5250:
4611:
7472:
64:
791:
2591:
3449:
639:
996:
51:. In some cases there may be two different problems with the same solution, yet one is not stiff and the other is. The phenomenon cannot therefore be a property of the exact solution, since this is the same for both problems, and must be a property of the differential system itself. Such systems are thus known as
2858:
manner, even for the restricted class of linear constant coefficient systems. We shall also see several qualitative statements that can be (and mostly have been) made in an attempt to encapsulate the notion of stiffness, and state what is probably the most satisfactory of these as a "definition" of stiffness.
8041:
4120:
If the numerical method also exhibits this behaviour (for a fixed step size), then the method is said to be A-stable. A numerical method that is L-stable (see below) has the stronger property that the solution approaches zero in a single step as the step size goes to infinity. A-stable methods do not
46:
displays much variation and to be relatively large where the solution curve straightens out to approach a line with slope nearly zero. For some problems this is not the case. In order for a numerical method to give a reliable solution to the differential system sometimes the step size is required
2885:
There are other characteristics which are exhibited by many examples of stiff problems, but for each there are counterexamples, so these characteristics do not make good definitions of stiffness. Nonetheless, definitions based upon these characteristics are in common use by some authors and are
6299:
2857:
In this section we consider various aspects of the phenomenon of stiffness. "Phenomenon" is probably a more appropriate word than "property", since the latter rather implies that stiffness can be defined in precise mathematical terms; it turns out not to be possible to do this in a satisfactory
2834:
2434:
3283:
5931:
786:{\displaystyle {\begin{aligned}\mathrm {A} \xrightarrow {0.04} &\mathrm {B} \\\mathrm {B} +\mathrm {B} \xrightarrow {3\cdot 10^{7}} &\mathrm {C} +\mathrm {B} \\\mathrm {B} +\mathrm {C} \xrightarrow {1\cdot 10^{4}} &\mathrm {A} +\mathrm {C} \end{aligned}}}
4863:
3803:
7648:
1503:
6672:
5424:
7879:
797:
2424:
555:
7890:
6976:
7249:
39:, unless the step size is taken to be extremely small. It has proven difficult to formulate a precise definition of stiffness, but the main idea is that the equation includes some terms that can lead to rapid variation in the solution.
4600:
8094:
barrier; it restricts the usefulness of linear multistep methods for stiff equations. An example of a second-order A-stable method is the trapezoidal rule mentioned above, which can also be considered as a linear multistep method.
3147:
6484:
7763:
5725:
6823:
6182:
5582:
4528:
2586:{\displaystyle {\bigl |}\operatorname {Re} ({\overline {\lambda }}){\bigr |}\geq {\bigl |}\operatorname {Re} (\lambda _{t}){\bigr |}\geq {\bigl |}\operatorname {Re} ({\underline {\lambda }}){\bigr |},\qquad t=1,2,\ldots ,n}
3906:) requires a very small step size until well into the smooth part of the solution curve, resulting in an error much smaller than required for accuracy. Thus the system also satisfies statement 2 and Lambert's definition.
2739:
1661:
1127:
5051:
2052:
5809:
3523:
2725:
276:
2666:
1375:
8089:
Explicit multistep methods can never be A-stable, just like explicit Runge–Kutta methods. Implicit multistep methods can only be A-stable if their order is at most 2. The latter result is known as the second
4310:
168:
2272:
2140:
3444:{\displaystyle \mathbf {A} ={\begin{pmatrix}0&1\\-1000&-1001\end{pmatrix}},\qquad \mathbf {f} (t)={\begin{pmatrix}0\\0\end{pmatrix}},\qquad \mathbf {x} (0)={\begin{pmatrix}x_{0}\\0\end{pmatrix}},}
1188:
2331:
2199:
1301:
5977:
the exact solution does. Nevertheless, the trapezoidal method does not have perfect behavior: it does damp all decaying components, but rapidly decaying components are damped only very mildly, because
5820:
2877:, is forced to use in a certain interval of integration a step length which is excessively small in relation to the smoothness of the exact solution in that interval, then the system is said to be
1732:
802:
644:
5175:
1895:
2997:. These are all examples of a class of problems called stiff (mathematical stiffness) systems of differential equations, due to their application in analyzing the motion of spring and mass
4665:
3594:
4231:
4118:
7083:
4934:
5936:
This region contains the left half-plane, so the trapezoidal method is A-stable. In fact, the stability region is identical to the left half-plane, and thus the numerical solution of
3672:
3658:
is even larger. (while "large" is not a clearly-defined term, but the larger the above quantities are, the more pronounced will be the effect of stiffness.) The exact solution to (
7485:
1391:
7364:
5210:
5122:
4009:
991:{\displaystyle {\begin{aligned}{\dot {x}}&=-0.04x+10^{4}y\cdot z\\{\dot {y}}&=0.04x-10^{4}y\cdot z-3\cdot 10^{7}y^{2}\\{\dot {z}}&=3\cdot 10^{7}y^{2}\end{aligned}}}
380:
342:
5971:
5464:
4657:
4441:
4167:
6503:
6088:
5266:
2989:
Problems involving rapidly decaying transient solutions occur naturally in a wide variety of applications, including the study of spring and damping systems, the analysis of
2087:
1820:
1789:
1577:
8079:
7774:
616:
6039:
6010:
2961:
7425:
7333:
6361:
6324:
6174:
6152:
6114:
4975:
4393:
4080:
1949:
1922:
1846:
1758:
1210:
4054:
3865:
1546:
618:. Applying this method instead of Euler's method gives a much better result (blue). The numerical results decrease monotonically to zero, just as the exact solution does.
