FORMULAE FOR FINITE DIFFERENCE APPROXIMATIONS, QUADRATURE AND LINEAR MULTISTEP METHODS
|
|
- June Franklin
- 5 years ago
- Views:
Transcription
1 Jourl o Mtemtcl Sceces: Advces d Alctos Volume Number - Pges -9 FRMULAE FR FINITE DIFFERENCE APPRXIMATINS QUADRATURE AND LINEAR MULTISTEP METHDS RAMESH KUMAR MUTHUMALAI Dertmet o Mtemtcs D G Vsv College Arumbm Ce-66 Tml Ndu Id e-ml: rm_@oocom Abstrct We derve te derece ormule or romtg ger order dervtves wc re lcble to evel or uevel sced grds wt rbtrr order o ccurc Usg tese ormule we descrbe qudrture ormule d ler multste metods or romtg tegrls d umercl soluto o deretl equtos Itroducto Fte derece romtos re ote used or romtg dervtves to solve deretl equtos [] Te recurso reltos to clculte te wegts o te derece ormule wt rbtrr order o ccurc re gve [7] K et l ve reseted drect ormule or te elct orwrd bcwrd d cetrl derece ormule o te derece romtos wt rbtrr order or rst dervtve d Mtemtcs Subect Clsscto: A 6D 6D 6L 6L6 6M6 Kewords d rses: brcetrc dervtves te derece ormule mlct romtos qudrture ormule ler multste metods Receved Setember 6 Scetc Advces Publsers
2 RAMESH KUMAR MUTHUMALAI te cetrl derece romtos or ger dervtes [-] Geerl elct ormule or rst d ger order dervtves o te bss o te geerlzed Vdermode determt wt rbtrr order o ccurc or romtg rst d ger order dervtves re gve [] Some smle d covetol eressos o cetrl derece ormule or rst d secod dervtves re oud [6] Ts metod o udetermed coecets s lmted to evel sced grds A clss o ler multste metods suc s Adms-Bsort d Adms-Moult metods or te umercl soluto o te tl vlue roblem s bsed o te rcle o umercl tegrto [ 9] Te rge o umercl tegrto ormule bsed o deret teroltg olomls s lmted s rule to te ots o terolto I te reset stud we reset te derece ormule terms o brcetrc wegts or romtg dervtves d costructo o ler multste metods to solve deretl equtos wt smle d coveet eressos Elct Fte Derece Aromtos Recursve ormul or umercl deretto [ b] Let ( ) re dstct umbers te tervl togeter wt corresodg ( ) Let be umber dstct rom ec d [ ] deotes -t order dvded derece o t te ots We ow tt ( )! [ ] tmes () Edg rgt d sde o () b Lgrge terolto ormul te ( )! [ ] l E () tmes
3 FRMULAE FR FINITE DIFFERENCE APPRXIMATINS were d l : E ( ) ( ξ) l ( )! were l ( ) d ξ [ b] te oter d usg recursve ormul o dvded derece [ ] we ve ( [ ] ) ( )! ( ) ( ) () tmes Substtutg () () d ter smlcto we d tt ( ) l! ( ) ( ) l ( ) E () l Settg ρr r : we obt te ollowg r ( ) recursve ormul coectg uctol vlues : d dervtves u to order t : ( ) ρ! ( ) l ( ) E () Note tt te recursve ormul () c be led ol we s deret rom ec Suose tt cocde wt oe o (e ) we c stll use te recursve ormul to estmte dervtves u to order s ollows:
4 RAMESH KUMAR MUTHUMALAI were r ( r ) l ( ) r : r! r ρr (6) ( )! ρm l ( ) m m : r (7) d l : () Elct te derece ormule wt Lgrg coecets We ote eed romto o dervtves or desgg derece scemes to solve deretl equtos Tble sows some emles o elct te derece romtos [7] Tble Emles o elct te derece ormule Cse Descrto Emles o elct te derece ormule Error level Cetered regulr grd Stggered regulr grd [ ( ) ( ) ( ) ( )] 9 [ ( ) ( ) 9 ( ) ( )] ( ) ( ) Regulr grd ( ) [ ( ) ( ) ( ) ( ) ( )] Suose oe use (6) to d t-t dervtve o t we get te ollowg orm o te elct romto:
5 FRMULAE FR FINITE DIFFERENCE APPRXIMATINS () t t! tχ Mt s s tχ (9) s were χ () d M ( ) ( ξ ) l l () ( ) ( )! d s s uow ucto Now substtutg ll M s (9) d ter smlcto we obt () t t! l tχ ED ( ; ) () t ( ) tχ were E D ( ; ) l tχ m m ( t m ) ( ξm ) ( t m)! or To d ll s set () we t we d d t l t s ( ) s t t l Rerrgg bove equto d settg ρr r : t r ( ) we d tt
6 6 RAMESH KUMAR MUTHUMALAI t ρt () Sce ll oter uow s c be oud recursvel rom () For equll sced grds I ew secl cses (eg or elct romtos o equsced grds) te otml wegts re ow closed orm Cses to Tble sow tree suc emles Let ( m ) re equll sced grds Settg () d relcg Lgrg coecets l b ( m ) A te ( m )!! m A m!! () It ollows tt ρ r ( m ) r A r m Settg ρ r ( ) W r r b ve () d ter smlcto we were gves Wr t t b b W () ( m ) A m r Tus or evel sced grds () t () t t! ( m ) b t A bt ( ) t (6) t m
7 FRMULAE FR FINITE DIFFERENCE APPRXIMATINS 7 Elct te derece ormule wt brcetrc wegts Te ormul gve () c be moded to eve more elegt orm wt brcetrc wegts Dvde l o bot sdes o () were w l l (7) w : () We d tt ρr w r r l δ s (9) Now dvdg () b l d usg (9) ter smlcto tt t δt () Dvdg bot umertor d deomtor o () b l usg (7) d ter smlcto we obt () t tχ w E t! t tχ δ () ( ) All te uow s c be oud rom () Qudrture Formule d Ler Multste Metods Tble sows some emles o qudrture ormule d ler multste metods Cses - d Cse re emles o Newto-Cotes tegrto ormule d Adms ormule or ler multste metods
