线性代数方程组的数值解法 (1) gauss 消去法
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线性代数方程组的数值解法 (1)
Gauss 消去法
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(Demos in Matlab: airfoil in 2D)
线性代数方程组的数值解法直接法： Gauss 消去法， SuperLU
迭代法：定常迭代（ Jacobi, GS, SOR, SSOR ）
Krylov 子空间方法（ CG, MINRES , GMRES, QMR, BiCGStab ）
The Landscape of Ax=b Solvers
Pivoting
LU
GMRES, QMR, …
Cholesky
Conjugate gradient
DirectA = LU
Iterativey’ = Ay
Non-symmetric
Symmetricpositivedefinite
More Robust Less Storage
More Robust
More General
刘徽 （约 220-280 ）
Gaussian elimination, which first appeared in the text Nine Chapters on the Mathematical Art written in 200 BC, was used by Gauss in his work which studied the orbit of the asteroid Pallas. Using observations of Pallas taken between 1803 and 1809, Gauss obtained a system of six linear equations in six unknowns. Gauss gave a systematic method for solving such equations which is precisely Gaussian elimination on the coefficient matrix. (The MacTutor History of Mathematics, http://www-history.mcs.st-andrews.ac.uk/history/index.html)
Gauss （ 1777-1855 ）
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zyx今有上禾三秉，中禾二秉，下禾一秉，实三十九斗；上禾二秉，中禾三秉，下禾一秉，实三十四斗；上禾一秉，中禾二秉，下禾三秉，实二十六斗。问上、中、下禾实一秉各几何？答曰：上禾一秉九斗四分斗之一。中禾一秉四斗四分斗之一。下禾一秉二斗四分斗之三。 ------- 《九章算术》
一个两千年前的例子
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Basic idea: Add multiples of each row to later rows to make A upper triangular
一个两千年前的例子 (2)
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Solving linear equations is not trivial.
Forsythe (1952)
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Gauss 消去过程图示
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用矩阵变换表达消去过程
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用矩阵变换表达消去过程 (2)
单位下三角形
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用 Gauss 消去法求解 A x=b
--- LU 分解 A = L U (cost = 2/3 n3 flops)
--- 求解 L y = b (cost = n2 flops) --- 求解 U x = y (cost = n2 flops)
版本一
for k = 1 to n-1 for i = k+1 to n m = A(i,k)/A(k,k) for j = k to n A(i,j) = A(i,j) - m * A(k,j)
算法实现： Gauss Elimination Algorithm
for k = 1 to n-1 … 对第 k 列 , 消去对角线以下元素 … ( 通过每行加上第 k 行的倍数) for i = k+1 to n … 对第 k 行以下的每一行 i for j = k to n … 第 k 行的倍数加到第 i 行 A(i,j) = A(i,j) - (A(i,k)/A(k,k)) * A(k,j)
版本二 : 在内循环中去掉常量 A(i,k)/A(k,k) 的计算
上一版本
Gauss Elimination Algorithm (2)
for k = 1 to n-1 for i = k+1 to n m = A(i,k)/A(k,k) for j = k to n A(i,j) = A(i,j) - m * A(k,j)
for k = 1 to n-1 for i = k+1 to n m = A(i,k)/A(k,k) for j = k +1 to n A(i,j) = A(i,j) - m * A(k,j)
版本三 : 第 k 列对角线以下为 0, 无需计算
上一版本
Gauss Elimination Algorithm (3)
for k = 1 to n-1 for i = k+1 to n m = A(i,k)/A(k,k) for j = k +1 to n A(i,j) = A(i,j) - m * A(k,j)
for k = 1 to n-1 for i = k+1 to n A(i,k) = A(i,k)/A(k,k) for j = k +1 to n A(i,j) = A(i,j) - A(i,k) * A(k,j)
版本四 : 将乘子 m 存储在对角线以下备用
上一版本for k = 1 to n-1 for i = k+1 to n A(i,k) = A(i,k)/A(k,k) for j = k+1 to n A(i,j) = A(i,j) - A(i,k) * A(k,j)
