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=5cb7ce80837f4bb7d8b21e8f6e1414b3aaa6dcc4;hb=86049fdfa3f79a188f08ec37a527772c2ba95223;hpb=d7793b97a19a2e8ae038305a4e162253f005bbb9 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 5cb7ce8..c5a751f 100644 --- a/scilab/modules/linear_algebra/help/en_US/factorization/householder.xml +++ b/scilab/modules/linear_algebra/help/en_US/factorization/householder.xml @@ -2,13 +2,16 @@ @@ -16,10 +19,14 @@ Householder orthogonal reflexion matrix - Calling Sequence - u=householder(v [,w]) + Syntax + + householder() // demo + u = householder(v [,w]) + [u, H] = householder(v [,w]) + - + Arguments @@ -32,29 +39,165 @@ w - real or complex column vector with same size as v. Default value is eye(v) + real or complex column vector with same size as v. + Default value is eye(v) ((Ox) axis). u - real or complex column vector + + 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 - 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 . + householder(..) computes the unit vector u + lying in the (v,w) plane and orthogonal to the bisectrix of + (v,w). - w default value is eye(v). In this case vector (eye()-2*u*u')*v is the - vector eye(v)*norm(v). + 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 @@ -67,4 +210,17 @@ + + History + + + 6.0 + + Householder reflexion matrix added as second output parameter. + Demo householder() added. Help page reviewed. + + + + +