8 RAMESH KUMAR MUTHUMALAI Formul or umercl tegrto Let [ ] d re dstct umbers o te closed tervl ( C ) [ ] Te roblem o umercl tegrto s to romte te dete tegrl () t dt Sce olomls re es to tegrte b usg Tlor seres we d tt ( () ) t dt ( ) () ( )! Relcg ger order dervtves ( ) : usg () we ve χ () t dt M χ Tble Emles o qudrture ormule d ler multste metods Cse Descrto Emles o qudrture ormule Error level Smso rule Newto-Cotes oe ormul Adms-Moult corrector ormul d [ ( ) ( )] d [ ( ) ( ) ( )] [ 9 ] ( ) ( ) ( ) m Settg γ m : d ter smlcto m m we d tt
9 FRMULAE FR FINITE DIFFERENCE APPRXIMATINS 9 Substtutg ll te vlues o obt () t dt γχ M γ ( ) M s rom () d ter smlcto we γ () t dt γ χ l EI ( γ; ) χ () were χ s stll deed b () d E I ( γ; ) l mχ ( m ) ( ξm ) γ ( m )! m ( ) Now dvdg l o bot umertor d deomtor o () usg dett (7) d ter smlcto we get γ () t dt γχ w E ( ; ) I γ () δ ( ) Newto-Cotes ormule χ Let ( ) re equll sced grds Now relcg b d b () d ter smlcto we obt were α d α () α m m m m Te Equto () s ( ) -t ot ormul or Newto-Cotes closed tegrto ormul
10 RAMESH KUMAR MUTHUMALAI Sce te oe qudrture ormule do ot requre te uctol vlues t te lmt ots o tegrto ssume tt d Now relcg b d b () d ter smlcto we get β d ( ) () were β m m m m Te Equto () s ( ) -t ot ormul or Newto-Cotes oe tegrto ormul Ler multste metods Let re dstct umbers o te tervl [ ] were I we tegrte te deretl equto ( ) rom to usg () d ter smlcto we d tt ξ ξ δ w (6) ( ) were ξ m m m m Te m re stll deed b () t te ots Ts d o multste ormul (6) s ow s ( ) -ot redctor ormul
11 FRMULAE FR FINITE DIFFERENCE APPRXIMATINS I we tegrte te deretl equto ( ) rom to usg () d ter smlcto we d tt τ τ δ w ( ) were (7) τ m m m m Ts d o multste ormul (7) s ow s ( ) -ot corrector ormul Comrso wt ter Formule I ts secto we comre te ew elct te derece ormule qudrture ormule d redctor-corrector ormule wt ormer ormule o vrous ot dstrbutos Elct te derece romtos Algortm For rbtrr sced grds < < < < d ow te ucto vlues ( ) t ( ) oe les ( ) -ot ormul to estmte te -t dervtve o t te tere re ve smle stes Ste For : clculte w w Ste For m : clculte δm m ( ) r Ste Set d or r : clculte r δ r
12 RAMESH KUMAR MUTHUMALAI Ste For : clculte c tχ t ( ) were χ Ste Clculte () t t! t χ δ wc Te Algortm requres cost o los Te quttes tt ve to be comuted oertos do ot deed o te dt e c esl obt more umercl deretto ormule or rbtrr sced grds usg Algortm We rovde ollowg 6-ot umercl deretto ormule o evel sced grds s emles or rst secod d trd order dervtves wose error levels re ( ) ( ) d ( ) resectvel ( 7 ) 7 6 ( 6 6 ) ( ) ( ) 6 6 ( 6 6 ) ( 7 ) ( ) ( 6 6 ) ( ) () () ()
13 FRMULAE FR FINITE DIFFERENCE APPRXIMATINS Comrg te ormer ormule gve [6 7 -] wt te corresodg orwrd bcwrd d cetrl derece ormule () () d () resectvel t could be oud tt te re equvlet Te bove metoed ormule re lso oud rom (7) b vrg te vlues o m d I rtculr m d m te (7) gves orwrd bcwrd d cetrl derece romtos o rst d ger order dervtves I ct te rst d secod order dervtve ormule [6] re secl cses o (7) or evel sced ots For uevel sced grds we coose Cebsev ot dstrbutos Te ollowg re some emles o te -ots d 7-ots ormule or rst secod d trd order dervtves o Cebsev ots o secod d s ollows: () () Te ormule gve [] estmte ger order dervtves ol t te smlg odes were te ew ormul gve () used to estmte dervtves eve we te uctol vlues t smlg odes re ow or ot ow Ule recurso ormule or clculto o wegts gve [7] ts metod o romto gves elct ormule tt use gve ucto vlues t smlg odes drectl d esl to clculte umercl romtos o rbtrr order t smlg dt or te rst d ger dervtves Also t eed less clcultos burde
14 RAMESH KUMAR MUTHUMALAI comutg tme d storge sce to estmte te dervtves t te oter metods stted bove It c be drectl used or desgg derece scemes o DEs d PDEs d solvg tem Qudrture ormule d ler multste metods Algortm For rbtrr sced grds < < < < d ow te ucto vlues ( ) t ( ) oe les ( ) -ot ormul to estmte te tegrl () t dt te tere re s stes Ste For : clculte w w Ste For m : clculte δm m ( ) r Ste For r : clculte r δr m Ste For : clculte γ m m m Ste For : clculte d χ γ ( ) were χ Ste 6 Clculte () t dt γ χ wd δ Te Algortm requres some quttes deedet o wt cost o ( ) los but do ot deed o te dt Usg ts lgortm oe c esl get more ormule or rbtrr sced grds wt degree o
15 FRMULAE FR FINITE DIFFERENCE APPRXIMATINS ccurc Te ollowg re some emles o Newto-Cotes closed tegrto ormule: d d d d (6) Smlrl we c obt Newto-Cotes oe qudrture ormule some o tem re lsted below: d d d d (7) For uevel sced ots we coose Cebsev ot dstrbutos Te ollowg re te -ots -ots d -ots ormule o Cebsev ots o secod d: 7 9 d d d () Smlrl te ollowg re te -ot ots d -ots ormule o Cebsev ots o rst d:
16 RAMESH KUMAR MUTHUMALAI d d d (9) Te ollowg re te some secl ormule or ler multste metods I we te we get Adms-Bsort redctor ormule o -ots -ots -ots d -ots resectvel () Smlrl we te we obt Adms-Bsort redctor ormule o -ots -ots -ots d -ots resectvel c c c c () Also we c geerte ormule or rbtrr sced grds Te ollowg re -ots -ots d -ots redctor ormule obted b tg resectvel to redct t
