5 Solving systems of non-linear equations

Transcription

1 umercal Methods n Chemcal Engneerng 5 Solvng systems o non-lnear equatons 5 Solvng systems o non-lnear equatons Overvew assng unctons D ewtons Method somethng you dd at school ewton's method n more than one dmenson Solvng the system o two coupled equatons Soluton trajectores The reduced ewton step method roblems wth ewton's Method A chemcal engneerng eample - equlbrum The element balance equatons The equlbrum equatons Thermodynamc data Solvng the equatons Full soluton to the equlbrum problem n Matlab Convergence o the ewton-method Calculatng the Jacoban Matr Fnte derences Broydens Method Takng advantage o sparsty Trust regon method Used by solve Summary Amendments Updated verson o ewtond Updated verson o ewton

2 umercal Methods n Chemcal Engneerng 5. Overvew ow we wll look at solvng systems o non-lnear equatons. We wll look at one method n partcular, ewton's method. We are gong to wrte our own routne or solvng non-lnear equatons usng ewton's method. Matlab has ts own routnes or solvng systems o non-lnear equatons e.g. solve, whch s loosely based on ewton's method. You are encouraged to use these Matlab routnes as they are lkely to be more robust than anythng we can wrte n a lecture! Addtonal readng or ths lecture: Chapter o "umercal Methods or Chemcal Engneers wth Matlab Applcatons" by Constantndes and Mostou Look up the routnes solve and zero n the Matlab help le - -

3 umercal Methods n Chemcal Engneerng 5. assng unctons In Matlab, uncton names can be passed as arguments to unctons. Ths means we can wrte routnes whch act on unctons. E.g. we want to ntegrate under the curve y, we could wrte a uncton whch, gven would return a value o y. We could then pass ths uncton to another routne whch does the numercal ntegraton. The mechansm Matlab provde or ths s a uncton handle other programmng languages would call ths a ponter. E.g. to solve - numercally, we wrte a uncton whch returns the value o -. uncton [resdual] Myunc resdual ^ * return We then call a solvng routne e.g. zero - look t up n the help le Answer s a handle to the uncton MyFunc, whch means that t can be used as an alas or MyFunc e.g. we do the ollowng The varable Functonandle can then be used as an alas or MyFunc,.e. y Functonandle s the same as y MyFunc Another command you can use s y evalfunctonandle, Ths latter command has the advantage o also be able to take a tet strng.e. the name o the uncton n place o a uncton handle and evaluate t. The ablty to store unctons n a varable means that t s possble to pass unctons as arguments to other unctons. - -

4 umercal Methods n Chemcal Engneerng 5. D ewtons Method somethng you dd at school Lets solve - usng ewton's Method. In ewton's method, we set - take an ntal guess or the soluton and calculate. We then etrapolate assumng s lnear to a new value o whch wll make. d.e. new o new o d o or new o o d d o The procedure s then repeated untl the soluton s reached..e. 5 y y Fgure. The soluton to - by ewton's method A Matlab uncton to perorm ths teratve procedure mght look lke: - -

5 umercal Methods n Chemcal Engneerng uncton [soluton] ewtondmyfunc,gradent,guess,tol % solves the non-lnear vector equaton F % set usng ewton Raphson teraton % IUTS: % Myunc andle to the uncton whch calculates F % Gradent andle to the uncton whch calculates F'X % Guess Intal Guess % Tol Tolerance % % OUTUTS: % soluton A soluton to the set o equatons % Descrpton % Solves a D non-lnear equaton by ewtons method Guess; %set the error *tol to make sure the loop runs at least once error *tol whle error > tol %calculate the uncton values at the current teraton F evalmyfunc,; %calculate the Gradent G evalgradent,; %calculate the update d -F/G; %update the value d; %calculate the error F evalmyfunc,; error absf; end %whle loop soluton ; return - 5 -

6 umercal Methods n Chemcal Engneerng 5. ewton's method n more than one dmenson Lets look at an eample: We must rst arrange our equatons n the orm: F Where s the vector o unknowns and F s a vector o uncton values. e.g. or ths eample, F I we start wth a guessed value, we can nd a better guess by etrapolaton. ear our guessed value, the uncton can be epanded M M M d d d d d d d d 5- ote that these dervatves are evaluated at the M L L M M M M M. d d d d d

