1、第九章 线性与非线性方程组的迭代解法 /* iteration methods for the solution of linear or nonlinear systems */,Linear systems：,A x = b,Ax=bA x* =b,How to design the iteration formula?,Ax=b,x=Bx+f,L,D,U,Jacobi iteration,Matrix form,Component form,Convenient in programming,Gauss-Seidel iteration,Component form,Convenient
2、 in programming,Matrix form,comparison,Convergence of iteration,Convergence of matrix,Error vector of iteration,example,Jacobi iteration,G-S iteration,How to check if a certain iteration system converges or not?,Conditions of convergence,Not flexible to use actually,Posterior error,Prior error,Proof
3、: from Jordan standard form of B, we know,j+1,j+1,Proof:,From above, we have,Summary:Bp或(B)越小，迭代法的收敛速度越快；若事先给出误差精度，由事先误差估计式可得到迭代次数的估计实际计算中，若Bp不太接近于1，利用事后误差估计作为控制迭代停止的条件，即当 x(k)-x(k-1)p 时，迭代终止，并取x(k)作为近似解；迭代法是否收敛主要取决于迭代矩阵的性质，可用条件2，3判断迭代法是否收敛。,example,Jacobi iteration matrix B=D-1(L+U),G-S iteration m
4、atrix G= - (D+L)-1 U,order r,order n-r,A reducible, else A not reducible.,Strictly diagonally dominant,Lemma 1: A is strictly diagonally dominant A is not singular.,Proof of lemma 1 (proof by contradiction):A singularAx=0 has one nonzero solution,Contradictory to A strictly diagonally dominant,Lemma
5、 2: A is weakly diagonally dominant and not reducible A is not singular.,Proof: (by contradiction): A singularAx=0 has one nonzero solution,A is weakly diagonally dominant,Define a subset of the index,M is not empty, else,Observe the kth equation of the system,Contradictory!,Example 3,Jacobi iterati
6、on diverges,G-S iteration diverges,Strictly diagonally dominantJ,G-S iteration converge,Strictly diagonally dominantJ,G-S iteration converge,Euavalent reformation:2*(1)+(2)(1)+(3),!note: if the given linear system doesnt satisfy the convergence condition, we can modify the order of the equations or
7、do some linear combinations to get an equivalent system satisfying the convergence condition.,Example 4,Jacobi or G-S iteration can be used to solve linear systems but sometimes it converges very slowly, how to accelerate it?,SOR:successive over relaxed methodacceleration of G-S iteration,Suppose ha
8、s been found by using G-S, now we have to find,SORsuccessive over relaxed methodacceleration of G-S iteration,SOR,W1,over relaxed; W=1,G-S,Matrix form,Convergency of SOR,Proof:,Conditions of convergence,necessary condition. In order for SOR to converge, we must choose 0w2,else it diverges.,(Th7) Giv
9、en Ax=b, where A is symmetric and positive definite and 0w2 SOR converges.,Example 5,Solve the following linear system using J,G-S,and SOR(w=1.15). Iteration halts when,Solution: take,1.Jacobi iteration,2.Gauss-Seidel iteration,3.SOR,!note: w chosed well, SOR converges very fast. As for J and G-S, t
10、he convergence speed depends on the spectral radius of the iteration matrix.,!NOTE: The key problem in SOR is how to choose such a w that SOR converges fastest-the problem of how to choose the best relaxed factor w. Presently, the problem has been solved for a few special matrices. For the general c
11、ase, successive searching method is used. At the start, choose one or more different w to try SOR. Then modify w according to the speed of convergence and successively find the best w. Finally fix w and continue iteration. In theory, by iteration we can get approximate solution to any accuracy expec
12、ted. Actually, however, due to the limit of computer word length, we cant arrive at any accuracy but the machine accuracy at most. So when we use to control iteration halting, we must be careful in choosing in that machine accuracy or less results in dead loop.,iteration methods for the solution of
13、nonlinear systems,Nonlinear system:,Basic idea of iteration for the solution of 1-dim nonlinear function f(x)=0:,Condition for convergence:,f1, f2, fn-nonlinear functions,Iteration for nonlinear system:,Condition for convergence:,Sufficient(not necessary) condition for the iteration converging,Taylo
14、r series of f(x) at xk,mathematically,Newton iteration formula,Newtons method(Newton-Raphson method)root finding for nonlinear function y=f(x),y=f(x),root x*f(x*)=0,x,y,Tangent line to the curve y=f(x):y=f(x0)+f (x0)(x- x0), root,Newtons method for nonlinear systems,F(x)=0,Jacobi matrix of F(x),Cond
15、ition for convergence,F(x) is continuously differential( fi(x1, x2, xn)(i=1,2,n) and their first partial derivatives are all continuous) in an open convex set D, and there exists x* s.t. F(x* )=0 and F(x) is not singular, then there must exist a closed sphere (over linearly converges).If F(x) satisf
16、ies Lipschitz condition near x*, or there exists a positive constant k, s.t. then x(k) quadratically converges to x*.,Simplified Newtons method for nonlinear systems,Secant methodroot finding for nonlinear function y=f(x),Uses secant line between (x0,f(x0 ) and (x1,f(x1 ) to the curve y=f(x) to approximate the tangent line. root,Discrete Newtons method for nonlinear systemsgeneralization of secant method in n-d,
