数值分析第五版李庆扬王能超易大义主编课后习题答案.docx
- 文档编号:4808447
- 上传时间:2023-05-07
- 格式:DOCX
- 页数:26
- 大小:20.66KB
数值分析第五版李庆扬王能超易大义主编课后习题答案.docx
《数值分析第五版李庆扬王能超易大义主编课后习题答案.docx》由会员分享,可在线阅读,更多相关《数值分析第五版李庆扬王能超易大义主编课后习题答案.docx(26页珍藏版)》请在冰点文库上搜索。
数值分析第五版李庆扬王能超易大义主编课后习题答案
数值分析第五版_李庆扬_王能超_易大义主编课后习题答案(Thefiftheditionofnumericalanalysis_liqingyang_yidayiisthechiefeditor)
ChapterIintroduction
Therelativeerrorofthesetistheerroroffinding.
Solution:
therelativeerrorofapproximatevaluesis
Andtheerroris
Andthenthereare
2.Therelativeerrorofthesetis2%,therelativeerroroftherequest.
Solution:
set,thefunction'sconditionnumberis
Again,
again
Andforthe2
3.ThefollowingaretheapproximateNumbersthathavebeenroundedandrounded,i.e.theerrorlimitisnotmorethanhalfofthelastone,andthetestindicatesthattheyareseveralvaliddigits:
Solution:
fivevaliddigits;
It'stwovaliddigits;
Fourvaliddigits;
Fivevaliddigits;
It'stwovaliddigits.
4.Useformula(2.3)tofindthefollowingerrorlimits
(1),
(2),(3).
ThesearetheNumbersgiveninquestion3.
Solution:
5.Whencalculatingthevolumeoftheball,therelativeerrorlimitis1.WhatistherelativeerrorlimitallowedwhenmeasuringtheradiusR?
Solution:
spherevolumeis
Theconditionnumberofthefunctionis
again
Therefore,therelativeerrorlimitallowedwhenradiusRismeasured
6.Set,presstheformula(n=1,2,...)
Tothecalculation.Ifyoutake(5validdigits),howmucherrordoyouhaveinthetest?
Solution:
......
Andthenyouplugin,youhave
Thatis,
Iftake,
Theerrorlimitis.
7.Findtworootsoftheequationandmakeithaveatleastfourvaliddigits().
Solution:
Sotherootoftheequationshouldbe
Therefore,
Ithasfivevaliddigits
Ithasfivevaliddigits
8.WhatdoyoudowhentheNissufficientlylarge?
solution
Set.
the
9.Thelengthofthesquareisabout100cm,sohowcantheareaerrorbenotexceeded?
Solution:
thesquareareafunctionis
.
When,if,
the
Therefore,theareaerrorcannotexceed0.005cm
Let'ssaythatthegisaccurate,andthemeasurementoftisaseconderror,provingthattheabsoluteerrorofSincreaseswhentincreases,buttherelativeerrordecreases.
Solution:
Theabsoluteerrorincreaseswhenincreasing
Whentheincreaseismaintained,therelativeerrorisreduced.
11.Thesequencesatisfiestherecursiverelationship(n=1,2,...).,
If(threevaliddigits),howlargeistheerror?
Isthiscomputationalprocessstable?
Solution:
again
again
Thecalculationofthetimeerroristhatthecalculationprocessisunstable.
12.Calculate,take,usethefollowingequationtocalculate,whichgetthebestresult?
,,.
Solution:
set,
Ifso,then.
Ifyoucomputetheyvalue,then
Ifyoucomputetheyvalue,then
Ifyoucomputetheyvalue,then
Theresultisbestcalculated.
13.Thevalueoftherequest.Soifwesquareitwith6tables,whatistheerrorinthelogarithmictime?
Ifyouswitchtoanotherequivalentformula.
Whatistheerrorinthelog?
solution
set
the
Therefore,
Ifyouusetheequivalentformula
the
Atthispoint,
Chaptertwointerpolationmethod
1.Quadraticinterpolationpolynomialintime.
