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1 NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING Fernando Casas Universitat Jaume I, Castellón, Spain (on sabbatical leave at DAMTP, University of Cambridge) Edinburgh, June 2004

2 ...approach suggested by Arieh Iserles NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 2

3 Outline of the talk 1. Some (well known) Lie group methods for linear problems (Fer and Magnus expansions). 2. Schemes based on triangular matrices (splitting + solvability). 3. Some methods and practical issues in their construction NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 3

4 1 Lie group methods (linear problems) Let us consider a linear matrix ODE evolving in a Lie group G Y = A(t)Y, Y (t 0 ) = Y 0 G (0) with A : [t 0, [ G g smooth enough. g: Lie algebra associated with G Examples of G: SL(n), O(n), SU(n), Sp(n), SO(n),... Y (t) Lie group G if A(t) Lie algebra g There are several schemes preserving this feature (Magnus, Fer, Cayley,...) NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 4

5 1.1 Magnus expansion For the equation Y = A(t)Y, Y (t 0 ) = I, Magnus (1954) proposed Y (t) = e Ω(t), Ω(t) = Ω k (t) (1) k=1 NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 5

6 1.1 Magnus expansion For the equation Y = A(t)Y, Y (t 0 ) = I, Magnus (1954) proposed Y (t) = e Ω(t), Ω(t) = Ω k (t) (1) k=1 with log(y (t)) satisfying Ω = d exp 1 Ω A(t) = k=0 B k k! adk Ω A(t), Ω(t 0) = 0, (2) NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 6

7 1.1 Magnus expansion (II) Here and B k are Bernoulli numbers. ad 0 Ω A = A ad k Ω A = [Ω,adk 1 Ω A] [Ω,A] ΩA AΩ NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 7

8 1.1 Magnus expansion (III) First terms in the expansion (A i A(t i )): Ω 1 (t) = t Ω 2 (t) = 1 2 Ω 3 (t) = 1 6 t 0 A(t 1 )dt 1 t t 0 dt 1 t t1 t 0 dt 2 [A 1,A 2 ] t1 t2 dt 1 dt 2 dt 3 ([A 1,[A 2,A 3 ]] + [A 3,[A 2,A 1 ]]) t 0 t 0 t 0 e Ω(t) G even if the series Ω is truncated NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 8

9 1.1 Magnus expansion (III) First terms in the expansion (A i A(t i )): Ω 1 (t) = t Ω 2 (t) = 1 2 Ω 3 (t) = 1 6 t 0 A(t 1 )dt 1 t t 0 dt 1 t t1 t 0 dt 2 [A 1,A 2 ] t1 t2 dt 1 dt 2 dt 3 ([A 1,[A 2,A 3 ]] + [A 3,[A 2,A 1 ]]) t 0 t 0 t 0 e Ω(t) G even if the series Ω is truncated * Expansion widely used in Quantum Mechanics, NMR spectroscopy, infrared divergences in QED, control theory,... NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 9

10 1.1 Magnus expansion (IV) Magnus as a numerical integration method (Iserles & Nørsett, 1997) NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 10

11 1.1 Magnus expansion (IV) Magnus as a numerical integration method (Iserles & Nørsett, 1997) Two critical factors in the computational cost of the resulting algorithms: NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 11

12 1.1 Magnus expansion (IV) Magnus as a numerical integration method (Iserles & Nørsett, 1997) Two critical factors in the computational cost of the resulting algorithms: (1) Evaluation of exp(ω) (Moler & Van Loan, Celledoni & Iserles,...) NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 12

13 1.1 Magnus expansion (IV) Magnus as a numerical integration method (Iserles & Nørsett, 1997) Two critical factors in the computational cost of the resulting algorithms: (1) Evaluation of exp(ω) (Moler & Van Loan, Celledoni & Iserles,...) (2) Number of commutators involved in the expansion NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 13

14 1.1 Magnus expansion (IV) Magnus as a numerical integration method (Iserles & Nørsett, 1997) Two critical factors in the computational cost of the resulting algorithms: (1) Evaluation of exp(ω) (Moler & Van Loan, Celledoni & Iserles,...) (2) Number of commutators involved in the expansion To reduce this number is particularly useful the concept of graded free Lie algebra (Munthe-Kaas, Owren 1999) NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 14

