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Modifying Pivot Elements in Gaussian Eliminatiom

Modifying Pivot Elements in Gaussian Eliminatiom. Name ： YI-JHOU LIN Life-time Distinguished Professor ： Jeng-Tzong Chen. outline. LU decomposition: Modifying Pivot Elements 高斯消去法流程圖 數值計算的問題. Introduction. LU decomposition:. Modifying Pivot Elements. Start. Input A(N,N+1).

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Modifying Pivot Elements in Gaussian Eliminatiom

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  1. Modifying Pivot Elements in Gaussian Eliminatiom Name：YI-JHOU LIN Life-time Distinguished Professor：Jeng-Tzong Chen

  2. outline • LU decomposition: • Modifying Pivot Elements • 高斯消去法流程圖 • 數值計算的問題

  3. Introduction LU decomposition:

  4. Modifying Pivot Elements

  5. Start Input A(N,N+1) Final Output Output: Singular Matrix STOP DO I=1,N YES YES A(I,I)=0 I=N? NO NO NO Exist K>I such That A(K,I)≠0? DO J≠I DO K=1,N+1 YES Exchange I-th and K-th Lines A(J,K)=A(J,K)-A(I,K)/A(I,I) 高斯消去法流程圖

  6. 當aii=0 時 • 由於在消去的過程中，我們會用到 ajk – aik/aii，要是aii = 0 時，該如何處置？ • 只要找到第 k 列的 aki不為零，與第 i 列對調，即可得到新的不為零的 aii • 但在有限精度下可能產生的問題: • 理論上高斯消去法可以精確地解出任何聯立線性方程式之解，然而電腦的精確度有限，在某些情況下可能會發生問題 • 由於在對第 k 列做第 i 列高斯消去法時必須用到mki=aki/aii，當軸元素（pivot element, 即 aii）為零而造成發散的情形，我們已經以列交換的方式處理了 • 然而當 aii 雖然不為零，但是非常小時，小到接近電腦精確度時，mki會變得非常大，這時候可能也會有誤差發生

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