Copyright c 2008 Kevin Long

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1 Lecture 4 Numerical solution of initial value problems Te metods you ve learned so far ave obtained closed-form solutions to initial value problems. A closedform solution is an explicit algebriac formula tat you can write down in a finite number of elementary operations. In te real world, suc problems are te exception rater tan te rule: most initial value problems don t ave closed-form solutions. Even wen a problem can be solved in closed form, te solution may be extremely difficult to obtain or just plain inconvenient to work wit. In suc cases, we will need to compute approximate solutions. In very broad terms ere are two kinds of approximations to consider: Analytical approximations we compute a formula tat approximates te true solution to te problem. Numerical approximations we compute numerical values tat approximate te numerical values of te true solution. In tis introductory course we ll not consider analytical approximation metods, but will ave a sort unit on numerical approximations. A word about terminology: it s common to talk about numerical solution of differential equations. Strictly speaking, we never compute numerical solutions: we only compute numerical approximations to solutions. But numerical approximation to te solution to a differential equation is a moutful, so we like everyone else will refer to numerical solution. Just keep in mind at all times tat a numerical solution is actually an approximation to a solution. Te idea of numerical approximation is very old, dating back (at least) to Arcimedes metod of exaustion for systematically refining approximations to te numerical value of π. Numerical solutions to initial value problems ave been done for several centuries. In Euler s day, computations would ave been done by and wit te aid of log tables. Mecanical computers ( adding macines ) and finally digital computers ave let numerical solutions be carried out very efficiently, and numerical metods are now one of te most widely used ways of solving differential equations. Te central callenge in numerical approximation is to find metods tat produce good approximations to a true solution in a reasonable amount of work. I will lead you troug an examination of te approximation error and ten a study of ow te computational cost depends on te desired amount of accuracy. 4.1 Approximation by Taylor Polynomials Te starting point for virtually all numerical metods for solving IVPs is Taylor s Teorem. Recall Taylor polynomials from Calculus II: to approximate a function y(x) in te vicinity of a point a, form an N-t 1

2 degree polynomial T N y (x, a) = y(a) + y (a) (x a) y (a) (x a) N! y(n) (a) (x a) N, (4.1) wic in summation notation is written T N y (x, a) = N k=0 1 k! y(k) (a) (x a) k. (4.2) Te rater complicated notation T N y (x, a) means tat T is an N-t degree approximation to a function y(x), expanded about te point a. Were does te Taylor polynomial come from? Te N-order Taylor polynomial approximation to y(x) is te unique polynomial wose first N derivatives exactly matc tose of y(x) at te point a. Oter sorts of polynomial approximations are possible: te Taylor approximation turns out to be a useful one for initial value problems, so we will concentrate on it ere Te error in a Taylor approximation Except in cases were y(x) is already a polynomial of degree less tan or equal to N, a Taylor polynomial will never be an exact fit to y. Te error in a Taylor polynomial approximation is given by te Lagrange remainder formula, y(x) Ty N (x, a) = Ry N 1 (x, a) = (N + 1)! y(n+1) (c) (x a) N+1 were c is an undetermined number somewere in te interval [a, x]. At first glance, te Lagrange remainder migt appear to be useless because we don t know c and terefore don t know te exact value of te remainder. Te importance of te Lagrange remainder is twofold: first, it tells us rougly ow te error depends on x a, te distance from te point a; second, it tells us rougly ow te error will cange as N increases. I say rougly because te value of c depends (in some complicated and undetermined way) on te value of x; in practice, as long as we don t wander too far from a te beavior of te remainder is dominated by te factor (x a) N N 1 N 5 sin x 2 N 3 N 7 4 Figure 4.1: Taylor polynomial approximations to sin x. Figure sows Taylor approximations to te sine function, expanded about te point a = 0. As te order of te approximation increases, te approximation closely follows te sine function over a greater 2