17 FRMULAE FR FINITE DIFFERENCE APPRXIMATINS 7 ( 7 7 ) ( ) ( ) 76 () Smlrl te ollowg re -ots -ots d -ots corrector ormule obted b tg resectvel to correct t ( 6 ) 7 c ( 6 6 ) 9 c ( ) c 6 () It s esl oud tt tese emle ormule re te sme s to tose ow corresodg umercl tegrto ormule bsed o teroltg olomls or rbtrr sced grds [ 9] Cocluso I cocluso we ote tt te derece ormule qudrture ormule d ler multste metods terms o brcetrc wegts ve bee develoed ts rtcle Frstl we ve troduced recursve ormul or romtg ger order dervtves Usg ts recursve ormul we ve derved elct te derece ormule or romtg ger order dervtves terms o Lgrge coecets d brcetrc wegts or rbtrr sced grds Moreover usg ts ew elct ormul d Tlor seres we ve derved umercl tegrto ormul or romtg dete tegrls Newto-Cotes ormule d Adms ormule re secl cses o ts ew ormul
18 RAMESH KUMAR MUTHUMALAI Secodl we ve comred te ew elct te derece ormule wt oter ormule We ve see tt ts ew ormul or romtg dervtves requres ver less comutto tme d storge sce We ve sow tt te orwrd bcwrd d cetrl derece romtos re secl cses o ts ew ormul Smlrl we ve sow te comrso o ew tegrto ormul or qudrture ormule d costructo o ler multste metods wt ormer ormule Former ormule or tegrto d ler multste metods re lmted to evel sced grds but ew ormul reseted ere s qute useul o rbtrr sced grds Reereces [] K E Atso A Itroducto to Numercl Alss d Edto Jo Wle & Sos New Yor 99 [] J P Berrut d L N Trete Brcetrc Lgrge terolto SIAM Rev 6() () -7 [] R L Burde d J D Fres Numercl Alss 7t Edto Broos d Cole Pcc Grove CA [] S C Cr d R P Cle Numercl Metods or Egeers rd Edto McGrw-Hll New Yor 99 [] S D Cote d Crl de boor Elemetr Numercl Alss rd Edto McGrw- Hll New Yor 9 [6] M Dvorov Formule or umercl deretto JCAAM (7) 77- e-rtrv:mtna/69 [7] B Forberg Clculto o wegts te derece ormuls SIAM Rev () (99) 6-69 [] N J Hgm Te umercl stblt o brcetrc Lgrge terolto IMAJNA () 7-6 [9] F B Hldebrd Itroducto to Numercl Alss d Edto McGrw-Hll New Yor 97 [] I R K d R b Closed-orm eressos or te te derece romtos o rst d ger dervtves bsed o Tlor seres J Comut Al Mt 7 (999) 79-9 [] I R K d R b New te derece ormuls or umercl deretto J Comut Al Mt 6 () 69-76
19 FRMULAE FR FINITE DIFFERENCE APPRXIMATINS 9 [] I R K R b d N Hozum Mtemtcl roo o closed orm eressos or te derece romtos bsed o Tlor seres J Comut Al Mt () -9 [] I R K d R b Tlor seres bsed te derece romtos o ger degree dervtves J Comut Al Mt () - [] J L Geerl elct derece ormuls or umercl deretto J Comut Al Mt () 9- g
Some Unbiased Classes of Estimators of Finite Population Mean
Itertol Jourl O Mtemtcs Ad ttstcs Iveto (IJMI) E-IN: 3 4767 P-IN: 3-4759 Www.Ijms.Org Volume Issue 09 etember. 04 PP-3-37 ome Ubsed lsses o Estmtors o Fte Poulto Me Prvee Kumr Msr d s Bus. Dertmet o ttstcs,
More informationMathematically, integration is just finding the area under a curve from one point to another. It is b
Numerl Metods or Eg [ENGR 9] [Lyes KADEM 7] CHAPTER VI Numerl Itegrto Tops - Rem sums - Trpezodl rule - Smpso s rule - Rrdso s etrpolto - Guss qudrture rule Mtemtlly, tegrto s just dg te re uder urve rom
More informationChapter Simpson s 1/3 Rule of Integration. ( x)
Cpter 7. Smpso s / Rule o Itegrto Ater redg ts pter, you sould e le to. derve te ormul or Smpso s / rule o tegrto,. use Smpso s / rule t to solve tegrls,. develop te ormul or multple-segmet Smpso s / rule
More informationCS321. Introduction to Numerical Methods
CS Itroducto to Numercl Metods Lecture Revew Proessor Ju Zg Deprtmet o Computer Scece Uversty o Ketucky Legto, KY 6 6 Mrc 7, Number Coverso A geerl umber sould be coverted teger prt d rctol prt seprtely
More informationChapter Runge-Kutta 2nd Order Method for Ordinary Differential Equations
Cter. Runge-Kutt nd Order Metod or Ordnr Derentl Eutons Ater redng ts cter ou sould be ble to:. understnd te Runge-Kutt nd order metod or ordnr derentl eutons nd ow to use t to solve roblems. Wt s te Runge-Kutt
More informationNumerical Differentiation and Integration
Numerl Deretto d Itegrto Overvew Numerl Deretto Newto-Cotes Itegrto Formuls Trpezodl rule Smpso s Rules Guss Qudrture Cheyshev s ormul Numerl Deretto Forwrd te dvded deree Bkwrd te dvded deree Ceter te
More informationCS321. Numerical Analysis
CS Numercl Alyss Lecture 4 Numercl Itegrto Proessor Ju Zg Deprtmet o Computer Scece Uversty o Ketucky Legto, KY 456 6 Octoer 6, 5 Dete Itegrl A dete tegrl s tervl or tegrto. For ed tegrto tervl, te result
More informationThe Derivation of Implicit Second Derivative Method for Solving Second-Order Stiff Ordinary Differential Equations Odes.
