Numerical Methods 數值方法概說. Daniel Lee. Nov. 1, 2006

Transcription

1 Numercal Methods 數值方法概說 Danel Lee Nov. 1, 2006

2 Outlnes Lnear system : drect, teratve Nonlnear system : Newton-lke Interpolatons : polys, splnes, trg polys Approxmatons (I) : orthogonal polys Approxmatons (II) : specal func Numercal quadratures : Smpson s, Gaussan Numercal ODE (IVP) : RK, PC Numercal ODE (BVP) : shootng, FD, MOL Numercal ODE (BVP) : collocaton, spectral Numercal PDE (IBVP) : selected topcs

3 Course Foc 1/3 Lnear system (I) : GE, Trd, GS_SOR Lnear system (II) : CG, BCGstab, GMRes Nonlnear sys (I) : secant_1d, damped Newton Nonlnear sys (II) : Broyden s, Inexact Newton Interpolatons (I) : polynomals, splnes Interpolatons (II) : trg_polys Approxmatons (I) : orthogonal polynomals, L2 theory Approxmatons (II) : Chebyshev polys, mnmax theory Approxmatons (III) : specal functons, Bessel s, Numercal ntegraton : Smpson s, Gaussan quadrature

4 Course Foc 2/3 Numercal ODE, IVP : rkf45, N-dm Numercal ODE, IVP : PC, predct-correct Numercal ODE, BVP : MOL, 1D/2D Numercal ODE, TPBVP : FD Numercal ODE, TPBVP : Collocaton/splne Numercal ODE, TPBVP : Collocaton/Tchebyshev Numercal ODE, TPBVP : Spectral/Tchebyshev Numercal ODE, TPBVP : Spectral/Bessel Numercal ODE, TPBVP : Shoottng

5 Course Foc 3/3 Posson 3D Heat 2D Wave 2D Incompressble NS 2D

6 Lnear (Algebrac) System Matrx types : dense or sparse Methods : drect or teratve

7 Gaussan Elmnaton wthout Pvotng A 4-by-4 example (on page 264) Phase 1 : forward elmnatons Phase 2 : back substtutons 6x1 2x2 + 2x3 + 4x4 = 16 12x1 8x2 + 6x3 + 10x4 = 26 3x1 13x2 + 9x3 + 3x4 = 19 6x1 + 4x2 + x3 18x4 = 34

8 Forward Elmnatons (1) 6x1 2x2 + 2x3 + 4x4 = 16 12x1 8x2 + 6x3 + 10x4 = 26 3x1 13x2 + 9x3 + 3x4 = 19 6x1 + 4x2 + x3 18x4 = 34 6x1 2x2 + 2x3 + 4x4 = 16 x1 4x2 + 2x3 + 2x4 = 6 12x2 + 8x3 + x4 = 27 2x2 + 3x3 14x4 = 18

9 Forward Elmnatons (2) 6x1 2x2 + 2x3 + 4x4 = 16 x1 4x2 + 2x3 + 2x4 = 6 12x2 + 8x3 + x4 = 27 2x2 + 3x3 14x4 = 18 6x1 2x2 + 2x3 + 4x4 = 16 4x2 + 2x3 + 2x4 = 6 2x3 5x4 = 9 4x3 13x4 = 21

10 Forward Elmnatons (3) 6x1 2x2 + 2x3 + 4x4 = 16 4x2 + 2x3 + 2x4 = 6 2x3 5x4 = 9 4x3 13x4 = 21 6x1 2x2 + 2x3 + 4x4 = 16 4x2 + 2x3 + 2x4 = 6 2x3 5x4 = 9 3x4 = 3

11 Back Substtutons 6x1 2x2 + 2x3 + 4x4 = 16 4x2 + 2x3 + 2x4 = 6 2x3 5x4 = 9 3x4 = 3? x = 1 3 x = 2 1 x 2 3 = x = 4 1

12 GE --- Key Ponts An upper trangular matrx s easy to solve Elementary row operatons on a matrx GE as a seq of matrx left-multplcatons

13 GE --- Notatons and Termnologes Gaussan Elmnaton ( GE ) Upper trangular matrx Pvot element, pvot equaton Multplers Resdual vector r = Ax b Error vector e = x_sol x_computed Condton Num cond(a)= A A_nv No/partal/complete pvotng

14 GE --- More Thoughts Q1: Upper Trangular vs. Lower Trangular? Q2: Elementary column operatons?

15 GE --- wth Partal Pvotng Choose the pvot elements ( and eqs ) Interchange rows accordngly Compute multplers and do elmnatons Back substtutons as before In practce, store pvots and multplers Amng at lots of RHS and same matrx Routnes : LU_Decomp(), LU_Solve()