7 umercal Methods n Chemcal Engneerng d d J d M M or F d F J d 5-5 ow we want our new vector d to be a better appromaton to the soluton, so we set F n Eq 5-5. Then, F J d 5-6 where J s called the Jacoban matr. The procedure or ewton's method s: calculate the uncton values at the guessed value o [,, ] T calculate the Jacoban matr usng the current guess or the soluton solve the lnear system F J d or the values o d update the guessed value d Ths procedure should be repeated, usng the updated value o as the guess, untl the values o F are sucently close to zero. F s a vector o resdual errors. For sucently close to zero we could use Ma{,,,, 5... } < Tolerance or the norm o the vector F, / F 5-7 A uncton whch wll perorm ths procedure s gven overlea. otce how smlar t s to the one dmensonal code, and that a set o lnear equatons must be solved! - 7 -

8 umercal Methods n Chemcal Engneerng uncton [soluton] ewtonmyfunc,jacoban,guess,tol % solves the non-lnear vector equaton F % set usng ewton Raphson teraton % IUTS; % Myuncandle to the uncton whch returns the vector F % Jacobanandle to the uncton whch returns the Jacoban Matr % Guess Intal Guess a vector % tol Tolerance % % OUTUTS % soluton The soluton to F Guess; %set the error *tol to make sure the loop runs at least once error *tol whle error > tol %calculate the uncton values at the current teraton F evalmyfunc,; %calculate the jacoban matr J evaljacoban,; %calculate the update solve the lnear system d J\-F; %update the value d; %calculate the error F evalmyfunc,; error maabsf; end %whle loop soluton ; return - 8 -

9 umercal Methods n Chemcal Engneerng 5.. Solvng the system o two coupled equatons The equatons we want to solve are: The Jacoban s J All we need to do to solve our equatons s to wrte two unctons: that returns the vector o uncton values uncton y Func and that returns the Jacoban matr uncton J Jacoban

10 umercal Methods n Chemcal Engneerng uncton man %man uncton whch call the ewton solver %call the solver soluton ewton@func,@jac,[;],e-6 return uncton y Func %the uncton whch returns the values o F y.^.^ ; y.^ -.^; y y'; return uncton J Jac %The uncton that returns the Jacoban matr J, *^; J, *; J, *; J, -*^; return - -

11 umercal Methods n Chemcal Engneerng 5.. Soluton trajectores From our denton o the norm 5-7 F So at a soluton we must have the smallest possble value o F, and the soluton s a global mnmum n F. For a gven set o equatons F, we can plot the values o F at all values o however, anythng more complcated than D s a bt dcult to nterpret. Returnng to our eample, - A plot o the norm o [, ] T s gven below 5-5- [, ] T Fgure. F There are solutons located at [,] T and [-,] T - -

12 umercal Methods n Chemcal Engneerng Soluton at [,] T Fgure. F and the trajectory taken by the ewton method Fgure shows the trajectory ollowed when the solver s started rom [-,.] T. The trajectory the soluton ollows can be very erratc, especally when the solver s stated a long way rom the soluton. In act, Fgure shows that the error F actually ncreases on the rst teraton. - -

13 umercal Methods n Chemcal Engneerng 5.. The reduced ewton step method We don't have to take a ull ewton step, nstead we can search along the drecton o the ewton step, or a pont where the error s less than the error at the startng pont. An algorthm to do ths s smply: Calculate ewton step, Work out F.5 s F.5 < F o Yes ew.5 - -

14 umercal Methods n Chemcal Engneerng.5 Trajectory taken by ewton method.5 Trajectory taken we search back along the lne o a ull ewton step -.5 Soluton Fgure. F and the trajectory taken by the ewton method wth lne searchng to reduce the step Much more robust! 5.. roblems wth ewton's Method ewton's method wll al, when the Jacoban s sngular, but we are not at the soluton. When the Jacoban s sngular, we cannot nd a soluton to F J so we cannot nd our update vector. Ths s analogous to what happens wth the dmensonal ewton method, and a mamum or mnmum n the uncton.e. zero gradent s encountered. - -