Solution:
ThequadraticLagrangianinterpolationpolynomialis
2.Numericaltable
X0.40.50.60.70.8
LNX-0.916291-0.693147-0.510826-0.356675-0.223144
Approximationoflinearinterpolationandquadraticinterpolation.
Solution:
byform,
Ifyouuselinearinterpolation,
the
Ifthequadraticinterpolationmethodisused,
3.Tocompletethefunctiontable,thesteplengthfunctiontablehasfivevaliddigits,andthetotalerrorboundswhenthelinearinterpolationisusedtofindtheapproximatevalue.
Solution:
whensolvingapproximatevalues,theerrorcanbedividedintotwoparts.Ontheonehand,xisapproximateandhasfivevaliddigits,resultinginacertainerrorpropagationinthesubsequentcalculation.Ontheotherhand,usinginterpolationmethodtofindtheapproximatevalueofthefunction,theinterpolationmethodwithlinearinterpolationmethodisnot0,andtherewillbesomeerror.Therefore,thecalculationoftotalerrorboundsshouldbecombinedwiththeabovetwofactors.
When,
make
take
make
the
When,linearinterpolationpolynomialis
Theinterpolationoftheremainderis
Whensettingupthefunctiontable,thedatainthetablehasfivevaliddigits,andtheerrorpropagationprocessisinthecalculation.
Totalerrorbounds
Settothedifferentnodes,please:
(1)
(2)
prove
(1)the
Iftheinterpolationnodeis,thesubinterpolationpolynomialofthefunctionis.
Theinterpolationoftheremainderis
again
Wecanseefromtheaboveconclusion
Havetopass.
5.
Solution:
orderistheinterpolationnode,andthelinearinterpolationpolynomialis
=
Theinterpolationiszero
6.Inthetableofequidistantnodefunctionsgiven,iftheapproximationofquadraticinterpolationisused,thetruncationerrorshouldnotbeexceeded,andhowmuchshouldthesteplengthofthefunctiontablebeused?
Solution:
iftheinterpolationnodeisand,theinterpolationofthequadraticinterpolationpolynomialis
Thesteplengthish,thatis
Ifthetruncationerrordoesnotexceed,then
7.If,
Solution:
accordingtothedefinitionoftheforwarddifferenceoperatorandthecentraldifferenceoperator.
8.Ifitismdegreepolynomial,rememberthatthek-orderdifferenceispolynomial,and(positiveinteger).
Solution:
thedisplayofthefunctionis
Amongthem
It'sapolynomialofthenumberoftimes
Polynomialoforder
Polynomialoforder
Inthisprocess,it'sapolynomial
Isconstant
Whenit'sapositiveinteger,
9.Toprove
prove
Havetothe
10.Prove
Proof:
itisknownfromtheaboveconclusion
Havetopass.
11.Toprove
prove
Havetopass.
12.Ifthereisadifferentroot,
Proof:
Proof:
thereisadifferentrealroot
and
make
the
while
make
the
again
Havetopass.
13.Thefollowingpropertiesoftheprooforderare:
(1)if
(2)if,then
Proof:
(1)
Havetopass.
+
Havetopass.
14.Please.
Solution:
if
the
15.ItisshownthattheremainderoftheHermiteinterpolationoftwopointsis
Solution:
If,theinterpolationpolynomialsatisfiesthecondition
Theinterpolationiszero
Theinterpolationconditioncanbeseen
and
Canbewrittenas
It'sabouttheundeterminedfunction,
Nowlet'sthinkofafixedpointasafunction
Accordingtothepropertiesoftheremainder,yes
Therolle'stheoremtellsusthatthereareandthat
Thatis,therearefourdifferentzeros.
Accordingtorolle'stheorem,atleastonezeropointbetweenthetwozeros,
Sothereareatleastthreedifferentzerosintheinside,
Withthiskindofpush,thereisatleastonezeroinside.