16 GE --- HW Functon ludecomp( ), page 285 Functon lusolve( ), page 287 Man : test

17 Nonlnear Solve--- Hghlghts 1D zero fnder N-dm Newton s method : Bascs Practcal N-dm Newton s method Two-pont Newton Methods Inexact Newton method Broyden s method

18 1-dm Zero Fnders Bsecton method Newton s method, quadratc convergence Secant method, superlnear convergence Other methods More readng assgnment

19 N-dm Newton Basc Issues (Crtcal) ntal start Expensve Jacoban evaluaton Approxmate Jacoban needed n practce Effcent lnear solve

20 Practcal N-dm Newton Modfed newton Damped newton Two-grd newton Inexact newton

21 Interpolaton --- Hghlghts Algebrac polynomal nterpolaton Cubc splne nterpolaton Trgonometrc polynomal nterpolaton

22 Polynomal Interpolatons Problem formulaton Lagrange bass functon approach Newton s recursve approach Ptfall of unform nodes : The Runge func Dvded dfference : general knots Newton s approach va dvded dfference Interpolaton at Chebyshev nodes

23 Splne Interpolatons (General) splnes as pecewse polynomals (Smooth) Cubc splnes Count of defnng coeffcents Count of constrants The degree-of-freedom Varous boundary condtons Matrx forms Convergence and stablty

24 Trg-polynomal Interpolatons Even/odd degree A trgonometrc dentty The bass functon

25 Approxmatons --- Hghlghts Orthogonal polynomals : weghted - 2 Chebyshev polynomals : L - theory (Non-poly) Specal Functons : Bessel s L

26 Orthogonal Polynomals Fnte/sem-nfnte/nfnte ntervals Weghted Remann ntegrals Weghted L 2 - norms Polynomals as subspace Gram-Schmdt orthogonalzaton Typcal examples

27 Chebyshev Polynomals T ( x) = cos( nθ ), wth x = cos n A few examples : explct forms A few examples : the graphs Three-term recurson Extremes and zeros θ T, T, L, T More recursve relatons The mn-max characterzaton Varous applcatons n numercal analyss

28 Numercal Integratons 1/2 Remann ntegrals : as upper/lower lmts One-pont quadrature : md-pont rule Two-pont quadrature : trapezod rule Three-pont quadrature: Smpson s rule (Composte) Smpson s rule General Newton-Cotes formulae Gaussan type : nodes and weghts Software resources : Quadpack/netlb

29 Md-pont rule b a n 1 1 ( ) ( ) ( + ) = 0 f x dx x x f x x Trapezod rule b a n 1 1 f ( x) dx ( x + 1 x ) f ( x ) + f ( x + 1) 2 = 0 [ ]

30 (Composte) Smpson s Rule b a n / 2 h f ( x) dx f ( a) + f ( b) + 4 f a + (2 1) h 3 = 1 Gaussan type : [ ] [ ] ( n 2)/ f ( a + 2 h) = 1 b f x dx A f x + A f x + A f x a L + n n ( ) ( ) ( ) ( ) nodes weghts x, x, L, x 0 1 n A, A, L, A 0 1 n

31 Numercal Integratons 2/2 Runge-Kutta approach Lower order RK methods : RK2 RK3 method : Varous RK4 methods : RK5, RK6, RK7, RK8 : Acceleraton : halvng and combnng rkf45, rk56 : Software resources : rksute/netlb

32 RK2 where x( t + h) = x( t) + ( K + K ) K = hf ( t, x) 1 K = hf ( t + h, x + K ) 2 1

33 RK3 x( t + h) = x( t) + (2K + 3K + 4 K ) where K1 = hf ( t, x) 1 1 K2 = hf ( t + h, x + K 2 2 1) 3 3 K3 = hf ( t + h, x + K 4 4 2)

34 RK4 x( t + h) = x( t) + ( K + 2K + 2 K + K ) where K1 = hf ( t, x) 1 1 K2 = hf ( t + h, x + K 2 2 1) 1 1 K3 = hf ( t + h, x + K 2 2 2) K4 = hf ( t + h, x + K3)

35 RK5 x( t + h) = x( t) + K + K + K K where K1 = hf ( t, x) 1 1 K2 = hf ( t + h, x + K 4 4 1) K3 = hf ( t + h, x + K K 32 2) K4 = hf ( t + h, x + K K K ) K5 = hf ( t + h, x + K K2 + K K )

36 Numercal ODE --- Hghlghts Types : IVP, BVP Doman geometry : regular or not, 1D/2D/ Numercal methods : mostly sem-mplct Dfferencng : explct, mplct, 3-level Spatal dfferences : FD, FV (cell-centered) Other methods for BVP : MOL, Collocaton Package approach : rksute, odepack, DAE :

