1、數(shù)學(xué)實(shí)驗(yàn)題目5 相對(duì)Gauss列主元消去法摘要由一般線性方程組在使用Gauss消去法求解時(shí)，從求解過(guò)程中可以清楚地看到，若，必須施以行交換的手續(xù)，才能使消去過(guò)程繼續(xù)下去。有時(shí)既使，但其絕對(duì)值很小，由于舍入誤差的影響，消去過(guò)程也會(huì)出現(xiàn)不穩(wěn)定現(xiàn)象。因此，為使這種不穩(wěn)定現(xiàn)象發(fā)生的可能性減至最小，在施行消去過(guò)程時(shí)每一步都要選主元素，即要尋找行，使并將第行與第行交換，以使的當(dāng)前值（即的數(shù)值）遠(yuǎn)大于0。這種列主元消去法的主要步驟如下：1消元過(guò)程對(duì)，做1 選主元，記若，說(shuō)明方程組系數(shù)矩陣奇異，則停止計(jì)算，否則進(jìn)行2。2 交換（增廣矩陣）的兩行元素3 計(jì)算2回代過(guò)程對(duì)，計(jì)算前言利用Gauss列主元消去法、顯式

2、相對(duì)Gauss列主元消去法、隱式相對(duì)Gauss列主元消去法求解線性方程組 程序設(shè)計(jì)流程問(wèn)題1（1） 程序運(yùn)行如下：x = GaussSysSolve(Mat1_1,b1_1x = GaussExpSysSolve(Mat1_1,b1_1x = GaussIneSysSolve(Mat1_1,b1_1（2） 程序運(yùn)行如下：x = GaussSysSolve(Mat1_2,b1_2x = GaussExpSysSolve(Mat1_2,b1_2x = GaussIneSysSolve(Mat1_2,b1_2（3） 程序運(yùn)行如下：x = GaussSysSolve(Mat1_3,b1_3x = Ga

3、ussExpSysSolve(Mat1_3,b1_3x = GaussIneSysSolve(Mat1_3,b1_3（4） 程序運(yùn)行如下：x = GaussSysSolve(Mat1_4,b1_4x = GaussExpSysSolve(Mat1_4,b1_4x = GaussIneSysSolve(Mat1_4,b1_4問(wèn)題2（1） 程序運(yùn)行如下：= GaussSysSolve(Mat2_1,b2_1x = GaussExpSysSolve(Mat2_1,b2_1x = GaussIneSysSolve(Mat2_1,b2_1（2） 程序運(yùn)行如下：x = GaussSysSolve(Mat2

4、_2,b2_2x = GaussExpSysSolve(Mat2_2,b2_2x = GaussIneSysSolve(Mat2_2,b2_2（3） 程序運(yùn)行如下：x = GaussSysSolve(Mat2_3,b2_3x = GaussExpSysSolve(Mat2_3,b2_3x = 1 1 1x = GaussIneSysSolve(Mat2_3,b2_3（4） 程序運(yùn)行如下：x = GaussSysSolve(Mat2_4,b2_4x = 1 1 1x = GaussExpSysSolve(Mat2_4,b2_4x = GaussIneSysSolve(Mat2_4,b2_4x =

5、 1 1 1使用的函數(shù)function x = GaussSysSolve(A, b% GaussSysSolve 用Gauss消去法解線性方程組 Ax = b% Synopsis: x = GaussSysSolve(A, b% Input: A = 系數(shù)矩陣% b = 方程組右端% Output: x = 線性系統(tǒng)的解向量m,n = size(A;b = b(:; %將b變?yōu)榱邢蛄縤f m = n %A必須為方陣error(Argument matrix A must be square!;elseif m = length(b %b的長(zhǎng)度應(yīng)與A維度相同error(The dimentio

6、ns of A and b do not agree!;endAb = A b; %構(gòu)造增廣矩陣for i = 1:namax, imax = max(Ab(i:n, i; %選擇主元if amax = 0 %主元為0,矩陣奇異error(Tne Linear System is singular!;elseif i = imax+i-1 %主元行數(shù)與i不同時(shí)，交換這兩行Ab(i imax+i-1,: = Ab(imax+i-1 i, :;endfor j = i+1:n %向下消元Ab(j,: = Ab(j,: - Ab(i,: * Ab(j,i/amax;endendx = zeros(n

