Ejericicio 1 Demostrar que (: , +1: ) forman una base ortonormal del espacio (()).Demostracin.- Sea = una factorizacin QR completa. [1|2| . |

] = [12

+1.

][

1

2

00]

= Multiplicamos por la transpuesta conjugada = = Se tiene [

1

2

+1

]

[1|2| . |

] =[

1

2

00]

[

1 1

2 1

1 2

2 2

1

1

1

2

+1 1

1

+1 2

+1 2

+1

]

=[

1

2

00]

Entonces:

+1

= 0

= 0 = 1, . . . Por lo anterior: {+1, . ,

} (()) Se observa que +1, . ,

son vectores ortogonales a cualquier columna de A. Por otro lado, vamos obtener la dimensin del espacio (()) Definimos (()) como: (())= {

/

= 0, = 1, . . , } dim(()) +dim((())) = +dim((())) = dim((())) = Por ltimo,definimos un espacio W como: = +1, ,

Alser{+1, . ,

}unconjuntodevectoresortogonales,podemos asegurar que son linealmente independiente. Entonces {+1, . ,

} es una base de W. Se saba que{+1, . ,

} (()) Entonces (()) Adems dim() = = dim(())Por lo tanto = (()) Concluimos que {+1, . ,

} forman una base ortonormal de (()). Ejercicio 3.-Cuestion 3.2 libro de demmel Datos de entrada para el cdigo QRStability.m m,n= nmero de filas y columnas muestras= nmero de matrices muestra cnd=nmero de condicionamiento Ejemplo 1 >> m=6 >> n=4 >> samples=20 >> cnd=10000000 Grafico que representa la diferencia

. La ortogonalidad Se observa que el error ms pequeo y estable esta dado por el mtodo de Householder, cumple que

~() Se observa que para el Mtodo Gram Schmidt modificado el erro es ms grande que para Householder pero menor que el mtodo CGS.

~()2() Se observa que para el Mtodo Gram Schmidt clsico cumple:

~()2()2 max( norm(A-Q*R)/norm(A) ) for Householder QR = 7.3938e-16 max( norm(A-Q*R)/norm(A) ) for CGS= 1.3381e-16 max( norm(A-Q*R)/norm(A) ) for MGS= 1.1465e-16 Los tres metodos son backward estable. Podemos observer que se cumple: ~() % Matlab code QRStability.m % For "Applied Numerical Linear Algebra",Question 3.2 % Written by James Demmel, Aug 2, 1996; %Modified, Jun 2, 1997 % % Compare the numerical stability of 3 algorithms for computing % the QR decomposition:% QR using Householder transformations % CGS (Classical Gram-Schmidt) % MGS (Modified Gram-Schmidt) % % Inputs: % % m, n = numbers of rows, columns in test matrices %m should be at least n % samples = number of test matrices to generate % samples should be at least 1 % cnd = condition number of test matrices to generate % (ratio of largest to smallest singular value) % cnd should be at least 1 % % Outputs: % % For each algorithm, the maximum value of the residual % norm(A-Q*R)/norm(A), which should be around% macheps = 1e-16 for a stable algorithm % % For each algorithm, a plot (versus sample number) % of the orthogonality norm(Q'*Q-I), which should% also be around macheps if the computed Q is orthogonal % and the algorithm is stable % residual = []; orth = []; % Generate samples random matrices for exnum = 1:samples, % Generate random matrix A, starting with the SVD of a random matrix A=randn(m,n); [u,s,v]=svd(A); % Let singular values range from 1 to cnd, with % uniformly distributed logarithms sd = [1, cnd, exp(rand(1,n-2)*log(cnd))]; s = diag(sd); A=u(:,1:n)*s*v'; % Perform CGS (A = Qcgs*Rcgs) and MGS (A = Qmgs*Rmgs) Rcgs=[]; Rmgs=[]; Qcgs=[]; Qmgs=[]; for i=1:n, Qcgs(:,i)=A(:,i); Qmgs(:,i)=A(:,i); for j=1:i-1, Rcgs(j,i) = (Qcgs(:,j))'*A(:,i); Qcgs(:,i) = Qcgs(:,i) - Rcgs(j,i)*Qcgs(:,j); Rmgs(j,i) = (Qmgs(:,j))'*Qmgs(:,i); Qmgs(:,i) = Qmgs(:,i) - Rmgs(j,i)*Qmgs(:,j); end Rcgs(i,i) = norm(Qcgs(:,i)); Rmgs(i,i) = norm(Qmgs(:,i)); Qcgs(:,i) = Qcgs(:,i) / Rcgs(i,i); Qmgs(:,i) = Qmgs(:,i) / Rmgs(i,i); end % Perform Householder QR [Qhouse,Rhouse]=qr(A,0); % Compute residuals for each algorithm (cnd = norm(A)) residual(exnum,1:3) = ... [ norm(A-Qhouse*Rhouse), norm(A-Qcgs*Rcgs), norm(A-Qmgs*Rmgs) ]/cnd; % Compute orthogonality of Q for each algorithmee = eye(n); orth(exnum,1:3) = ... [ norm(Qhouse'*Qhouse-ee), norm(Qcgs'*Qcgs-ee), norm(Qmgs'*Qmgs-ee) ]; end % Limit orth to 1 (any larger error is as bad) so it shows up on plot orth = min(orth,ones(size(orth))); % Print max residuals disp(['max( norm(A-Q*R)/norm(A) ) for Householder QR = ', ... num2str(max(residual(:,1)))]) disp(['max( norm(A-Q*R)/norm(A) ) for CGS= ', ... num2str(max(residual(:,2)))]) disp(['max( norm(A-Q*R)/norm(A) ) for MGS= ', ... num2str(max(residual(:,3)))]) % Plot orthogonalities clf % make sure orthogonalities are at least eps to avoid log(0) orth = max(orth,eps); semilogy((1:samples),orth(:,1),'b*', ... (1:samples),orth(:,2),'r*', ... (1:samples),orth(:,3),'g*') axis([1 samples 1e-16 1]) grid ylabel('orthogonality') xlabel(['test matrix number, condition num =',num2str(cnd)]) title('Orthogonality of Q for CGS (red), MGS (green), Householder (blue)')