IOSR Jourl o Mtetcs (IOSR-JM) e-issn: - p-issn: 9-6X. Volue Issue Ver. I (Mr. - Apr. ) PP - www.osrourls.org Te Dervto o Iplct Secod Dervtve Metod or Solvg Secod-Order St Ordr Deretl Equtos Odes. Y. Skwe
More informationIn Calculus I you learned an approximation method using a Riemann sum. Recall that the Riemann sum is
Mth Sprg 08 L Approxmtg Dete Itegrls I Itroducto We hve studed severl methods tht llow us to d the exct vlues o dete tegrls However, there re some cses whch t s ot possle to evlute dete tegrl exctly I
More informationAn Extended Mixture Inverse Gaussian Distribution
Avlble ole t htt://wwwssstjscssructh Su Sudh Scece d Techology Jourl 016 Fculty o Scece d Techology, Su Sudh Rjbht Uversty A Eteded Mture Iverse Guss Dstrbuto Chookt Pudrommrt * Fculty o Scece d Techology,
More informationChapter 12-b Integral Calculus - Extra
C - Itegrl Clulus Cpter - Itegrl Clulus - Etr Is Newto Toms Smpso BONUS Itroduto to Numerl Itegrto C - Itegrl Clulus Numerl Itegrto Ide s to do tegrl smll prts, lke te wy we preseted tegrto: summto. Numerl
More informationDifferential Entropy 吳家麟教授
Deretl Etropy 吳家麟教授 Deto Let be rdom vrble wt cumultve dstrbuto ucto I F s cotuous te r.v. s sd to be cotuous. Let = F we te dervtve s deed. I te s clled te pd or. Te set were > 0 s clled te support set
More informationOutline. Finite Difference Grids. Numerical Analysis. Finite Difference Grids II. Finite Difference Grids III
Itrodcto to Nmercl Alyss Mrc, 9 Nmercl Metods or PDEs Lrry Cretto Meccl Egeerg 5B Semr Egeerg Alyss Mrc, 9 Otle Revew mdterm soltos Revew bsc mterl o mercl clcls Expressos or dervtves, error d error order
More informationIntegration by Parts for D K
Itertol OPEN ACCESS Jourl Of Moder Egeerg Reserc IJMER Itegrto y Prts for D K Itegrl T K Gr, S Ry 2 Deprtmet of Mtemtcs, Rgutpur College, Rgutpur-72333, Purul, West Begl, Id 2 Deprtmet of Mtemtcs, Ss Bv,
More information12 Iterative Methods. Linear Systems: Gauss-Seidel Nonlinear Systems Case Study: Chemical Reactions
HK Km Slghtly moded //9 /8/6 Frstly wrtte t Mrch 5 Itertve Methods er Systems: Guss-Sedel Noler Systems Cse Study: Chemcl Rectos Itertve or ppromte methods or systems o equtos cosst o guessg vlue d the
More information4. Runge-Kutta Formula For Differential Equations. A. Euler Formula B. Runge-Kutta Formula C. An Example for Fourth-Order Runge-Kutta Formula
NCTU Deprme o Elecrcl d Compuer Egeerg Seor Course By Pro. Yo-Pg Ce. Ruge-Ku Formul For Derel Equos A. Euler Formul B. Ruge-Ku Formul C. A Emple or Four-Order Ruge-Ku Formul
More information4. Runge-Kutta Formula For Differential Equations
NCTU Deprme o Elecrcl d Compuer Egeerg 5 Sprg Course by Pro. Yo-Pg Ce. Ruge-Ku Formul For Derel Equos To solve e derel equos umerclly e mos useul ormul s clled Ruge-Ku ormul
More informationUNIVERSITI KEBANGSAAN MALAYSIA PEPERIKSAAN AKHIR SEMESTER I SESI AKADEMIK 2007/2008 IJAZAH SARJANAMUDA DENGAN KEPUJIAN NOVEMBER 2007 MASA : 3 JAM
UNIVERSITI KEBANGSAAN MALAYSIA PEPERIKSAAN AKHIR SEMESTER I SESI AKADEMIK 7/8 IJAZAH SARJANAMUDA DENGAN KEPUJIAN NOVEMBER 7 MASA : 3 JAM KOD KURSUS : KKKQ33/KKKF33 TAJUK : PENGIRAAN BERANGKA ARAHAN :.
More informationEuropean Journal of Mathematics and Computer Science Vol. 3 No. 1, 2016 ISSN ISSN
Euroe Jour of Mthemtcs d omuter Scece Vo. No. 6 ISSN 59-995 ISSN 59-995 ON AN INVESTIGATION O THE MATRIX O THE SEOND PARTIA DERIVATIVE IN ONE EONOMI DYNAMIS MODE S. I. Hmdov Bu Stte Uverst ABSTRAT The
More informationME 501A Seminar in Engineering Analysis Page 1
Mtr Trsformtos usg Egevectors September 8, Mtr Trsformtos Usg Egevectors Lrry Cretto Mechcl Egeerg A Semr Egeerg Alyss September 8, Outle Revew lst lecture Trsformtos wth mtr of egevectors: = - A ermt
More informationIntroduction to mathematical Statistics
Itroducto to mthemtcl ttstcs Fl oluto. A grou of bbes ll of whom weghed romtely the sme t brth re rdomly dvded to two grous. The bbes smle were fed formul A; those smle were fed formul B. The weght gs
More informationCalculating the Values of Multiple Integrals by Replacing the Integrands Interpolation by Interpolation Polynomial
Jour o omputtos & Modeg vo o -5 ISSN: 79-75 prt 79-5 oe Scepress Ltd cutg te Vues o Mutpe Itegrs by Repcg te Itegrds Iterpoto by Iterpoto Poyom S Nzrov d bduzzov bstrct Te or des t te costructo o mutdmeso
More informationPatterns of Continued Fractions with a Positive Integer as a Gap
IOSR Jourl of Mthemtcs (IOSR-JM) e-issn: 78-578, -ISSN: 39-765X Volume, Issue 3 Ver III (My - Ju 6), PP -5 wwwosrjourlsorg Ptters of Cotued Frctos wth Postve Iteger s G A Gm, S Krth (Mthemtcs, Govermet
More informationStats & Summary
Stts 443.3 & 85.3 Summr The Woodbur Theorem BCD B C D B D where the verses C C D B, d est. Block Mtrces Let the m mtr m q q m be rttoed to sub-mtrces,,,, Smlrl rtto the m k mtr B B B mk m B B l kl Product
More informationCURVE FITTING LEAST SQUARES METHOD
Nuercl Alss for Egeers Ger Jord Uverst CURVE FITTING Although, the for of fucto represetg phscl sste s kow, the fucto tself ot be kow. Therefore, t s frequetl desred to ft curve to set of dt pots the ssued
More informationICS141: Discrete Mathematics for Computer Science I
Uversty o Hw ICS: Dscrete Mthemtcs or Computer Scece I Dept. Iormto & Computer Sc., Uversty o Hw J Stelovsy bsed o sldes by Dr. Be d Dr. Stll Orgls by Dr. M. P. Fr d Dr. J.L. Gross Provded by McGrw-Hll
More informationMTH 146 Class 7 Notes
7.7- Approxmte Itegrto Motvto: MTH 46 Clss 7 Notes I secto 7.5 we lered tht some defte tegrls, lke x e dx, cot e wrtte terms of elemetry fuctos. So, good questo to sk would e: How c oe clculte somethg
More informationCHAPTER 6 CURVE FITTINGS
CHAPTER 6 CURVE FITTINGS Chpter 6 : TOPIC COVERS CURVE FITTINGS Lest-Squre Regresso - Ler Regresso - Poloml Regresso Iterpolto - Newto s Dvded-Derece Iterpoltg Polomls - Lgrge Iterpoltg Polomls - Sple
More informationNumerical Analysis Third Class Chemical Engineering Department University of Technology Assoc. Prof. Dr. Zaidoon Mohsin shakor
Numercl Alss Thrd Clss Chemcl Egeerg Deprtmet Uverst o Techolog Assoc. Pro. Dr. Zdoo Mohs shkor Itroducto to Numercl Alss U. Alss versus Numercl Alss The word lss mthemtcs usull mes who to solve problem
More informationМИНИСТЕРСТВО ОБРАЗОВАНИЯ И НАУКИ РОССИЙСКОЙ ФЕДЕРАЦИИ. Численные методы. Учебно-методическое пособие
МИНИСТЕРСТВО ОБРАЗОВАНИЯ И НАУКИ РОССИЙСКОЙ ФЕДЕРАЦИИ Нижегородский государственный университет им. Н.И. Лобачевского Численные методы К.А.Баркалов Учебно-методическое пособие Рекомендовано методической
More informationKeywords: Heptic non-homogeneous equation, Pyramidal numbers, Pronic numbers, fourth dimensional figurate numbers.