37 Numercal BVP-FD 1st dervatve, 2-pont 1st order dfference 1st dervatve, 2-pont 2nd order central 2nd dervatve, 3-pont 2nd order central 2nd dervatve, 3-pont one-sded 1st order Combnaton of two lower order schemes 2D Cartesan type fve-pont stencls General geometry

38 Numercal BVP-FV Dvergence theorem and conservaton law Concept of fnte volumes Center/Vertex/Edge based FV methods Boundary-ftted or not Combnatons of dfferent types of FV Relatons to FD and FEM

39 Numercal BVP-MOL Sem-mplct dscretzaton Contnuous n tme, dscrete n spatal var Results n system of ODE/IVP Good IVP package/solver/routne expected Easy 1D for learnng May not be effcent for 3D applcatons

40 Numercal BVP-Collocaton The Fourer case Chebyshev polys and seres Bessel functons and seres Legendre polys and seres Model examples

41 Fnte Dfference for PDE Implct tme dfferencng u ( n+ 1) ( n) u x ( 1), j u x n+, j t ( x, j ) ( ) ( ) t Explct/Implct spatal dfferencng ( α = n, n + 1) u α α u x ( n 1) + 1, j u x + 1, j x ( x, j ) ( ) ( ) 2 x

42 FD --- Unform Cartesan Grd y y y h y y 2y + y h

43 FV --- General Geometry y p = Ap y p + Aw yw + Ae ye + As ys + An yn conservaton law plus Dvergence Theorem

44 Numercal BVP MOL (1) D.E. ut = uxx, 0 x 1, t 0 I.C. B.C. t = 0, u( x,0) = f ( x) x = 0,1, u(0, t) = u ( t), u(1, t) = u ( t) 0 n+ 1 Model : (contnuous n t, dscrete n x) Spatal Dscretzaton : 0 = x0 < x1 < L < x < Lxn < x n + 1 = 1 u( x, t) u ( t)

45 MOL (2) Dscrete System : = 1,2, L, n u ( t) = u ( t), u ( t) 0 n+ 1 u+ 1( t) u 1( t) 2 x : gven Problem : Solve system of IVP n t

46 FD -Frst Dervatve Approxmatons y x y( x ) y( x ) + 1 ( ) x+ 1 x y( x ) y( x 1) y ( x ) x x 1 y( x + 1) y( x 1) y ( x ) x x + 1 1

47 FD - 2nd Dervatve Approxmatons y 2y + y y ( x ) = h y, y, y y, y, y

48 Dervaton of Central Dfference of y h h h y y hy y y y (4) + 1 = h h h y y hy y y y (4) 1 = h 5 y+ 1 y 1 = 2 hy + y + O( h ) 3 2 y+ 1 y 1 h 4 y = + y + O( h ) 2h 6

49 Dervaton of Central Dfference of y h h h y y hy y y y (4) + 1 = h h h y y hy y y y (4) 1 = y+ 1 2y + y 1 h (4) 4 y + y ( ) 2 + O h h 12

50 One-sded Dfference for y 2 3 h h ''' y+ 1 = y + hy + y + y h 8h y + 2 = y + 2hy + y + y 2 6 ''' 4h y y y hy y 6 3 ''' = y + 2y y 2 2 h = y h y 3

51 One-sded Dfference for y h h y y hy y y ''' + 1 = h 8h y y hy y y ''' + 2 = h 6h y y y y y ''' = + y 2y + y h = y + hy

52 Combnng Two FD Schemes 2 y+ 1 2y + y 1 h (4) 4 y + y ( ) 2 + O h h 12 2 y+ 2 2y + y 2 4h (4) 4 y + y ( ) 2 + O h 4h 12 y 16y + 30y 16y + y + 4h y O( h ) 2

53 Two-dmensonal FD Applcatons u j j ( u ) x ( u ) u y xx = u( x, y ) j j + u yy u u u + 1, j 1, j 2h, j+ 1, j 1 2h x u y u + u + u + u 4u + 1, j 1, j, j+ 1, j 1 j 4

54 數值習題自習手冊李天佑東海大學數學系 Nov. 1, 2006

55 線性系統 Q1 : 何謂二階段陽春高斯消去法? Q2 : 陽春型前進消去階段之可能障礙為何? 解決方法為何? Q3 : 前述階段是否必定成功? Q4 : 消去階段結束, 該矩陣主對角元素乘積之理論意義為何? Q5 : 前述判斷之風險為何? Q6 : 後退代入階段之理想假設為何? Q7 : 試述部分列篩選型高斯消去法 Q8 : 詳述高斯消去法之 LU 分解理論 Q9 : 當高斯 ( 消去 ) 遇見對稱 ( 矩陣 ),, 請補實續完 Q10 : 重新開始, 如選擇第 n 個方程式與第 n 個變數, 消去其餘方程之, 再利用第 n-1 個方程式之此類推, 是為後退消去階段, 變數, 消去其餘方程 () 之變數係數, 以所得矩陣有何特徵? 本階段有何風險? 如何解決? Q11 : ( 續前 ) 繼以前進代入, 總結為 UL 型高斯消去法程式習作 : P1 : 設計並測試副程式 ludecomp( ), lusolve( )

