Let
K
{\mathbb {K}}
be either the real, complex, or quaternion number system and let
O
(
K
)
{\mathbb {O}}({\mathbb {K}})
be the corresponding integers. Let
x
=
(
x
1
,
…
,
x
n
)
x = (x_{1}, \dots , x_{n})
be a vector in
K
n
{\mathbb {K}}^{n}
. The vector
x
x
has an integer relation if there exists a vector
m
=
(
m
1
,
…
,
m
n
)
∈
O
(
K
)
n
m = (m_{1}, \dots , m_{n}) \in {\mathbb {O}}({\mathbb {K}})^{n}
,
m
≠
0
m \ne 0
, such that
m
1
x
1
+
m
2
x
2
+
…
+
m
n
x
n
=
0
m_{1} x_{1} + m_{2} x_{2} + \ldots + m_{n} x_{n} = 0
. In this paper we define the parameterized integer relation construction algorithm PSLQ
(
τ
)
(\tau )
, where the parameter
τ
\tau
can be freely chosen in a certain interval. Beginning with an arbitrary vector
x
=
(
x
1
,
…
,
x
n
)
∈
K
n
x = (x_{1}, \dots , x_{n}) \in {\mathbb {K}}^{n}
, iterations of PSLQ
(
τ
)
(\tau )
will produce lower bounds on the norm of any possible relation for
x
x
. Thus PSLQ
(
τ
)
(\tau )
can be used to prove that there are no relations for
x
x
of norm less than a given size. Let
M
x
M_{x}
be the smallest norm of any relation for
x
x
. For the real and complex case and each fixed parameter
τ
\tau
in a certain interval, we prove that PSLQ
(
τ
)
(\tau )
constructs a relation in less than
O
(
n
3
+
n
2
log
M
x
)
O(n^{3} + n^{2} \log M_{x})
iterations.