3 distance from a. Also notice tat for x 3, as N increases te error gets smaller: wit N = 1 te error is large, but at N = 7 te polynomial approximation and te sine function are ard to distinguis by eye. Far from a, say, x 3 4, all of te approximations sown are wildly off. It would take a very ig-order Taylor polynomial to get a good approximation to te sine function so far away from a = 0. To understand tis beavior, recall tat te N-t order Taylor polynomial was derived by fitting te first N derivatives of y at te point a. All of te information used in building te Taylor approximation came from te point a. It sould not be surprising tat te approximation is very good near a, but gets very bad once you wander too far from tat point. Box 4.1 Approximations based on Taylor polynomials get better te smaller te step you take away from te point a, and tis improvement is more rapid te iger te polynomial order of te approximation. Upper bounds on te approximation error Te Lagrange remainder formula for an N-t order Taylor polynomial approximation to y(x) near a is Ry N 1 (x, a) = (N + 1)! y(n+1) (c) (x a) N+1. Again, we don t know wat c is and so can never compute Ry N exactly. However, we can often put an upper bound on te error. If y (N+1) (x) is bounded so tat tere is some maximum value ( upper bound ) M > 0 suc tat y (N+1) (x) M(N + 1)! ten we can in turn put an upper bound on te Lagrange remainder: R N y = 1 y (N+1) (x) x a N+1 (4.3) (N + 1)! M x a N+1. (4.4) We still don t necessarily know a value for M, but we do know tat as we take x closer and closer to a, te upper bound on te Lagrange remainder gets better and better. For example: if N = 1 and x = a + 1, ten R 1 y M 2 1. If we get less adventurous and stay closer to a, say, taking a step one-tent as far away from a, x = a , ten R 1 y M2 1. If we step a tent as far out from a, we reduce te maximum value of te Lagrange remainder by a factor of 100. Repeat tis argument wit N = 3, i.e. a cubic Taylor polynomial, and find tat reducing te step by ten reduces te bound on te Lagrange remainder by Order notation for error estimates In numerical solution of IVPs we ll usually use to represent a small step in te independent variable x away from te expansion point a of a Taylor polynomial. Te Lagrange remainder tells us tat provided y (N+1) (x) is bounded, tere is an M suc tat te absolute value of te error will be less tan M N+1. It s 3

4 cumbersome to go around saying tere exists an M > 0 suc tat te asolute value of te error is less tan or equal to N+1 so we ll develop some sortand notation for describing errors. Tis order notation is used for approximations of all sorts, bot numerical and analytical. Tis section contains definitions and proofs, and you migt lose sigt of te forest for te trees. Te bottom line is a set of simple rules for working wit error estimates, igligted in several gray boxes. We ll write inequalities suc as E M p for M > 0 in te sortand notation E = O( p ), read as E is of order p. Te symbol O is te letter O often called big O not te number zero. Te notation E = O( p ) does not mean tat E is given by some function O acting on p. We ll sometimes abbreviate even furter and write E p for E = O( p ). Box 4.2 Order notation: O( p ) is sortand for te statement tat tere exists an M > 0 suc tat te quantity E is bounded from above by M p. Wen we see O-notation in an equation suc as a = b + O( 3 ) we mean tat a is equal to b witin an approximation error of order 3. In oter words, tere is a positive number M suc tat a b M 3. We can write te error in a Taylor approximation using tis notation. For example, te cubic Maclaurin 1 approximation for sin x is sin x = x 1 6 x3 + O(x 5 ), wic we interpret to mean tat te error in te cubic approximation to sin x is of order x 5, or again, more precisely, tere exists a positive number M suc tat x 1 6 x3 sin x M x 5. We ll next look at ow error estimates propagate troug matematical calculations. Multiplying an error estimate by a constant If E = O( p ), ten any constant times E is also O( p ). Consequently, wen you see an expression suc as you can immediately rewrite it as a = b + π 137 O(p ) a = b + O( p ). Tis saves us from dragging unnecessary constants wit us trougout a messy error calculation. It is similar in spirit to noticing tat if C is a constant of integration, ten 2 C is also just a constant. 1 A Maclaurin approximation is a Taylor approximation wit te expansion point a = 0. 4

