Introd uction to Num erical Analysis for Eng ineers
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1 Introd uction to Num erical Analysis for Eng ineers System s of Linear Equations Cram er s Rule Gaussian Elim ination Num erical im plem entation Num erical stability: Partial Pivoting, Equilibration, Full Pivoting Multiple rig ht hand sides Com p utation count LU factorization Error Analysis for Linear System s Condition Num ber Sp ecial Matrices Iterative Method s Jacob i s m ethod Gauss-Seid el iteration Converg ence Successive Overrelaxation Method Grad ient Method s Numerical Marine Hydrodynamics 87
2 Linear System s of Equations Iterative Method s Sparse, Full-bandwidth Systems Rewrite Equations 0 x x 0 x 0 0 x 0 x 0 x x 0 0 x x 0 x Iterative, Recursive Methods Jacobi s Method Gauss-Seidels s Method Numerical Marine Hydrodynamics 87
3 Linear System s of Equations Iterative Method s Convergence Jacobi s Method Iteration Matrix form Iteration Matrix form Decompose Coefficient Matrix Convergence Analysis with / Note: NOT LU-factorization / Numerical Marine Hydrodynamics Sufficient Convergence Condition 87
4 Linear System s of Equations Iterative Method s Sufficient Convergence Condition Stop Criterion for Iteration +_ Jacobi s Method Sufficient Convergence Condition Diagonal Dominance Numerical Marine Hydrodynamics 87
5 Linear System s of Equations Tri-diag onal System s Forced Vibration of a String f(x,t) Finite Difference x i Harmonic excitation f(x,t) = f(x) cos(ωt) y(x,t) Discrete Difference Equations y i!1 + ((kh) 2! 2)y i + y i+1 = h 2 f (x i ) Matrix Form Differential Equation Boundary Conditions Tridiagonal Matrix kh < 1 or kh > 3 Symmetric, positive definite: No pivoting needed Numerical Marine Hydrodynamics 87
6 Linear System s of Equations Tri-diag onal System s Finite Difference Discrete Difference Equations y i!1 + ((kh) 2! 2)y i + y i+1 = h 2 f (x i ) Matrix Form Tridiagonal Matrix kh > 2! h > 2 k Diagonally Dominance Numerical Marine Hydrodynamics 87
7 vib_ string.m n=99; L=1.0; h=l/(n+1); k=2*pi; kh=k*h x=[h:h:l-h]'; a=zeros(n,n); f=zeros(n,1); o=1 a(1,1) =kh^2-2; a(1,2)=o; Off-diagonal values for i=2:n-1 a(i,i)=a(1,1); a(i,i-1) = o; a(i,i+1) = o; end a(n,n)=a(1,1); a(n,n-1)=o; % Hanning windowed load nf=round((n+1)/3); nw=round((n+1)/6); nw=min(min(nw,nf-1),n-nf); nw1=nf-nw; nw2=nf+nw; f(nw1:nw2) = h^2*hanning(nw2-nw1+1); figure(1) hold off subplot(2,1,1); plot(x,f,'r'); % Exact solution y=inv(a)*f; subplot(2,1,2); plot(x,y,'b'); % Iterative solution using Jacobi and Gauss-Seidel b=-a; c=zeros(n,1); for i=1:n b(i,i)=0; for j=1:n b(i,j)=b(i,j)/a(i,i); c(i)=f(i)/a(i,i); end end nj=100; xj=f; xgs=f; figure(2) nc=6 col=['r' 'g' 'b' 'c' 'm' 'y'] hold off for j=1:nj xj=b*xj+c; xgs(1)=b(1,2:n)*xgs(2:n) + c(1); for i=2:n-1 xgs(i)=b(i,1:i-1)*xgs(1:i-1) + b(i,i+1:n)*xgs(i+1:n) +c(i); end xgs(n)= b(n,1:n-1)*xgs(1:n-1) +c(n); cc=col(mod(j-1,nc)+1); subplot(2,1,1); plot(x,xj,cc); hold on; subplot(2,1,2); plot(x,xgs,cc); hold on; hold on end Numerical Marine Hydrodynamics 87
8 vib_ string.m o= 1.0,, k= 2 *pi, h=.0 1, kh< 2 Exact Solution Iterative Solutions Coefficient Matrix Not Strictly Diagonally Dominant Numerical Marine Hydrodynamics 87
9 vib_ string.m o= 1.0,, k= 2 *pi*3 1, h=.0 1, kh< 2 Exact Solution Iterative Solutions Coefficient Matrix Not Strictly Diagonally Dominant Numerical Marine Hydrodynamics 87
10 vib_ string.m o= 1.0, k= 2 *pi*3 3, h=.0 1, kh> 2 Exact Solution Iterative Solutions Coefficient Matrix Strictly Diagonally Dominant Numerical Marine Hydrodynamics 87