15 1.1 Magnus expansion (V) As a result, NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 15

16 1.1 Magnus expansion (V) As a result, * Numerical schemes based on Magnus up to order 8 have been constructed involving the minimum number of commutators in terms of quadratures and/or univariate integrals. NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 16

17 1.1 Magnus expansion (V) As a result, * Numerical schemes based on Magnus up to order 8 have been constructed involving the minimum number of commutators in terms of quadratures and/or univariate integrals. * Efficient in applications NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 17

18 1.2 Other schemes Fer expansion. Built by Francis Fer (1958). NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 18

19 1.2 Other schemes Fer expansion. Built by Francis Fer (1958). Referred erroneously in the (mathematical physics) literature (e.g., Wilcox 1967), but... NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 19

20 1.2 Other schemes Fer expansion. Built by Francis Fer (1958). Referred erroneously in the (mathematical physics) literature (e.g., Wilcox 1967), but... proposed (as an exercise!) by R. Bellman, Introduction to Matrix Analysis, 1960, page 204: NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 20

21 1.2 Other schemes Fer expansion. Built by Francis Fer (1958). Referred erroneously in the (mathematical physics) literature (e.g., Wilcox 1967), but... proposed (as an exercise!) by R. Bellman, Introduction to Matrix Analysis, 1960, page 204: The solution of dx/dt = Q(t)X, X(0) = I, can be put in the form e P e P 1 e Pn, where P = t 0 Q(s)ds, and P n = t 0 Q nds, with Q n = e P n 1 Q n 1 e P n The infinite product converges it t is sufficiently small. NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 21 0 e sp n 1 Q n 1 e sp n 1 ds (See also Mathematical Reviews , review done by R. Bellman)

22 1.2 Other schemes (II) used as an (analytic) procedure in perturbation theory in Quantum Mechanics by Klarsfeld & Oteo (1989), but... NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 22

23 1.2 Other schemes (II) used as an (analytic) procedure in perturbation theory in Quantum Mechanics by Klarsfeld & Oteo (1989), but... numerical integration method built by Iserles (1984). NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 23

24 1.2 Other schemes (II) used as an (analytic) procedure in perturbation theory in Quantum Mechanics by Klarsfeld & Oteo (1989), but... numerical integration method built by Iserles (1984). This class of methods can actually be built from Magnus. NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 24

25 1.2 Other schemes (II) used as an (analytic) procedure in perturbation theory in Quantum Mechanics by Klarsfeld & Oteo (1989), but... numerical integration method built by Iserles (1984). This class of methods can actually be built from Magnus. They require the computation of several matrix exponentials. NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 25

26 1.3 Methods based on the Cayley transform Let us suppose that Y = A(t)Y is defined in a J-orthogonal Lie group, J: constant matrix O J (n) = {A GL n (R) : A T JA = J}, NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 26

27 1.3 Methods based on the Cayley transform Let us suppose that Y = A(t)Y is defined in a J-orthogonal Lie group, O J (n) = {A GL n (R) : A T JA = J}, J: constant matrix Examples: orthogonal group (J = I), symplectic group, Lorentz group (J = diag(1, 1, 1, 1)). Solution: Y (t) = (I 12 C(t) ) 1 (I + 12 C(t) ) NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 27

28 1.3 Methods based on the Cayley transform (II) with C(t) o J (n) satisfying (Iserles 2001) dc dt = A 1 2 [C,A] 1 4 CAC, t t 0, C(t 0 ) = 0. efficient methods without matrix exponentials! NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 28

29 1.3 Methods based on the Cayley transform (II) with C(t) o J (n) satisfying (Iserles 2001) dc dt = A 1 2 [C,A] 1 4 CAC, t t 0, C(t 0 ) = 0. efficient methods without matrix exponentials! In fact, one can also combine Magnus with Padé to avoid the use of matrix exponentials in J-orthogonal groups! NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 29

30 1.3 Methods based on the Cayley transform (II) with C(t) o J (n) satisfying (Iserles 2001) dc dt = A 1 2 [C,A] 1 4 CAC, t t 0, C(t 0 ) = 0. efficient methods without matrix exponentials! In fact, one can also combine Magnus with Padé to avoid the use of matrix exponentials in J-orthogonal groups! * It is possible to construct methods which are more efficient than those based on the Cayley transform (Blanes, C., Ros 2002). NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 30

