mtrix equtions Au = f when A is symmetric (A = A T nd positive-definite (u T Au > 0, u 0. (We will refer to mtrices hving both properties s SPD. This problem brings together severl different topics we hve discussed in lecture in 70A, including itertive methods for mtrix equtions, mnipultions with inner-products nd norms in vector spce, together with some bsic ides from pproximtion nd orthogonl polynomils tht will be exmined more completely in 70B. 1. The Cyley-Hmilton Theorem sttes tht squre n n mtrix M stisfies its own chrcteristic eqution: P n (M = 0. Using this result, prove tht if M is lso nonsingulr, then the mtrix M 1 cn be written s mtrix polynomil of degree n 1 in M, or M 1 = Q n 1 (M.. Consider now the mtrix eqution: Au = f, where A is n n n SPD mtrix, nd u nd f re n-vectors. It is common to precondition such n eqution before ttempting numericl solution, by multiplying by n pproximte inverse opertor B A 1 nd then solving the preconditioned system: BAu = Bf. If A nd B re both SPD, under wht conditions is BA lso SPD? Show tht if A nd B re both SPD, then BA is A-SPD (symmetric nd positive in the A-inner-product. 3. Given n initil guess u 0 to the solution of BAu = Bf, we cn form the initil residuls: r 0 = f Au 0, s 0 = Br 0 = Bf BAu 0. Do simple mnipultion to show tht the solution u cn be written s: u = u 0 + Q n 1 (BAs 0, where Q( is the mtrix polynomil representing (BA 1. In other words, you hve estblished tht the solution u lies in trnslted Krylov spce: u u 0 + V n 1 (BA, s 0, where V n 1 (BA, s 0 = spn{s 0, BAs 0, (BA s 0,..., (BA n 1 s 0 }. Note tht we cn view the Krylov spces s sequence of expnding subspces V 0 (BA, s 0 V 1 (BA, s 0 V n 1 (BA, s We will now try to construct n itertive method (the CG method for finding u. The lgorithm determines the best pproximtion u k to u in subspce V k (BA, s 0 t ech step k of the lgorithm, by forming u k+1 = u k + α k p k, where p k is such tht p k V k (BA, s 0 t step k, but p k V j (BA, s 0 for j < k. In ddition, we wnt to enforce minimiztion of the error in the A-norm, e k+1 A = u u k+1 A, t step k of the lgorithm. The next itertion expnds the subspce to V k+1 (BA, s 0, finds the best pproximtion in the expnded spce, nd so on, until the exct solution in V n 1 (BA, s 0 is reched. To relize this lgorithm, let s consider how to construct the required vectors p k in n efficient wy. Let p 0 = s 0, nd consider the construction of n A-orthogonl bsis for V n 1 (BA, s 0 using the stndrd Grm-Schmidt procedure: p k+1 = BAp k k i=0 (BAp k, p i A (p i, p i A p i, k = 0,..., n. 3

4 At ech step of the procedure, we will hve generted n A-orthogonl (orthogonl in the A-inner-product bsis {p 0,..., p k } for V k (BA, s 0. Now, note tht by construction, (p k, v A = 0, v V j (BA, s 0, j < k. Using this fct, nd the fct you estblished previously tht BA is A-self-djoint, show tht the Grm-Schmidt procedure hs only three non-zero terms in the sum, nmely p k+1 = BAp k (BApk, p k A (p k, p k A p k (BApk, p k 1 A (p k 1, p k 1 A p k 1, k = 0,..., n 1. Thus, there exists n efficient three-term recursion for generting the A-orthogonl bsis for the solution spce. Note tht this three-term recursion is possible due to the fct tht we re working with orthogonl (mtrix polynomils. 5. We cn nerly write down the CG method now, by ttempting to expnd the solution in terms of our cheply generted A-orthogonl bsis. However, we need to determine how fr to move in ech conjugte direction p k t step k fter we generte p k from the recursion. As remrked erlier, we would like to enforce minimiztion of the quntity e k+1 A = u u k+1 A t step k of the itertive lgorithm. Show tht this is equivlent to enforcing (e k+1, p k A = 0. (I.e., show tht minimizing the error in finding n pproximtion from subspce is mthemticlly equivlent to enforcing tht the pproximtion error be orthogonl to tht subspce. 6. Let s ssume we hve somehow enforced (e k, p i A = 0, i < k, t the previous step of the lgorithm. We hve t our disposl p k V k (BA, s 0, nd let s tke our new pproximtion t step k + 1 s: u k+1 = u k + α k p k, for some step-length α k R in the direction p k. Thus, the error in the new pproximtion is simply: e k+1 = e k + α k p k. Show tht in order to enforce (e k+1, p k A = 0, we must choose α k to be α k = (rk, p k (p k, p k A. 7. The Conjugte Grdient Algorithm you hve derive bove is: (The Conjugte Grdient Algorithm Let u 0 H be given. r 0 = f Au 0, s 0 = Br 0, p 0 = s 0. Do k = 0, 1,... until convergence: α k = (r k, p k /(p k, p k A u k+1 = u k + α k p k r k+1 = r k α k Ap k s k+1 = Br k+1 β k+1 = (BAp k, p k A /(p k, p k A γ k+1 = (BAp k, p k 1 A /(p k 1, p k 1 A p k+1 = BAp k + β k+1 p k + γ k+1 p k 1 End do. Show tht equivlent expressions for some of the prmeters in CG re: ( α k = (r k, s k /(p k, p k A (b δ k+1 = (r k+1, s k+1 /(r k, s k (c p k+1 = s k+1 + δ k+1 p k Remrk: The CG lgorithm tht ppers in textbooks is usully formulted to employ these equivlent expressions due to the reduction in computtionl work of ech itertion. 4

