Relative distance—an error measure in round-off error analysis
Authors: Ziv Abraham 
Published: June 2010

Olver (SIAM J. Numer. Anal., v. 15, 1978, pp. 368-393) suggested relative precision as an attractive substitute for relative error in round-off error analysis. He remarked that in certain respects the error measure
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d
(

x
¯

,
x
)
=
min
{
α

|

1
−
α
⩽
x

/

x
¯

⩽
1

/

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1
−
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)
}

d(\bar x,x) = \min \{ \alpha |1 - \alpha \leqslant x/\bar x \leqslant 1/(1 - \alpha )\}

,

x
¯

≠
0

\bar x \ne 0

,

x

/

x
¯

>
0

x/\bar x > 0

is even more favorable, through it seems to be inferior because of two drawbacks which are not shared by relative precision: (i) the inequality

d
(

x
¯

k

,

x
k

)
⩽

|

k

|

d
(

x
¯

,
x
)

d({\bar x^k},{x^k}) \leqslant |k|d(\bar x,x)

is not true for

0
>

|

k

|

>
1

0 > |k| > 1

. (ii)

d
(

x
¯

,
x
)

d(\bar x,x)

is not defined for complex

x
¯

,
x

\bar x,x

. In this paper the definition of

d
(
⋅
,
⋅
)

d( \cdot , \cdot )

is replaced by

d
(

x
¯

,
x
)
=

|

x
¯

−
x

|

/

max
{

|

x
¯

|

,

|

x

|

}

d(\bar x,x) = |\bar x - x|/\max \{ |\bar x|,|x|\}

. This definition is equivalent to the first in case

x
¯

≠
0

\bar x \ne 0

,

x

/

x
¯

>
0

x/\bar x > 0

, and is free of (ii). The inequality

d
(

x
¯

k

,

x
k

)
⩽

|

k

|

d
(

x
¯

,
x
)

d({\bar x^k},{x^k}) \leqslant |k|d(\bar x,x)

is replaced by the more universally valid inequality
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d
(

x
¯

k

,

x
k

)
⩽

|

k

|

d
(

x
¯

,
x
)

/

(
1
−
δ
)
,
δ
=
max
{
d
(

x
¯

,
x
)
,

|

k

|

d
(

x
¯

,
x
)
}

d({\bar x^k},{x^k}) \leqslant |k|d(\bar x,x)/(1 - \delta ),\delta = \max \{ d(\bar x,x),|k|d(\bar x,x)\}

. The favorable properties of

d
(
⋅
,
⋅
)

d( \cdot , \cdot )

are preserved in the complex case. Moreover, its definition may be generalized to linear normed spaces by
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d
(

x
¯

,
x
)
=

‖

x
¯

−
x

‖

/

max
{

‖

x
¯

‖

,

‖
x
‖

}

d(\bar x,x) = \left \| {\bar x - x} \right \|/\max \{ \left \| {\bar x} \right \|,\left \| x \right \|\}

. Its properties in such spaces raise the possibility that with further investigation it might become the basis for error analysis in some vector, matrix, and function spaces.