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Chapter 3 Multiplicative Error Reduction
Xmath Model Reduction Module 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
GG
ˆ
G
ˆ
2
G
ˆ
3
G
1–
GG
ˆ
3
–()G
1–
GG
ˆ
–()=
G
1–
G
ˆ
G
ˆ
1–
G
ˆ
G
ˆ
2
–()+
G
1–
G
ˆ
G
ˆ
1–
G
ˆ
2
G
ˆ
2
1–
G
ˆ
2
G
ˆ
3
–()+
G
1–
G
ˆ
G
ˆ
1–
G
ˆ
G–()I+=
1 v
ns
+≤
G
ˆ
1–
G
ˆ
2
1 v
ns 1–
+≤ G
ˆ
2
1–
G
ˆ
3
1 v
ns 2–
+≤,
G
1–
GG
ˆ
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
insr1+=
ns
∑
2
v
i
insr1+=
ns
∑
for mulhank( ) for bst(