X-Git-Url: http://gitweb.scilab.org/?p=scilab.git;a=blobdiff_plain;f=scilab%2Fmodules%2Flinear_algebra%2Fhelp%2Fen_US%2Ffactorization%2Fhouseholder.xml;h=c5a751fb819d075acdcb17088fd25ea6ab30827b;hp=2572e90e6e1b409a4cfb03491d094668c72cdb07;hb=86049fdfa3f79a188f08ec37a527772c2ba95223;hpb=a846576255304dd81d6c449ecb577121c6454d94 diff --git a/scilab/modules/linear_algebra/help/en_US/factorization/householder.xml b/scilab/modules/linear_algebra/help/en_US/factorization/householder.xml index 2572e90..c5a751f 100644 --- a/scilab/modules/linear_algebra/help/en_US/factorization/householder.xml +++ b/scilab/modules/linear_algebra/help/en_US/factorization/householder.xml @@ -1,70 +1,226 @@ - - - - - householder - Householder orthogonal reflexion matrix - - - Calling Sequence - u=householder(v [,w]) - - - Arguments - - - v - - real or complex column vector - - - - w - - - real or complex column vector with same size as v. Default value is eye(v) - - - - - u - - real or complex column vector - - - - - - Description - - given 2 column vectors v, w of same size, householder(v,w) returns a unitary - column vector u, such that (eye()-2*u*u')*v is proportional to w. - (eye()-2*u*u') is the orthogonal Householder reflexion matrix . - - - w default value is eye(v). In this case vector (eye()-2*u*u')*v is the - vector eye(v)*norm(v). - - - - See Also - - - qr - - - givens - - - - + + + householder + Householder orthogonal reflexion matrix + + + Syntax + + householder() // demo + u = householder(v [,w]) + [u, H] = householder(v [,w]) + + + + Arguments + + + v + + real or complex column vector + + + + w + + + real or complex column vector with same size as v. + Default value is eye(v) ((Ox) axis). + + + + + u + + + unit vector lying in the (v,w) plane and orthogonal + to the bisectrix of (v,w). + Column of size(v) of real or complex numbers. + + + + + H + + + Orthogonal Householder reflexion matrix: H= eye() - 2*u*u'. + H is such that inv(H)==H, + H'==H, and det(H)==-1. + + + If v and w are real, + H*v is proportional to w. + + + + + + + Description + + householder(..) computes the unit vector u + lying in the (v,w) plane and orthogonal to the bisectrix of + (v,w). + + + If v and w are proportional: + + + + If they are opposite, u= v/|v| is returned. + + + + If they are real and have the same direction, u is set + in the (xOy) plane with a priori u(1)>0, and orthogonal to + v (u'*v==0). However, + + + If they are along (Ox), u = (Oy+) is returned instead. + + + If v and w are scalars with same + signs, the orthogonal sub-space is restricted to {0} + that can't be normalized: + u and H are then set to + %nan. + + + + + + + If the related reflexion matrix H is computed, for any point A + of column coordinates a, H*a are the coordinates of + the reflected image of A (see the example below). + + + If v or/and w are in row, they are priorly + transposed into columns. + + + If v or/and w are [], + [] is returned for u and H. + + + + Examples + + + + Application: Reflected image of an object w.r. to a given plane + + + + + See Also + + + qr + + + givens + + + + + History + + + 6.0 + + Householder reflexion matrix added as second output parameter. + Demo householder() added. Help page reviewed. + + + + + +