Post on 18-Dec-2015
3
Taylor’s Expansion
)()(2
1)()()(
)()(2
1)()()(
)()(2
1)()()(
2
32
32
hOhxfh
xfhxfxf
hOhxfxfhxfhxf
hOhxfhxfxfhxf
4
Forward Difference Formula for
)(xf
)( )()(
)( hOerrorh
xfhxfxf
)(
)()()( hOerror
h
xfhxfxf
Geometrically
h
)()( xfhxf
)(xf
hx x
)(xf
5
Backward Difference Formula for )(xf
)(
)()()(
hOerrorh
hxfxfxf
)(
)()()(
hOerrorh
hxfxfxf
Similarly
)()(2
1)()()( 32 hOhxfhxfxfhxf
h
)(xf
xxhx
)()( hxfxf
Geometrically
6
Central Difference Formula for
)()(!4
1)(
6
1)(
2
1)()()( 54)4(32 hOhxfhxfhxfhxfxfhxf
)(xf
)()(!4
1)(
6
1)(
2
1)()()( 54)4(32 hOhxfhxfhxfhxfxfhxf –)
)()( hxfhxf )(2 xfh 3)(3
1hxf )( 5hO
)()(3
1)()()(2 53 hOhxfhxfhxfxfh
7
Central Difference Formula for )(xf
h
hxfhxfxf
2
)()()(
)( 2hOerror
)(xf
x
2h
)()( hxfhxf
hx x hx
)(xf
Geometrically
9
Example (cont)
h=0.1
h=0.1
h=0.1
h=0.05
h=0.05
h=0.05
31.31.0
)1()1.1()1(
ff
f
1525.305.0
)1()05.1()1(
ff
f
71.21.0
)9.0()1()1(
ff
f
8453.205.0
)95.0()1()1(
ff
f
01.32.0
)9.0()1.1()1(
ff
f
00250.31.0
)95.0()05.1()1(
ff
f
31.0error
1525.0error
29.0error
1547.0error
01.0error
00250.0error
herror
2herror
FD:
BD:
CD:
10
Example (cont)
• Remarks:– FD, BD, CD each involves 2 function calls, 1
subtraction, and 1 division: same computation time
– CD is the most accurate (hence, the most recommended method)
– However, sometimes, CD cannot be applied
11
Forward Difference Formula for )(xf
)()2)((6
1)2)((
2
12)()()2( 432 hOhxfhxfhxfxfhxf
)()(6
1 )(
2
1 )( )()( 432 hOhxfhxfhxfxfhxf –2)
)(]3
1
3
4[)(]
2
122[)()()(2)2( 432 hOhxfhxfxfhxfhxf
)()()()(2)2()( 432 hOhxfxfhxfhxfhxf
2
)()(2)2()(
h
xfhxfhxfxf
)(hOerror
12
Backward Difference Formula for )(xf
)()2)((6
1)2)((
2
12)()()2( 432 hOhxfhxfhxfxfhxf
)()(6
1 )(
2
1 )( )()( 432 hOhxfhxfhxfxfhxf –2)
)(]3
1
3
4[)(]
2
122[)()()(2)2( 432 hOhxfhxfxfhxfhxf
)()()()(2)2()( 432 hOhxfxfhxfhxfhxf
2
)2()(2)()(
h
hxfhxfxfxf
)(hOerror
13
Central Difference Formula for
)()(!4
1)(
6
1)(
2
1)()()( 54)4(32 hOhxfhxfhxfhxfxfhxf
)(xf
)()(!4
1)(
6
1)(
2
1)()()( 54)4(32 hOhxfhxfhxfhxfxfhxf +)
)()(12
1 )( )(2)()( 64)4(2 hOhxfhxfxfhxfhxf
)()(2)()(
)( 22
hOh
xfhxfhxfxf
)(
)(2)()()( 2
2hO
h
xfhxfhxfxf
Similar remark on the selection of FD|BD|CD applies for f”(x)
14
More Accurate FD Formula for )(xf
)()(2
1)()()( 32 hOhxfhxfxfhxf
)()]()()(2)2(
[2
1)()()( 3
22 hOhO
h
xfhxfhxfhhxfxfhxf
)()(2
1)()2(
2
1)()( 3hOxfhxfhxfhxfxf
)()(
23
)(2)2(21
)( 2hOh
xfhxfhxfxf
)(2
)(3)(4)2()( 2hO
h
xfhxfhxfxf
)(
2
)(3)(4)2()( 2hO
h
xfhxfhxfxf
15
• Better accuracy can be achieved using this formula
• But, it involves more computations: – 3 function calls, two +/–, one division
• Trade-off: – More computation is the price you paid for better
accuracy
• Similar idea applies to more accurate BD formula
More Accurate FD Formula (cont)
16
Richardson Extrapolation
Idea:• exact
= computed+ error
• The truncation error is of the form: chk
– where c is some constant
• Use different h to estimate the truncation error
• Use extrapolation to get more accurate result
17
• Example: CD for f’(x)
• Using Different h (h1, h2):
• c1 and c2 could be different
Richardson Extrapolation (cont)
2111 hcRexact
2222 hcR
)(2
)()()( 2hO
h
hxfhxfxf
18
• If
Richardson Extrapolation (cont)
ccc 21
exactchRchR 222
211
1222
21 RRhhc 2
22
1
12
hh
RRc
22
22
212
2
122
222
22
1
122
1
h
hh
h
RRRh
hh
RRRexact
12
122
2
1
hh
RRRexact
19
• f(x) = x3. Use CD with Richardson extrapolation to compute f’(1)
Example
05.0
1.0
2
1
h
h
2
1
0025.3)1(
01.3)1(
Rf
Rf
0000.3
105.01.0
01.30025.30025.3 2
exact
Magic?Coincidence
?
20
Revisit CD Formula for )(xf
5)5(4)4(32 )(!5
1)(
!4
1)(
!3
1)(
2
1)()()( hxfhxfhxfhxfhxfxfhxf
5)5(4)4(32 )(!5
1)(
!4
1)(
!3
1)(
2
1)()()( hxfhxfhxfhxfhxfxfhxf–)
5)5(3 )(!5
2)(
!3
2)(2)()( hxfhxfxfhhxfhxf
4)5(2 )(!5
2)(
!3
2
2
)()()( hxfhxf
h
hxfhxfxf
Change notation:
)()()( 642
21 hOhahahFxf
21
Error Analysis
64
2
2
1 22~
2~)()
2(
hO
ha
haxf
hF
)()()( 642
21 hOhahaxfhF
2211~ and ~ aaaa Assuming
Eliminate a1 to get better accuracy
22
Error Analysis (cont)
)(4
1)()
2(4
3
1)(
)(16
141)(3)()
2(4
642
642
hOhahFh
Fxf
hOhaxfhFh
F
64
2
2
1 222)()
2(
hO
ha
haxf
hF
)()()( 642
21 hOhahaxfhF
4
–)
O(h4)
23
Revisit Previous Example
• f(x) = x3. Use CD with Richardson extrapolation to compute f’(1)
• a2 involves f(5)(x), hence, the exact solution is no surprise.
24
Remark
• How much effort did we use to get this level of accuracy?– F(h): f(x+h), f(x-h); one –, one – F(h/2): f(x+h/2), f(x-h/2); one –, one – R.E.: two , one –