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[ÇóÖú] Çó¾ØÕóQR·Ö½âµÄÔ´´úÂë³ÌÐò ÒÑÓÐ2È˲ÎÓë

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hppdyx

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CODE:
function [Q,R]=qrgv(A)
% »ùÓÚGivens±ä»»£¬½«·½ÕóA·Ö½âΪA=QR£¬ÆäÖÐQΪÕý½»¾ØÕó£¬RΪÉÏÈý½ÇÕó
%
% ²ÎÊý˵Ã÷
% A£ºÐèÒª½øÐÐQR·Ö½âµÄ·½Õó
% Q£º·Ö½âµÃµ½µÄÕý½»¾ØÕó
% R£º·Ö½âµÃµ½µÄÉÏÈý½ÇÕó
%
% ʵÀý˵Ã÷
% A=[-12 3 3;3 1 -2;3 -2 7];
% [Q,R]=qr(A) % µ÷ÓÃMATLAB×Ô´øµÄQR·Ö½âº¯Êý½øÐÐÑéÖ¤
% [q,r]=qrgv(A) % µ÷Óñ¾º¯Êý½øÐÐQR·Ö½â
% q*r-A % ÑéÖ¤ A=QR
% q'*q % ÑéÖ¤qµÄÕý½»ÐÔ
% norm(q) % ÑéÖ¤qµÄ±ê×¼»¯£¬¼´¶þ·¶ÊýµÈÓÚ1
%
% ÏßÐÔ´úÊý»ù´¡ÖªÊ¶
% 1.B=P*A*inv(P)£¬³ÆAÓëBÏàËÆ£¬ÏàËÆ¾ØÕó¾ßÓÐÏàͬµÄÌØÕ÷Öµ
% 2.Q*Q'=I£¬³ÆQΪÕý½»¾ØÕó£¬Õý½»¾ØÕóµÄ³Ë»ýÈÔΪÕý½»¾ØÕó
%
% by dynamic of Matlab¼¼ÊõÂÛ̳
% see also http://www.matlabsky.com
% contact me matlabsky@gmail.com
% 2010-01-17 22:51:18
%
n=size(A,1);
R=A;
Q=eye(n);
for i=1:n-1
    for j=2:n-i+1
        x=R(i:n,i);
        rt=givens(x,1,j);
        r=blkdiag(eye(i-1),rt);
        Q=Q*r';
        R=r*R;
    end
end

function [R,y]=givens(x,i,j)
% Çó½â±ê×¼Õý½»µÄGiven±ä»»¾ØÕóR£¬Ê¹ÓÃRx=y£¬ÆäÖÐy(j)=0£¬y(i)=sqrt(x(i)^2+x(j)^2)
%
% ²ÎÊý˵Ã÷
% x£ºÐèÒª½øÐÐGivens±ä»»µÄÁÐÏòÁ¿
% i£º±äΪsqrt(x(i)^2+x(j)^2)µÄÔªËØÏ±ê
% j£º±äΪ0µÄÔªËØµÄϱê
% R£ºGivens±ä»»¾ØÕó
% y£ºGivens±ä»»½á¹û
%
% ʵÀý˵Ã÷
% x=[1 3 5 9 6]'; % ½«3µÈЧµ½9ÉÏ
% [R,y]=givens(x,4,2) % ×¢Òâ3µÄϱêΪ2£¬9µÄϱêΪ4
% R*x-y % ÑéÖ¤Rx=y
% R'*R % ÑéÖ¤Õý½»ÐÔ
% norm(R) % ÑéÖ¤±ê×¼ÐÔ£¬¾ÍÊÇ·¶ÊýΪ1
%
% ¹ØÓÚGivens±ä»»ËµÃ÷
% 1.Givens¾ØÕóÊDZê×¼Õý½»¾ØÕó£¬Ò²½ÐÆ½ÃæÐýת¾ØÕó£¬ËüÊÇͨ¹ý×ø±êÐýתµÄÔ­Àí½«ÔªËØjµÄÊýÖµµÈЧµ½ÔªËØiÉÏ
% 2.Givens±ä»»Ã¿´ÎÖ»Äܽ«Ò»¸öÔªËØ±äΪ0£¬¶øHouseholder±ä»»ÔòÒ»´Î¿ÉÒÔ½«ÈÎÒâ¸öÔªËØ±äΪ0
% 3.Givens±ä»»³£ÓÃÓÚ½«¾ØÕóA±äΪ¶Ô½ÇÕó
%
xi=x(i);
xj=x(j);
r=sqrt(xi^2+xj^2);
cost=xi/r;
sint=xj/r;
R=eye(length(x));
R(i,i)=cost;
R(i,j)=sint;
R(j,i)=-sint;
R(j,j)=cost;
y=x(:);
y([i,j])=[r,0];

