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94 lines
2.4 KiB
Matlab
Executable File
94 lines
2.4 KiB
Matlab
Executable File
format long
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% Choose the maximum number of iterations of the Inverse Power Method
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% that will be performed.
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k = 20;
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% Choose the eigenvalues
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disp( 'input a vector of eigenvalues. e.g.: [ 4; 3; 2; 1 ] ' );
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eigs = input('');
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n = size( eigs, 1 );
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% Make sure that the last entry is equal to smallest in absolute value
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[ eig_min ] = min( abs( eigs( 1:n-1 ) ) );
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if eig_min < abs( eigs( n ) )
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abort( 'The last entry should be equal to smallest in absolute value.' );
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end
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[ A, V ] = CreateMatrixForEigenvalueProblem( eigs );
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disp( 'Matrix A:' )
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disp( A );
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% Print V
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disp( 'Matrix of eigenvectors:' );
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disp( V );
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% Print eigenvalues
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disp( 'Eigenvalues:' );
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disp( eigs );
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% Ask for a shift
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sigma = input( 'enter a shift to use: (a number close to the smallest eigenvalue) ' );
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% Create a random initial vector x, and normalize it to have unit length
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x = rand( n, 1 );
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x = x / norm( x );
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disp( 'Initial random vector:' )
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disp( x )
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disp( 'iteration' );
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disp( 0 );
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% Print the length of the component of x orthogonal to the eigenvector
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% associated with the smallest eigenvalue (in magnitude)
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disp( 'The length of the component of x orthogonal to V( :, n ) is ' );
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disp( norm( x - x' * V( :, n ) * V( :, n ) ) );
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% Compute the Rayleigh quotient (no need to divide by x' * x since x is
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% of unit length)
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disp( 'Rayleigh quotient:' );
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disp( x' * A * x );
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% Compute the LU factorization (with pivoting) of A - \sigma I
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[ L, U, P ] = lu( A - sigma * ( eye( size( A ) ) ));
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cont = 1;
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% Perform at most k steps of the Shifted Inverse Power Method
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for i=1:k
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cont = input( 'continue? (0=NO, return = YES)' );
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if cont == 0
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break;
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end
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disp( 'iteration' );
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disp( i );
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% Next step of Inverse Power Method
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% x = ( A - sigma I ) \ x; Let's use the LU factorization instead.
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x = P * x;
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x = L \ x;
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x = U \ x;
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x = x / norm( x );
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% Print the length of the component of x orthogonal to the eigenvector
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% associated with the smallest eigenvalue (in magnitude)
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disp( 'The length of the component of x orthogonal to V( :, n ) is ' );
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disp( norm( x - x' * V( :, n ) * V( :, n ) ) );
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% Compute the Rayleigh quotient (no need to divide by x' * x since x is
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% of unit length)
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disp( 'sigma (Rayleigh quotient):' );
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disp( x' * A * x );
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end
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disp( 'Final vector x:' );
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disp( x );
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sigma = x' * A * x;
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disp( 'A * x - sigma * x (should equal, approximately, the zero vector)' );
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disp( A * x - sigma * x );
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