31 1.4 Summary These methods require the evaluation of one or several matrix exponentials NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 31

32 1.4 Summary These methods require the evaluation of one or several matrix exponentials They are expensive when n is very large NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 32

33 1.4 Summary These methods require the evaluation of one or several matrix exponentials They are expensive when n is very large In some cases, if the exponential is approximated by rational functions the method does not preserve the Lie group structure, in particular, when G = SL(n) NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 33

34 1.4 Summary These methods require the evaluation of one or several matrix exponentials They are expensive when n is very large In some cases, if the exponential is approximated by rational functions the method does not preserve the Lie group structure, in particular, when G = SL(n) = Another class of methods is required. NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 34

35 2 Solvability + splitting The procedure For the linear system Y = A(t)Y, Y (0) = I, we denote Y 0 Y, A 0 A and suppose that A 0 (t) = A 0+ (t) + A 0 (t), where A 0+ n is strictly upper-triangular A 0 n is weakly lower-triangular. NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 35

36 2 Solvability + splitting (II) The idea is to write the solution as a product of upper and lower triangular matrices. NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 36

37 2 Solvability + splitting (II) The idea is to write the solution as a product of upper and lower triangular matrices. More specifically, we propose the following factorization: Y 0 (t) = L 0 (t)u 0 (t)y 1 (t) such that L 0 = A 0 (t)l 0, L 0 (0) = I NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 37

38 2 Solvability + splitting (II) The idea is to write the solution as a product of upper and lower triangular matrices. More specifically, we propose the following factorization: Y 0 (t) = L 0 (t)u 0 (t)y 1 (t) such that L 0 = A 0 (t)l 0, L 0 (0) = I Observe then that L 0 (t) can be obtained by quadratures and L 0 (t) n. NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 38

39 2 Solvability + splitting (III) Now we form the matrix C 0 = L 1 0 A 0 + L 0 NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 39

40 2 Solvability + splitting (III) Now we form the matrix C 0 = L 1 0 A 0 + L 0 which can also be split as where C 0 (t) = C 0+ (t) +C 0 (t), C 0+ n is weakly upper-triangular C 0 n is strictly lower-triangular. NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 40

41 2 Solvability + splitting (IV) Next we choose U 0 as the solution of U 0 = C 0+ (t)u 0, U 0 (0) = I so that U 0 (t) can also be obtained by quadratures. NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 41

42 2 Solvability + splitting (IV) Next we choose U 0 as the solution of U 0 = C 0+ (t)u 0, U 0 (0) = I so that U 0 (t) can also be obtained by quadratures. It is easy to show that Y 1 satisfies Y 1 = A 1 (t)y 1, Y 1 (0) = I, with A 1 = U 1 0 C 0 U 0. NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 42

43 2 Solvability + splitting (V) This gives a single step of the solvable cycle, which we repeat with A 1. A 1 = A 1+ + A 1, A 1+ n, A 1 n Y 1 = L 1 U 1 Y 2 L 1 = A 1 L 1, L 1 (0) = I etc. NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 43

44 2 Solvability + splitting (VI) In this way one has the following algorithm: with (k = 0,1,2,...) Y Y 0 = L 0 U 0 L 1 U 1 L k U k Y k+1 A k = A k+ + A k, A k+ n, A k n L k = A k L k, L k (0) = I C k Lk 1 A k+ L k = C k+ +C k C k+ n, C k n U k = C k+ U k, U k (0) = I NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 44

45 2 Solvability + splitting (VII) and finally A k+1 U 1 k C k U k, Y k+1 = A k+1 Y k+1 NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 45

46 2 Solvability + splitting (VII) and finally A k+1 U 1 k C k U k, Y k+1 = A k+1 Y k+1 Usually the factorization is truncated by taking Y k+1 = I. NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 46

47 2 Solvability + splitting (VII) and finally A k+1 U 1 k C k U k, Y k+1 = A k+1 Y k+1 Usually the factorization is truncated by taking Y k+1 = I. In what follows we will analyse the main features of this procedure as a numerical integrator. NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 47

48 2.1 Order of the method Suppose that A(t) = ε(a 0 + a 1 t + a 2 t 2 + ) for some parameter ε > 0. NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 48

