《数字信号处理》教学参考资料（Numerical Recipes in C，The Art of Scientific Computing Second Edition）Chapter 16.2

714 Chapter 16.Integration of Ordinary Differential Equations dv=vector(1,nvar); for(1=1;1<=nvar;i++){ Load starting values. v[i]-vstart[i]; y[i][1]=v[1]; 2 xx[1]=x1; x=x1; h=(x2-x1)/nstep; for (k=1;k<=nstep;k++){ Take nstep steps. (*derivs)(x,v,dv); rk4(v,dv,nvar,x,h,vout,derivs); if ((float)(x+h)==x)nrerror("Step size too small in routine rkdumb"); X+=h; xx[0k+1]=x; Store intermediate steps. for (i=1;i<=nvar;i++){ v[i]-vout[i]: y[i][k+1]=v[i]; 2 2 令 1-800 NUMERI ICAL free_vector(dv,1,nvar); free_vector(vout,1,nvar); free_vector(v,1,nvar); RECIPES I (North America computer, Press. CITED REFERENCES AND FURTHER READING: Abramowitz,M.,and Stegun,I.A.1964,Handbook of Mathematical Functions,Applied Mathe- matics Series,Volume 55 (Washington:National Bureau of Standards;reprinted 1968 by 9 Programs Dover Publications,New York),825.5.[1] Gear,C.W.1971,Numerical Initial Value Problems in Ordinary Differential Equations(Englewood 、核 OF SCIENTIFIC Cliffs,NJ:Prentice-Hall),Chapter 2.[2] Shampine,L.F.,and Watts,H.A.1977,in Mathematica/Software /l,J.R.Rice,ed.(New York:Aca- 61 demic Press),pp.257-275:1979,Applied Mathematics and Computation,vol.5,pp.93- 121.[3] Rice,J.R.1983,Numerical Methods,Software,and Analysis (New York:McGraw-Hill),89.2. Numerica 10.621 16.2 Adaptive Stepsize Control for Runge-Kutta 431 Recipes A good ODE integrator should exert some adaptive control over its own progress, (outside making frequent changes in its stepsize.Usually the purpose of this adaptive stepsize 首 control is to achieve some predetermined accuracy in the solution with minimum computational effort.Many small steps should tiptoe through treacherous terrain. while a few great strides should speed through smooth uninteresting countryside. The resulting gains in efficiency are not mere tens of percents or factors of two; they can sometimes be factors of ten,a hundred,or more.Sometimes accuracy may be demanded not directly in the solution itself,but in some related conserved quantity that can be monitored. Implementation of adaptive stepsize control requires that the stepping algorithm signal information about its performance,most important,an estimate of its truncation error.In this section we will learn how such information can be obtained.Obviously

714 Chapter 16. Integration of Ordinary Differential Equations Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). dv=vector(1,nvar); for (i=1;i<=nvar;i++) { Load starting values. v[i]=vstart[i]; y[i][1]=v[i]; } xx[1]=x1; x=x1; h=(x2-x1)/nstep; for (k=1;k<=nstep;k++) { Take nstep steps. (*derivs)(x,v,dv); rk4(v,dv,nvar,x,h,vout,derivs); if ((float)(x+h) == x) nrerror("Step size too small in routine rkdumb"); x += h; xx[k+1]=x; Store intermediate steps. for (i=1;i<=nvar;i++) { v[i]=vout[i]; y[i][k+1]=v[i]; } } free_vector(dv,1,nvar); free_vector(vout,1,nvar); free_vector(v,1,nvar); } CITED REFERENCES AND FURTHER READING: Abramowitz, M., and Stegun, I.A. 1964, Handbook of Mathematical Functions, Applied Mathe￾matics Series, Volume 55 (Washington: National Bureau of Standards; reprinted 1968 by Dover Publications, New York), §25.5. [1] Gear, C.W. 1971, Numerical Initial Value Problems in Ordinary Differential Equations (Englewood Cliffs, NJ: Prentice-Hall), Chapter 2. [2] Shampine, L.F., and Watts, H.A. 1977, in Mathematical Software III, J.R. Rice, ed. (New York: Aca￾demic Press), pp. 257–275; 1979, Applied Mathematics and Computation, vol. 5, pp. 93– 121. [3] Rice, J.R. 1983, Numerical Methods, Software, and Analysis (New York: McGraw-Hill), §9.2. 16.2 Adaptive Stepsize Control for Runge-Kutta A good ODE integrator should exert some adaptive control over its own progress, making frequent changes in its stepsize. Usually the purpose of this adaptive stepsize control is to achieve some predetermined accuracy in the solution with minimum computational effort. Many small steps should tiptoe through treacherous terrain, while a few great strides should speed through smooth uninteresting countryside. The resulting gains in efficiency are not mere tens of percents or factors of two; they can sometimes be factors of ten, a hundred, or more. Sometimes accuracy may be demanded not directly in the solution itself, but in some related conserved quantity that can be monitored. Implementation of adaptive stepsize control requires that the stepping algorithm signal information about its performance,most important, an estimate of its truncation error. In this section we will learn how such information can be obtained. Obviously

