Gaussian Elimination Thm.2.4(Elementary Row Operations).The following operations applied to the augmented matrix yield an equivalent linear system. Interchanges:The order of two rows can be changed. Scaling:Multiplying a row by a nonzero constant. Replacement:The row can be replaced by the sum of that row and a nonzero multiple of any other row;that is row,=row,-mrpX rowp. It is common to use the last operation by replacing a row with the difference of that row and a multiple of another row. 华南师范大学数学科学学院谢删玲
Gaussian Elimination ◼ Thm. 2.4(Elementary Row Operations). The following operations applied to the augmented matrix yield an equivalent linear system. • Interchanges: The order of two rows can be changed. • Scaling: Multiplying a row by a nonzero constant. • Replacement: The row can be replaced by the sum of that row and a nonzero multiple of any other row; that is : rowr =rowr -mrp×rowp . ◼ It is common to use the last operation by replacing a row with the difference of that row and a multiple of another row. 华南师范大学数学科学学院 谢骊玲
Gauss Elimination Forward Elimination-multiple of one equation is subtracted from another to eliminate unknowns Back Substitution-last equation yields unknown,substituting back into the other equations yields the rest 华南师范大学数学科学学院谢珊玲
Gauss Elimination ◼ Forward Elimination - multiple of one equation is subtracted from another to eliminate unknowns ◼ Back Substitution – last equation yields unknown, substituting back into the other equations yields the rest 华南师范大学数学科学学院 谢骊玲
Gaussian Elimination Process a+a2x2 +a3x3+..+aiwxy =b a21x1+a22x2+a23x3+…+a2Nxw=b2 a21/a11 a1x+a322+a333+…+a3Nxw=b avx1 an2x2 an3x3+...aNNxy by ax1+a2x2+a13x3+...+anxy=b 0a22x2+asx3++aiyx=b2 a31x1+a32x2+a33x3+…+a3wxv=b3 anx+an2x2 +an3x3+...+aNNxN=bv 华南师范大学数学科学学院谢删玲
Gaussian Elimination Process 华南师范大学数学科学学院 谢骊玲 21 11 a /a N N N NN N N N N N N N N a x a x a x a x b a x a x a x a x b a x a x a x a x b a x a x a x a x b + + + + = + + + + = + + + + = + + + + = 1 1 2 2 3 3 31 1 32 2 33 3 3 3 21 1 22 2 23 3 2 2 11 1 12 2 13 3 1 1 11 1 12 2 13 3 1 1 22 2 23 3 2 2 31 1 32 2 33 3 3 3 1 1 2 2 3 3 0 N N N N N N N N N NN N N a x a x a x a x b a x a x a x b a x a x a x a x b a x a x a x a x b + + + + = + + + + = + + + + = + + + + =
Gaussian Elimination Process-Cont ax+a2x2 +a3x3+...+awxy =b a31/a11 a2 +a22x2 +a23x3+..+a2wxN =b2 as /au a3x+a32x2+a33x3+..+aswxN =b3 aw+aN2x2+aw3X3+·+awxN=b时 au1+a2x2 +a3x3 +.+aiwxy =b a2x32+a33+…+awxw=b a2x2+ax3+…+avxw=b av2X2 an3x3+.+aNxy =by 华南师范大学数学科学学院谢卿玲
Gaussian Elimination Process-Cont. 华南师范大学数学科学学院 谢骊玲 31 11 a /a 1 11 a /a N N N NN N N N N N N N N a x a x a x b a x a x a x b a x a x a x b a x a x a x a x b + + + = + + + = + + + = + + + + = 2 2 3 3 32 2 33 3 3 3 22 2 23 3 2 2 11 1 12 2 13 3 1 1 N N N NN N N N N N N N N a x a x a x a x b a x a x a x a x b a x a x a x a x b a x a x a x a x b + + + + = + + + + = + + + + = + + + + = 1 1 2 2 3 3 3 1 1 3 2 2 3 3 3 3 3 2 1 1 2 2 2 2 3 3 2 2 1 1 1 1 2 2 1 3 3 1 1
Gaussian Elimination Process-Cont a+a2x2 +a3x3 +..aiwxy=b a2x3+d33+…+axxw=b d2/d2 a32X2+ag3x3+..+aiwxy =b av2X2 +aw3X3+.+anwxy b a411x1+a12x2+a13X3+…+a1wxw=b d22+a3x3+…+axxw=b a3x3+…+ayxw=bg ONNXN =bN 华南师范大学数学科学学院谢删玲
Gaussian Elimination Process-Cont. 华南师范大学数学科学学院 谢骊玲 ... ... 33 3 3 3 22 2 23 3 2 2 11 1 12 2 13 3 1 1 NN N N N N N N N N a x b a x a x b a x a x a x b a x a x a x a x b = + + = + + + = + + + + = N N NN N N N N N N N N a x a x a x b a x a x a x b a x a x a x b a x a x a x a x b + + + = + + + = + + + = + + + + = 2 2 3 3 3 2 2 3 3 3 3 3 2 2 2 2 3 3 2 2 1 1 1 1 2 2 1 3 3 1 1 2 22 / j a a