for k = 1 to n-1 for i = k+1 to n A(i,k) = A(i,k)/A(k,k) for i = k+1 to n for j = k+1 to n A(i,j) = A(i,j) - A(i,k) * A(k,j)
版本五 : Split loop
Gauss Elimination Algorithm (4)
上一版本
版本六 : 用矩阵运算
for k = 1 to n-1 for i = k+1 to n A(i,k) = A(i,k)/A(k,k) for i = k+1 to n for j = k+1 to n A(i,j) = A(i,j) - A(i,k) * A(k,j)
Gauss Elimination Algorithm (5)
for k = 1 to n-1 A(k+1:n,k) = A(k+1:n,k) / A(k,k) … BLAS 1 (scale a vector) A(k+1:n,k+1:n) = A(k+1:n , k+1:n ) - A(k+1:n , k) * A(k , k+1:n) … BLAS 2 (rank-1 update)
What we haven’t told you
选主元策略 ( 当主元 A(k)(k,k) 为 0 或很小时 )
向后误差分析 并行技术 块算法 Sparse LU, Band LU 最新进展 (F.Gustavson & S.Toledo, Recursive Algorithm) 还可用于矩阵求逆 , 求行列式 , 秩
定理 : 主元 A(k)(k,k) 不为 0 的充要条件是顺序主子矩阵非奇异 定理 : 分解的存在性和唯一性
Matlab 中的相应函数
Linpack 中对应的函数
inv
lu
\
C, Fortran, Matlab 代码
sgea.f
sgefa.f
function x = lsolve(A, b)% x = lsolve(A, b) returns the solution to the equation Ax = b,% where A is an n-by-b matrix and b is a column vector of% length n (or a matrix with several such columns).% Gaussian elimination with partial pivoting[n, n] = size(A);for k = 1 : n-1% find index of largest element below diagonal in column kmax = k; for i = k+1 : nif abs(A(i, k)) > abs(A(max, k))max = i;endend % swap with row kA([k max], :) = A([max k], :);b([k max]) = b([max k]); % zero out entries of A and b using pivot A(k, k)A(k+1:n,k)=A(k+1:n,k)/A(k,k);b(k+1:n)=b(k+1:n)-A(k+1:n,k)*b(k);A(k+1:n,k+1:n)=A(k+1:n,k+1:n)-A(k+1:n,k)*A(k,k+1:n);%for i = k+1 : n%alpha = A(i, k) / A(k, k);%b(i) = b(i) - alpha * b(k);%A(i, :) = A(i, :) - alpha * A(k, :);%end end% back substitutionx = zeros(size(b));for i = n : -1 : 1j = i+1 : n;x(i) = (b(i) - A(i, j) * x(j)) / A(i, i);end
/* Computer Soft/c2-1.c Gauss Elimination */#include <stdio.h>#include <stdlib.h>#include <math.h>#define TRUE 1/* a[i][j] : matrix element, a(i,j) n : order of matrix eps : machine epsilon det : determinant */void main(){int i, j, _i, _r;static n = 3;static float a_init[10][11] = {{1, 2, 3, 6}, {2, 2, 3, 7}, {3, 3, 3, 9}};static double a[10][11];void gauss();/*static int _aini = 1; */
printf( "\nComputer Soft/C2-1 Gauss Elimination \n\n" ); printf( "Augmented matrix\n" ); for( i = 1; i <= n; i++ ){ for( j = 1; j <= n+1; j++ ) {
a[i][j]=a_init[i-1][j-1]; printf( " %13.5e", a[i][j] ); } printf( "\n" ); } gauss( n, a ); printf( " Solution\n" ); printf( "-----------------------------------------\n" ); printf( " i x(i)\n" ); printf( "-----------------------------------------\n" ); for( i = 1; i <= n; i++ ) printf( " %5d %16.6e\n", i, a[i][n+1] ); printf( "-----------------------------------------\n\n" ); exit(0);}