16.2 Adaptive Stepsize Control for Runge-Kutta 715 the calculation of this information will add to the computational overhead,but the investment will generally be repaid handsomely. With fourth-order Runge-Kutta,the most straightforward technique by far is step doubling (see,e.g.,[11).We take each step twice,once as a full step,then, independently,as two half steps(see Figure 16.2.1).How much overhead is this, say in terms of the number of evaluations of the right-hand sides?Each of the three separate Runge-Kutta steps in the procedure requires 4 evaluations,but the single and double sequences share a starting point,so the total is 11.This is to be compared not to 4,but to 8 (the two half-steps),since-stepsize control aside-we are achieving the accuracy of the smaller(half)stepsize.The overhead cost is therefore 81 a factor 1.375.What does it buy us? Let us denote the exact solution for an advance from x to x+2h by y(x+2h) and the two approximate solutions by y1(one step 2h)and y2(2 steps each of size h).Since the basic method is fourth order,the true solution and the two numerical 智 approximations are related by y(x+2h)=1+(2h)φ+O(h)+.. RECIPES (16.2.1) y(x+2h)=2+2(h)p+O(h)+.. 9 where,to order h5,the value remains constant over the step.[Taylor series expansion tells us the is a number whose order of magnitude is y(5)()/5!.]The first expression in(16.2.1)involves(2h)5 since the stepsize is 2h,while the second 9 expression involves 2(h5)since the error on each step is h5.The difference between 里刀。 the two numerical estimates is a convenient indicator of truncation error △三2一1 (16.2.2) It is this difference that we shall endeavor to keep to a desired degree of accuracy, neither too large nor too small.We do this by adjusting h. It might also occur to you that,ignoring terms of order h6 and higher,we can solve the two equations in (16.2.1)to improve our numerical estimate of the true solution y(x +2h),namely, Numerica 10.621 (红+2h)=2+5 +O(h) (16.2.3) 三%e步 This estimate is accurate to fifth order,one order higher than the original Runge- Kutta steps.However,we can't have our cake and eat it:(16.2.3)may be fifth-order accurate,but we have no way of monitoring its truncation error.Higher order is not always higher accuracy!Use of(16.2.3)rarely does harm,but we have no way of directly knowing whether it is doing any good.Therefore we should use A as the error estimate and take as"gravy"any additional accuracy gain derived from (16.2.3).In the technical literature,use of a procedure like (16.2.3)is called "local extrapolation." An alternative stepsize adjustment algorithm is based on the embedded Runge- Kutta formulas,originally invented by Fehlberg.An interesting fact about Runge- Kutta formulas is that for orders M higher than four,more than M function evaluations(though never more than M+2)are required.This accounts for the

16.2 Adaptive Stepsize Control for Runge-Kutta 715 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). the calculation of this information will add to the computational overhead, but the investment will generally be repaid handsomely. With fourth-order Runge-Kutta, the most straightforward technique by far is step doubling (see, e.g., [1]). We take each step twice, once as a full step, then, independently, as two half steps (see Figure 16.2.1). How much overhead is this, say in terms of the number of evaluations of the right-hand sides? Each of the three separate Runge-Kutta steps in the procedure requires 4 evaluations, but the single and double sequences share a starting point, so the total is 11. This is to be compared not to 4, but to 8 (the two half-steps), since — stepsize control aside — we are achieving the accuracy of the smaller (half) stepsize. The overhead cost is therefore a factor 1.375. What does it buy us? Let us denote the exact solution for an advance from x to x + 2h by y(x + 2h) and the two approximate