4367:
3946:
3898:
8036:{\displaystyle \left\{z\in \mathbb {C} \ \left|\ \left|{\tfrac {1}{2}}\left(1+{\tfrac {3}{2}}z\pm {\sqrt {1+z+{\tfrac {9}{4}}z^{2}}}\right)\right|<1\right.\right\}.}
7045:
2340:
1236:
400:
7461:
5239:
2981:
5618:
6834:
3981:
3241:
3215:
6388:
5083:
4330:
1051:
7091:
7008:
2886:
good clues as to the presence of stiffness. Lambert refers to these as "statements" rather than definitions, for the aforementioned reasons. A few of these are:
8585:
8395:
4533:
3273:
3189:
7384:
7292:
7272:
3656:
3636:
2219:
1972:
3024:
6393:
7659:
6486:. A method is A-stable if and only if its stability function has no poles in the left-hand plane and its order star contains no purely imaginary numbers.
6294:{\displaystyle \phi (z)={\frac {\det \left(\mathbf {I} -z\mathbf {A} +z\mathbf {e} \mathbf {b} ^{\mathsf {T}}\right)}{\det(\mathbf {I} -z\mathbf {A} )}},}
5626:
6683:
2829:{\displaystyle {\frac {{\bigl |}\operatorname {Re} ({\overline {\lambda }}){\bigr |}}{{\bigl |}\operatorname {Re} ({\underline {\lambda }}){\bigr |}}}.}
1053:
there is no problem in numerical integration. However, if the interval is very large (10 say), then many standard codes fail to integrate it correctly.
5472:
8680:
5258:
4454:
386:
32:
1587:
4980:
8558:
8526:
8464:
8429:
5736:
3470:
2671:
1977:
1072:
194:
2612:
42:
When integrating a differential equation numerically, one would expect the requisite step size to be relatively small in a region where the
1306:
4236:
3900:
term, even with a small coefficient, is enough to make the numerical computation very sensitive to step size. Stable integration of (
8489:
8288:
2228:
2096:
8118:, an extension of the notion of differential equation that allows discontinuities, in part as way to sidestep some stiffness issues
1148:
5926:{\displaystyle \left\{z\in \mathbb {C} \ \left|\ \left|{\frac {1+{\frac {1}{2}}z}{1-{\frac {1}{2}}z}}\right|<1\right.\right\}.}
2277:
2145:
1241:
8104:
82:
8046:
This region is shown on the right. It does not include all the left half-plane (in fact it only includes the real axis between
8546:
1062:
624:
8357:
On Padé approximations to the exponential function and A-stable methods for the numerical solution of initial value problems
1682:
8441:
8121:
5177:, which is outside the stability region. Indeed, the numerical results do not converge to zero. However, with step size
5127:
1851:
4858:{\displaystyle y_{n+1}=y_{n}+h\cdot f(t_{n},y_{n})=y_{n}+h\cdot (ky_{n})=y_{n}+h\cdot k\cdot y_{n}=(1+h\cdot k)y_{n}.}
2910:
6363:
and thus, their stability function is a polynomial. It follows that explicit Runge–Kutta methods cannot be A-stable.
47:
to be at an unacceptably small level in a region where the solution curve is very smooth. The phenomenon is known as
3914:
The behaviour of numerical methods on stiff problems can be analyzed by applying these methods to the test equation
3533:
4172:
4085:
7050:
4871:
8280:
3798:{\displaystyle x(t)=x_{0}\left(-{\frac {1}{999}}e^{-1000t}+{\frac {1000}{999}}e^{-t}\right)\approx x_{0}e^{-t}.}
301:
The figure (right) illustrates the numerical issues for various numerical integrators applied on the equation.
5249:
8393:
Gear, C. W. (1981), "Numerical solution of ordinary differential equations: Is there anything left to do?",
7643:{\displaystyle y_{n+1}=y_{n}+h\left({\tfrac {3}{2}}f(t_{n},y_{n})-{\tfrac {1}{2}}f(t_{n-1},y_{n-1})\right).}
6494:
5241:
which is just inside the stability region and the numerical results converge to zero, albeit rather slowly.