5 Let s prove tat tis is true. By definition, E = O( p ) means tat M > 0 suc tat E M p. If true, ten for any constant A (constant in te sense of not depending on ) clearly But AM is just anoter positive constant. Call it M 2, and we ave AE AM p (4.5) AM p. (4.6) AE M 2 p. Terefore, tere exists a constant M 2 > 0 suc tat AE M 2 p, wic means AE = O( p ). Box 4.3 Multiplicative constants don t affect order of accuracy. If you see an expression suc as you can simplify it to Errors in addition of approximate quantities a = b + 42 O( 2 ), a + b = O( 2 ). Suppose tat u 1 = ũ 1 + O( p ) and u 2 = ũ 2 + O( q ). If we approximate u 1 + u + 2 by adding ũ 1 + ũ 2, wat is te error in tat calculation? Te error E in an approximate calculation u 1 + u 2 ũ 1 ũ 2 will be E = u 1 + u 2 ũ 1 ũ 2. By te triangle inequality, 2 E u 1 ũ 1 + u 2 ũ 2, and ten because we ve stipulated tat u 1 ũ 1 = O( p ) and u 2 ũ 2 = O( q ), E M 1 p + M 2 q were M 1 and M 2 are positive constants. Let s arbitrarily coose p q, or p = q + r were r 0. Let M be te larger of te two constants, M = max(m 1, M 2 ). Ten E M 1 p + M 2 q (4.7) M q+r + M q (4.8) M q r + M q (4.9) q (1 + r ) M { (4.10) 2M q 1 2M p > 1. (4.11) In te context of Taylor approximation, we re interested in te case were 1, so in tis context we ll always use te first case, E 2M q. Because 2M is just a constant multiplier and could be renamed M, we ave E = O( q ). 2 See your calculus textbook if you don t remember te triangle inequality. 5

6 Box 4.4 Wen adding a quantity wit error O( p ) to anoter quantity wit error O( q ), te error in te sum is O( min(p,q) ). Te sum of two errors is dominated by te error wit te lower order. Incidentally, in oter contexts suc as analysis of algoritms and asymptotic approximation, tis same order notation is used to describe te beavior for large. In tose contexts, a sum is dominated by te term wit te larger order. Errors in coefficients of p In te development of te Improved Euler and Runge-Kutta metods we ll encounter expressions suc as A p were A is independent of. We will eventually want to approximate A by A = à + O(q ). Te approximation to A p is ten Ãp + O( p+q ). You migt try to prove tis statement using te definition of O( q ) in terms of error bounds. Box 4.5 If te coefficient A of an expression A p is approximated by à wit an error O( q ), te error in Ãp is ten O( p+q ). Errors in function arguments Suppose te argument u of some function g(u) is approximated to O( p ). Given tat error in u, wat is te approximation error in te computed value g(u)? Let ũ be te approximate value of u, and E u = u ũ = O( p ) be te absolute error in u. Make a zerot-order Taylor polynomial approximation to g(u) expanded about te point ũ, using order notation for te Lagrange remainder, g(u) = g(ũ) + O(E u ) (4.12) = g(ũ) + O( p ). (4.13) Box 4.6 If u is approximated by ũ to O( p ), ten g(ũ) is accurate to O( p ). 4.2 Euler s Metod Euler s metod is te simplest sceme for computing approximate numerical solutions to initial value problems y = f(x, y), y(x 0 ) = y 0. Te idea is intuitive: near a point (x 0, y 0 ), approximate te solution by a line wit slope y = f(x 0, y 0 ), y(x) y 0 + f(x 0, y 0 ) (x x 0 ). Notice tat because of te ODE y = f, tis is equivalent to y(x) y(x 0 ) + y (x 0 ) (x x 0 ). 6

