Error bounds, Error bounds -20 – National Instruments NI MATRIXx Xmath User Manual
Page 66
Chapter 3
Multiplicative Error Reduction
3-20
ni.com
Error Bounds
The error bound formula (Equation 3-3) is a simple consequence of
iterating (Equation 3-5). To illustrate, suppose there are three reductions
→ →
→ , each by degree one. Then,
Also,
Similarly,
Then:
The error bound (Equation 3-3) is only exact when there is a single
reduction step. Normally, this algorithm has a lower error bound than
bst( )
; in particular, if the
ν
i
are all distinct and
, the error
bounds are approximately
G
Gˆ
Gˆ
2
Gˆ
3
G
1
–
G Gˆ
3
–
(
)
G
1
–
G Gˆ
–
(
)
=
G
1
–
GˆGˆ
1
–
Gˆ Gˆ
2
–
(
)
+
G
1
–
GˆGˆ
1
–
Gˆ
2
Gˆ
2
1
–
Gˆ
2
Gˆ
3
–
(
)
+
G
1
–
Gˆ
Gˆ
1
–
Gˆ G
–
(
) I
+
=
1 v
ns
+
≤
Gˆ
1
–
Gˆ
2
1 v
ns 1
–
+
≤
Gˆ
2
1
–
Gˆ
3
1 v
ns 2
–
+
≤
,
G
1
–
G Gˆ
3
–
(
)
v
ns
1 v
ns
+
(
)v
ns 1
–
1 v
ns 1
–
+
(
)v
ns 2
–
+
+
≤
1 v
ns
+
(
) 1 v
ns 1
–
+
(
) 1 v
ns 2
–
+
(
)
=
1
–
v
nsr 1
+
1
«
v
i
i
nsr 1
+
=
ns
∑
2
v
i
i
nsr 1
+
=
ns
∑
for
mulhank( )
for
bst(