1498:{\displaystyle \mathbf {y} (x)=\sum _{t=1}^{n}\kappa _{t}e^{\lambda _{t}x}\mathbf {c} _{t}+\mathbf {g} (x),}
390:
6129:
4129:
8654:
8364:
8115:
8499:
Lambert, J. D. (1977), D. Jacobs (ed.), "The initial value problem for ordinary differential equations",
67:
Explicit numerical methods exhibiting instability when integrating a stiff ordinary differential equation
6667:{\displaystyle y_{n+1}=\sum _{i=0}^{s}a_{i}y_{n-i}+h\sum _{j=-1}^{s}b_{j}f\left(t_{n-j},y_{n-j}\right).}
6117:
5419:{\displaystyle y_{n+1}=y_{n}+{\tfrac {1}{2}}h\cdot {\bigl (}f(t_{n},y_{n})+f(t_{n+1},y_{n+1}){\bigr )},}
3014:
2930:
2874:
72:
28:
7874:{\displaystyle w={\tfrac {1}{2}}\left(1+{\tfrac {3}{2}}z\pm {\sqrt {1+z+{\tfrac {9}{4}}z^{2}}}\right),}
7341:
5180:
5092:
3986:
2897:
Stiffness occurs when stability requirements, rather than those of accuracy, constrain the step length.
350:
312:
4610:
4398:
8518:
8481:
8381:
6048:
2061:
1794:
1763:
1551:
8341:
8049:
2913:, the term "stiff" is used because such systems correspond to tight coupling between the driver and
8658:
7471:
7011:
6015:
5980:
2998:
2936:
2914:
2870:
36:
7389:
7297:
6344:
6307:
6157:
6135:
6093:
4939:
4372:
4059:
2419:{\displaystyle {\overline {\lambda }},{\underline {\lambda }}\in \{\lambda _{t},t=1,2,\ldots ,n\}}
1927:
1900:
1825:
1737:
1193:
550:{\displaystyle y_{n+1}=y_{n}+{\tfrac {1}{2}}h{\bigl (}f(t_{n},y_{n})+f(t_{n+1},y_{n+1}){\bigr )},}
8623:
8328:
7463:. Again, if this set contains the left half-plane, the multi-step method is said to be A-stable.
7338:
The region of absolute stability for a multistep method of the above form is then the set of all
5939:
5432:
4625:
4135:
4014:
3830:
2926:
2866:
1524:
295:
6367:
4339:
3870:
63:
8107:, a family of implicit methods especially used for the solution of stiff differential equations
7017:
6971:{\displaystyle \left(1-b_{-1}z\right)y_{n+1}-\sum _{j=0}^{s}\left(a_{j}+b_{j}z\right)y_{n-j}=0}
1215:
575:
8554:
8522:
8485:
8460:
8456:
8425:
8298:
8284:
8091:
7430:
6338:
6327:
5215:
2994:
2966:
628:
382:, produces a solution within the graph boundaries, but oscillates about zero (shown in green).
306:
8665:
7244:{\displaystyle \Phi (z,w)=w^{s+1}-\sum _{i=0}^{s}a_{i}w^{s-i}-z\sum _{j=-1}^{s}b_{j}w^{s-j}.}
5590:
8615:
8606:
Wanner, Gerhard; Hairer, Ernst; Nørsett, Syvert (1978), "Order stars and stability theory",
8594:
8404:
8318:
8310:
8110:
4595:{\displaystyle {\bigl \{}z\in \mathbb {C} \,{\big |}\,\operatorname {Re} (z)<0{\bigr \}}}
3951:
3220:
3194:
6373:
5059:
4315:
3917:
1018:
8635:
8421:
8355:
7479:
Let us determine the region of absolute stability for the two-step Adams–Bashforth method
6984:
2900:
Stiffness occurs when some components of the solution decay much more rapidly than others.
8580:
3252:
3168:
8385:
8504:
7369:
7277:
7257:
3641:
3621:
3142:{\displaystyle m{\ddot {x}}+c{\dot {x}}+kx=0,\qquad x(0)=x_{0},\qquad {\dot {x}}(0)=0,}
3002:
2990:
2918:
2204:
1957:
43:
6479:{\displaystyle {\bigl \{}z\in \mathbb {C} \,{\big |}\,|\phi (z)|>|e^{z}|{\bigr \}}}
8674:
8475:
8418:
Solving ordinary differential equations II: Stiff and differential-algebraic problems
8377:
8332:
8274:
8651:
7758:{\displaystyle \Phi (w,z)=w^{2}-\left(1+{\tfrac {3}{2}}z\right)w+{\tfrac {1}{2}}z=0}
7475:
The pink region is the stability region for the second-order Adams–Bashforth method.