solutions by y1 (one step 2h) and y2 (2 steps each of size h). Since the basic method is fourth order, the true solution and the two numerical approximations are related by y(x + 2h) = y1 + (2h) 5φ + O(h6) + ... y(x + 2h) = y2 + 2(h5)φ + O(h6) + ... (16.2.1) where, to order h5, the value φ remains constant over the step. [Taylor series expansion tells us the φ is a number whose order of magnitude is y (5)(x)/5!.] The first expression in (16.2.1) involves (2h)5 since the stepsize is 2h, while the second expression involves 2(h5)since the error on each step is h5φ. The difference between the two numerical estimates is a convenient indicator of truncation error ∆ ≡ y2 − y1 (16.2.2) It is this difference that we shall endeavor to keep to a desired degree of accuracy, neither too large nor too small. We do this by adjusting h. It might also occur to you that, ignoring terms of order h6 and higher, we can solve the two equations in (16.2.1) to improve our numerical estimate of the true solution y(x + 2h), namely, y(x + 2h) = y2 + ∆ 15 + O(h6) (16.2.3) This estimate is accurate to fifth order, one order higher than the original Runge￾Kutta steps. However, we can’t have our cake and eat it: (16.2.3) may be fifth-order accurate, but we have no way of monitoring its truncation error. Higher order is not always higher accuracy! Use of (16.2.3) rarely does harm, but we have no way of directly knowing whether it is doing any good. Therefore we should use ∆ as the error estimate and take as “gravy” any additional accuracy gain derived from (16.2.3). In the technical literature, use of a procedure like (16.2.3) is called “local extrapolation.” An alternative stepsize adjustment algorithm is based on the embedded Runge￾Kutta formulas, originally invented by Fehlberg. An interesting fact about Runge￾Kutta formulas is that for orders M higher than four, more than M function evaluations (though never more than M + 2) are required. This accounts for the

716 Chapter 16. Integration of Ordinary Differential Equations big step ● ● two small steps Figure 16.2.1.Step-doubling as a means for adaptive stepsize control in fourth-order Runge-Kutta Points where the derivative is evaluated are shown as filled circles.The open circle represents the same derivatives as the filled circle immediately above it,so the total number of evaluations is I 1 per two steps. Comparing the accuracy of the big step with the two small steps gives a criterion for adjusting the stepsize on the next step,or for rejecting the current step as inaccurate. popularity of the classical fourth-order method:It seems to give the most bang 透 for the buck.However.Fehlberg discovered a fifth-order method with six function evaluations where another combination of the six functions gives a fourth-order method.The difference between the two estimates of y(+h)can then be used as an estimate of the truncation error to adjust the stepsize.Since Fehlberg's original RECIPES I 2 formula,several other embedded Runge-Kutta formulas have been found. Many practitioners were at one time wary of the robustness of Runge-Kutta- Fehlberg methods.The feeling was that using the same evaluation points to advance Press. the function and to estimate the error was riskier than step-doubling,where the error ART estimate is based on independent function evaluations.However,experience has shown that this concern is not a problem in practice.Accordingly,embedded Runge- Progra Kutta formulas,which are roughly a factor of two more efficient,have superseded algorithms based on step-doubling. 小三绿 OF SCIENTIFIC( The general form of a fifth-order Runge-Kutta formula is 61 k1 hf(In;yn) k2 hf(In +a2h,yn +b21k1) 1920 COMPUTING (ISBN (16.2.4) k6 hf(In +a6h,yn +b61k1 +...+b65ks) 10621 ym+1=n+C11+C2k2+C3k3+c4k4+c5+c6k6+O(h6) Numerical Recipes 43108 The embedded fourth-order formula is (outside Un+1=Un +cik1 c2k2 c3k3+cak4+csks +cok6+O(h) (16.2.5) North and so the error estimate is △三yn+1-y+1 -c)ki (16.2.6) The particular values of the various constants that we favor are those found by Cash and Karp [21,and given in the accompanying table.These give a more efficient method than Fehlberg's original values,with somewhat better error properties