²»ÒÔ·çɧ¾ªÌìÏ£¬µ«ÇóÒùµ´¶¯ÊÀÈË
2Â¥2013-12-13 13:10:24
ÒÑÔÄ   »Ø¸´´ËÂ¥   ¹Ø×¢TA ¸øTA·¢ÏûÏ¢ ËÍTAºì»¨ TAµÄ»ØÌû

fish.yfyh

Í­³æ (СÓÐÃûÆø)

¡¾´ð°¸¡¿Ó¦Öú»ØÌû

ÎÒÌṩһ¸öFortran°æ±¾µÄ´úÂë:
======================FortranÔ´Âë=============================
      subroutine qr_matrix_decomp(m,n,A, Q,R)
      ! ²ÉÓÃGram-SchmitÕý½»»¯·½·¨½øÐÐQR·Ö½â
      implicit real*8 (a-h,o-z)
      integer ::m,n
      real*8  :: A(m,n),Q(m,m),R(m,n),vec_tmp(m)

      R(1,1) = dsqrt(dot_product(A(:,1),A(:,1)))
      Q(:,1) = A(:,1)/R(1,1)

      do k=2,n
          do j=1,k-1
              R(j,k) = dot_product(Q(:,j),A(:,k))
          end do
          vec_tmp = A(:,k)
          do j=1,k-1
              vec_tmp = vec_tmp - Q(:,j)*R(j,k)
          end do
          R(k,k) = dsqrt(dot_product(vec_tmp,vec_tmp))
          Q(:,k) = vec_tmp/R(k,k)
      end do
      return
      end subroutine


      program qr_main
      implicit real*8 (a-h,o-z)
      integer :: i
      real*8  :: A(4,4),Q(4,4),R(4,4)
      
      A = reshape( (/1.0, 2.0,-1.0, 1.0, &
                      2.0, 5.0, 0.0, 3.0, &
                      1.0, 7.0, 9.0, 2.0, &
                      8.0,-1.0,-2.0, 1.0 /),(/4,4/))
      
      call qr_matrix_decomp(4,4,A, Q,R)
      
      print*, 'Q='
      do i = 1,4
          print '(4(f14.8,x))',Q(i,
      end do
      print*, 'R='
      do i = 1,4
          print '(4(f14.8,x))',R(i,
      end do
      
      stop
      end program
============================================================
±àÒëºóÔËÐУ¬½á¹ûΪ
Q=
    0.37796447    -0.05902813     0.14256649     0.91287093
    0.75592895     0.29514067     0.45621276    -0.36514837
   -0.37796447     0.88542200     0.19959308     0.18257419
    0.37796447     0.35416880    -0.85539892     0.00000000
R=
    2.64575131     5.66946710     3.02371578     3.40168026
    0.00000000     2.42015348    10.68409218    -2.18404094
    0.00000000     0.00000000     3.42159569    -0.57026595
    0.00000000     0.00000000     0.00000000     7.30296743
MatlabµÄ½á¹ûΪ£º
>> A

A =

     1     2    -1     1
     2     5     0     3
     1     7     9     2
     8    -1    -2     1

>> [Q,R]=qr(A')

Q =

    0.3780   -0.0590    0.1426    0.9129
    0.7559    0.2951    0.4562   -0.3651
   -0.3780    0.8854    0.1996    0.1826
    0.3780    0.3542   -0.8554    0.0000


R =

    2.6458    5.6695    3.0237    3.4017
    0.0000    2.4202   10.6841   -2.1840
    0.0000    0.0000    3.4216   -0.5703
    0.0000    0.0000    0.0000    7.3030
============================================================
3Â¥2013-12-13 14:13:08
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