5720:{\displaystyle y_{n+1}={\frac {1+{\frac {1}{2}}hk}{1-{\frac {1}{2}}hk}}\cdot y_{n}.}
8661:
8627:
8564:
8439:
Hirshfelder, J. O. (1963), "Applied
Mathematics as used in Theoretical Chemistry",
6818:{\displaystyle y_{n+1}=\sum _{i=0}^{s}a_{i}y_{n-i}+hk\sum _{j=-1}^{s}b_{j}y_{n-j},}
4619:
2909:
The origin of the term "stiffness" has not been clearly established. According to
8646:
5577:{\displaystyle y_{n+1}=y_{n}+{\tfrac {1}{2}}h\cdot \left(ky_{n}+ky_{n+1}\right).}
6042:
4523:{\displaystyle {\bigl \{}z\in \mathbb {C} \,{\big |}\,|\phi (z)|<1{\bigr \}}}
20:
6366:
The stability function of implicit Runge–Kutta methods is often analyzed using
8323:
6331:
4530:. The method is A-stable if the region of absolute stability contains the set
2891:
1656:{\displaystyle \operatorname {Re} (\lambda _{t})<0,\qquad t=1,2,\ldots ,n,}
8140:
Robertson, H. H. (1966). "The solution of a set of reaction rate equations".
8647:
An
Introduction to Physically Based Modeling: Energy Functions and Stiffness
3006:
5053:
which is the disk depicted on the right. The Euler method is not A-stable.
5046:{\displaystyle {\bigl \{}z\in \mathbb {C} \,{\big |}\,|1+z|<1{\bigr \}}}
344:
oscillates wildly and quickly exits the range of the graph (shown in red).
4121:
exhibit the instability problems as described in the motivating example.
3618:) then certainly satisfies statements 1 and 3. Here the spring constant
2047:{\textstyle \sum _{t=1}^{n}\kappa _{t}e^{\lambda _{t}x}\mathbf {c} _{t}}
8619:
8314:
5804:{\displaystyle \phi (z)={\frac {1+{\frac {1}{2}}z}{1-{\frac {1}{2}}z}}}
3525:. Both eigenvalues have negative real part and the stiffness ratio is
3518:{\displaystyle {\overline {\lambda }}=-1000,{\underline {\lambda }}=-1}
2720:{\displaystyle \kappa _{t}e^{{\underline {\lambda }}x}\mathbf {c} _{t}}
1122:{\displaystyle \mathbf {y} '=\mathbf {A} \mathbf {y} +\mathbf {f} (x),}
271:{\displaystyle y(t)=e^{-15t},\quad y(t)\to 0{\text{ as }}t\to \infty .}
2661:{\displaystyle \kappa _{t}e^{{\overline {\lambda }}x}\mathbf {c} _{t}}
8598:
8408:
8301:(1963), "A special stability problem for linear multistep methods",
1370:{\displaystyle \mathbf {c} _{t}\in \mathbb {C} ^{n},t=1,2,\ldots ,n}
745:
690:
656:
8374:
Numerical
Initial-Value Problems in Ordinary Differential Equations
5253:
The pink region is the stability region for the trapezoidal method.
7470:
5248:
4609:
62:
8581:"A user's view of solving stiff ordinary differential equations"
4305:{\displaystyle y_{n}={\bigl (}\phi (hk){\bigr )}^{n}\cdot y_{0}}
8545:
Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007).
4614:
The pink disk shows the stability region for the Euler method.
2267:{\displaystyle \left|\operatorname {Re} (\lambda _{t})\right|}
2135:{\displaystyle \left|\operatorname {Re} (\lambda _{t})\right|}
1183:{\displaystyle \mathbf {y} ,\mathbf {f} \in \mathbb {R} ^{n}}
2890:
A linear constant coefficient system is stiff if all of its
2326:{\displaystyle \kappa _{t}e^{\lambda _{t}x}\mathbf {c} _{t}}
2194:{\displaystyle \kappa _{t}e^{\lambda _{t}x}\mathbf {c} _{t}}
1296:{\displaystyle \lambda _{t}\in \mathbb {C} ,t=1,2,\ldots ,n}
1015:
If one treats this system on a short interval, for example,
8022:
6116:. The trapezoidal method is A-stable but not L-stable. The
5912:
4977:. The region of absolute stability for this method is thus
163:{\displaystyle \,y'(t)=-15y(t),\quad t\geq 0,\quad y(0)=1.}
2921:. According to Richard. L. Burden and J. Douglas Faires,
2894:
have negative real part and the stiffness ratio is large.
8343:
Stability analysis for systems of differential equations
8553:(3rd ed.). New York: Cambridge University Press.