56 多項式插值法 Q1 : 何謂多項式插值問題? Q2 : 相關之 Lagrange 基底函數為何? Q3 : 為何 b-orthogonalty 意謂線性獨立? Q4 : 試以 Lagrange 基底表示內插解 Q5 : 牛頓之遞迴建構法為何? Q6 : 試詳述 ( 高階 ) 差分? Q7 : 內插解之差分形式為何? 程式習作 : P1 : Lagrange 形式插值解之簡易測試 P2 : 牛頓差分型插值解之軟體設計

57 三角多項式插值法 Q1 : 何謂三角多項式插值問題? Q2 : 相關之基底函數為何? 程式習作 : P1 : 軟體設計與測試

58 三次樣條函數內插法 Q1 : 何謂片段多項式? Q2 : 何謂樣條函數? Q3 : 何謂三次樣條插值問題? Q4 : 節點與結點差異為何? Q5 : 樣條插值一般是否有解? Q6 : 可能之邊界條件為何? Q7 : 實用性之考量因素為何? 程式習作 : P1 : 設計副程式 splne3_coef(), splne3_evalu8() 與測試用之主程式

59 數值積分 Q1 : 何謂中點法, 梯形法, 與辛普森法? Q2 : 試述 Newton-Cotes 之想法? Q3 : 高斯積分公式之想法為何? Q4 : 試比較前述二種構想 Q5 : 調適型辛普森法之想法為何? 程式習作 : P1 : 設計辛普森法副程式與測試用之主程式 P2 : 針對次數為 4, 6, 8 之高斯積分公式, 設計副程式與測試用之主程式

60 Lnear System Readng assgnment : norms, analyss, prob Q1 : work out by hands an example va QL P1 : routne and test trd_dag (n,a_vec,b_vec,c_vec,x_vec,rhs_vec) P2 : lu_decomp( ), lu_solve( ) and test P3 : gs_sor( ) and test P4 : cg() and test

61 Newton s Method 1/2 How s Newton s method related to Taylor expanson? How s the newton step mplemented n practce? Descrbe reasonable stoppng crteron(s) How s Newton s method related to functonal teratons? Can accelerated Newton s method accelerate? Does damped newton s method damp somethng?

62 Newton s method 2/2 Desgn approxmate Jacoban n varous ways. What can a two-grd setup help Newton s method n one way or the other? Desgn a modfed Newton s method at a double root.of a two-by-two system

63 Interpolatons --- polynomals Readng assgnment : Q1 : What s a polynomal nterpolaton problem? Q2 : What are the Lagrange bass functons? Q3 : Why b-orthogonalty mples ndependence? Q4 : Express the nterpolant n terms of Lagrange bass. Q5 : What s the recursve constructon of Newton s? Q6 : Descrbe dvded dfferences n detals? Q7 : Express the nterpolant n terms of dvded dfferences. P1 : Smple test of Lagrange form. P2 : Work out the nterpolant n newton s form

64 Interpolatons --- Trg polys Odd and/or even degree. Some trg denttes. The bass functons. P1 : Desgn the routne(s) and test.

65 Interpolatons --- Cubc splnes What s a pecewse polynomal functon? What s a (polynomal) splne functon? What s a cubc splne nterpolaton problem? Dfferences bewteen nodes and knots? What s a rule of thurm for solvablty? Some possble boundary condtons? Practcal ssues and dscussons. P1 : routnes splne3_coef(), splne3_evalu8(), test

66 Numercal Quadratures Descrbe mdpont rule, trapezod s, and smpson s What s the dea of Newton-Cotes? Why s the Newton-Cotes restrctve, compared to Gaussan quadrature? What s the dea behnd adaptve smpson s? P1 : smpson s and test P2 : gaussan quadrature of degree 4, 6, 8.

67 Numercal ODE --- IVP P1 : rk4 of general dm, and test. Predctor-Corrector method : low order case Exercse on usng rkf45 : fnd cycle problem from your ODE text.

68 Numercal ODE --- BVP P0 : Implement the Shootng method for 1D P1 : Implement the FD method for 1D/2D P2 : Implement the MOL method for 1D/2D P3 : Implement Collocaton va cubc splne

Numerical Methods 數 值 方 法 概 說. Daniel Lee. Nov. 1, 2006

Numerical Methods 數值方法概說. Daniel Lee. Nov. 1, 2006