49 2.1 Order of the method Suppose that A(t) = ε(a 0 + a 1 t + a 2 t 2 + ) for some parameter ε > 0. Then A j = t n j ε n j ( εα 1 +t(εα 2 + ε 2 α 3 ) + O(t 2 ) ) A j+ = t m j ε m j ( εβ 1 +t(εβ 2 + ε 2 β 3 ) + O(t 2 ) ) for j = 1,2,..., so that L j (t) = I + 1 n j + 1 (tε)n j+1 α n j + 2 tn j+2 ε n j (εα 2 + ε 2 α 3 ) + U j (t) = I + 1 m j + 1 (tε)m j+1 β m j + 2 tm j+2 ε m j (εβ 2 + ε 2 β 3 ) + NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 49

50 2.1 Order of the method (II) Furthermore, n j+1 = n j + m j + 1 m j+1 = n j + 2m j + 2 j = 1,2,... NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 50

51 2.1 Order of the method (II) Furthermore, n j+1 = n j + m j + 1 m j+1 = n j + 2m j + 2 j = 1,2,... j n j m j NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 51

52 2.1 Order of the method (III) In consequence, NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 52

53 2.1 Order of the method (III) In consequence, (1) This algorithm could be useful for problems of the form Y = (B 0 + εb 1 )Y if the system Y = B 0 Y can be solved exactly. NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 53

54 2.1 Order of the method (III) In consequence, (1) This algorithm could be useful for problems of the form Y = (B 0 + εb 1 )Y if the system Y = B 0 Y can be solved exactly. (2) The order of approximation is... NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 54

55 2.1 Order of the method (IV) Y 0 L 0 U 0 is order 1 Y 0 L 0 U 0 L 1 2 Y 0 L 0 U 0 L 1 U 1 4 Y 0 L 0 U 0 L 1 U 1 L 2 7 Y 0 L 0 U 0 L 1 U 1 L 2 U 2 12 Y 0 L 0 U 0 L 1 U 1 L 2 U 2 L 3 20 Y 0 L 0 U 0 L 1 U 1 L 2 U 2 L 3 U 3 33 NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 55

56 2.1 Order of the method (IV) Y 0 L 0 U 0 is order 1 Y 0 L 0 U 0 L 1 2 Y 0 L 0 U 0 L 1 U 1 4 Y 0 L 0 U 0 L 1 U 1 L 2 7 Y 0 L 0 U 0 L 1 U 1 L 2 U 2 12 Y 0 L 0 U 0 L 1 U 1 L 2 U 2 L 3 20 Y 0 L 0 U 0 L 1 U 1 L 2 U 2 L 3 U 3 33 With only 4 solvable cycles we get order 33! NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 56

57 2.1 Order of the method (IV) Y 0 L 0 U 0 is order 1 Y 0 L 0 U 0 L 1 2 Y 0 L 0 U 0 L 1 U 1 4 Y 0 L 0 U 0 L 1 U 1 L 2 7 Y 0 L 0 U 0 L 1 U 1 L 2 U 2 12 Y 0 L 0 U 0 L 1 U 1 L 2 U 2 L 3 20 Y 0 L 0 U 0 L 1 U 1 L 2 U 2 L 3 U 3 33 With only 4 solvable cycles we get order 33!...if we can compute L k and U k up to this order... NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 57

58 2.2 Questions Several problems involved NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 58

59 2.2 Questions Several problems involved (1) Does the approximate solution evolve in the Lie group if A is in the Lie algebra, i.e., is it a Lie group method? NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 59

60 2.2 Questions Several problems involved (1) Does the approximate solution evolve in the Lie group if A is in the Lie algebra, i.e., is it a Lie group method? (2) Solve explicitly the systems L k = A k L k and U k = C k + U k NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 60

61 2.2 Questions Several problems involved (1) Does the approximate solution evolve in the Lie group if A is in the Lie algebra, i.e., is it a Lie group method? (2) Solve explicitly the systems L k = A k L k and U k = C k + U k (3) Approximate efficiently the (multiple) integrals involved. NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 61

62 3 Practical issues (1) Preservation of the Lie-group structure If A(t) sl(n), the algorithm provides by construction approximations to Y (t) in SL(n). NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 62