15 umercal Methods n Chemcal Engneerng For the reduced step ewton algorthm, a ewton step s calculated and then a search s perormed along the drecton o the ewton step to nd a step whch wll reduce the error. When the update step s almost perpendcular to the drecton o the gradent o the orm, F.e. parallel to the contours o F, the value o F whch we are tryng to make as small as possble doesn't change much along the lne we are searchng. The solver wll then take very small steps

16 umercal Methods n Chemcal Engneerng 5.5 A chemcal engneerng eample - equlbrum Consder the reacton o methane, wth water at hgh temperatures C O Steam reormng o methane has been proposed as one way to produce clean ydrogen or the 'ydrogen economy'. In any reactor contanng and, the sht reacton wll also occur. O So we have a reactor as ollows O C C O and we want to know, what the composton o the outlet stream s, the system s allowed to reach equlbrum. To solve ths problem we can use the method o mnmum reactons. There are 5 speces o nterest.e. 5 unknown lows leavng the reactor we need to solve or. The amounts o, C and O enterng the system must be equal to the amounts leavng; ths generates three elemental balance equatons. Also, there are two reactons sht and reormaton, these wll yeld two equatons, each nvolvng an equlbrum constant

17 umercal Methods n Chemcal Engneerng 5.5. The element balance equatons j j E ν 5-8 where E j and are the number o moles o element j enterng and speces leavng the system the system per unt tme; ν the number o atoms o j n one molecule o speces. e.g. C E C 5-9 C O E 5- O E O 5- Ths can be represented by a stochometrc matr equaton OUT O C I O C O C O C O C O C O C E E E 5.5. The equlbrum equatons For the steam reormng reacton C O o C O o o C o O o o p y y y y K - 7 -

18 umercal Methods n Chemcal Engneerng Takng natural logs Ln K p Ln o Ln Ln Ln O Ln C For the water-gas sht 5- O o o y O y K p y y o o O Takng natural logs Ln K p Ln O Ln Ln Ln Thermodynamc data The values o K p as a uncton o temperature gven below LnKp y.66e-5-7.9e- 6.5E R 9.979E- y -6.95E-6.69E- -.6E R 9.97E- LnKp LnKp Temperature K Fgure 5, Ln K p as a uncton o temperature - 8 -

19 umercal Methods n Chemcal Engneerng A uncton I have wrtten to calculate these values s: uncton [LnKp,LnKp] LnKpT return LnKp.66e-5*T^ - 7.9e-*T 6.5e; LnKp -6.95e-6*T^.69e-*T -.6e; 5.5. Solvng the equatons The equaton set we have to solve s C - E C 5-9 C O - E 5- O - E O 5- Ln Ln Kp Ln Ln Ln Ln 5- o O C Ln Ln Ln Ln Ln Kp 5 5- O We need to wrte a uncton whch returns the let hand sde o these equatons uncton [resdual] global Stochometry global E global T global _o %the element balance equatons resdual: Stochometry* - E %the equlbrum constants [LnKp,LnKp] LnKpT %equlbrum equatons resdual -LnKp log_o/sum*log... log-*log5-log resdual5- LnKp log5log- log- log return - 9 -

20 umercal Methods n Chemcal Engneerng We also need the Jacoban: The rst rows o the Jacoban are just the stochometry matr. The row or equaton 5-,.e. the th row s made up o the ollowng terms., 5 5,, The 5th row o the Jacoban s made up o 5, 5, 5, 5, uncton [Jac] jac global Stochometry Jac:,: Stochometry; %row or the steam reormng reacton Jac, -/ - /sum; Jac, - /sum; Jac, / - /sum; Jac, / - /sum; Jac,5 -/5 - /sum; %row 5 or the water-gas sht Jac5, ; Jac5, /; Jac5, -/; Jac5, -/; Jac5,5 /5; return - -

21 umercal Methods n Chemcal Engneerng Full soluton to the equlbrum problem n Matlab %MAI FUCTIO %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% uncton man global T global E global Stochometry global _o %set the nlet lows [C;;;;O] n [;;;;]; %set the temperature K T 9; %set the pressure o bar _o ; %set the stocometry matr Stochometry [,,,, ;...,,,, ;...,,,, ]; %Calculate the lows o elements enterng the system E Stochometry*n; %solve the system o equatons [] ewtonreduced@,@jac,[,,,,]',*eps; %prnt the soluton dsp return %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Functon that computes the Jacoban %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% uncton [Jac] jac global Stochometry Jac:,: Stochometry; %row or the steam reormng reacton Jac, -/ - /sum; Jac, - /sum; Jac, / - /sum; Jac, / - /sum; Jac,5 -/5 - /sum; - -

