数值分析(英文版)

§0.1FoundationofLinearAlgebraChapter0FoundationofAlgebraandCalculus1,n-dimensionalvectorx=(x1,x2,…,xn),y=(y1,y2,…,yn).operations:x+y,x-y,cx.innerproductoftwovectors(x,y)lengthofavector|x|avectorspaceV,asubspaceofVlinearindependentofvectors12024/10/62,Matrix

mrows,ncolumnsoperations:A+B,A-B,cA,AB,AT|A|(determinantofsquarematrix)singularornonsingularmatrixA-1inverseofasquarematrixHowtodecidewhetherAisinvertibleornot?HowtocalculateA-1?22024/10/6(Continue)somespecialmatrices(1)unitmatrix(2)diagonalmatrix(3)symmetricmatrix(4)orthogonalmatrix(5)triangularmatrix(upperorlower)(6)elementarymatrix

3,EigenvalueandEigenvectorofmatrix

32024/10/63,EigenvalueandEigenvectorofmatrixEigenpolynomialofmatrixEigenvalueofmatrixEigenvectorofmatrixsimilarmatrixdiagonalizationofmatrix4,Quadraticformquadraticformcanonicalformofmatrixpositive(negative)definitematrixhalfpositivedefinitematrix

42024/10/6Exercise1,x=(2,1,4)T,y=(1,1,1)T,pleaseshowx+2y,3x-4y,|x-y|,(2y,x).2,

PleasegiveAT,2A,|A|,A-1,Rank(A),BC,AB-B.52024/10/63,PleasedescribeCramerRuleaboutthelinearsystemofequations.4,Solvethefollowinglinearsystemofequations

5,Pleaseshowalleigenvaluesandeigenvectorsofthem.

and62024/10/66,PleasechangeAintocanonicalform,anddecidewhetherAispositivedefiniteornot.

7,PleaseshowtheconditionunderwhichAispositivedefinite.

72024/10/6§0.2FoundationofCalculus1,derivativeanddifferentiationderivativeofelementaryfunctions(sinx)’,(cosx)’,(xa)’,(lnx)’,(ax)’derivativeofcompositefunction(sin2x+a2x)’(tan(1+x2))’(f(g(x)))’highorderderivativeoffunctionstheTaylor’sexpansionofafunctionf(x)showtheTaylor’sexpansionthefollowingfunctionsaboutthepointx=0:sinx,1/(1+x),ex82024/10/6§1.2FoundationofCalculus1,multivariablesfunctionanditspartialderivativez=f(x,y),wecandefine

showthepartialderivativesofthefollowingfunctions.z=x2+y2,z=x2y2+sinxy,z=1/(x+y)TheTaylor’sexpansionoftwovariablesfunction

92024/10/6WhatisaTaylorseries?SomeexamplesofTaylorserieswhichyoumusthaveseen102024/10/6GeneralTaylorSeriesThegeneralformoftheTaylorseriesisgivenbyprovidedthatallderivativesoff(x)arecontinuousandexistintheinterval[x,x+h]WhatdoesthismeaninplainEnglish?AsArchimedeswouldhavesaid,“Givemethevalueofthefunctionatasinglepoint,andthevalueofall(first,second,andsoon)itsderivativesatthatsinglepoint,andIcangiveyouthevalueofthefunctionatanyotherpoint”112024/10/6Example—TaylorSeriesFindthevalueofgiventhatandallotherhigherorderderivativesofatarezero.Solution:122024/10/6Example(cont.)Solution:(cont.)Sincethehigherorderderivativesarezero,Notethattofindexactly,weonlyneedthevalueofthefunctionandallitsderivativesatsomeotherpoint,inthiscase132024/10/6DerivationforMaclaurinSeriesforexDerivetheMaclaurinseriesTheMaclaurinseriesissimplytheTaylorseriesaboutthepointx=0142024/10/6Derivation(cont.)SinceandtheMaclaurinseriesisthenSo,152024/10/6ErrorinTaylorSerieswheretheremainderisgivenbywherethatis,cissomepointinthedomain[x,x+h]TheTaylorpolynomialofordernofafunctionf(x)with(n+1)continuousderivativesinthedomain[x,x+h]isgivenby162024/10/6Example—errorinTaylorseriesTheTaylorseriesforatpointisgivenbyItcanbeseenthatasthenumberoftermsusedincreases,theerrorbounddecreasesandhenceabetterestimateofthefunctioncanbefound.Howmanytermswoulditrequiretogetanapproximationofe1withinamagnitudeoftrueerroroflessthan10-6.172024/10/6Example—(cont.)Solution:UsingtermsofTaylorseriesgiveserrorboundofSince182024/10/6Example—(cont.)