[Gol 5: M 0] ISSN: 77-9655 IJEST INTENTIONL JOUNL OF ENGINEEING SCIENCES & ESECH TECHNOLOGY O the Hetc No-Hoogeeous Euto th Four Ukos z 6 0 M..Gol * G.Suth S.Vdhlksh * Dertet of MthetcsShrt Idr Gdh CollegeTrch
More informationNumerical differentiation (Finite difference)
MA umercl Alyss Week 4 To tk d wrte portle progrms we eed strct model of ter computtol evromet. Ftful models do est ut tey revel tt evromet to e too dverse forcg portle progrmmers to lot eve te smplest
More informationSpectrum Estimation by Several Interpolation Methods
IJCSNS Itertol Jourl o Computer Scece d Network Securty VOL.6 No.A Februry 006 05 Spectrum Estmto by Severl Iterpolto Metods Mbu Isr Oym Ntol College o Tecology Oym-S Tocg 33-0806 JAPAN Summry I ts pper
More informationMATLAB PROGRAM FOR THE NUMERICAL SOLUTION OF DUHAMEL CONVOLUTION INTEGRAL
Bullet of te Trslv Uversty of Brşov Vol 5 (54-1 Seres 1: Specl Issue No 1 MATLAB PROGRAM FOR THE NUMERICAL SOLUTION OF DUHAMEL CONVOLUTION INTEGRAL M BOTIŞ 1 Astrct: I te ler lyss of structures troug modl
More informationDATA FITTING. Intensive Computation 2013/2014. Annalisa Massini
DATA FITTING Itesve Computto 3/4 Als Mss Dt fttg Dt fttg cocers the problem of fttg dscrete dt to obt termedte estmtes. There re two geerl pproches two curve fttg: Iterpolto Dt s ver precse. The strteg
More informationMATRIX AND VECTOR NORMS
Numercl lyss for Egeers Germ Jord Uversty MTRIX ND VECTOR NORMS vector orm s mesure of the mgtude of vector. Smlrly, mtr orm s mesure of the mgtude of mtr. For sgle comoet etty such s ordry umers, the
More informationi+1 by A and imposes Ax
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING CAMBRIDGE, MASSACHUSETTS 09.9 NUMERICAL FLUID MECHANICS FALL 009 Mody, October 9, 009 QUIZ : SOLUTIONS Notes: ) Multple solutos
More informationORDINARY DIFFERENTIAL EQUATIONS
6 ORDINARY DIFFERENTIAL EQUATIONS Introducton Runge-Kutt Metods Mult-step Metods Sstem o Equtons Boundr Vlue Problems Crcterstc Vlue Problems Cpter 6 Ordnr Derentl Equtons / 6. Introducton In mn engneerng
More informationSequences and summations
Lecture 0 Sequeces d summtos Istructor: Kgl Km CSE) E-ml: kkm0@kokuk.c.kr Tel. : 0-0-9 Room : New Mleum Bldg. 0 Lb : New Egeerg Bldg. 0 All sldes re bsed o CS Dscrete Mthemtcs for Computer Scece course
More informationAnalytical Approach for the Solution of Thermodynamic Identities with Relativistic General Equation of State in a Mixture of Gases
Itertol Jourl of Advced Reserch Physcl Scece (IJARPS) Volume, Issue 5, September 204, PP 6-0 ISSN 2349-7874 (Prt) & ISSN 2349-7882 (Ole) www.rcourls.org Alytcl Approch for the Soluto of Thermodymc Idettes
More informationSUM PROPERTIES FOR THE K-LUCAS NUMBERS WITH ARITHMETIC INDEXES
Avlble ole t http://sc.org J. Mth. Comput. Sc. 4 (04) No. 05-7 ISSN: 97-507 SUM PROPERTIES OR THE K-UCAS NUMBERS WITH ARITHMETIC INDEXES BIJENDRA SINGH POOJA BHADOURIA AND OMPRAKASH SIKHWA * School of
More informationIntroduction to Numerical Differentiation and Interpolation March 10, !=1 1!=1 2!=2 3!=6 4!=24 5!= 120
Itroducto to Numercal Deretato ad Iterpolato Marc, Itroducto to Numercal Deretato ad Iterpolato Larr Caretto Mecacal Egeerg 9 Numercal Aalss o Egeerg stems Marc, Itroducto Iterpolato s te use o a dscrete
More informationObjective of curve fitting is to represent a set of discrete data by a function (curve). Consider a set of discrete data as given in table.