7982:
7950:
7927:
7840:
7808:
7785:
7735:
7709:
7577:
7530:
6370:. The order star for a method with stability function
5509:
5303:
5191:
5147:
5103:
3410:
3363:
3300:
1980:
1727:{\displaystyle e^{\lambda _{t}x}\mathbf {c} _{t}\to 0}
437:
361:
323:
8052:
7893:
7777:
7662:
7488:
7433:
7392:
7372:
7344:
7300:
7280:
7260:
7094:
7053:
7020:
6987:
6837:
6686:
6506:
6396:
6376:
6347:
6310:
6185:
6160:
6138:
6096:
6051:
6018:
5983:
5942:
5823:
5739:
5629:
5593:
5475:
5435:
5269:
5218:
5183:
5130:
5095:
5062:
4983:
4942:
4874:
4668:
4628:
4536:
4457:
4401:
4375:
4342:
4318:
4239:
4175:
4138:
4088:
4062:
4017:
3989:
3954:
3920:
3873:
3833:
3675:
3644:
3624:
3536:
3473:
3286:
3255:
3223:
3197:
3171:
3027:
2969:
2939:
2742:
2674:
2615:
2437:
2343:
2280:
2231:
2207:
2148:
2099:
2064:
1960:
1930:
1903:
1854:
1828:
1797:
1766:
1740:
1685:
1590:
1554:
1527:
1394:
1309:
1244:
1218:
1196:
1151:
1075:
1021:
800:
642:
578:
403:
353:
315:
197:
85:
7254:
All solutions converge to zero for a given value of
16:
Differential equation exhibiting unusual instability
8515:
Numerical
Methods for Ordinary Differential Systems
8240:
2933:when the exact solution contains terms of the form
1579:is a particular integral. Now let us suppose that
8551:Numerical Recipes: The Art of Scientific Computing
8073:
8035:
7873:
7757:
7642:
7455:
7419:
7378:
7358:
7327:
7286:
7266:
7243:
7077:
7039:
7002:
6970:
6817:
6666:
6478:
6382:
6355:
6318:
6293:
6168:
6146:
6108:
6082:
6033:
6004:
5965:
5925:
5803:
5719:
5612:
5576:
5458:
5418:
5233:
5204:
5169:
5116:
5077:
5045:
4969:
4928:
4857:
4651:
4594:
4522:
4435:
4387:
4361:
4324:
4304:
4225:
4161:
4112:
4074:
4048:
4003:
3975:
3940:
3892:
3859:
3797:
3650:
3630:
3588:
3517:
3443:
3267:
3235:
3209:
3183:
3141:
2975:
2955:
2828:
2719:
2660:
2585:
2418:
2325:
2266:
2213:
2193:
2134:
2081:
2046:
1974:to be time (as it often is in physical problems),
1966:
1943:
1916:
1889:
1840:
1814:
1783:
1752:
1726:
1655:
1571:
1540:
1497:
1369:
1303:(assumed distinct) and corresponding eigenvectors
1295:
1230:
1204:
1182:
1121:
1045:
990:
785:
610:
549:
374:
336:
270:
162:
8081:) so the Adams–Bashforth method is not A-stable.
7047:of the recurrence relation converge to zero when
5170:{\displaystyle z=-15\times {\tfrac {1}{4}}=-3.75}
2925:Significant difficulties can occur when standard
1890:{\displaystyle e^{\lambda _{t}x}\mathbf {c} _{t}}
7467:Example: The second-order Adams–Bashforth method
6260:
6204:
3827:behaves quite similarly to a simple exponential
1063:linear constant coefficient inhomogeneous system
623:One of the most prominent examples of the stiff
8273:Burden, Richard L.; Faires, J. Douglas (1993),
4618:Consider the Euler methods above. The explicit
8244:
6471:
6418:
6399:
6045:: a method is L-stable if it is A-stable and
5408:
5322:
5038:
5005:
4986:
4587:
4558:
4539:
4515:
4479:
4460:
4278:
4255:
3589:{\displaystyle {\frac {|-1000|}{|-1|}}=1000,}
2983:is a complex number with negative real part.