716 Chapter 16. Integration of Ordinary Differential Equations Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). two small steps big step x Figure 16.2.1. Step-doubling as a means for adaptive stepsize control in fourth-order Runge-Kutta. Points where the derivative is evaluated are shown as filled circles. The open circle represents the same derivatives as the filled circle immediately above it, so the total number of evaluations is 11 per two steps. Comparing the accuracy of the big step with the two small steps gives a criterion for adjusting the stepsize on the next step, or for rejecting the current step as inaccurate. popularity of the classical fourth-order method: It seems to give the most bang for the buck. However, Fehlberg discovered a fifth-order method with six function evaluations where another combination of the six functions gives a fourth-order method. The difference between the two estimates of y(x + h) can then be used as an estimate of the truncation error to adjust the stepsize. Since Fehlberg’s original formula, several other embedded Runge-Kutta formulas have been found. Many practitioners were at one time wary of the robustness of Runge-Kutta￾Fehlberg methods. The feeling was that using the same evaluation points to advance the function and to estimate the error was riskier than step-doubling, where the error estimate is based on independent function evaluations. However, experience has shown that this concern is not a problem in practice. Accordingly, embedded Runge￾Kutta formulas, which are roughly a factor of two more efficient, have superseded algorithms based on step-doubling. The general form of a fifth-order Runge-Kutta formula is k1 = hf(xn, yn) k2 = hf(xn + a2h, yn + b21k1) ··· k6 = hf(xn + a6h, yn + b61k1 + ··· + b65k5) yn+1 = yn + c1k1 + c2k2 + c3k3 + c4k4 + c5k5 + c6k6 + O(h6) (16.2.4) The embedded fourth-order formula is y∗ n+1 = yn + c∗ 1k1 + c∗ 2k2 + c∗ 3k3 + c∗ 4k4 + c∗ 5k5 + c∗ 6k6 + O(h5) (16.2.5) and so the error estimate is ∆ ≡ yn+1 − y∗ n+1 =
6 i=1 (ci − c∗ i )ki (16.2.6) The particular values of the various constants that we favor are those found by Cash and Karp [2], and given in the accompanying table. These give a more efficient method than Fehlberg’s original values, with somewhat better error properties

16.2 Adaptive Stepsize Control for Runge-Kutta 717 Cash-Karp Parameters for Embedded Runga-Kutta Method i ai b ci c 1 嘉 器 2 0 0 3 品 斋 品 翯 18575 4838 4 35 品 ~品 器 器 11 5 一 》 夢 0 277 14336 83g 6 1631 575 44275 55296 噩 13824 110592 器 14 2 t 4 5 1-00 Now that we know,at least approximately,what our error is,we need to consider how to keep it within desired bounds.What is the relation between A and h?According to (16.2.4)-(16.2.5),A scales as h5.If we take a step h and produce an error A1,therefore,the step ho that would have given some other value Ao is readily estimated as 需 10.2 △0 ho △1 (16.2.7) Henceforth we will let Ao denote the desired accuracy.Then equation (16.2.7)is used in two ways:If A1 is larger than Ao in magnitude,the equation tells how 拿 much to decrease the stepsize when we retry the present (failed)step.If A is 景 可 smaller than Ao,on the other hand,then the equation tells how much we can safely increase the stepsize for the next step.Local extrapolation consists in accepting the fifth order value yn+1,even though the error estimate actually applies to the fourth order valuey+1 Our notation hides the fact that Ao is actually a vector of desired accuracies. one for each equation in the set of ODEs.In general,our accuracy requirement will be that all equations are within their respective allowed errors.In other words,we ridge.org Fuurggoglrion Numerical Recipes 10.621 43108 will rescale the stepsize according to the needs of the"worst-offender"equation. How is Ao,the desired accuracy,related to some looser prescription like"get a solution good to one part in 106"?That can be a subtle question,and it depends on (outside exactly what your application is!You may be dealing with a set of equations whose North Software. dependent variables differ enormously in magnitude.In that case,you probably want to use fractional errors,Ao =ey,where e is the number like 10-6 or whatever. On the other hand,you may have oscillatory functions that pass through zero but are bounded by some maximum values.In that case you probably want to set Ao equal to e times those maximum values. A convenient way to fold these considerations into a generally useful stepper routine is this:One of the arguments of the routine will of course be the vector of dependent variables at the beginning of a proposed step.Call that y[1..n].Let us require the user to specify for each step another,corresponding,vector argument yscal[1..n],and also an overall tolerance level eps.Then the desired accuracy

