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146 lines
3.9 KiB
Matlab
146 lines
3.9 KiB
Matlab
## Algorithm 4 from P. Maponi,
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## p. 283, doi:10.1016/j.laa.2006.07.007
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clc ## Clear the screen
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% ## Define the matrix to be inverted. This is example 8 from the paper
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% ## In the future this matrix needs to be read from the function call arguments
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% A=[1,1,-1; ...
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% 1,1,0; ...
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% -1,0,-1];
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%## The modified example that gives all singular updates at some point
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%A=[1,1,1; ...
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% 1,1,0; ...
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% -1,0,-1];
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## A square uniform distributed random integer matrix with entries in [-1,1]
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do
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A=randi([-5,5],5,5);
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until (det(A)!=0) ## We need matrix non-singular
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% ## A square uniform distributed random float matrix with entries in (0,1)
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% do
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% A=rand(5);
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% until (det(A)!=0) ## We need matrix non-singular
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nCols=columns(A); ## The number of coluns of A (M in accompanying PDF)
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Id=eye(nCols);
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% d=norm(A);
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% d=max(eig(A));
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d=0.1
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A0=d*eye(nCols);
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Ar=A-A0; ## The remainder of A
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A0inv=eye(nCols);
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Ainv=zeros(nCols,nCols);
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ylk=zeros(nCols,nCols,nCols);
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breakdown=zeros(nCols,1);
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cutOff=1e-10;
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p=zeros(nCols,1);
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P=zeros(nCols);
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## Calculate the inverse of A0 and populate p-vector
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for i=1:nCols
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A0inv(i,i) = 1 / A0(i,i);
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p(i)=i;
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endfor
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A,A0,Ar,A0inv
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printf("Determinant of A is: %d\n",det(A))
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printf("Determinant of A0 is: %d\n",det(A0))
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printf("Determinant of A0inv is: %d\n",det(A0inv))
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## Calculate all the y0k in M^2 multiplications instead of M^3
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for k=1:nCols
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for i=1:nCols
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ylk(i,k,1) = A0inv(i,i) * Ar(i,k);
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% printf("ylk(%d,%d,1) = A0inv(%d,%d) * Ar(%d,%d)\n",i,k,i,i,i,k);
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endfor
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endfor
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## Calculate all the ylk from the y0k calculated previously
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for l=2:nCols
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el=Id(:,l-1);
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## Calculate break-down conditions and put in a vector
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for j=l-1:nCols
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%breakdown(j) = abs( el(j) + ylk(j,l-1,l-1) ) ## Condition from Maponi. Probably not correct!
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l
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ylk(j,j,l-1);
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breakdown(j) = abs( el(j) + ylk(j,j,l-1) )
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% printf("|el(%d) + ylk(%d,%d,%d)|\n", j, j, l-1, l-1);
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endfor
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[val, s] = max(breakdown) ## Find the index of the max value
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% printf("l = %d\ns = %d\n",l-1,s);
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breakdown=zeros(nCols,1); ## Reset the entries to zero for next l-round
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if (s!=l-1) ## Apply partial pivoting
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## Swap yl-1,k(r) and yl-1,k(s) for all k=l,l+1,...,M
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r=l-1;
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tmp=p(r);
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p(r)=p(s);
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p(s)=tmp;
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for k=l-1:nCols
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tmp=ylk(r,k,l-1);
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ylk(r,k,l-1)=ylk(s,k,l-1);
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% printf("ylk(%d,%d,%d)=ylk(%d,%d,%d)\n",r,k,l-1,s,k,l-1);
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ylk(s,k,l-1)=tmp;
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endfor
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## Modify yl-1,r and yl-1,s
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er=Id(:,r);
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es=Id(:,s);
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ylk(:,r,l-1) = ylk(:,r,l-1) + es - er;
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ylk(:,s,l-1) = ylk(:,s,l-1) + er - es;
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endif
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## Compute finally the yl,k
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for k=l:nCols
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for i=1:nCols
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ylk(i,k,l) = ylk(i,k,l-1) - ylk(l-1,k,l-1) / (1 + ylk(l-1,l-1,l-1)) * ylk(i,l-1,l-1);
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% printf("ylk(%d,%d,%d) = ylk(%d,%d,%d) - (ylk(%d,%d,%d) / (1+ylk(%d,%d,%d) * (ylk(%d,%d,%d)\n",i,k,l,i,k,l-1,l-1,k,l-1,l-1,l-1,l-1,i,l-1,l-1);
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endfor
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endfor
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endfor
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## Construct A-inverse from A0-inverse and the ylk
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Ainv=A0inv;
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for l=1:nCols
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Ainv=(Id - ylk(:,l,l) * transpose(Id(:,l)) / (1 + ylk(l,l,l))) * Ainv;
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P(:,l)=Id(:,p(l)); ## Construct permutation matrix to swap columns of Ainv
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% printf("Ainv=(Id - ylk(:,%d,%d) * transpose(Id(:,%d)) / (1 + ylk(%d,%d,%d))) * Ainv\n",l,l,l,l,l,l);
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endfor
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Ainv=transpose(P*transpose(Ainv)); ## Swap the
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## Test if the inverse found is really an inverse (does not work if values are floats)
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IdTest=A*Ainv;
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if (IdTest==eye(nCols))
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printf("\n");
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printf("Inverse of A^{-1} FOUND!\n");
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Ainv
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else
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printf("\n");
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printf("Inverse of A^{-1} NOT found yet.\nRunning another test...\n");
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for i=1:nCols
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for j=1:nCols
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if (abs(IdTest(i,j))<cutOff)
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IdTest(i,j)=0;
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elseif (abs(IdTest(i,j))-1<cutOff)
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IdTest(i,j)=1;
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endif
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endfor
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endfor
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if (IdTest==eye(nCols))
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printf("\n");
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printf("Inverse of A^{-1} FOUND!\n");
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Ainv
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else
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printf("\n");
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printf("Still not found. Giving up!\n");
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Ainv,IdTest
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endif
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endif |