2929:are applied to approximate the solution of a
2815:
2786:
2777:
2748:
2547:
2518:
2508:
2479:
2469:
2440:
539:
453:
8:
8189:
7034:
7021:
6326:denotes the vector with all ones. This is a
4226:{\displaystyle y_{n+1}=\phi (hk)\cdot y_{n}}
4113:{\displaystyle \operatorname {Re} (k)<0.}
2861:J. D. Lambert defines stiffness as follows:
2413:
2370:
8178:
7078:{\displaystyle \operatorname {Re} (z)<0}
4929:{\displaystyle y_{n}=(1+hk)^{n}\cdot y_{0}}
8501:The State of the Art in Numerical Analysis
7014:. The method is A-stable if all solutions
6677:Applied to the test equation, they become
8322:
8257:
8214:
8051:
7997:
7981:
7967:
7949:
7926:
7906:
7905:
7892:
7884:thus the region of absolute stability is
7855:
7839:
7825:
7807:
7784:
7776:
7734:
7708:
7688:
7661:
7617:
7598:
7576:
7564:
7551:
7529:
7512:
7493:
7487:
7442:
7434:
7432:
7391:
7371:
7352:
7351:
7343:
7299:
7279:
7259:
7226:
7216:
7206:
7192:
7170:
7160:
7150:
7139:
7120:
7093:
7052:
7028:
7019:
6986:
6950:
6932:
6919:
6904:
6893:
6874:
6853:
6836:
6800:
6790:
6780:
6766:
6741:
6731:
6721:
6710:
6691:
6685:
6644:
6625:
6607:
6597:
6583:
6561:
6551:
6541:
6530:
6511:
6505:
6470:
6469:
6464:
6458:
6449:
6441:
6424:
6423:
6417:
6416:
6415:
6411:
6410:
6398:
6397:
6395:
6375:
6348:
6346:
6311:
6309:
6277:
6266:
6246:
6245:
6240:
6234:
6223:
6212:
6201:
6184:
6161:
6159:
6139:
6137:
6095:
6069:
6052:
6050:
6017:
5982:
5941:
5885:
5864:
5855:
5836:
5835:
5822:
5785:
5764:
5755:
5738:
5708:
5682:
5658:
5649:
5634:
5628:
5598:
5592:
5554:
5538:
5508:
5499:
5480:
5474:
5434:
5407:
5406:
5391:
5372:
5350:
5337:
5321:
5320:
5302:
5293:
5274:
5268:
5217:
5190:
5182:
5146:
5129:
5102:
5094:
5061:
5037:
5036:
5025:
5011:
5010:
5004:
5003:
5002:
4998:
4997:
4985:
4984:
4982:
4941:
4920:
4907:
4879:
4873:
4846:
4812:
4787:
4771:
4746:
4730:
4717:
4692:
4673:
4667:
4627:
4586:
4585:
4563:
4557:
4556:
4555:
4551:
4550:
4538:
4537:
4535:
4514:
4513:
4502:
4485:
4484:
4478:
4477:
4476:
4472:
4471:
4459:
4458:
4456:
4422:
4402:
4400:
4374:
4347:
4341:
4317:
4296:
4283:
4277:
4276:
4254:
4253:
4244:
4238:
4217:
4180:
4174:
4137:
4087:
4061:
4037:
4016:
3997:
3996:
3988:
3953:
3919:
3878:
3872:
3848:
3838:
3832:
3783:
3773:
3752:
3738:
3723:
3709:
3695:
3674:
3643:
3623:
3569:
3558:
3551:
3540:
3537:
3535:
3496:
3474:
3472:
3417:
3405:
3388:
3358:
3341:
3295:
3287:
3285:
3254:
3222:
3196:
3170:
3110:
3109:
3099:
3050:
3049:
3032:
3031:
3026:
2968:
2944:
2938:
2814:
2813:
2800:
2785:
2784:
2776:
2775:
2762:
2747:
2746:
2743:
2741:
2711:
2706:
2690:
2689:
2679:
2673:
2652:
2647:
2631:
2630:
2620:
2614:
2546:
2545:
2532:
2517:
2516:
2507:
2506:
2497:
2478:
2477:
2468:
2467:
2454:
2439:
2438:
2436:
2377:
2357:
2344:
2342:
2317:
2312:
2300:
2295:
2285:
2279:
2250:
2230:
2206:
2185:
2180:
2168:
2163:
2153:
2147:
2118:
2098:
2065:
2063:
2038:
2033:
2021:
2016:
2006:
1996:
1985:
1979:
1959:
1935:
1929:
1908:
1902:
1881:
1876:
1864:
1859:
1853:
1827:
1798:
1796:
1767:
1765:
1739:
1712:
1707:
1695:
1690:
1684:
1604:
1589:
1555:
1553:
1532:
1526:
1478:
1469:
1464:
1452:
1447:
1437:
1427:
1416:
1395:
1393:
1331:
1327:
1326:
1316:
1311:
1308:
1259:
1258:
1249:
1243:
1217:
1197:
1195:
1174:
1170:
1169:
1160:
1152:
1150:
1102:
1094:
1089:
1077:
1074:
1020:
978:
968:
940:
939:
929:
919:
891:
860:
859:
840:
806:
805:
801:
799:
774:
766:
756:
736:
728:
719:
711:
701:
681:
673:
664:
647:
643:
641:
577:
538:
537:
522:
503:
481:
468:
452:
451:
436:
427:
408:
402:
360:
352:
322:
314:
251:
217:
196:
86:
84:
8451:Iserles, Arieh; Nørsett, Syvert (1991),
8226:The definition of L-stability is due to
8201:
5814:and the region of absolute stability is
347:Euler's method with half the step size,
8547:"Section 17.5. Stiff Sets of Equations"
8442:American Mathematical Society Symposium
8416:Hairer, Ernst; Wanner, Gerhard (1996),
8166:
8154:
8132:
4443:. This motivates the definition of the
6247:
2142:is large, then the corresponding term
1679:which implies that each of the terms
627:(ODEs) is a system that describes the
186:The exact solution (shown in cyan) is
8579:Shampine, L. F.; Gear, C. W. (1979),
6120:is an example of an L-stable method.