22 umercal Methods n Chemcal Engneerng %row 5 or the water-gas sht Jac5, ; Jac5, /; Jac5, -/; Jac5, -/; Jac5,5 /5; return %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Functon that computes equlbrum constants %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% uncton [LnKp,LnKp] LnKpT return LnKp.66e-5*T^ - 7.9e-*T 6.5e; LnKp -6.95e-6*T^.69e-*T -.6e; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Functon that computes the uncton values set o equatons to be solved %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% uncton [resdual] global Stochometry global E global T global _o %the element balance equatons resdual: Stochometry* - E; %the equlbrum constants [LnKp,LnKp] LnKpT; %equlbrum equatons resdual -LnKp log_o/sum*log log-*log5-log; resdual5 - LnKp log5log- log- log; resdual resdual'; return - -

23 umercal Methods n Chemcal Engneerng 5.6 Convergence o the ewton-method The error or a gven teraton can be taken as e - * where * s the true soluton. ow assumng we are close to the soluton, we can epand the uncton F about *, and use ths epanson n place o F F F * J * * O[ * ] 5- F J * * O[ * ] 5-5 F J * e O[ e ] 5-6 Substtutng ths nto our update ormula F gves J * e J * e J O[ e O[ e ] J ] J e e I we are sucently close to the soluton, then J J*, and O [ e ] J * e 5-9 Thus, the convergence s quadratc ths o-course only apples when the Jacoban s sensble. I the Jacoban s sngular, or nearly sngular near the soluton, the error can be very large. [In act, when you have a sngular Jacoban at the root tsel, convergence becomes lnear.] - -

24 umercal Methods n Chemcal Engneerng 5.7 Calculatng the Jacoban Matr The methods presented so ar assume that an analytcal orm o the Jacoban was avalable. When we have a large system o equatons and the unctons are 'epensve' to evaluate, calculatng the Jacoban can become necent. Ecent calculaton o the Jacoban then becomes mportant Fnte derences The Jacoban s just a matr o dervatves: J L. L M M M Each o these dervatves can be calculated usng nte derences j δ δ 5- The value o δ cannot be made too small, because computers are only able to store numbers to a nte precson. The workng precson o the computer can be ound wth the Matlab command eps whch s equal to.e-6 on my computer. A common value o δ s eps. Calculatng the Jacoban n ths way s less ecent than an analytcal orm o the Jacoban was suppled. The Matlab Routne 'solve' wll use nte derences to calculate the Jacoban, you don't supply a uncton or the Jacoban s constant, a matr to do t. - -

25 umercal Methods n Chemcal Engneerng 5.7. Broydens Method It s not necessary to calculate the eact value o the Jacoban at each teraton. Broyden's method provdes a way o updatng an estmate o the Jacoban. Generally, the resdual uncton values at the th and th teraton are lnked by J F F 5- So we replace the eact Jacoban by an estmate o the Jacoban at teraton, B, we desre ths appromate Jacoban to obey B F F 5- I we use B the estmate o the Jacoban at teraton, wth a ewton step to get our updated, B B F 5- Usng 5- n 5- B F B B F B ost multply both sdes by T T T T B F B T B F B T F B B 5- So on the th teraton we can calculate an updated estmate o the Jacoban, rom normaton calculated n the prevous teraton, and the value o F, whch we would have to calculate anyway. Usng ths type o update results n a Quas-ewton scheme, whch has lnear convergence

