高阶方向导数及其应用

36820108JOURNALOFBEIJINGUNIVERSITYOFTECHNOLOGYVol．36No．8Aug．2010100124．1234．5Taylor．Taylor46G05A0254－0037201008－1135－062010-05-12．1938—．01-6．1xlFig．1Thechangeratesofthetwo-dimensionalfunctionatapointxinadirectionl．《》．1fx=fx1x2x、l=cosθsinθTfl=limα→0fx+αl－fx‖αl‖x≥01．1fl=limα→0fx+αl－fx‖αl‖=limα→0fx1+αcosθx2+αsinθ－fx1x2α=limα→0fx1+αcosθx2+αsinθ－fx1x2+αsinθ+fx1x2+αsinθ－fx1x2α=fx1cosθ+fx2sinθ[=fx1fx]2cosθsin{}θ=ΔTflΔfx[=fxx1fxx]2Tfx．．xl∈Enfx∈C1‖l‖=1α∈E1α≥0fxlfl=limα→0fx+αl－fx‖αl‖=limα→0fx+αl－fxα2010fx+αl=Fαfl=limα→0Fα－F0α=F'α|α=0=∑ni=1fxi+αli·dxi+αlidα|α=0=ΔTflΔf[=fx1…fx]nTL'Hospitalfl=limα→0fx+αl－fxα=limα→[0dfx+αldαdαd]α=∑ni=1fxxili=ΔTfl11.1xl∈Enfx∈C2fx/lfxxlfx2fx/l2fx/lxl．1.2fxl=ΔTfxl=lTΔfx2fxl2=(lfx)l=lT(Δfx)l=lTΔlTΔfx=lTΔlTΔfx+ΔΔTfxl=lTΔ2fxl21lΔlT=02ΔaTb=ΔaTb+ΔbTaΔaTb=(Δ∑ni=1aib)i=∑ni=1Δaibi=∑ni=1Δaibi+aiΔbi=Δa1…Δanb+Δb1…Δbna=ΔaTb+ΔbTa．fxl=∑ni=1fxxili2fxl2=∑nj=1x(j∑ni=1fxxil)ilj=∑nj=1∑ni=12fxxixjlilj=lTΔ2fxl1.3xl∈Enfx∈Cmm－1fx/lm－1fxxlm－1fxmmfx/lmm－1m－1fx/lm－1xl．m－1fxlm－1=∑ni1=1…∑nim－1=1m－1fxxi1…xim－1li1…lim－163118mfxlm=∑nim=1xi(m∑ni1=1…∑nim－1=1m－1fxxi1…xim－1li1…lim)－1lim=∑ni1=1…∑nim=1mfxxi1…ximli1…lim2．22.12Fig．2Multivariatefunctionatagivenpointalongaanydirectionisregardedasfunctionwithsimplevariable．fxxlx+αlαnfxα2．Fα=fx+αl3－n+1x+αlfxFα．Fαl．dFαdα=∑ni1=1fx+αlxi1+αli1dxi1+αli1dα=∑ni1=1fx+αlxi1+αli1li1d2Fαdα2=∑ni2=1[dFαd]αxi2+αli2li2=∑ni1=1∑ni2=12fx+αlxi1+αli1xi2+αli2li1li2dm－1Fαdαm－1=∑ni1=1…∑nim－1=1m－1fx+αlxi1+αli1…xim－1+αlim－1li1．．．lim－1dmFαdαm=∑ni1=1…∑nim=1mfx+αlxi1+αli1…xim+αlimli1．．．lim2α=0dmFαdαmα=0=fmxlmm=01…44α=0．2.23fx．3a、bfxfx3c、dfxf'x=0x．1fxxFαα=0F'α|α=0=fx/l=0ΔTfl=0l731120103、Fig．3ExtremeconditionsofquadraticandthricefunctionswithsimplevariableΔfx=0…0Tfxx．fxxxlfxl=0．2fxxFαα=0F'α|α=0=0F″α|α=0＞0fx/l=02fx/l2＞0Δfx=0…0TlTΔ2fxl＞0lΔ2fx．Δfx=0…0TΔ2fxfxx．fxxxlfx/l=02fx/l2＞0．2.37-8n×nAnx≠0xTAx≥0A．．fx=xTAx/2x∈En5l∈Enfx/l=Ax2fx/l2=lTΔ2fxl=lTAlA2fx/l2≥0．“”6l∈En．312Δ2fxfx383118．5、．5、．2.4Ax=bfx=12xTAx－bTx6fxl=Ax－bTlAx=bfx/l=0Ax=bx=A－1b．Afxx=A－1bAfxx=A－1b．6．2.5TaylorTaylor．TaylorTaylor．mx0mTaylorx≈x|x=0+1x|x=0x+…+1mmx|x=0xm77x∈Enfxx0l∈Enα∈E1lx=x0+αlfx0+αlαmTaylorfx=fx0+αl=Fα≈F0+dFαdαα=0α+12d2Fαdα2α=02+…+1mdmFαdαmαm848fx≈fx0+fxlx=x0α+122fxl2x=x0α2+…+1mmfxlmx=x0αm9x=x0+αlαl=x－x0αli=xi－x0i29mTaylorfx≈fx0+∑ni=1fxxix=x0xi－x0i+12∑ni1=1∑ni2=12fxxi1xi2x=x0xi1－x0i1xi2－x0i2+…+1m∑ni1=1…∑nim=1mfxxi1…ximx=x0xi1－x0i1…xim－x0imm=2fx≈fx0+ΔTfx|x=x0x－x0+12x－x0TΔ2fx|x=x0x－x03931120101、、23．．1RUDINW．PrinciplesofmathematicalanalysisM．3rded．S．l．McGraw-HillScienceEngineering1976．2．M．1990．3SPIVAKM．CalculusM．3rded．S．l．GilbertStrang1994．4THOMASGBWEIRMDHASSJetal．Thomas'calculusM．11thed．BostonAddisonWesley2004．5ZAKONE．MathematicalanalysisM．S．l．TheTrilliaGroup2004．6．、M．6．2007．7LEONSJ．LinearalgebrawithapplicationsM．．2004．8．M．5．2007．AnExtensiontoHighOrderforDirectionalDerivativeofMultivariateFunctionSUIYun-kangTheMechanicalandElectricalEngineeringCollegeBeijingUniversityofTechnologyBeijing100124ChinaAbstractThedirectionalderivativeconceptofthemultivariatefunctionisexpandedfromfirstordertohighorderinthispaper．Aftergettingthedefinitionofsecondorderdirectionalderivativeandthecalculationformulathehighorderdirectionalderivativehasbeengiven．Applicationsofthehighorderdirectionalderivativeareproposedasfollowing1Ageneralwaytoexpandingsimplevariablefunctioncharacteristicstomultivariatefunctionispresented．2Thenecessaryconditionsnecessaryandsufficientconditionsofextremevaluecriterionoffunctionareeasilyobtained．3Thegeometricalmeaningofsemi-positiveandsemi-negativedefiniteisexplainedaccordingtosecond-orderdirectionalderivatives．4Itisrevealedthatthereisextremevalueproblemofthefunctionwhenthematrixofthelinearequationsispositiveornegativedefinite．5Taylor'sexpansionformulaofthemultivariatefunctioniseasilydeduced．Keywordsdirectionalderivativehighorderdirectionalderivativesemi-positivedefinitematrixsemi-negativedefinitematrixtaylor'sexpansionofmultivariatefunction0411