5 Exercise 3.3. (Properties of the Conjugte Grdient Method. In this problem, we will estblish some simple properties of the CG method derived in Problem 1. (Although this nlysis is stndrd, you will hve difficulty finding ll of the pieces in one text. 1. Show tht the error in the CG lgorithm propgtes s: e k+1 = [I BAp k (BA]e 0, where p k P k, the spce of polynomils of degree k.. By construction, we know tht the polynomil bove is such tht e k+1 A = min p k P k [I BAp k (BA]e 0 A. Now, since BA is A-SPD, we know tht it hs rel positive eigenvlues λ j σ(ba, nd further, tht the corresponding eigenvectors v j of BA re orthonorml. Using the expnsion of the initil error estblish the inequlity: e 0 = Σ n j=1α j v j, ( [ ] e k+1 A min mx 1 λ jp k (λ j e 0 A. p k P k λ j σ(ba The polynomil which minimizes the mximum norm bove is sid to solve mini-mx problem. 3. It is well-known in pproximtion theory tht the Chebyshev polynomils T k (x = cos(k rccos x solve mini-mx problems of the bove type, in the sense tht they devite lest from zero (in the mx-norm sense in the intervl [ 1, 1], which cn be shown is due to their unique equi-oscilltion property. (These fcts cn be found in ny introductory numericl nlysis text, such s the one used in Mth 170C. If we extend the Chebyshev polynomils outside the intervl [ 1, 1] in the nturl wy (outlined in clss, it be shown tht shifted nd scled forms of the Chebyshev polynomils solve the bove mini-mx problem. In prticulr, the solution is simply ( λ T mx+λ min λ k+1 λ mx λ min 1 λp k (λ = p k+1 (λ = (. λ T mx+λ min k+1 λ mx λ min Use n obvious property of the polynomils T k+1 (x to conclude e k+1 A [ ( ] 1 λmx + λ min T k+1 e 0 A. λ mx λ min 4. Use one of the Chebyshev polynomil results given in the lst problem below to refine this inequlity to: λmx(ba e k+1 λ A 1 min(ba λmx(ba λ + 1 min(ba k+1 e 0 A. 5. Using the fct tht BA is A-self-djoint nd positive definite, show tht the bove error reduction inequlity cn be written more simply s e k+1 A ( k+1 ( κa (BA 1 e κa (BA A = κ A (BA k+1 e 0 A. 6. Assume tht we would like to chieve the following ccurcy in our itertion fter some number of steps n: e n+1 A e 0 A < ɛ. 5

6 Using the pproximtion: ln ( ( ( 1/ = ln = ( 1/ [ ( show tht we cn chieve this error tolernce if n stisfies ( n = O κ 1/ ln A (BA ɛ. ( ( ] <, 7. Mny types of mtrices hve O(1 non-zeros per row (the finite difference nd finite element discretiztions of ordinry nd prtil differentil equtions we look t next qurter lwys generte such mtrices. Therefore, the cost of one itertion of CG will be O(n, where n is the dimension of the problem, i.e., A is n n n mtrix. Wht is the overll complexity (in terms of n nd κ A (BA to solve the problem to given tolernce ɛ? If κ A (BA cn be bounded by constnt, independent of the problem size n, wht is the complexity? Is this then n optiml method? 8. As noted erlier, mny types of mtrices hve O(1 non-zeros per row (the finite difference nd finite element discretiztions of ordinry nd prtil differentil equtions we looked t briefly in 70A, nd will look t more in 70C, lwys generte such mtrices. Assume tht the cost to store nd pply B is O(n. Prove tht the cost of one itertion of CG will be O(n, where n is the dimension of the problem, i.e., A is n n n mtrix. 9. Wht is the overll complexity (in terms of n nd κ A (BA to solve the problem with CG to given tolernce ɛ? If κ A (BA cn be bounded by constnt, independent of the problem size n, wht is the complexity? Is this then n optiml method? Exercise 3.4. (Properties of the Chebyshev Polynomils. The Chebyshev polynomils re defined s: t n (x = cos(n cos 1 x, n = 0, 1,,.... Tking t 0 (x = 1, t 1 (x = x, it cn be show tht the Chebyshev polynomils re n orthogonl fmily tht cn be generted by the stndrd recursion (which holds for ny orthogonl polynomil fmily: t n+1 (x = t 1 (xt n (x t n 1 (x, n = 1,, 3,.... Prove the following extremely useful reltionships: t k (x = 1 [ ( x + k ( x 1 + x ] k x 1, x, (3.1 ( α + 1 t k > 1 ( k α + 1, α 1 α 1 α > 1. (3. (These two results re fundmentl in the convergence nlysis of the conjugte grdient method in the erlier prts of the homework. Hint: For the first result, use the fce tht cos kθ = (e ikθ + e ikθ /. The second result will follow from the first fter lot of lgebr. 6

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