Remembertokeepthe
again
Dependson
Ifthenodeissetatthreetimes,thesteplengthis
Intheneighborhood
FindapolynomialP(x)thatisnomorethan4timestosatisfyit
Solution:
usetheemirinterpolationtoobtainapolynomialofnohigherthan4
set
Where,Aistheundeterminedconstant
thus
17.Setupandtakeapiecewiselinearinterpolationfunctionattheisometricnodetocalculatethevalueanderrorofthemid-pointofeachnode.
Solution:
if
Thesteplength
Intheinterplot,thepiecewiselinearinterpolationfunctionis
Thevalueatthemiddlepointofeachnodeis
When,
When,
When,
When,
When,
error
again
make
Thestagnationpointisthesum
18.Findthelinearinterpolationfunctionintheuppersegmentandestimatetheerror.
Solution:
Ontheinterval,
Thefunctionisalinearinterpolationfunctionintheintersections
Erroris
19.Pleaseinterpolatetheemmettintheuppersectionandestimatetheerror.
Solution:
Ontheinterval,
make
ThefunctionisasegmentHermiteinterpolationfunctionintheinterval
Erroris
again
20.Thegivendatatableisasfollows:
Xj0.300.390.450.53
Yj0.54770.62450.67080.7280
Trythreesplineinterpolationandmeettheconditions:
Solution:
Thematrixformofthesystemis
21M0
2M1
2M2
2M3
12M4
Solvethissystem
Thecubicsplineexpressionis
Toplugin
Thesystemofthematrixstartsfromthismatrixis
Solvethissystem
Andthenwehavethreemoresplineexpressions
Toplugin
21.Ifitisacubicsplinefunction,theproof:
If,intheformofinterpolationnode,and,then
Proof:
Thusthereare
Thefunctionapproximationandthecurvematch
1.,
GivetheBernsteinpolynomials.
Solution:
Bernsteinpolynomialis
Amongthem
When,
When,
Whenyouare,askforevidence
Proof:
If,
Theprooffunctionislinearlyindependent
Proof:
if
Taketheinnerproductoftheupperandtheupperends,respectively
ThecoefficientmatrixofthissystemisHilbertmatrix,andthesymmetryispositiveandnonsingular,
Sowehavezerosolutionaisequalto0.
Thefunctionislinearlyindependent.
4.Calculatethefollowingfunctions:
Mandnarepositiveintegers,
Solution:
If,
It'smonotonicallyincreasing
If,
Ifmandnarepositiveintegers
When,
When,
It'sdecreasingintheinnermonotone
When,
It'sdecreasingintheinnermonotone.
if
When,
It'sdecreasingintheinnermonotone.
5.prove
Proof:
6.Todefine,
Askthemiftheyforminnerproduct.
Solution:
(Cisconstant,and)
the
while
Thisiscontradictorywhenandonlywhen
Cannotconstitutetheinnerproduct.
If,
If,
And,
Thatis,whenandonlywhen.
Soyoucanformtheinnerproduct.
7.So,thetestistheorthogonalpolynomialsoftheupperright.
Solution:
If,
Order,then,andtherefore
Andthenchebyshevpolynomialsareorthogonaltoeachotherontheinterval,and
It'sanorthogonalpolynomialwithright.
again
8.Fortherightfunction,theinterval,trytofindtheorthogonalpolynomialof1
Solution:
If,theintervalistheinnerproduct
Definition,
Amongthem
9.Itisprovedthatthesecondtypeofchebyshevpolynomialsgivenbythetextbookformulaareorthogonalpolynomialsofupperbelt
- 配套讲稿:
如PPT文件的首页显示word图标,表示该PPT已包含配套word讲稿。双击word图标可打开word文档。
- 特殊限制:
部分文档作品中含有的国旗、国徽等图片,仅作为作品整体效果示例展示,禁止商用。设计者仅对作品中独创性部分享有著作权。
- 关 键 词:
- 数值 分析 第五 版李庆扬王能超易 大义 主编 课后 习题 答案