1 nonlinear.mcd Find solution root to nonlinear algebraic equation f(x)=0. Instructor: Nam Sun Wang
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1 nonlinermc Fin solution root to nonliner lgebric eqution ()= Instructor: Nm Sun Wng Bckgroun In science n engineering, we oten encounter lgebric equtions where we wnt to in root(s) tht stisies given eqution An eqution is nonliner, i it is not liner Liner lgebric eqution: A b Generl (nonliner) lgebric eqution: ( ) O course, liner lgebric eqution is specil cse o the generl/nonliner lgebric eqution Below re some common nonliner lgebric equtions Emple: squre root (which is probbly the very irst nonliner lgebric eqution mny o us got epose to in n lgebr clss) which is equivlent to (or eine s) number, which when squre, results in : Emple: qurtic eqution (which is probbly the secon nonliner lgebric eqution mny o us ever got epose to) b c Emple: cubic root Emple: cubic eqution b c Emple: polynomil n n Emple: logrithmic unction ln( ) log( ) log bse ( ) which is equivlent to (or eine s) number, which when eponentite, results in : e bse Emple: rc sine or inverse sine n bunch o other trigonometric unctions sin( ) which is equivlent to (or eine s) number, whose sine is : sin( ) Emple: when we got beyon lgebr n strte tking clculus, we were introuce to integrls n erivtives in clculus courses g( t) t t g( t) evlute_t_
2 nonlinermc Stnr nottion: ()= To evelop systemtic pproch to solving nonliner lgebric equtions, we use the "stnr" ()= nottion To put given eqution into this stnr orm, it is simple mtter o moving everything to one sie o the eqution, n it oes not mtter which sie (LHS or RHS) Below re some emples or re-writing lgebric equtions in the "stnr" ()= orm squre root: ( ) cubic root: ( ) logrithm: e ( ) e inverse sine sin( ) ( ) sin( ) einite integrl g( t) t ( ) g( t) t Bisection Metho We strt with two points n b to brcket the root, where () n (b) hve ierent signs, ie, () (b)< I () n (b) hve ierent signs, the given unction () must pss through (ie, intersect with the horizontl is) t some points within [ b] We then choose the mi point between n b s the net guess I ( ) hs the sme sign s (), becomes the new let intervl Otherwise, becomes the new right intervl b The sme process is repete or the net guess, n so orth, until the intervl [ b] is suiciently smll Ech itertion hlves the intervl [ b] n gins one signiicnt binry igit Thus, every three itertions reuce the intervl by /8, which is roughly / This correspons to gining one signiicnt eciml igit An vntge o bisection metho is tht it lwys converges to root, lbeit somewht slower thn the other methos (eg, Newton's metho) tht re introuce lter here A chie isvntge is tht we hve to come up with two points o ierent signs to get the itertion process strte net guess is the mi point between n b: b 5 Bisection Metho ter unction evlution t : ter unction evlution t : b ter unction evlution t : b ter unction evlution t 4 : b b () () b
3 nonlinermc Metho o Chors As in bisection metho, we strt with two points n b, where () n (b) hve ierent signs, ie, () (b)< Inste o choosing the mi point between n b s the net guess, we rw chor to connect n b, n the intersection o the chor with the horizontl is becomes the net guess I ( ) hs the sme sign s (), becomes the new let intervl Otherwise, becomes the new right intervl b The sme process is repete or the net guess, n so orth net guess is where the chor intersects with horizontl is: ( ) b ( b) ( ) b ( b) b ( b) ( ) 5 Metho o Chors () () b Metho o Secnt We strt with ny two points n Unlike the metho o chors, these two initil points o not necessrily hve to yiel ierent signs or Inste o rwing the chor to connect n one o the en points (either or b, epening on the sign o the unction), we rw secnt tht connects the lst two points n - An the intersection o the secnt with the horizontl is becomes the net guess + Bsicll we use the lst two points to estimte the slope o the given unction; n bse on this estimte slope, we in where this secnt intersects with the horizontl is net guess is where the secnt intersects with horizontl is:
4 4 nonlinermc 5 Secnt Metho () () Newton's Metho We ssume tht () n its erivtive '() re both given We strt with ny one point initil guess Unlike the secnt metho, the slope t the present point is not estimte by connecting the lst two points, but nlyticlly evlute through '( ), which is given We rw tngent through the unction n the intersection o the tngent with the horizontl is becomes the net guess + slope ' verticl_rise horizontl_run ' 5 Newton's Metho () ()
5 Emple: squre root ( ) '( ) ' i strt with = 55 = 659 = 96 5 nonlinermc 4 = 4 6 compre = 6 Moiictions There re mny moiictions to the Newton's metho For emple, the bove squre root emple shows bit o oscilltion prior to convergence We sometimes tepily tke smll step α, inste o ull step Emple: squre root As beore, strt with iterte or times: i α 5 i i α ' i i Convergence is slower, but there is less oscilltion
