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线性代数方程组的数值解法 (1) Gauss 消去法. (Demos in Matlab: airfoil in 2D). Direct A = LU. Iterative y’ = Ay. More General. Non- symmetric. Symmetric positive definite. More Robust. More Robust. Less Storage. 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 )

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slide4

Direct

A = LU

Iterative

y’ = Ay

More General

Non-

symmetric

Symmetric

positive

definite

More Robust

More Robust

Less Storage

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

slide6

Gauss(1777-1855)

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)

slide7

一个两千年前的例子

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》

slide8

一个两千年前的例子(2)

  • Basic idea: Add multiples of each row to later rows to make A upper triangular
slide13

Gauss 消去过程图示

After k=1 After k=2 After k=3 After k=n-1

slide15

用矩阵变换表达消去过程(2)

利用Gauss 变换矩阵的性质:

单位下三角形

slide16

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)
slide17

算法实现: Gauss Elimination Algorithm

  • 版本一

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

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)

slide18

Gauss Elimination Algorithm (2)

  • 上一版本

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)

  • 版本三: 第k列对角线以下为0,无需计算

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)

slide19

Gauss Elimination Algorithm (3)

  • 上一版本

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)

  • 版本四: 将乘子 m 存储在对角线以下备用

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)

slide20

Gauss Elimination Algorithm (4)

  • 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)

  • 版本五: Split loop

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)

slide21

Gauss Elimination Algorithm (5)

  • 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)

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)

slide22

What we haven’t told you

  • 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
  • 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
slide23

Matlab中的相应函数

inv

lu

\

Linpack中对应的函数

sgea.f

sgefa.f

C, Fortran, Matlab代码

slide24

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 k

max = k;

for i = k+1 : n

if abs(A(i, k)) > abs(A(max, k))

max = i;

end

end

% swap with row k

A([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 substitution

x = zeros(size(b));

for i = n : -1 : 1

j = i+1 : n;

x(i) = (b(i) - A(i, j) * x(j)) / A(i, i);

end

slide25

/* 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);

}

slide26

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;

}

}

slide27

C

C PAGE 220-223: NUMERICAL MATHEMATICS AND COMPUTING, CHENEY/KINCAID, 1985

C

C FILE: GAUSS.FOR

C

C 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)

RETURN

10 FORMAT(5X,3(F10.5,2X))

END

slide28

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