CURVE FITTING Obectve curve ttg s t represet set dscrete dt b uct curve. Csder set dscrete dt s gve tble. 3 3 = T use the dt eectvel, curve epress s tted t the gve dt set, s = + = + + = e b ler uct plml
More information4 Round-Off and Truncation Errors
HK Km Slgtly moded 3//9, /8/6 Frstly wrtte at Marc 5 4 Roud-O ad Trucato Errors Errors Roud-o Errors Trucato Errors Total Numercal Errors Bluders, Model Errors, ad Data Ucertaty Recallg, dv dt Δv v t Δt
More informationThe z-transform. LTI System description. Prof. Siripong Potisuk
The -Trsform Prof. Srpog Potsuk LTI System descrpto Prevous bss fucto: ut smple or DT mpulse The put sequece s represeted s ler combto of shfted DT mpulses. The respose s gve by covoluto sum of the put
More informationComputational Modelling
Computtol Modellg Notes orfeeg Desg d Computg Oct t v.d Computtol Modellg b Pro Smo J. Co sjc@soto.c.uk Wt s Computtol Modellg? Computtol modellg d te use o ssocted umercl metods s bout coosg te rgt lgortm
More informationAssignment 7/MATH 247/Winter, 2010 Due: Friday, March 19. Powers of a square matrix
Assgmet 7/MATH 47/Wter, 00 Due: Frday, March 9 Powers o a square matrx Gve a square matrx A, ts powers A or large, or eve arbtrary, teger expoets ca be calculated by dagoalzg A -- that s possble (!) Namely,
More informationAvailable online through
Avlble ole through wwwmfo FIXED POINTS FOR NON-SELF MAPPINGS ON CONEX ECTOR METRIC SPACES Susht Kumr Moht* Deprtmet of Mthemtcs West Begl Stte Uverst Brst 4 PrgsNorth) Kolt 76 West Begl Id E-ml: smwbes@yhoo
More informationINTERPOLATION(2) ELM1222 Numerical Analysis. ELM1222 Numerical Analysis Dr Muharrem Mercimek
ELM Numerl Alss Dr Murrem Merme INTEROLATION ELM Numerl Alss Some of te otets re dopted from Luree V. Fusett Appled Numerl Alss usg MATLAB. rete Hll I. 999 ELM Numerl Alss Dr Murrem Merme Tod s leture
More informationThe Computation of Common Infinity-norm Lyapunov Functions for Linear Switched Systems
ISS 746-7659 Egd UK Jour of Iformto d Comutg Scece Vo. 6 o. 4. 6-68 The Comutto of Commo Ifty-orm yuov Fuctos for er Swtched Systems Zheg Che Y Go Busess Schoo Uversty of Shgh for Scece d Techoogy Shgh
More informationSOME REMARKS ON HORIZONTAL, SLANT, PARABOLIC AND POLYNOMIAL ASYMPTOTE
D I D A C T I C S O F A T H E A T I C S No (4) 3 SOE REARKS ON HORIZONTAL, SLANT, PARABOLIC AND POLYNOIAL ASYPTOTE Tdeusz Jszk Abstct I the techg o clculus, we cosde hozotl d slt symptote I ths ppe the
More informationArea and the Definite Integral. Area under Curve. The Partition. y f (x) We want to find the area under f (x) on [ a, b ]
Are d the Defte Itegrl 1 Are uder Curve We wt to fd the re uder f (x) o [, ] y f (x) x The Prtto We eg y prttog the tervl [, ] to smller su-tervls x 0 x 1 x x - x -1 x 1 The Bsc Ide We the crete rectgles
More informationChapter 2 Intro to Math Techniques for Quantum Mechanics
Wter 3 Chem 356: Itroductory Qutum Mechcs Chpter Itro to Mth Techques for Qutum Mechcs... Itro to dfferetl equtos... Boudry Codtos... 5 Prtl dfferetl equtos d seprto of vrbles... 5 Itroducto to Sttstcs...
More informationChapter 5. Curve fitting
Chapter 5 Curve ttg Assgmet please use ecell Gve the data elow use least squares regresso to t a a straght le a power equato c a saturato-growthrate equato ad d a paraola. Fd the r value ad justy whch
More informationChapter Gauss-Seidel Method
Chpter 04.08 Guss-Sedel Method After redg ths hpter, you should be ble to:. solve set of equtos usg the Guss-Sedel method,. reogze the dvtges d ptflls of the Guss-Sedel method, d. determe uder wht odtos
More informationPOWERS OF COMPLEX PERSYMMETRIC ANTI-TRIDIAGONAL MATRICES WITH CONSTANT ANTI-DIAGONALS
IRRS 9 y 04 wwwrppresscom/volumes/vol9issue/irrs_9 05pdf OWERS OF COLE ERSERIC I-RIIGOL RICES WIH COS I-IGOLS Wg usu * Q e Wg Hbo & ue College of Scece versty of Shgh for Scece d echology Shgh Ch 00093
More informationClosed Form Evaluations of Some Exponential Sums N. A. Carella, March 2011.
Close Form Evlutos o Some Eoetl Sums N. A. Crell Mrch. Abstrct: Ths ote roves ew close orm evlutos o ew clsses o eoetl sums ssocte wth elltc curves hyerelltc curves. Keywors: Chrcter sums Eoetl sums Elltc
More informationComputation of Fifth Degree of Spline Function Model by Using C++ Programming
www.ijci.org 89 Computton o Ft Degree o plne Functon Model b Usng C Progrmmng Frdun K. Hml, Aln A. Abdull nd Knd M. Qdr Mtemtcs Dept, Unverst o ulmn, ulmn, IRAQ Mtemtcs Dept, Unverst o ulmn, ulmn, IRAQ
More informationCalculation of Effective Resonance Integrals
Clculo of ffecve Resoce egrls S.B. Borzkov FLNP JNR Du Russ Clculo of e effecve oce egrl wc cludes e rel eerg deedece of euro flux des d correco o e euro cure e smle s eeded for ccure flux deermo d euro
More informationMathematical models for computer systems behaviour
Mthemtcl models for comuter systems ehvour Gols : redct comuter system ehvours - erformces mesuremets, - comrso of systems, - dmesog, Methodology : - modellg evromet (stochstc rocess) - modellg system
More informationPubH 7405: REGRESSION ANALYSIS REGRESSION IN MATRIX TERMS
PubH 745: REGRESSION ANALSIS REGRESSION IN MATRIX TERMS A mtr s dspl of umbers or umercl quttes ld out rectgulr rr of rows d colums. The rr, or two-w tble of umbers, could be rectgulr or squre could be
More informationSolutions Manual for Polymer Science and Technology Third Edition
Solutos ul for Polymer Scece d Techology Thrd Edto Joel R. Fred Uer Sddle Rver, NJ Bosto Idols S Frcsco New York Toroto otrel Lodo uch Prs drd Cetow Sydey Tokyo Sgore exco Cty Ths text s ssocted wth Fred/Polymer
More informationNumerical Integration - (4.3)
Numericl Itegrtio - (.). Te Degree of Accurcy of Qudrture Formul: Te degree of ccurcy of qudrture formul Qf is te lrgest positive iteger suc tt x k dx Qx k, k,,,...,. Exmple fxdx 9 f f,,. Fid te degree
More informationChapter Simpson s 1/3 Rule of Integration. ( x)
Cper 7. Smpso s / Rule o Iegro Aer redg s per, you sould e le o. derve e ormul or Smpso s / rule o egro,. use Smpso s / rule o solve egrls,. develop e ormul or mulple-segme Smpso s / rule o egro,. use
More informationCS473-Algorithms I. Lecture 3. Solving Recurrences. Cevdet Aykanat - Bilkent University Computer Engineering Department
CS473-Algorthms I Lecture 3 Solvg Recurreces Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet Solvg Recurreces The lyss of merge sort Lecture requred us to solve recurrece. Recurreces re lke solvg
More informationGENERALIZED OPERATIONAL RELATIONS AND PROPERTIES OF FRACTIONAL HANKEL TRANSFORM
S. Res. Chem. Commu.: (3 8-88 ISSN 77-669 GENERLIZED OPERTIONL RELTIONS ND PROPERTIES OF FRCTIONL NKEL TRNSFORM R. D. TYWDE *. S. GUDDE d V. N. MLLE b Pro. Rm Meghe Isttute o Teholog & Reserh Bder MRVTI
More informationGEOMETRY OF JENSEN S INEQUALITY AND QUASI-ARITHMETIC MEANS
IJRRS ugust 3 wwwararesscom/volumes/volissue/ijrrs 4d GEOMETRY OF JENSEN S INEQULITY ND QUSI-RITHMETIC MENS Zlato avć Mechacal Egeerg Facult Slavos Brod Uverst o Osje Trg Ivae Brlć Mažurać 35 Slavos Brod
More informationOn Signed Product Cordial Labeling
Appled Mathematcs 55-53 do:.436/am..6 Publshed Ole December (http://www.scrp.or/joural/am) O Sed Product Cordal Label Abstract Jayapal Baskar Babujee Shobaa Loaatha Departmet o Mathematcs Aa Uversty Chea
More informationInterpolation. 1. What is interpolation?
