Q10-91例:10-9①27x2=1, x=1!Note: thlum's pivoting elimination is lessstabln complete pivoting elimination[1109109010°10°例= x2=1, x=0①-10°-10°2XNote: the two equations arestrictly mathematicalequivalent上页下页返面
上页 下页 返回 例： − 1 1 2 10 1 1 9 x2 = 1 , x1 = 1 0 1 1 1 1 2 − 10 1 1 1 1 2 9 ✓ Note: the colum's pivoting elimination is less stable than complete pivoting elimination . 例： x2 = 1 , x1 = 0 1 1 2 1 10 10 9 9 − − 9 9 9 9 0 10 10 1 10 10 Note: the two equations are strictly mathematical equivalent
In practicalapplication,theresultsTheresults Combined with theofsolving3-Ranksystemoflinearcomplete Gauss pivoting elimination:equationsby directlycallingtheGausseliminationmethod:主#上页下页返圆
上页 下页 返回 In practical application, the results of solving 3-Rank system of linear equations by directly calling the Gauss elimination method: The results Combined with the complete Gauss pivoting elimination:
>Gauss-Jordan eliminationmethodThe main differences compared to Gauss elimination method:Don'tcompute miin every step,but turn currently principalcomponent:ak() into 1;Turn the upper and lower elements of akk(k) 's colum into O;= I x=A-bAx=bX=A-b上页下页返圆
上页 下页 返回 ➢ Gauss-Jordan elimination method The main differences compared to Gauss elimination method: Don't compute mik in every step,but turn currently principal component: akk (k) into 1; Turn the upper and lower elements of akk (k) 's colum into 0; A x b = I x A b −1 = x A b −1 =
3. operandBecause the consuming time of multiplication and Division is far bigger thanaddition and subtraction on computer,when estimating an algorithm of operation,weoften estimate times of multiplication and Division, and usually turn the maximumpower of orders of magnitude about times of multiplication and Division as the index.(n-k)((n-k)timesCGausseliminatinStepk: If a(k)compallthetimes of multiplicationand Division:2)5h31orders ofmagnitude:operand2+nn333.plication andall n-1 s.iunx,=b(" /am)Nn(n-k)(n-k+21+(n+if1)"1k=1b9-Zxi=ltimesn?h3上页5j=i+(i=n- ,., I)x, =+an下页326返圆
上页 下页 返回 3、operand Because the consuming time of multiplication and Division is far bigger than addition and subtraction on computer,when estimating an algorithm of operation ,we often estimate times of multiplication and Division, and usually turn the maximum power of orders of magnitude about times of multiplication and Division as the index . Gauss elimination method: Step k：If , factors: compute 0 ( ) k akk / ( 1, ., ) ( ) ( ) m a a i k n k kk k ik = ik = + ( , 1, ., ) ( 1) ( ) ( ) ( 1) ( ) ( ) i j k n b b m b a a m a k ik k k i k i k ik kj k ij k ij = + = − = − + + all n − 1 steps = ( ) (2) 2 (1) 1 2 1 ( ) (2) 2 (2) 22 (1) 1 (1) 12 (1) 11 . . . . . . . . . . . . n n n n nn n n b b b x x x a a a a a a ( ) ( ) / n nn n xn = bn a ( 1, .,1) ( ) 1 ( ) ( ) = − − = = + i n a b a x x i ii n j i j i ij i i i (n − k) times (n − k) 2 times (n − k) times (n − k) (n − k + 2) times n n n n k n k n k 6 5 3 2 ( )( 2) 3 2 1 1 = + − − − + − = times of multiplication and Division 1 time (n − i +1) times 2 2 1 ( 1) 1 2 1 n n n i n i + − + = + − = substitution times: all the times of multiplication and Division: operand orders of magnitude: n n n 3 1 3 2 3 + − 3 3 n
CThe complete pivoting elimination method:nCompared with gauss elimination method,more:0,butitis stable.3CThe colum pivoting elimination method:Compared with gauss elimination method,more:O,a bittime consuming,But3does not guarantee stability.CCGauss-Jordan elimination methodOperand is o(n /2), It is often used to seek inverse matrices, and is not usedfor soluting systems of equations. seeking Inverse matrix: [A] ] = [1|A-"]上页下页返圆
上页 下页 返回 The complete pivoting elimination method: Compared with gauss elimination method,more: ,but it is stable. 3 3 n O The colum pivoting elimination method: Compared with gauss elimination method,more: ,a bit time consuming,But does not guarantee stability. 3 2 n O Gauss-Jordan elimination method Operand is , It is often used to seek inverse matrices , and is not used for soluting systems of equations. seeking Inverse matrix: ( 2 ) 3 O n | | . −1 A I I A