63 3 Practical issues (1) Preservation of the Lie-group structure If A(t) sl(n), the algorithm provides by construction approximations to Y (t) in SL(n). Proof. A k = A k+ + A k, with A k+ n, A k n. In fact A k belongs to a solvable subalgebra of sl(n). Therefore the solution of L k = A k L k, L k (0) = I L k (t) SL(n) (in fact, a solvable subgroup of). NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 63

64 3 Practical issues (1) Preservation of the Lie-group structure If A(t) sl(n), the algorithm provides by construction approximations to Y (t) in SL(n). Proof. A k = A k+ + A k, with A k+ n, A k n. In fact A k belongs to a solvable subalgebra of sl(n). Therefore the solution of L k = A k L k, L k (0) = I L k (t) SL(n) (in fact, a solvable subgroup of). Tr(A k+ ) = 0, and the trace is invariant under similarity, so that Tr(C k ) = Tr(Lk 1 A k+ L k ) = Tr(A k+ ) = 0 C k sl(n) NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 64

65 3 Practical issues (II) Next, C k = C k+ +C k, with C k+ n, C k n and U k, solution of U k = C k + U k, U k (0) = I belongs to SL(n). Finally A k+1 U 1 k C k U k sl(n) and the process is repeated. NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 65

66 3 Practical issues (II) Next, C k = C k+ +C k, with C k+ n, C k n and U k, solution of U k = C k + U k, U k (0) = I belongs to SL(n). Finally A k+1 U 1 k C k U k sl(n) and the process is repeated. Other properties (i.e., orthogonality) are preserved only up to the order of the method. NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 66

67 3 Practical issues (III) (2a) Explicit solution of L k = A k L k NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 67

68 3 Practical issues (III) (2a) Explicit solution of L k = A k L k Consider k = 0 and denote A 0 (t) = (a i j ), i, j = 1,...,n, L 0 (t) = (L i j ), j i A ii (t) t 0 a ii (t 1 )dt 1. Then the solution of L 0 = A 0 (t)l 0, L 0 (0) = I is L ii (t) = e Aii(t), i = 1,...,n (3) ( ) t i 1 L i j (t) = e A ii(t) e A ii(t 1 ) a ik (t 1 )L k j (t 1 ) dt 1 0 k= j i = 2,...,n, j = 1,...,i 1. NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 68

69 3 Practical issues (IV) (2b) Explicit solution of U k = C k + U k NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 69

70 3 Practical issues (IV) (2b) Explicit solution of U k = C k + U k Consider k = 0 and denote C 0 (t) = (c i j ), i, j = 1,...,n, U 0 (t) = (U i j ), j i C ii (t) t 0 c ii (t 1 )dt 1. Then the solution of U 0 = C 0 + (t)u 0, U 0 (0) = I is U ii (t) = e Cii(t), i = 1,...,n (4) ) U i j (t) = e C ii(t) t 0 ( j e C ii(t 1 ) c ik (t 1 )U k j (t 1 ) k=i+1 dt 1 i = 1,...,n 1, j = i + 1,...,n. NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 70

71 3 Practical issues (V) Explicit expressions for the elements of L k and U k in terms of multivariate integrals. NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 71

72 3 Practical issues (V) Explicit expressions for the elements of L k and U k in terms of multivariate integrals. They can be evaluated in sequence as follows: L ii i = 1,...,n U ii i = 1,...,n L i,i 1 i = 2,...,n U i,i+1 i = 1,...,n 1 L i,i 2 i = 3,...,n U i,i+2 i = 1,...,n 2.. L n1 U 1n NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 72

73 3 Practical issues (VI) In principle, the integrals appearing in L k and U k can be approximated by quadrature rules. NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 73

74 3 Practical issues (VI) In principle, the integrals appearing in L k and U k can be approximated by quadrature rules. To minimise the computational cost this has to be done by using the minimum number of A evaluations in each integration step. NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 74

75 3 Practical issues (VI) In principle, the integrals appearing in L k and U k can be approximated by quadrature rules. To minimise the computational cost this has to be done by using the minimum number of A evaluations in each integration step. Question: Is it possible to approximate all the nested integrals with the evaluations required to compute A ii = t 0 a ii (t 1 )dt 1, i.e., à la Magnus? NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 75