7 Tis is simply a first-order Taylor polynomial approximation! Now tis is probably a very poor approximation to te solution to te IVP; owever, refer back to figure and notice tat wile te N = 1 Taylor approximation to te sine function is very poor at large x, it is quite good wen x 0.5. Even a low-order Taylor polynomial can be a very good local approximation provided we stay sufficiently close to te point x 0. Euler s insigt was to use a Taylor approximation s good beavior near a point x 0 wile avoiding its bad beavior far from x 0 by making a sequence of sort steps of some small size rater tan one big step. Let s define some notation: x n = x 0 + n, i.e, x n is te value of x reaced after n steps of size. y n is te Euler approximation to te solution y(x) at x = x n. An Euler step from x n to x n+1 is a first-order Taylor approximation to y(x n+1 ) starting at te point x n. Note tat x n+1 x n =, so y n+1 = y n + f(x n, y n ). (4.14) Euler s metod is to repeat sort Euler steps until a desired point x N is reaced: Te local error y 1 = y 0 + f(x 0, y 0 ) (4.15) y 2 = y 1 + f(x 1, y 1 ) (4.16) y 3 = y 2 + f(x 2, y 2 ) (4.17). (4.18) y N = y N 1 + f(x N 1, y N 1 ) (4.19) We can use te Lagrange remainder formula to estimate te error in te first Euler step: recalling tat x 1 x 0 =, te error will be 1 2! y (c) 2 or simply O( 2 ). Tis is te error in te first step. Te next step to x 2 will ave a similar error, plus an additional error resulting from aving started te second step at an approximately-computed y 1. Te step to x 3 will ten ave its own error, plus te error resulting from starting at a value of y 2 tat s itself based on two approximate steps. If you re incredibly lucky wic you almost never will be all of tose errors will cancel. In te more typical case we need to figure out ow all tose single-step errors accumulate. We ll call te error in a single step te local error. Te error accumulated after many steps will be called te global error, wic we examine next Te global error Let s ignore for te moment te compounding effect of building eac step on previous steps tat are temselves only approximate, and look at te effect of taking N steps, eac wit an error O( 2 ), to traverse an interval [x 0, x N ]. To cover tat interval wit steps of size, you must take N = x N x 0 steps. If eac step as error O( 2 ), te error after N steps will be x N x 0 O( 2 ) = O(). Tus, te global error is one power of worse tan te local error. Te effect of compounding is more difficult to analyze, but it can be sown tat te effect of compounding approximations leads to te same conclusion as above: to get te global error, reduce te local error s exponent on by one. For Euler s metod, te local error O( 2 ) becomes a global error O(). A metod wit O( 5 ) local error becomes O( 4 ) globally, and so on. 7

8 4.2.3 Computational cost of numerical solution Suppose you ave used a stepsize 1 to compute a numerical solution wit absolute global error E( 1 ). You can use tis information to estimate te stepsize 2 needed to reac a desired error E( 2 ). Because E M p wit a p-t order numerical solution metod, te worst-case error of a p-t order metod wit stepsize is E() = M p. Ten and after eliminating te unknown M and solving for 2, ( ) 1/p E(2 ) 2 = 1. E( 1 ) Tus, to reduce te error by a factor R = E2 E 1 E( 1 ) = M p 1 (4.20) E( 2 ) = M p 2 (4.21) (not to be confused wit te letter R as used in te Lagrange remainder; tink R for reduction in tis context) you must cange te stepsize from 1 to 2 = 1 R 1/p. Te number of steps of size required to traverse an interval of fixed size L will be N = L/, so if you ave taken N 1 steps of size 1, you will need N 2 = 1 2 N 1 steps of size 2 to traverse te same interval. Combining tis wit te previous result relating cange in stepsize to desired cange in error, we find N 2 = N 1 R. 1/p Now, in making tis argument I assumed tat we were always at te worst-case error, so tat te inequality E M p could be replaced by an equality E = M p. In practice, we won t necessarily it te worst-case error, so te true equalities don t old. It s best to tink of te formulas we ve derived as approximate equalities: ( ) 1/p E(2 ) 2 1 E( 1 ) and N 2 N 1 R. 1/p At tis point, te following clever idea migt occur to you: if we were assuming everyting was a worstcase error, if we take N 2 = N 1 R 1/p steps, won t we always do better tan te desired error reduction R? Te answer is no. Te reason is tat we assumed tat bot E( 1 ) and E( 2 ) used te worst-case M. Suppose it appened tat for some problem a calculation wit stepsize 1 gave a better approximation tan te worst case possible wit tat stepsize. Ten, going to a smaller stepsize migt not elp as muc as teoretically possible. Tese approximate rules for stepsize reduction will become more reliable as te stepsize is reduced. Box 4.7 To reduce te absolute global error in a p-t order approximation by a factor R, reduce te stepsize by a factor Tis will require a factor 2 1 R 1/p. N 2 1 N 1 R 1/p more steps. Tese approximate rules for stepsize reduction will become more reliable i.e closer to equalities as te stepsize is reduced. 8

