1671normsofvectorsandmatrices–matrixnorms-資料下載頁(yè)
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【導(dǎo)讀】由向量范數(shù)||·||p導(dǎo)出關(guān)于矩陣A?Frobenius范數(shù)不是算子范數(shù)。我們只關(guān)心有相容性的范數(shù)，算子范數(shù)總是相容的。即使A中元素全為實(shí)數(shù)，其特征根和相應(yīng)特征向量。/*eigenvector*/仍可能是復(fù)數(shù)。將上述定義中絕對(duì)值換。成復(fù)數(shù)模均成立。||·||v使得對(duì)任意A成立。定理對(duì)任意算子范數(shù)||·||有||||)(AA??求解時(shí)，A和的誤差對(duì)解有何影響？設(shè)A精確，有誤差，得到的解為，即b?A的條件數(shù)，記為cond,cond的具體大小與||·||的取法有關(guān)，但相。cond取決于A，與解題方法無(wú)關(guān)。A可逆，R正交，則cond2=cond2=cond2
【正文】 有例子表明： GaussSeidel法收斂時(shí)， Jacobi法可能不收斂；而 Jacobi法收斂時(shí)， GaussSeidel法也可能不收斂。 2 給出了例子。 收斂性分析將在下節(jié)課討論。 167。 3 Jacobi amp。 GaussSeidel Iterative Methods Lab 07. GaussSeidel Method Use the GaussSeidel method to solve a given n n linear system with an initial approximation and a given tolerance TOL. Input There are several sets of inputs. For each set: The 1st line contains an integer 100 ? n ? 0 which is the size of a matrix. n = ?1 signals the end of file. The following n lines contain the augmented matrix in the following format: The numbers are separated by spaces and new lines. The last line of each test case contains a real number TOL, which is the tolerance for || ||? norm, and an integer N ? 0 which is the maximum number of iteration. bxA ?? ? 0)0( ?? ?xnnnnnnbaabaabaa. . .. . .. . .. . .. . .. . .. . .122211111167。 3 Jacobi amp。 GaussSeidel Iterative Methods Output /* ? represents a space */ ? Each entry of the solution is to be printed as in the C fprintf: fprintf(outfile, %\n, x )。 ? If the matrix has a zero column, print the message “ Matrix? has? a? zero? column. ?? No ? unique? solution? exists.\n” . ? If the method fails to give a solution after N iterations, print the message “ Maximum? number? of? iterations? exceeded.\n” . ? If there is an entry of that is out of the range , print the message “ No? convergence.\n” . The outputs of two test cases must be seperated by a blank line. Sample Input (? represents a space) 3 10? –1? 0? 9 –1? 10? –2? 7 0? –4? 10? 6 ? 1000 3 3? –1? 0? 1 3? 6? 0? 0 3? 3? 0? 4 ? 1000 –1 )(kx? ]2,2[ 127127?Sample Output (? represents a space) ?? ?? ?? Matrix? has? a? zero? column.?? No? unique? solution? exists. 注意：檢查每個(gè) aii時(shí)，先向下 找 最大元 ，若非 0則 交換 到對(duì)角線(xiàn)上； 否則向上找 最大元 ，若非 0則 將該行 加 到第 i 行上。
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