Iterpoltio. Wht is iterpoltio? A uctio is ote give ol t discrete poits such s:.... How does oe id the vlue o t other vlue o? Well cotiuous uctio m e used to represet the + dt vlues with pssig through the
More informationITERATIVE METHODS FOR SOLVING SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS
Numercl Alyss for Egeers Germ Jord Uversty ITERATIVE METHODS FOR SOLVING SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS Numercl soluto of lrge systems of ler lgerc equtos usg drect methods such s Mtr Iverse, Guss
More informationA Damped Guass-Newton Method for the Generalized Linear Complementarity Problem
Itertol Jourl of Comuter d Iformto Techology (ISSN: 79 764 olume Issue 4 July 3 A Dmed Guss-Newto Method for the Geerlzed Ler Comlemetrty Problem Huju L Houchu Zhou Dertmet of Mthemtcs Ly Uversty L Shdog
More informationSoo King Lim Figure 1: Figure 2: Figure 3: Figure 4: Figure 5: Figure 6: Figure 7: Figure 8: Figure 9: Figure 10: Figure 11:
Soo Kg Lm 1.0 Nested Fctorl Desg... 1.1 Two-Fctor Nested Desg... 1.1.1 Alss of Vrce... Exmple 1... 5 1.1. Stggered Nested Desg for Equlzg Degree of Freedom... 7 1.1. Three-Fctor Nested Desg... 8 1.1..1
More informationBasic Concepts in Numerical Analysis November 6, 2017
Basc Cocepts Nuercal Aalyss Noveber 6, 7 Basc Cocepts Nuercal Aalyss Larry Caretto Mecacal Egeerg 5AB Sear Egeerg Aalyss Noveber 6, 7 Outle Revew last class Mdter Exa Noveber 5 covers ateral o deretal
More informationContinuous Random Variables: Conditioning, Expectation and Independence
Cotuous Radom Varables: Codtog, xectato ad Ideedece Berl Che Deartmet o Comuter cece & Iormato geerg atoal Tawa ormal Uverst Reerece: - D.. Bertsekas, J.. Tstskls, Itroducto to robablt, ectos 3.4-3.5 Codtog
More informationObservations on the transcendental Equation
IOSR Jourl o Mecs IOSR-JM e-issn: 78-78-ISSN: 9-7 Volue 7 Issue Jul. - u. -7 www.osrjourls.or Oservos o e rscedel Euo M..Gol S.Vds T.R.Us R Dere o Mecs Sr Idr Gd Collee Trucrll- src: Te rscedel euo w ve
More informationNATIONAL OPEN UNIVERSITY OF NIGERIA COURSE CODE : MTH 307 COURSE TITLE: NUMERICAL ANALYSIS II
NATIONAL OPEN UNIVERSITY OF NIGERIA COURSE CODE : MTH 7 COURSE TITLE: NUMERICAL ANALYSIS II Corse Code: MTH 7: Corse Ttle: NUMERICAL ANALYSIS II Corse Developer/Wrter: Corse Edtor: Progrmme Leder: Corse
More informationSystems of second order ordinary differential equations
Ffth order dgolly mplct Ruge-Kutt Nystrom geerl method solvg secod Order IVPs Fudzh Isml Astrct A dgolly mplct Ruge-Kutt-Nystróm Geerl (SDIRKNG) method of ffth order wth explct frst stge for the tegrto
More informationDensity estimation II
CS 750 Mche Lerg Lecture 6 esty estmto II Mlos Husrecht mlos@tt.edu 539 Seott Squre t: esty estmto {.. } vector of ttrute vlues Ojectve: estmte the model of the uderlyg rolty dstruto over vrles X X usg
More informationDifferentiation and Numerical Integral of the Cubic Spline Interpolation
JOURNAL OF COMPUTER VOL. NO. OCTOBER 7 Deretto d Nuercl Itegrl o te Cuc ple Iterpolto g Go cool o Coputer cece d Tecolog Jgsu Uverst o cece d Tecolog Zejg C El: go_sg@otl.co Zue Zg d Cuge Co Ke Lortor
More informationPATTERNS IN CONTINUED FRACTION EXPANSIONS
PATTERNS IN CONTINUED FRACTION EXPANSIONS A THESIS SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY SAMUEL WAYNE JUDNICK IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR
More informationCOMPLEX NUMBERS AND DE MOIVRE S THEOREM
COMPLEX NUMBERS AND DE MOIVRE S THEOREM OBJECTIVE PROBLEMS. s equl to b d. 9 9 b 9 9 d. The mgr prt of s 5 5 b 5. If m, the the lest tegrl vlue of m s b 8 5. The vlue of 5... s f s eve, f s odd b f s eve,
More informationβ (cf Khan, 2006). In this model, p independent
Proc. ICCS-3, Bogor, Idoes December 8-4 Vol. Testg the Equlty of the Two Itercets for the Prllel Regresso Model Bud Prtko d Shhjh Kh Dertmet of Mthemtcs d Nturl Scece Jederl Soedrm Uversty, Purwokerto,
More information02/15/04 INTERESTING FINITE AND INFINITE PRODUCTS FROM SIMPLE ALGEBRAIC IDENTITIES
0/5/04 ITERESTIG FIITE AD IFIITE PRODUCTS FROM SIMPLE ALGEBRAIC IDETITIES Thomas J Osler Mathematcs Departmet Rowa Uversty Glassboro J 0808 Osler@rowaedu Itroducto The dfferece of two squares, y = + y
More informationGeneral Method for Calculating Chemical Equilibrium Composition
AE 6766/Setzma Sprg 004 Geeral Metod for Calculatg Cemcal Equlbrum Composto For gve tal codtos (e.g., for gve reactats, coose te speces to be cluded te products. As a example, for combusto of ydroge wt
More informationORDINARY DIFFERENTIAL EQUATIONS EULER S METHOD
Numercal Analss or Engneers German Jordanan Unverst ORDINARY DIFFERENTIAL EQUATIONS We wll eplore several metods o solvng rst order ordnar derental equatons (ODEs and we wll sow ow tese metods can be appled
More informationTopic 6b Finite Difference Approximations
/8/8 Course Instructor Dr. Rymond C. Rump Oice: A 7 Pone: (95) 747 6958 E Mil: rcrump@utep.edu Topic 6b Finite Dierence Approximtions EE 486/5 Computtionl Metods in EE Outline Wt re inite dierence pproximtions?
