MATH 317 Lecture 14: 317-1018
31 Oct 2018
School
Department
Course
Professor
Roud
off error is large at small h
Truncation error is large at large h
Say we have an analytical fix and an approximate fand
rounding error uxkfxl
fxl.flxothl
flxothltelxothlflxo
hkflxo hltelx.tn
Centred Differences
fix flxothl 批以1䇢f引
Scheme truncationerror
Error we make in 1St derivative is
fix 1挔叶
竺
舆0坦
舆
答f吲
Now say elxothl elx.tn is boundedby Eand that
fGl is bounded by Mf somerange of xthen
work 𠵯1
号
川
Optimum his not too big not too small
Richardson extrapolation
say we have an approx Nth to anumber Meg
fXo Mexact unknown
Nlhkflxothy approximate computable
Now use Taylor
fo th flxoltflxo ht fx冷tf业
6
Nch 我0fNf则f如等
M
Document Summary
Say we have an analytical fix and an approximate rounding error uxkfxl fxl. flxothl flxothltelxothlflxo hkflxo hltelx. tn. Centred differences fix flxothl 1 f f and. Error we make in 1st derivative is fix 1 (cid:1896) . Now say elxothl f gl is bounded by mf somerange of x then elx. tn is boundedby e and that work 1(cid:1897) . Optimum h is not too big not too small. Richardson extrapolation say we have an approx nth to a number meg f xo m exact. Nch 0 flxoltflxo ht f x t f . Nlhltkhtkz. li t evaluate na x it at two different value of hand use at hand . Theidea the result to construct a more accurate approximation nuh. Forward euler forward differences was yltnnj yltnltfltnynlltnn tnfhyn yltnl. fr tilt at eachstepyou can. Anexample of an explicit scheme calculate yltmu as function of quantities that are already. Could approximate integral more accurately yltnthkyltnukylt. nl h h 2h k.
