HP MLIB User's Guide Vol. 1 7th Ed.

Chapter 2 Basic Vector Operations 175
Generate Householder transform F_SGEN_HOUSE/F_DGEN_HOUSE/F_CGEN_HOUSE/F_ZGEN_HOUSE
Name F_SGEN_HOUSE/F_DGEN_HOUSE/F_CGEN_HOUSE/F_ZGEN_HOUSE
Generate Householder transform
Purpose F_xGEN_HOUSE generates an elementary reflector H of order n, s such that
where α and β are scalars, and x is an (n - 1)-element vector. β is always a real
scalar. H is represented in the following form:
where τ is a scalar and υ is an (n - 1)-element vector. τ is called the Householder
scalar and the Householder vector.
Note that when x is a complex vector, H is not Hermitian. If the elements of x
are zero, and α is real, then τ is equal to zero and H is the unit matrix.
Otherwise, the real part of τ is greater than or equal to one, and less than or
equal to two. Moreover, the absolute value of the quantity (τ - 1) is less than or
equal to one.
On exit, the scalar argument alpha is overwritten with the value of the scalar β.
Similarly, the vector argument x is overwritten with the vector v. If n is less
than or equal to zero, this function returns immediately with the output scalar
tau set to zero.
H
α
x
β
0
and H
∗
HI==
HIτ
1
υ
1 υ
T
()–=
1
υ