6 6 nonlinermc Emple: ivision without oing ivision opertion At the CPU level, some computers cn perorm ition n multipliction opertions but not ivision opertions ( ) ' strt with 5 ( ) i = 5 = = 4 4 = '( ) = converge! strt with 8 = = 94 = = = 58 4 iverge! In generl, some initil guesses converge to n swer, while others iverge Or, when there re multiple solutions, the converge nswer epens on the initil guess Rigorous Derivtion o Newton's Metho or sclr cse We strt with n initil guess o, n we epn () roun with Tylor's series '' ( ) '! '''! We mke n pproimtion by keeping only the st orer term n ignoring the n orer n higher orer terms, n we hope to mke ( )= ' ' Once we hve, We repet the sme process to in Then rom, we in, etc Thus, we rrive t the ollowing itertion ormul ' or i=,,,, until convergence
7 nonlinermc Rigorous Derivtion o Newton's Metho or vector cse Let us strt with n lgebric equtions (, n ) contining n unknown vribles ( ) These equtions re couple n cnnot be solve iniviully in sequentil mnner Anlogy: A =b, n couple liner lgebric equtions contining n unknown vribles ( ) : nlogy :,, n b,,, n b n n, n,, vector nottion: ( ) A b, n n We epn ech o these n unctions roun <> with Tylor's series We hve n equtions : + < > + n_erivtive_n_higher_orer_terms + < > + n_erivtive_n_higher_orer_terms b n n n n n + < > + n_erivtive_n_higher_orer_terms n We mke n pproimtion by keeping only the st orer terms n ignoring the n orer n higher orer terms, n we hope to choose =(,,, ) to orce (,,, )=, (,,, )=, n (,,, )= simultneously + < n>
8 : n nonlinermc n n n In mtri-vector nottion, the bove n sclr equtions become: n n n n n n n < > < > < > It urther simpliies to Newton's ormul or couple set o nonliner lgebric equtions (linerize represention o ()): ( ) compre to liner eqn A b b Δ Δ b compre to liner eqn A b We wnt to choose <> such tht the bove eqution is stisie < > where ( ) n Newton's itertion ormul < i > n compre to liner eqn A b n n Jcobin mtri
9 9 nonlinermc Quick-Dirty Derivtion o Newton's Metho or vector cse Let us strt with how much chnges by Δ s chnges by Δ Δ Δ < i > < i> Sme s bove with some vector epne < i > Δ Δ Δ n Δ Δ Δ where j j j how ech unction chgnes s j chnges j n n n n n n n < i > < i> < i >
10 Emple: couple pretor-prey ynmics rt( r, ) r b r t( r, ) c r vector nottion rt( r, ) t( r, ) r r b r c r g( ) rbbit (prey) o (pretor) g rt( r, ) g t( r, ) nonlinermc b c g ( ) g g g g ( rt) r ( t) r ( rt) ( t) b b r c r b b c Apply Newton's itertion ormul mnully or one itertion < i > g g r r b b r c r r b r c r with b c provie initil guess r r r 4 I we go or nother itertion, we substitute <> n in <> with the sme Newton's itertion ormul r We cn see tht, we hve lrey converge to root, becuse <> = <>
11 Emple: couple three equtions sin( ) y ln( nonlinermc y z y z 5 Step First, we epress the given three sclr equtions in stnr vector orm "()=" (, sin( ) y ln( (, z ) y z (, y z 5 Or in vector orm or the unction (, sin( ) y ln( y z y z 5 Or in vector orm or the unknown vrible s well ( vect) sin vect vect vect vect ln vect vect vect vect vect 5 where vect vect vect vect Step Fin the Jcobin mtri (ie, mtri o erivtives o ech unction wrt ech vrible) (, y y y z z z cos( ) y ln( ) y z z Step Provie n initil guess (z= is not goo initil guess becuse o ln( vect y z Step 4 Apply Newton's ormul y z vect y y, y, z, y, z z z
12 nonlinermc cos y sin y ln z vect y z y z ln( ) y z z y z y z 5 vect y z cos( ) ln( ) sin( ) ln( ) 5 vect y ln( ) z 4 4 Procee to the net itertion by pplying the sme Newton's ormul vect y y, y, z, y, z z z cos y sin y ln z vect y z y z ln( ) y z z y z y z 5 vect y z 68 4 cos( 68) ln( ) 4 4 sin( 68 ) ( ) ln( 4) vect y z Iterte likewise until cnovergence
13 nonlinermc Implement Newton's itertion ormul in computer Moel prmeters b c Keep own vribles rt( r, ) r b r t( r, ) c r Combine to orm vector eqution ( ) rbbit (prey) o (pretor) rt, t, ( ), ( r, ) b b r c r Strting with n initil guess o <>, iterte or number o times i 5 reset = < i > It seems we hve reche convergence Disply the lst columns s the solution cols( ) = Implement Newton's itertion ormul in computer (or the -eqution problem) Keep own vribles (, sin( ) y ln( (, z ) y z (, (, y z 5 cos( ) y ln( ) y z z Combine to orm vector eqution vect, vect, vect ( vect) vect, vect, vect ( ) vect, vect, vect vect vect, vect, vect Strting with n initil guess o vect <>, iterte or number o times i 5 vect vect i vect < i > vect i vect i vect = It seems we hve reche convergence Disply the lst columns s the solution 5 vect cols( ) =
x dx does exist, what does the answer look like? What does the answer to
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