Chapter 3 Multiplicative Error Reduction
Xmath Model Reduction Module 3-20 ni.com
Error Bounds
The error bound formula (Equation3-3) 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 (Equation3-3) is only exact when there is a single
reduction step. Normally, this algorithm has a lower error bound than
bst( ); in particular, if the νi are all distinct and , the error
bounds are approximately
GG
ˆG
ˆ2G
ˆ3
G1–GG
ˆ3
–()G1–GG
ˆ
–()=
G1–G
ˆG
ˆ1–G
ˆG
ˆ2
–()+
G1–G
ˆG
ˆ1–G
ˆ2G
ˆ2
1–G
ˆ2G
ˆ3
–()+
G1–G
ˆG
ˆ1–G
ˆG–()I+=
1vns
+≤
G
ˆ1–G
ˆ21vns 1–
+≤G
ˆ2
1–G
ˆ31vns 2–
+≤,
G1–GG
ˆ3
–()vns 1vns
+()vns 1–1vns 1–
+()vns 2–
++≤
1vns
+()1vns 1–
+()1vns 2–
+()=1–
vnsr 1+1«
vi
insr1+=
ns
∑2vi
insr1+=
ns
∑for mulhank( ) for bst(