XGitUrl: 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 subspace 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.
+
+
+
+
+
+