76 3 Practical issues (VI) In principle, the integrals appearing in L k and U k can be approximated by quadrature rules. To minimise the computational cost this has to be done by using the minimum number of A evaluations in each integration step. Question: Is it possible to approximate all the nested integrals with the evaluations required to compute A ii = t 0 a ii (t 1 )dt 1, i.e., à la Magnus? YES! NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 76

77 3.1 Example Illustration: method of order 4 with 2 A evaluations Step t = 0 t = h. NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 77

78 3.1 Example Illustration: method of order 4 with 2 A evaluations Step t = 0 t = h. 1- Approximate A ii (h), i = 1,...,n up to order 4 A ii (h) = h 0 a ii (t)dt = h 3( aii (0) + 4a ii (h/2) + a ii (h) ) + O(h 5 ) Ã ii (h) + O(h 5 ) and A ii (h/2), i = 1,...,n 1, up to order 3 (necessary to approximate L i j ): A ii (h/2) = h 24 (5a ii(0) + 8a ii (h/2) a ii (h)) + O(h 4 ) NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 78

79 3.1 Example (II) 2- L ii (h) = exp(ã ii (h)) + O(h 5 ) (i = 1,...,n) and L ii (h/2) = exp(ã ii (h/2)) + O(h 4 ) (i = 1,...,n 1). 3- Obtain an approximation to L i j (h), j < i, of order 4 and L i j (h/2) of order 3 with L i j (h) = e A ii(h) h 0 F i j (t)dt F i j (t) e A ii(t) i 1 a ik (t)l k j (t) k= j NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 79

80 3.1 Example (III) Then L i j (h) = h eãii(h) 3 (F i j(0) + 4F i j (h/2) + F i j (h)) + O(h 5 ) where F i j (0) = a i j (0) and F i j (h/2) and F i j (h) have to be approximated up to order h 3. The sequence of computation is (i = 2,...,n): (a) F i,i 1 (h/2) = e à ii (h/2) a i,i 1 (h/2)l i 1,i 1 (h/2) + O(h 4 ) (b) F i,i 1 (h) = e à ii (h/2) a i,i 1 (h)l i 1,i 1 (h) + O(h 5 ) (c) L i,i 1 (h), i = 2,...,n up to order 4 NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 80

81 3.1 Example (IV) (d) L i,i 1 (h/2) = eãii(h/2) h 24 (5a i,i 1(0) + 8F i,i 1 (h/2) F i,i 1 (h)) + O(h 4 ) (e) L i,i 2 (h), i = 3,...,n, up to order 4 and L i,i 2 (h/2) up to order 3...and so on. NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 81

82 3.1 Example (IV) (d) L i,i 1 (h/2) = eãii(h/2) h 24 (5a i,i 1(0) + 8F i,i 1 (h/2) F i,i 1 (h)) + O(h 4 ) (e) L i,i 2 (h), i = 3,...,n, up to order 4 and L i,i 2 (h/2) up to order 3...and so on. In this way we have L 0 (h) computed up to order O(h 5 ) and also L 0 (h/2) up to order O(h 4 ) with 2 evaluations of A(t). NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 82

83 3.1 Example (V) 3- Next we compute C 0 : C 0 (0) = A 0+ (0) error O(h 5 ) C 0 (h/2) = L 1 0 (h/2)a 0 + (h/2)l 0 (h/2) error O(h 4 ) C 0 (h) = L 1 0 (h)a 0 + (h)l 0 (h) error O(h 5 ) 4- C ii (h) = h 3( cii (0) + 4c ii (h/2) + c ii (h) ) + O(h 5 ) C ii (h/2) = h 24 (5c ii(0) + 8c ii (h/2) c ii (h)) + O(h 4 ) NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 83

84 3.1 Example (VI) 5- U i,i+1 (h), i = 1,...,n 1, up to order O(h 5 ); U i,i+1 (h/2), i = 1,...,n 1, up to order O(h 4 ); U i,i+2 (h), i = 1,...,n 2, up to order O(h 5 ); U i,i+2 (h), i = 1,...,n 2, up to order O(h 4 );... and so on. Thus we compute U 0 (h) with error O(h 5 ) and also U 0 (h/2) with error O(h 4 ). NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 84