26 umercal Methods n Chemcal Engneerng 5.7. Takng advantage o sparsty Oten, the set o equatons we wll be solvng wll be sparse. The Jacoban wll contan manly zeros. I we know that an element o the Jacoban s zero then we don't have to calculate t. Also, a set o lnear equatons s solved at each teraton, these lnear equatons are sparse then t should be possble to use a ecent soluton method recall handout.there are two ways to eplot sparsty when solvng non-lnear equatons n Matlab:. Use sparse matrces. I you wrte a uncton to calculate the Jacoban, make sure t returns a sparse matr. Ecent methods can then be used to solve the resultng sets o lnear equatons. ote that the Matlab \ command wll detect sparse matrces and try to use the most ecent method t can nd. I you are usng solve, t wll detect that the Jacoban s sparse and act accordngly.. I you are usng solve, but you don't want to calculate the Jacoban analytcally, you can nstead supply a Jacoban pattern. The Jacoban pattern s a sparse matr wth ones where the Jacoban s non-zero. Ths means that solve does not have to calculate every element o the Jacoban. It also means that t wll create a sparse Jacoban

27 umercal Methods n Chemcal Engneerng 5.8 Trust regon method Used by solve The update equaton or s F J 5-6 The soluton to ths set o lnear equatons, wll mnmse the value o J F 5-5 So, rather than solvng the set o lnear equatons, we can mnmse 5-5, wth respect to. A long way rom the soluton, the update can be erratc, so we constran to to be wthn a trust regon. We then solve the constraned optmsaton problem mnmse q J F wth respect to, subject to the constrant that < some value I the ewton step s wthn the trust regon, the update wll be a ull ewton step dentcal to we has solved 5-6, by e.g. elmnaton. I however, the ull ewton step s outsde the trust regon, then a mnmsaton routne s ree to vary both the drecton and length o to mnmse the cost uncton q. We wll get as close as possble to a soluton o 5-6, wthout steppng too ar, because the length o s constraned. Ths overcomes problems wth the reduced step method, n that both the drecton and the length o the update can be vared to keep the step wthn the trust regon. A dscusson o how you set the trust regon at each teraton s beyond the scope o these lectures

28 umercal Methods n Chemcal Engneerng 5.9 Summary The soluton o non-lnear equatons usng a computer wll generally requre some sort o teratve procedure. In ths handout the ewton method or solvng systems o non-lnear equatons was presented. Ths method s one o the most wdely used schemes or solvng non-lnear equatons. Matlab also contans routnes whch can be used to solve systems o non-lnear equaton; the most useul beng solve. When solvng a set o non-lnear equatons, at some pont n the procedure the soluton to a set o lnear equatons s oten requred. Thus, ecent soluton o non-lnear equatons requres the ablty to solve a set o lnear equatons. In prevous lectures we have seen that a set o lnear equatons s sparse, t can be solved ecently. Thus, when the Jacoban o a non-lnear system s a sparse matr, any non-lnear teratve routne should be wrtten to take advantage o ths

29 umercal Methods n Chemcal Engneerng 5. Amendments The ewton and ewton D routnes gven earler evaluate the unctons twce or each teraton. Ths s necent, and can be avoded by changng where n the routne the error s evaluated. Updated routnes are gven below. 5.. Updated verson o ewtond uncton [soluton] ewtondmyfunc,gradent,guess,tol % solves the non-lnear vector equaton F % set usng ewton Raphson teraton % IUTS: % Myunc andle to the uncton whch calculates F % Gradent andle to the uncton whch calculates F'X % Guess Intal Guess % Tol Tolerance % % OUTUTS: % soluton A soluton to the set o equatons % Descrpton % Solves a D non-lnear equaton by ewtons method Guess; %set the error *tol to make sure the loop runs at least once error *tol whle error > tol %calculate the uncton values at the current teraton F evalmyfunc,; %calculate the error error absf; %calculate the Gradent G evalgradent,; %calculate the update d -F/G; %update the value d; end %whle loop soluton ; return - 9 -

30 umercal Methods n Chemcal Engneerng 5.. Updated verson o ewton uncton [soluton] ewtonmyfunc,jacoban,guess,tol % solves the non-lnear vector equaton F % set usng ewton Raphson teraton % IUTS; % Myuncandle to the uncton whch returns the vector F % Jacobanandle to the uncton whch returns the Jacoban Matr % Guess Intal Guess a vector % tol Tolerance % % OUTUTS % soluton The soluton to F Guess; %set the error *tol to make sure the loop runs at least once error *tol whle error > tol %calculate the uncton values at the current teraton F evalmyfunc,; error maabsf; %calculate the jacoban matr J evaljacoban,; %calculate the update solve the lnear system d J\-F; %update the value d; end %whle loop soluton ; return - -