National Instruments NI MATRIXx Xmath - Manual (Page 66)

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
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(