9 Example To reduce an error E 1 = 0.01 to an error E 2 = 10 6 = 10 4 E 1 we need a reduction factor R = Euler s metod is first-order, so it will take N 2 N 1 R = 104 N 1 steps to acieve te desired reduction. Suppose you ave found a metod wit global error O( 2 ). Ten, to improve te error by R = 10 4 takes N 2 N 1 = 10 2 N 1 R steps. 4.3 Improving Euler s metod Te improved Euler metod is a simple extension of Euler s metod giving second-order accuracy. Te obvious way to get iger order of accuracy would be to take more terms in te Taylor approximation. Euler s metod used te first-order term; wy not try including te second-order terms as well? Ten we would ave y n+1 = y n + f(x n, y n ) y (x n ) 2 + O( 3 ) But wat is y (x n )? Wit y = f(x, y) and te multivariable cain rule, we can find and so y = f x + f y y y (x n ) = f x + f xn,y n y f. xn,y n Tis is somewat complicated to compute, and will be even more so wen tere is more tan one unknown function or wen an even iger-order metod is desired. Te second derivative is ard to compute, but do we really need it? Suppose you can approximate te second derivative y by some y (x n ), accurate to O( q ): y (x n ) = y (x n ) + O( q ) Ten ( ) y n+1 = y n + f(x n, y n ) y (x n ) + O( q ) + O( 3 ). To wat accuracy do we need to approximate te second derivative? Recall from tat if we approximate A by à wit accuracy O(q ), ten p A is accurate to O( p+q ). So, upon replacing y by its q-t order approximation ỹ, we ave y n+1 = y n + f(x n, y n ) y (x n ) + O( 2+q ) + O( 3 ). Te first error term, O( 2+q ), is te result of approximating y by ỹ. Te second, O( 3 ), is te result of copping te initial Taylor series after te second-derivative term. Using te rule for adding errors, te combined local error in y n+1 will be O( min(3,2+q) ). Tis tells us te accuracy requirement for te estimate ỹ : if we want to do better tan Euler s O( 2 ) local error, we need q 1 so tat we get 2 + q 3. Furtermore, tere s no point to getting y more accurately 9

10 tan q = 1, because te overall error would still be O( 3 ) just from te error of te original second-order Taylor polynomial. Terefore, we want q = 1. So wile it migt be ard to compute y, te good news is tat we don t actually need it! Any first-order approximation to y will contribute only O( 3 ) to te overall local error in te step, same as te error from stopping at second order terms to begin wit Estimating y How do we estimate y? It s easier tan you migt tink. In calculus, you learned tat te derivative of a function g(x) is te limit of a difference quotient g (x) = lim >0 g(x + ) g(x). It seems reasonable to approximate g by skipping te passage to te limit and simply using te difference quotient g g(x + ) g(x) (x) =. How accurate is tat approximation? As always, use Taylor s teorem: g(x + ) = g(x) + g (x) + O( 2 ), so g (x) = g(x) + g (x) + O( 2 ) g(x) (4.22) g (x) = g (x) + O(). (4.23) In oter words, te difference between g and g is O(), wic is exactly te accuracy needed for our purposes. To approximate y to O(), form a difference quotient But y (x) = f(x, y), so y (x n ) = y (x n+1 ) y (x n ). y (x n ) = f(x n+1, y n+1 ) f(x n, y n ). Putting tis approximate expression for y into te Taylor polynomial for y n+1 gives us y n+1 = y n + f(x n, y n ) + 2 f(x n+1, y n+1 ) f(x n, y n ) + O( 3 ) (4.24) 2 = y n + 2 [f(x n, y n ) + f(x n+1, y n+1 )] + O( 3 ). (4.25) I ve already absorbed te error in 2 y into te overall O( 3 ) error Estimating f(x n+1, y n+1 ) Tere s still one itc: te unknown y n+1 appears on te RHS of te formula for te step, as indicated wit a box: y n+1 = y n + [ ] f(x n, y n ) + f(x n+1, y n+1 ) + O( 3 ) 2 In principle, we could solve (algebraically) for y n+1, and in fact for certain problems tat s a good approac. However, if we keep in mind tat tis wole calculation is already only an approximation, wy not use an approximation for te quantity in te box rater tan te exact value of y n+1. Let s call tat approximation 10