7、,1;x(n = Ab(n,n+1/Ab(n,n;for k = n-1:-1:1 %計(jì)算xx(k = ( Ab(k,n+1 - Ab(k,k+1:n*x(k+1:n / Ab(k,k;endfunction x = GaussExpSysSolve(A, b% GaussExpSysSolve 用顯式Gauss列主元消去法解線性方程組 Ax = b% Synopsis: x = GaussExpSysSolve(A, b% Input: A = 系數(shù)矩陣% b = 方程組右端% Output: x = 線性系統(tǒng)的解向量m,n = size(A;b = b(:; %將b變?yōu)榱邢蛄縤f m =

8、n %A必須為方陣error(Argument matrix A must be square!;elseif m = length(b %b的長(zhǎng)度應(yīng)與A維度相同error(The dimentions of A and b do not agree!;endAb = A b; %構(gòu)造增廣矩陣for i = 1:n %顯式平衡技術(shù)s = max(Ab(i,1:n;Ab(i,: = Ab(i,:/s;endfor i = 1:namax, imax = max(Ab(i:n, i; %選擇主元if amax = 0 %主元為0,矩陣奇異error(Tne Linear System is sin

9、gular!;elseif i = imax+i-1 %主元行數(shù)與i不同時(shí)，交換這兩行Ab(i imax+i-1,: = Ab(imax+i-1 i, :;endfor j = i+1:n %向下消元Ab(j,: = Ab(j,: - Ab(i,: * Ab(j,i/amax;endendx = zeros(n,1;x(n = Ab(n,n+1/Ab(n,n;for k = n-1:-1:1 %計(jì)算xx(k = ( Ab(k,n+1 - Ab(k,k+1:n*x(k+1:n / Ab(k,k;endfunction x,det = GaussIneSysSolve(A, b% GaussIne

10、SysSolve 用隱式Gauss列主元消去法解線性方程組 Ax = b% Synopsis: x = GaussIneSysSolve(A, b% Input: A = 系數(shù)矩陣% b = 方程組右端% Output: x = 線性系統(tǒng)的解向量% det = 系數(shù)矩陣行列式的值m,n = size(A;b = b(:; %將b變?yōu)榱邢蛄縤f m = n %A必須為方陣error(Argument matrix A must be square!;elseif m = length(b %b的長(zhǎng)度應(yīng)與A維度相同error(The dimentions of A and b do not agr

11、ee!;endAb = A b; %構(gòu)造增廣矩陣det = 1; %初始化系數(shù)矩陣行列式為1for i = 1:n %隱式平衡技術(shù)s(i = max(abs(Ab(i,1:n;if s(i = 0error(Tne Linear System is singular!; %系數(shù)矩陣某行全為0時(shí)，矩陣奇異end ends = s(:;for k = 1:n-1 c, kmax = max(abs(Ab(k:n, k./s(k:n; %選擇主元 if c = 0 %主元為0,矩陣奇異 det = 0; error(Tne Linear System is singular! det(A = 0;

12、elseif k = kmax+k-1 %主元行數(shù)與k不同時(shí)，交換這兩行 s(k kmax+k-1 = s(kmax+k-1 k; Ab(k kmax+k-1,: = Ab(kmax+k-1 k, :; det = -det; endfor j = k+1:n %向下消元 Ab(j,: = Ab(j,: - Ab(k,: * Ab(j,k/Ab(k,k; enddet = Ab(k,k*det; endif Ab(n,n = 0 det = 0; error(Tne Linear System is singular!; %最后一行唯一非0元素為0時(shí)，矩陣奇異 endx = zeros(n,1;x(n = Ab(n,n+1/Ab(n,n;for k = n-1:-1:1 %計(jì)算xx(k = ( Ab(k,n+1 - Ab(k,k+1:n*x(k+1:n / Ab(k,k;enddet = Ab(n,n*det;思考題（1） 在各主元不是非常小的時(shí)候，三種方法結(jié)果一致（2） 隱式平