85 3.1 Example (VII) 6- A 1 : A 1 (0) = C 0 (0) error O(h 5 ) A 1 (h/2) = U 1 0 (h/2)c 0 (h/2)u 0 (h/2) error O(h 4 ) A 1 (h) = U 1 0 (h)c 0 (h)u 0 (h) error O(h 5 )... and the process is repeated again for the second cycle NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 85

86 3.1 Example (VII) 6- A 1 : A 1 (0) = C 0 (0) error O(h 5 ) A 1 (h/2) = U 1 0 (h/2)c 0 (h/2)u 0 (h/2) error O(h 4 ) A 1 (h) = U 1 0 (h)c 0 (h)u 0 (h) error O(h 5 )... and the process is repeated again for the second cycle it is possible to construct a method of order 4 with only 2 A(t) evaluations (3 for the first step). NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 86

87 3.2 Other possibilities One could use other quadrature rules instead, for instance Gauss Legendre, but... NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 87

88 3.2 Other possibilities One could use other quadrature rules instead, for instance Gauss Legendre, but... in that case there are not enough nodes to approximate all the multivariate integrals up to the required order. NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 88

89 3.2 Other possibilities One could use other quadrature rules instead, for instance Gauss Legendre, but... in that case there are not enough nodes to approximate all the multivariate integrals up to the required order. Remark. This is not the case with Newton Cotes rules, although the error introduced may be important for higher order (negative coefficients) NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 89

90 3.2 Other possibilities One could use other quadrature rules instead, for instance Gauss Legendre, but... in that case there are not enough nodes to approximate all the multivariate integrals up to the required order. Remark. This is not the case with Newton Cotes rules, although the error introduced may be important for higher order (negative coefficients) Solution: use G L with matrix evaluations in the previous/next step. NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 90

91 3.2 Other possibilities One could use other quadrature rules instead, for instance Gauss Legendre, but... in that case there are not enough nodes to approximate all the multivariate integrals up to the required order. Remark. This is not the case with Newton Cotes rules, although the error introduced may be important for higher order (negative coefficients) Solution: use G L with matrix evaluations in the previous/next step. method of order 4 with 2 evaluations (and 1 in the next step) NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 91

92 3.3 Some methods Order 4 Y L 0 U 0 L 1 U 1 Quadratures NC / GL, 2 matrix evaluations per step Order 6 Y L 0 U 0 L 1 U 1 L 2 order 6 with a 5 points NC quadrature (4 evaluations per step) order 7 with a 7 points NC (6 evaluations) Order 12 Y L 0 U 0 L 1 U 1 L 2 U 2 with a 11 points NC (or GL involving several steps). NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 92

93 3.4 Variable step size Local extrapolation technique is trivial to implement in this setting. NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 93

94 3.4 Variable step size Local extrapolation technique is trivial to implement in this setting. For instance, Y 1 L 0 U 0 L 1 Ŷ 1 L 0 U 0 L 1 U 1 = Y 1 U 1 Then Ŷ 1 Y 1 = Y 1 U 1 Y 1 = Y 1 (U 1 I) and Ŷ 1 Y 1 can be used as a measure of the error NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 94

95 3.5 Future work Analyse the convergence of the procedure NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 95

96 3.5 Future work Analyse the convergence of the procedure Consider numerical examples in SL(n) with (very) large n NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 96

97 3.5 Future work Analyse the convergence of the procedure Consider numerical examples in SL(n) with (very) large n Highly oscillatory problems (with special quadratures) NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 97

98 3.5 Future work Analyse the convergence of the procedure Consider numerical examples in SL(n) with (very) large n Highly oscillatory problems (with special quadratures) Analyse in practice the preservation of other structures (Blanes & Moan) NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 98

99 3.5 Future work Analyse the convergence of the procedure Consider numerical examples in SL(n) with (very) large n Highly oscillatory problems (with special quadratures) Analyse in practice the preservation of other structures (Blanes & Moan) Try to generalize to nonlinear problems NEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING 99

100 The End Copyright c 2004 Fernando Casas Universitat Jaume I, Castellón, Spain Fernando.Casas@uji.es (on sabbatical leave at DAMTP, University of Cambridge) Made with the ujislides document class c Sergio Barrachina (barrachi@icc.uji.es)