=dlbx+cx,=d.+b,x,ax+c,x,+b,x3=ds+Cxagx2+bn-IXn-1+cn-rX,= dn-an-1Xn-2+bx.=d.a,xn-1将x,=P-qx,代入，得：dCX+X(b, -qa,)x, +c, = d, -azPbbd.d,-a,PiCC2记：PiqiX+X3-b,b,b,-qa2b,-qia2[tz=bz-q1a2记d,-a,Pi242P2=xXi+qx2= Pit2t2X2+42X3=P2X3+q3X4=P3依此规律，进行到底！！....Xn-1+qn-x,=Pn-1X.=Pn
n nn n n n x qx p x qx p x qx p x qx p x p − − − ⎧ + = ⎪ + = ⎪ ⎪⎪ + = ⎨ ⎪ ⎪ + = ⎪ ⎪ = ⎩ 1 12 1 2 23 2 3 34 3 11 1 . . nn nn nn n nn nn n bx cx d ax bx cx d ax bx cx d ax bx cx d ax bx d −− −− − − − ⎧ + = ⎪ ++ = ⎪ ⎪⎪ ++ = ⎨ ⎪ ⎪ + += ⎪ ⎪ + = ⎩ 11 12 1 21 22 23 2 32 33 34 3 12 11 1 1 1 %% % c d x x b b + = 1 1 1 2 1 1 , c d q p b b = = 1 1 1 1 1 1 记： c d ap x x b qa b qa − ⇔+ = − − 2 2 21 2 3 2 12 2 12 将 代入，得： x p qx 1 1 12 = − ( ) b qa x c x d a p 2 12 2 2 3 2 2 1 − + =− t b qa c d ap q p t t ⎧ = − ⎪ ⎨ − = = ⎪ ⎩ 2 2 12 2 2 21 2 2 2 2 记 ， 依此规律，进行到底!!
中国矿亚大医CHINA UNIVERSITY OF MININGANDTECHNOLOGYd,C2Pi=bh>.计算规律：方-qk-akdk-akPk-l(k=2,.,n)CkAkPk =tktkx+qix2=PiX2+q2X3=P2X3+434=P3>.消元结束后：.Xn-1+qn-iX,=Pn-1X,=Pnx,=pn>.回代求解：(X,=Pk-qxk+ (k=n-1,",1)
CHINA UNIVERSITY OF MINING AND TECHNOLOGY , c d q p b b = = 1 1 1 1 1 1 ( ,., ) k k kk k k kk k k k k t b qa c d ap q p kn t t − − ⎧ = − ⎪ ⎨ − == = ⎪ ⎩ 1 1 ， 2 n nn n n n x qx p x qx p x qx p x qx p x p − − − ⎧ + = ⎪ + = ⎪ ⎪⎪ + = ⎨ ⎪ ⎪ + = ⎪ ⎪ = ⎩ 1 12 1 2 23 2 3 34 3 11 1 . . n n ⎧ x p = ⎨ ⎩ ¾ . 计算规律： ¾ .消元结束后： ¾ .回代求解： ( , ,) k k kk x p qx k n = − =− + 1 1 1
中国矿亚大警CHINAUNIVERSITY OF MININGAND TECHNOLOGY4追赶法成立条件（P45-定理1，了解、自学）若三对角方程组系数矩阵满足：1)所有abk,c,均不为零；2)1b,/≥la,+lc/（k=1,2,,n），且其中至少有一个取不等号。则追赶法计算过程中每步的分母t,=be-Ak-1a,满足[t/=b-qk-1ak2|b/-1a>0(=2,3,..,n)因此追赶法能进行到底
CHINA UNIVERSITY OF MINING AND TECHNOLOGY 4 追赶法成立条件(P45-定理1，了解、自学) , , | || || | | | ( , , ) kkk k kk k k kk k k kk k k abc b a ck n t b qa t b qa b a k n − − ≥ + = − =− ≥ − > = 1 1 0 23 " " 若三对角方程组系数矩阵满足： 1)所有 均不为零； 2)| | | | | |( =1,2, , ),且其中至少有一个取不等号。 则追赶法计算过程中每步的分母 满足 因此追赶法能进行到底
中国矿亚大警CHINAUNIVERSITY OFMININGANDTECHNOLOGYS2矩阵分解法
CHINA UNIVERSITY OF MINING AND TECHNOLOGY §2 矩阵分解法
中国矿亚天整CHINA UNIVERSITYOF MININGANDTECHNOLOGY容易求解的方程组Ax=bA为下三角结构00bab2-x=ba21a122X2b(31X032aax,(k =2,..,n):-anann3A为上三角结构双bala12a13Ln0b2a22X2a23a2n0，则b31=X3a33a3nEagx,)/ax(k=n-1,.,l)=（b-X.j=k+1-0
CHINA UNIVERSITY OF MINING AND TECHNOLOGY 一 容易求解的方程组Ax b = 11 1 1 21 22 2 2 31 32 33 3 3 123 00 0 0 0 0 , n n n nn n n a xb aa x b aaa x b aaa a x b ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ = ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦ " " " # # #%# # # " ⎧ ⎨ ⎩ 则 x1 = b1 ( 2, , ) 1 1 x b a x k n k j k = k − ∑ kj j = " − = ( ) / ( 1, ,1 ) 1 = − ∑ = − " = + x b a x a k n n j k k k kj j kk x n b n ann = / 11 12 13 1 1 1 22 23 2 2 2 33 3 3 3 0 0 0 , 000 n n n nn n n aaa a x b aa a x b a ax b ax b ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ = ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦ " " " # # #%# # # " ⎧ ⎨ ⎩ 则 " A.为下三角结构 " A.为上三角结构