void gauss(n, a)int n; double a[][11];{int i, j, jc, jr, k, kc, nv, pv;double det, eps, ep1, eps2, r, temp, tm, va; eps = 1.0; ep1 = 1.0 ; /* eps = Machine epsilon */ while( ep1 > 0 ){ eps = eps/2.0; ep1 = eps*0.98 + 1; ep1 = ep1 - 1; } eps = eps*2; eps2 = eps*2; printf( " Machine epsilon=%g \n", eps ); det = 1; /* Initialization of determinant */ for( i = 1; i <= (n - 1); i++ ){ pv = i; for( j = i + 1; j <= n; j++ ){ if( fabs( a[pv][i] ) < fabs( a[j][i] ) ) pv = j; } if( pv != i ){ for( jc = 1; jc <= (n + 1); jc++ ){ tm = a[i][jc]; a[i][jc] = a[pv][jc]; a[pv][jc] = tm; } det = -det; } if( a[i][i] == 0 ){ /* Singular matrix */ printf( "Matrix is singular.\n" ); exit(0); } for( jr = i + 1; jr <= n; jr++ ){ /* Elimination of below-diagonal. */ if( a[jr][i] != 0 ){ r = a[jr][i]/a[i][i]; for( kc = i + 1; kc <= (n + 1); kc++ ){ temp = a[jr][kc]; a[jr][kc] = a[jr][kc] - r*a[i][kc]; if( fabs( a[jr][kc] ) < eps2*temp ) a[jr][kc] = 0.0;/* If the result of subtraction is smaller than * 2 times machine epsilon times the original * value, it is set to zero. */ } } } } for( i = 1; i <= n; i++ ) { det = det*a[i][i]; /* Determinant is calculated. */ } if( det == 0 ){ printf( "Matrix is singular.\n" ); exit(0); } else{ /* Backward substitution starts. */ a[n][n+1] = a[n][n+1]/a[n][n]; for( nv = n - 1; nv >= 1; nv-- ){ va = a[nv][n+1]; for( k = nv + 1; k <= n; k++ ) {va = va - a[nv][k]*a[k][n+1];} a[nv][n+1] = va/a[nv][nv]; } printf( " Determinant = %g \n", det ); return; } }
CC PAGE 220-223: NUMERICAL MATHEMATICS AND COMPUTING, CHENEY/KINCAID, 1985CC FILE: GAUSS.FORCC GAUSSIAN ELIMINATION WITH SCALED PARTIAL PIVOTING (GAUSS,SOLVE,TSTGAUS)C DIMENSION A1(4,4),A2(4,4),A3(4,4),B1(4),B2(4),B3(4) DIMENSION L(4),S(4),X(4) DATA ((A1(I,J),I=1,4),J=1,4)/3.,1.,6.,0.,4.,5.,3.,0.,3.,-1.,7., A 0.,0.,0.,0.,0./ DATA (B1(I),I=1,4)/16.,-12.,102.,0./ DATA ((A2(I,J),I=1,4),J=1,4)/3.,2.,1.,0.,2.,-3.,4.,0.,-5.,1.,-1., A 0.,0.,0.,0.,0./ DATA (B2(I),I=1,4)/4.,8.,3.,0./ DATA ((A3(I,J),I=1,4),J=1,4)/1.,3.,5.,4.,-1.,2.,8.,2.,2.,1.,6., A 5.,1.,4.,3.,3./ DATA (B3(I),I=1,4)/5.,8.,10.,12./C CALL TSTGAUS(3,A1,4,L,S,B1,X) CALL TSTGAUS(3,A2,4,L,S,B2,X) CALL TSTGAUS(4,A3,4,L,S,B3,X) END SUBROUTINE TSTGAUS(N,A,IA,L,S,B,X) DIMENSION A(IA,N),B(N),X(N),S(N),L(N) PRINT 10,((A(I,J),J=1,N),I=1,N) PRINT 10,(B(I),I=1,N) CALL GAUSS(N,A,IA,L,S) CALL SOLVE(N,A,IA,L,B,X) PRINT 10,(X(I),I=1,N) RETURN10 FORMAT(5X,3(F10.5,2X)) END
SUBROUTINE GAUSS(N,A,IA,L,S) DIMENSION A(IA,N),L(N),S(N) DO 3 I = 1,N L(I) = I SMAX = 0.0 DO 2 J = 1,N SMAX = AMAX1(SMAX,ABS(A(I,J))) 2 CONTINUE S(I) = SMAX 3 CONTINUE DO 7 K = 1,N-1 RMAX = 0.0 DO 4 I = K,N R = ABS(A(L(I),K))/S(L(I)) IF(R .LE. RMAX) GO TO 4 J = I RMAX = R 4 CONTINUE LK = L(J) L(J) = L(K) L(K) = LK DO 6 I = K+1,N XMULT = A(L(I),K)/A(LK,K) DO 5 J = K+1,N A(L(I),J) = A(L(I),J) - XMULT*A(LK,J) 5 CONTINUE A(L(I),K) = XMULT 6 CONTINUE 7 CONTINUE RETURN END SUBROUTINE SOLVE(N,A,IA,L,B,X) DIMENSION A(IA,N),L(N),B(N),X(N) DO 3 K = 1,N-1 DO 2 I = K+1,N B(L(I)) = B(L(I)) - A(L(I),K)*B(L(K)) 2 CONTINUE 3 CONTINUE X(N) = B(L(N))/A(L(N),N) DO 5 I = N-1,1,-1 SUM = B(L(I)) DO 4 J = I+1,N SUM = SUM - A(L(I),J)*X(J) 4 CONTINUE X(I) = SUM/A(L(I),I) 5 CONTINUE RETURN END
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