16.2 Adaptive Stepsize Control for Runge-Kutta 717 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). Cash-Karp Parameters for Embedded Runga-Kutta Method i ai bij ci c∗ i 1 37 378 2825 27648 2 1 5 1 5 0 0 3 3 10 3 40 9 40 250 621 18575 48384 4 3 5 3 10 − 9 10 6 5 125 594 13525 55296 5 1 −11 54 5 2 −70 27 35 27 0 277 14336 6 7 8 1631 55296 175 512 575 13824 44275 110592 253 4096 512 1771 1 4 j = 123 4 5 Now that we know, at least approximately, what our error is, we need to consider how to keep it within desired bounds. What is the relation between ∆ and h? According to (16.2.4) – (16.2.5), ∆ scales as h5. If we take a step h1 and produce an error ∆1, therefore, the step h0 that would have given some other value ∆0 is readily estimated as h0 = h1 ∆0 ∆1 0.2 (16.2.7) Henceforth we will let ∆0 denote the desired accuracy. Then equation (16.2.7) is used in two ways: If ∆1 is larger than ∆0 in magnitude, the equation tells how much to decrease the stepsize when we retry the present (failed) step. If ∆1 is smaller than ∆0, on the other hand, then the equation tells how much we can safely increase the stepsize for the next step. Local extrapolation consists in accepting the fifth order value yn+1, even though the error estimate actually applies to the fourth order value y∗ n+1. Our notation hides the fact that ∆0 is actually a vector of desired accuracies, one for each equation in the set of ODEs. In general, our accuracy requirement will be that all equations are within their respective allowed errors. In other words, we will rescale the stepsize according to the needs of the “worst-offender” equation. How is ∆0, the desired accuracy, related to some looser prescription like “get a solution good to one part in 106”? That can be a subtle question, and it depends on exactly what your application is! You may be dealing with a set of equations whose dependent variables differ enormously in magnitude. In that case, you probably want to use fractional errors, ∆0 = y, where  is the number like 10−6 or whatever. On the other hand, you may have oscillatory functions that pass through zero but are bounded by some maximum values. In that case you probably want to set ∆ 0 equal to  times those maximum values. A convenient way to fold these considerations into a generally useful stepper routine is this: One of the arguments of the routine will of course be the vector of dependent variables at the beginning of a proposed step. Call that y[1..n]. Let us require the user to specify for each step another, corresponding, vector argument yscal[1..n], and also an overall tolerance level eps. Then the desired accuracy

718 Chapter 16.Integration of Ordinary Differential Equations for the ith equation will be taken to be △o=eps×yscal[i] (16.2.8) If you desire constant fractional errors,plug a pointer to y into the pointer to yscal calling slot(no need to copy the values into a different array).If you desire constant absolute errors relative to some maximum values,set the elements of yscal equal to those maximum values.A useful "trick"for getting constant fractional errors except "very"near zero crossings is to set yscal[i]equal to ly [i]+h x dydx[i] (The routine odeint,below,does this.) Here is a more technical point.We have to consider one additional possibility for yscal.The error criteria mentioned thus far are "local,"in that they bound the error of each step individually.In some applications you may be unusually sensitive about a "global"accumulation of errors,from beginning to end of the integration and in the worst possible case where the errors all are presumed to add with the same sign.Then,the smaller the stepsize h,the smaller the value Ao that you will need to impose.Why?Because there will be more