7:
8536:Mathews, John; Fink, Kurtis (1992),
8227:
6337:Explicit Runge–Kutta methods have a
3666:
3527:
3277:
3018:
2733:
2428:
1581:
1385:
1066:
633:
394:
188:
76:
8241:Wanner, Hairer & Nørsett (1978)
8144:. Academic Press. pp. 178–182.
8142:Numerical analysis: an introduction
7085:. The characteristic polynomial is
4056:. This solution approaches zero as
4011:. The solution of this equation is
7663:
7393:
7301:
7095:
6103:
6028:
5429:when applied to the test equation
4382:
4069:
3638:is large and the damping constant
1835:
1747:
775:
767:
737:
729:
720:
712:
682:
674:
665:
648:
262:
14:
8634:Stability of Runge-Kutta Methods
7653:The characteristic polynomial is
7359:{\displaystyle z\in \mathbb {C} }
5205:{\displaystyle h={\tfrac {1}{8}}}
5117:{\displaystyle h={\tfrac {1}{4}}}
4447:(sometimes referred to simply as
4004:{\displaystyle k\in \mathbb {C} }
3948:subject to the initial condition
2869:with a finite region of absolute
2274:is small, the corresponding term
375:{\displaystyle h={\tfrac {1}{8}}}
337:{\displaystyle h={\tfrac {1}{4}}}
298:that exhibits the same behavior.
8681:Numerical differential equations
8477:Advanced Engineering Mathematics
8105:Backward differentiation formula
6349:
6312:
6278:
6267:
6241:
6235:
6224:
6213:
6162:
6140:
5730:Thus, the stability function is
4602:, that is, the left half plane.
4436:{\displaystyle |\phi (hk)|<1}
3612:which is fairly large. System (
3389:
3342:
3288:
2727:the slowest. We now define the
2707:
2648:
2313:
2181:
2066:
2034:
1877:
1799:
1768:
1708:
1556:
1479:
1465:
1396:
1312:
1198:
1161:
1153:
1103:
1095:
1090:
1078:
6083:{\displaystyle |\phi (z)|\to 0}
3387:
3340:
3108:
3079:
2873:, applied to a system with any
2555:
2221:increases and is thus called a
2082:{\displaystyle \mathbf {g} (x)}
1815:{\displaystyle \mathbf {g} (x)}
1784:{\displaystyle \mathbf {y} (x)}
1622:
1572:{\displaystyle \mathbf {g} (x)}
1212:is a constant, diagonalizable,
625:ordinary differential equations
232:
141:
128:
8538:Numerical methods using MATLAB
8074:{\displaystyle -1\leq z\leq 0}
7678:
7666:
7629:
7591:
7570:
7544:
7443:
7435:
7408:
7396:
7316:
7304:
7110:
7098:
7066:
7060:
6465:
6450:
6442:
6438:
6432:
6425:
6282:
6263:
6195:
6189:
6100:
6074:
6070:
6066:
6060:
6053:
6022:
5996:
5993:
5987:
5749:
5743:
5403:
5365:
5356:
5330:
5026:
5012:
4952:
4946:
4904:
4888:
4839:
4821:
4777:
4761:
4736:
4710:
4576:
4570:
4503:
4499:
4493:
4486:
4423:
4419:
4410:
4403:
4379:
4353:
4272:
4263:
4207:
4198:
4101:
4095:
4066:
4027:
4021:
3964:
3958:
3685:
3679:
3570:
3559:
3552:
3541:
3399:
3393:
3352:
3346:
3243:, can be written in the form (
3127:
3121:
3089:
3083:
2810:
2797:
2772:
2759:
2542:
2529:
2503:
2490:
2464:
2451:
2333:decays slowly and is called a
2256:
2243:
2124:
2111:
2076:
2070:
1832:
1809:
1803:
1778:
1772:
1744:
1718:
1610:
1597:
1566:
1560:
1489:
1483:
1406:
1400:
1113:
1107:
1040:
1028:
605:
593:
534:
496:
487:
461:
259:
245:
242:
236:
207:
201:
151:
145:
122:
116:
101:
95:
1:
8666:doi:10.4249/scholarpedia.2855
8122:Explicit and implicit methods
6041:. This led to the concept of
6034:{\displaystyle z\to -\infty }
6005:{\displaystyle \phi (z)\to 1}
4622:applied to the test equation