11 q n+1, and say tat it s accurate to O( r ). Ten, by te rule in for te error in a function value given error in te function s argument, we get f(x n+1, y n+1 ) = f(x n+1, q n+1 ) + O( r ). Tis approximation to f arises in an expression tat is a coefficient of, so by section we know it will contribute O( 1+r ) to te overall error. We want O( 3 ) overall, so we must ave r = 2 meaning q n+1 must be a second-order approximation to y n+1. Luckily, we already know a way to get a second-order approximation to y n+1 : Euler s metod! A single Euler step to y n+1 as local error O( 2 ). Let s see ow to put all tis togeter Te improved Euler metod Te approximation developed in te preceding section defines an improved Euler step from y n to y n+1. We can encapsulate it as a procedure: Box 4.8 Improved Euler step: 1. Make a single Euler step to compute q n+1, a first approximation to y n+1 : q n+1 = y n + f(x n, y n ). 2. Advance te independent variable: x n+1 = x n Compute te average of te initial slope f(x n, y n ) and te slope at te (approximate) new point, f(x n+1, q n+1 ): s avg = 1 2 [f(x n, y n ) + f(x n+1, q n+1 )]. 4. Make a step from (x n, y n ) to (x n+1, y n+1 ) using s avg as te slope: y n+1 = y n + s avg Te resulting y n+1 will be accurate to a local error O( 3 ). As wit Euler s metod, te procedure is repeated for many steps. Wen doing multiple steps, errors will accumulate so tat te global error becomes O( 2 ). 4.4 Summary 1. A Taylor polynomial approximates a function near a point a. Te coefficients in te Taylor polynomial are cosen so tat its first N derivatives matc te first N derivatives of te function at a. Taylor polynomials can be very good approximations near a, but are increasingly poor approximations te farter you go from a. 2. Order notation keeps track of te dependence of an error estimate on an approximation parameter. Te notation E = O( p ) reads E is order p, or more briefly, E is p-t order. Saying x is accurate to p-t order is equivalent to saying te error in x is p-t order. 11

12 Box 4.9 Rules for working wit error estimates: (a) O( p ) is sortand for te statement tat tere exists an M > 0 suc tat te quantity E is bounded from above by M p. (b) Multiplicative constants don t affect order of accuracy. (c) Wen adding a quantity wit error O( p ) to anoter quantity wit error O( q ), te error in te sum is O( min(p,q) ). (d) If te coefficient A of an expression A p is approximated by à wit an error O( q ), te error in Ãp is ten O( p+q ). (e) If u is approximated by ũ to O( p ), ten g(ũ) is accurate to O( p ). 3. Euler s metod advances from (x n, y n ) to (x n +, y n+1 ) using a first-order Taylor polynomial approximation. One ten computes te first derivative at te new point, and advances from tere. Because it is based on a first-order polynomial, te local error in a single step is O( 2 ). Because errors accumulate from step to step and because te number of steps required is inversely proportional to, te global error is O(). 4. In principle, we could get an approximation of iger order of accuracy tan Euler s metod by retaining more terms in te Taylor polynomial used at eac step; in practice, owever, te computation of te additional derivatives appearing in te iger-order terms is difficult. Te insigt beind te improved Euler metod is tat we don t need to compute te derivatives appearing in tose iger-order terms exactly: we need only approximate tem to an accuracy sufficient to keep te total error in te step te same order as te error in te Taylor polynomial approximation. An improved Euler step approximates te second derivative y (x n ) by a difference quotient aving O() accuracy. Wen computing te step, te second derivative appears only in a term 2 y (x n ), so te O() error in te approximate y results in a O( 3 ) local error, exactly te same order as te error from truncating te Taylor polynomial. In computing te difference quotient approximation to y, te f(x n+1, y n+1 ) appears, and ten times tis quantity appears in te formula for te step to y n+1. But y n+1 isn t yet known until we take te step! However, if we approximate y n+1 to O( 2 ) accuracy by some q n+1, ten f(x n+1, q n+1 ) will be accurate to O( 3 ), no worse tan te previous approximations. We compute q n+1 using a step of Euler s metod, wit O( 2 ) local error. 5. If a metod is known to ave O( p ) global error, we can use tat information to estimate te stepsize needed to reduce te error by a given factor. To reduce te error by a factor R, te rule of tumb is to reduce te stepsize by R 1/p. Tis in turn increases te cost of te computation by a factor 1/R 1/p. 12

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