Mercurial > matrix-functions
diff toolbox/house.m @ 2:c124219d7bfa draft
Re-add the 1995 toolbox after noticing the statement in the ~higham/mctoolbox/ webpage.
author | Antonio Pino Robles <data.script93@gmail.com> |
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date | Thu, 07 May 2015 18:36:24 +0200 |
parents | 8f23314345f4 |
children |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/house.m Thu May 07 18:36:24 2015 +0200 @@ -0,0 +1,36 @@ +function [v, beta] = house(x) +%HOUSE Householder matrix. +% If [v, beta] = HOUSE(x) then H = EYE - beta*v*v' is a Householder +% matrix such that Hx = -sign(x(1))*norm(x)*e_1. +% NB: If x = 0 then v = 0, beta = 1 is returned. +% x can be real or complex. +% sign(x) := exp(i*arg(x)) ( = x./abs(x) when x ~= 0). + +% Theory: (textbook references Golub & Van Loan 1989, 38-43; +% Stewart 1973, 231-234, 262; Wilkinson 1965, 48-50). +% Hx = y: (I - beta*v*v')x = -s*e_1. +% Must have |s| = norm(x), v = x+s*e_1, and +% x'y = x'Hx =(x'Hx)' real => arg(s) = arg(x(1)). +% So take s = sign(x(1))*norm(x) (which avoids cancellation). +% v'v = (x(1)+s)^2 + x(2)^2 + ... + x(n)^2 +% = 2*norm(x)*(norm(x) + |x(1)|). +% +% References: +% G.H. Golub and C.F. Van Loan, Matrix Computations, second edition, +% Johns Hopkins University Press, Baltimore, Maryland, 1989. +% G.W. Stewart, Introduction to Matrix Computations, Academic Press, +% New York, 1973, +% J.H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University +% Press, 1965. + +[m, n] = size(x); +if n > 1, error('Argument must be a column vector.'), end + +s = norm(x) * (sign(x(1)) + (x(1)==0)); % Modification for sign(0)=1. +v = x; +if s == 0, beta = 1; return, end % Quit if x is the zero vector. +v(1) = v(1) + s; +beta = 1/(s'*v(1)); % NB the conjugated s. + +% beta = 1/(abs(s)*(abs(s)+abs(x(1)) would guarantee beta real. +% But beta as above can be non-real (due to rounding) only when x is complex.