数值分析英文版(29章).ppt

Definition 6. If two type of norms‖? ‖α and‖? ‖β satisify m‖? ‖α　≤ ‖? ‖β　≤ M ‖? ‖α where m and M are positive constants, we say ‖? ‖α is equivalent to‖? ‖β. （λ1 is the spectral radius of ATA.） The following norms are often used （column sun norm） （row sum norm） Theorem 1. (1) All norms in Rn are equivalent each other. (2) All norms in Rn×n are equivalent each other. * * Theorem 2. Definition 7. If the series of vectors satisfy Definition 8. If the series of matrices satisfy We say converge to the vector denote We say converge to denote * * 2.7.３Spectral Radius Definition 9. Suppose is the eigenvalue of A, we say Is the spectral radius of A. Theorem 4. Fro any square matrix, we have Theorem 3. Proof. Suppose λi is an eigenvalue of A and xi is the corresponding eigenvector, that is, Axi= λixi * * Passing to the norm, we have |λi|‖x i‖≤‖A‖‖x i‖. Because x i ≠ 0, thus‖x i‖0，we get |λi |≤‖A‖ i=1,2,…,n。 As a result, ρ(A)=max|λi |≤‖A‖. Theorem 5. Eg. 2 Let show‖A‖1,‖A‖2,‖A‖∞,ρ(A). Solution. Obviously, ‖A‖1 =4,‖A‖∞=4. * * Theorem 6. If A is a real symmetric matrix, holds. * * 2.7.2 Condition Number of Matrix and Error Analysis In this section, we will analyze the effect of error of A and b for solving system Ax=b. １．Perturbation δb Suppose that there exists a perturbation δb of b and matrix A is accurate，we analyze the error δx . Because Ax=b，so A δx= δb，further we have δx=A-1 δb * * ‖ δx‖≤‖A-1‖ ‖ δb‖ Because ‖b‖=‖Ax‖≤‖A‖‖x‖ we have And then Similarly, x=A-1b and A δx= δb，we have ‖ δb‖≤‖A‖ ‖ δx‖ ‖x‖=‖A-1b‖≤‖A-1‖‖b‖ So, * * ２．Perturbation δA Definition 10 If A is non-singular, then ‖A‖×‖A-1‖ is called the condition number of A, denoted