More informationNumerical Analysis Formulae Booklet
Numercal Aalyss Formulae Booklet. Iteratve Scemes for Systems of Lear Algebrac Equatos:.... Taylor Seres... 3. Fte Dfferece Approxmatos... 3 4. Egevalues ad Egevectors of Matrces.... 3 5. Vector ad Matrx
More informationShuai Dong. Isaac Newton. Gottfried Leibniz
Computatonal pyscs Sua Dong Isaac Newton Gottred Lebnz Numercal calculus poston dervatve ntegral v velocty dervatve ntegral a acceleraton Numercal calculus Numercal derentaton Numercal ntegraton Roots
More informationAsymptotic Dominance Problems. is not constant but for n 0, f ( n) 11. 0, so that for n N f
Asymptotc Domce Prolems Dsply ucto : N R tht s Ο( ) ut s ot costt 0 = 0 The ucto ( ) = > 0 s ot costt ut or 0, ( ) Dee the relto " " o uctos rom N to R y g d oly = Ο( g) Prove tht s relexve d trstve (Recll:
More informationAsymptotic Formulas Composite Numbers II
Iteratoal Matematcal Forum, Vol. 8, 203, o. 34, 65-662 HIKARI Ltd, www.m-kar.com ttp://d.do.org/0.2988/mf.203.3854 Asymptotc Formulas Composte Numbers II Rafael Jakmczuk Dvsó Matemátca, Uversdad Nacoal
More informationEcon 401A Three extra questions John Riley. Homework 3 Due Tuesday, Nov 28
Econ 40 ree etr uestions Jon Riley Homework Due uesdy, Nov 8 Finncil engineering in coconut economy ere re two risky ssets Plnttion s gross stte contingent return of z (60,80) e mrket vlue of tis lnttion
More informationA Technique for Constructing Odd-order Magic Squares Using Basic Latin Squares
Itertol Jourl of Scetfc d Reserch Publctos, Volume, Issue, My 0 ISSN 0- A Techque for Costructg Odd-order Mgc Squres Usg Bsc Lt Squres Tomb I. Deprtmet of Mthemtcs, Mpur Uversty, Imphl, Mpur (INDIA) tombrom@gml.com
More informationON NILPOTENCY IN NONASSOCIATIVE ALGEBRAS
Jourl of Algebr Nuber Theory: Advces d Applctos Volue 6 Nuber 6 ges 85- Avlble t http://scetfcdvces.co. DOI: http://dx.do.org/.864/t_779 ON NILOTENCY IN NONASSOCIATIVE ALGERAS C. J. A. ÉRÉ M. F. OUEDRAOGO
More informationNumerical Methods. Lecture 5. Numerical integration. dr hab. inż. Katarzyna Zakrzewska, prof. AGH. Numerical Methods lecture 5 1
Numeril Methods Leture 5. Numeril itegrtio dr h. iż. Ktrzy Zkrzewsk, pro. AGH Numeril Methods leture 5 Outlie Trpezoidl rule Multi-segmet trpezoidl rule Rihrdso etrpoltio Romerg's method Simpso's rule
More informationBasics of Information Theory: Markku Juntti. Basic concepts and tools 1 Introduction 2 Entropy, relative entropy and mutual information
: Markku Jutt Overvew Te basc cocepts o ormato teory lke etropy mutual ormato ad EP are eeralzed or cotuous-valued radom varables by troduc deretal etropy ource Te materal s maly based o Capter 9 o te
More informationA METHOD FOR THE RAPID NUMERICAL CALCULATION OF PARTIAL SUMS OF GENERALIZED HARMONICAL SERIES WITH PRESCRIBED ACCURACY
UPB c Bull, eres D, Vol 8, No, 00 A METHOD FOR THE RAPD NUMERAL ALULATON OF PARTAL UM OF GENERALZED HARMONAL ERE WTH PRERBED AURAY BERBENTE e roue o etodă ouă etru clculul rd l suelor rţle le serlor roce
More information3/20/2013. Splines There are cases where polynomial interpolation is bad overshoot oscillations. Examplef x. Interpolation at -4,-3,-2,-1,0,1,2,3,4
// Sples There re ses where polyoml terpolto s d overshoot oslltos Emple l s Iterpolto t -,-,-,-,,,,,.... - - - Ide ehd sples use lower order polyomls to oet susets o dt pots mke oetos etwee djet sples
More informationANALYSIS ON THE NATURE OF THE BASIC EQUATIONS IN SYNERGETIC INTER-REPRESENTATION NETWORK
Far East Joural of Appled Mathematcs Volume, Number, 2008, Pages Ths paper s avalable ole at http://www.pphm.com 2008 Pushpa Publshg House ANALYSIS ON THE NATURE OF THE ASI EQUATIONS IN SYNERGETI INTER-REPRESENTATION
More informationIdea is to sample from a different distribution that picks points in important regions of the sample space. Want ( ) ( ) ( ) E f X = f x g x dx
Importace Samplg Used for a umber of purposes: Varace reducto Allows for dffcult dstrbutos to be sampled from. Sestvty aalyss Reusg samples to reduce computatoal burde. Idea s to sample from a dfferet
More informationA stopping criterion for Richardson s extrapolation scheme. under finite digit arithmetic.
A stoppg crtero for cardso s extrapoato sceme uder fte dgt artmetc MAKOO MUOFUSHI ad HIEKO NAGASAKA epartmet of Lbera Arts ad Sceces Poytecc Uversty 4-1-1 Hasmotoda,Sagamara,Kaagawa 229-1196 JAPAN Abstract:
More information