steps between your starting and ending values of z.In such cases you will want to set yscal proportional to h,typically to something like △o=eh×dydx[i] (16.2.9) 9 This enforces fractional accuracy e not on the values ofy but(much more stringently) on the increments to those values at each step.But now look back at(16.2.7).If Ao SCIENTIFIC( has an implicit scaling with h,then the exponent 0.20 is no longer correct:When the stepsize is reduced from a too-large value,the new predicted value h will fail to meet the desired accuracy when yscal is also altered to this new h value.Instead of 0.20 =1/5,we must scale by the exponent 0.25 =1/4 for things to work out. The exponents 0.20 and 0.25 are not really very different.This motivates us to adopt the following pragmatic approach,one that frees us from having to know in advance whether or not you,the user,plan to scale your yscal's with stepsize. Whenever we decrease a stepsize,let us use the larger value of the exponent(whether we need it or not!),and whenever we increase a stepsize,let us use the smaller Numerica 10.621 exponent.Furthermore,because our estimates of error are not exact,but only 431 accurate to the leading order in h,we are advised to put in a safety factor S which is (outside Recipes a few percent smaller than unity.Equation (16.2.7)is thus replaced by 10.20 North △1 △0≥△1 ho 10.25 (16.2.10) △0 △0<△1 We have found this prescription to be a reliable one in practice Here,then,is a stepper program that takes one"quality-controlled"Runge- Kutta step

718 Chapter 16. Integration of Ordinary Differential Equations Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). for the ith equation will be taken to be ∆0 = eps × yscal[i] (16.2.8) If you desire constant fractional errors, plug a pointer to y into the pointer to yscal calling slot (no need to copy the values into a different array). If you desire constant absolute errors relative to some maximum values, set the elements of yscal equal to those maximum values. A useful “trick” for getting constant fractional errors except “very” near zero crossings is to set yscal[i] equal to |y[i]| + |h × dydx[i]|. (The routine odeint, below, does this.) Here is a more technical point. We have to consider one additional possibility for yscal. The error criteria mentioned thus far are “local,” in that they bound the error of each step individually. In some applications you may be unusually sensitive about a “global” accumulation of errors, from beginning to end of the integration and in the worst possible case where the errors all are presumed to add with the same sign. Then, the smaller the stepsize h, the smaller the value ∆0 that you will need to impose. Why? Because there will be more steps between your starting and ending values of x. In such cases you will want to set yscal proportional to h, typically to something like ∆0 = h × dydx[i] (16.2.9) This enforces fractional accuracy  not on the values of y but (much more stringently) on the increments to those values at each step. But now look back at (16.2.7). If ∆ 0 has an implicit scaling with h, then the exponent 0.20 is no longer correct: When the stepsize is reduced from a too-large value, the new predicted value h 1 will fail to meet the desired accuracy when yscal is also altered to this new h1 value. Instead of 0.20 = 1/5, we must scale by the exponent 0.25 = 1/4 for things to work out. The exponents 0.20 and 0.25 are not really very different. This motivates us to adopt the following pragmatic approach, one that frees us from having to know in advance whether or not you, the user, plan to scale your yscal’s with stepsize. Whenever we decrease a stepsize, let us use the larger value of the exponent (whether we need it or not!), and whenever we increase a stepsize, let us use the smaller exponent. Furthermore, because our estimates of error are not exact, but only accurate to the leading order in h, we are advised to put in a safety factor S which is a few percent smaller than unity. Equation (16.2.7) is thus replaced by h0 =    Sh1 ∆0 ∆1 0.20 ∆0 ≥ ∆1 Sh1 ∆0 ∆1 0.25 ∆0 < ∆1 (16.2.10) We have found this prescription to be a reliable one in practice. Here, then, is a stepper program that takes one “quality-controlled” Runge￾Kutta step.