11 vib_ string.m o= 1.0, k= 2 *pi*5 0, h=.0 1, kh> 2 Exact Solution Iterative Solutions Coefficient Matrix Strictly Diagonally Dominant Numerical Marine Hydrodynamics 87
12 vib_ string.m o = 0.5, k= 2 *pi, h=.0 1 Exact Solution Iterative Solutions Coefficient Matrix Strictly Diagonally Dominant Numerical Marine Hydrodynamics 87
13 Iterative Method s: General Princip les Major application: sparse m atrixes, unstructured m esh Key property: Self Correcting ( avoids accum ulations of errors unlike Gauss m ethods) More robust than direct m ethods Linear system s Usually converg ence independent of initial g uess General Fo rm ula Ax e = b x i+1 = B i x i + C i b i = 0,1,2,... Num erical converg ence stop: i! n max x i " x i"1! # r i " r i"1 r i! #! #, where r i = Ax i " b Numerical Marine Hydrodynamics 87
14 Converg ence of Jacobi and Gauss-Seidel General criteria: 1. x e = B i x e + C i b = B i x e + C i Ax e = (B i + C i A)x e! B i + C i A = I 2. lim i!" B i B i#1...b 2 B 1 B 0 = 0 Special case of stationary iterations: B i = B, C i = C i = 0,1,2,... Theorem : above converg ent for any g uess if spectral radius of B is sm aller than one (!(B) < 1 ). Def inition:!(b) = max j, j =1...n where " j = eigenvalue(b n#n ) Note: B < 1 in any matrix norm! "(B) < 1 but com m only use inf inity norm due to sim plicity B! = max i=1...n n (" b ij ) j =1 Numerical Marine Hydrodynamics 8 7
15 Converg ence of Jacobi and Gauss-Seidel Jacob i: Dx + (L + U)x = b x i+1 =!D!1 (L + U)x i + D!1 b Gauss-Seid el: (D + L)x + Ux = b x i+1 =!(D + L)!1 Ux i + (D + L)!1 b Both converg e for diag onally dom inant m atrixes Gauss-Seidel converg ent for positive def inite m atrix Also Jacobi converg ent for A if A sym m etric and { D, D+ L+ U, D-L-U} are all positive def inite Numerical Marine Hydrodynamics 8 7
16 Successive Over-relaxation ( SOR) Method k x i+1 Interpolate or extrapolate the Gauss-Seidel at each substep: k =! x i+1 + (1 "!)x k k k i, where!!!x i+1!!gauss " Seidel!guess for x i+1 Matrix form at: x i+1 =!(D + " L)!1 {"U! (1! ")D}x i + "(D + " L)!1 b! = 1 " SOR # Gauss $ Seidel For A sym m etric and positive def inite: Converges for any! "(0,2) Proper value of over-relaxation param eter (! ) leads to fast converg ence, but hard to f ind:! =! opt =? Numerical Marine Hydrodynamics 8 7
17 Grad ient Method s Applicable to physically im portant m atrixes: sym m etric and positive def inite ones Construct the equivalent optim ization problem Prop ose step rule Com m on m ethods Gauss-Seid el Steep est d escent Conjug ate g radient Q(x) = 1 2 xt Ax! x T b dq(x) = Ax! b dx dq(x opt ) = 0 " x opt = x e, where!!ax e = b dx x i+1 = x i +! i+1 v i+1 Numerical Marine Hydrodynamics 8 7
18 Steep est Descent Method Move exactly in the neg ative direction of Gradient dq(x) dx = Ax! b =!(b! Ax) =!r r : residual,!!r i = b! Ax i Step rule x i+1 = x i + r T i ir i r r T i i Ar i Q( x) reduces in each step, but not as effective as conjug ate g radient m ethod Numerical Marine Hydrodynamics 8 7
19 Conjug ate Grad ient Method A symmetric & positive definite: for i! j we say v i,v j orthogonal with!respect to A, if v i T Av j = 0 Proposed in so that directions v i are generated by the orthogonalization of residuum vectors. Alg orithm Numerical Marine Hydrodynamics 8 7
20 Conjug ate Grad ient Method Accurate solution w ith n iterations, but decent accuracy with m uch few er num ber of iterations Only m atrix or vector product Possib le variations for nonsym m etric nonsing ular m atrices: g eneralized m inim al residual ( stabilized) biconjug ate g radients quasi-m inim al residual, Numerical Marine Hydrodynamics 8 7
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