4132:applied to the test equation
2956:{\displaystyle e^{\lambda t}}
2853:Characterization of stiffness
2668:is the fastest transient and
35:for solving the equation are
8245:Iserles & Nørsett (1991)
7420:{\displaystyle \Phi (z,w)=0}
7328:{\displaystyle \Phi (z,w)=0}
6356:{\displaystyle \mathbf {A} }
6319:{\displaystyle \mathbf {e} }
6169:{\displaystyle \mathbf {b} }
6147:{\displaystyle \mathbf {A} }
6128:The stability function of a
6109:{\displaystyle z\to \infty }
4970:{\displaystyle \phi (z)=1+z}
4445:region of absolute stability
4388:{\displaystyle n\to \infty }
4075:{\displaystyle t\to \infty }
3902:
3823:
3660:
3614:
3479:
2767:
2636:
2459:
2349:
1944:{\displaystyle \lambda _{t}}
1924:is real and sinusoidally if
1917:{\displaystyle \lambda _{t}}
1897:will decay monotonically if
1841:{\displaystyle x\to \infty }
1753:{\displaystyle x\to \infty }
1548:are arbitrary constants and
1377:. The general solution of (
1205:{\displaystyle \mathbf {A} }
8420:(second ed.), Berlin:
6828:which can be simplified to
5966:{\displaystyle y'=k\cdot y}
5459:{\displaystyle y'=k\cdot y}
5245:Example: Trapezoidal method
5056:The motivating example had
4652:{\displaystyle y'=k\cdot y}
4336:. Thus, the condition that
4162:{\displaystyle y'=k\cdot y}
4049:{\displaystyle y(t)=e^{kt}}
3860:{\displaystyle x_{0}e^{-t}}
3245:
2911:Joseph Oakland Hirschfelder
1541:{\displaystyle \kappa _{t}}
1379:
8697:
8480:(3rd ed.), New York:
8281:Prindle, Weber and Schmidt
8213:This definition is due to
4606:Example: The Euler methods
4362:{\displaystyle y_{n}\to 0}
3893:{\displaystyle e^{-1000t}}
3867:, but the presence of the
8239:The definition is due to
8190:Burden & Faires (1993
7040:{\displaystyle \{y_{n}\}}
6390:is defined to be the set
6339:strictly lower triangular
1231:{\displaystyle n\times n}
611:{\displaystyle y'=f(t,y)}
8474:Kreyszig, Erwin (1972),
8279:(5th ed.), Boston:
7456:{\displaystyle |w|<1}
7335:lie in the unit circle.
6495:Linear multistep methods
5234:{\displaystyle z=-1.875}
2976:{\displaystyle \lambda }
1238:matrix with eigenvalues
389:(that is, the two-stage
8513:Lambert, J. D. (1992),
5613:{\displaystyle y_{n+1}}
1760:, so that the solution
8365:University of Waterloo
8340:Eberly, David (2008),
8116:Differential inclusion
8075:
8037:
7875:
7759:
7644:
7476:
7457:
7421:
7380:
7360:
7329:
7288:
7268:
7245:
7211:
7155:
7079:
7041:
7004:
6972:
6909:
6819:
6785:
6726:
6668:
6602:
6546:
6480:
6384:
6357:
6320:
6295:
6170:
6148:
6110:
6084:
6035:
6006:
5967:
5927:
5805:
5721:
5614:
5578:
5460:
5420:
5254:
5235:
5206:
5171:
5118:
5089:when taking step size
5079:
5047:
4971:
4930:
4859:
4653:
4615:
4596:
4524:
4437:
4389:
4363:
4326:
4306:
4227:
4163:
4114:
4076:
4050:
4005:
3977:
3976:{\displaystyle y(0)=1}
3942:
3894:
3861:
3799:
3652:
3632:
3590:
3519:
3445:
3269:
3237:
3236:{\displaystyle k=1000}
3211:
3210:{\displaystyle c=1001}
3185:
3143:
3011:
2977:
2957:
2883:
2830:
2721:
2662:
2587:
2420:
2327:
2268:
2215:
2201:will decay quickly as
2195:
2136:
2083:
2048:
2001:
1968:
1945:
1918:
1891:
1842:
1816:
1785:
1754:
1728:
1657:
1573:
1542:
1499:
1432:
1371:
1297:
1232:
1206:
1184:
1123:
1047:
992:
787:
612:
551:
376:
338:
272:
164:
68:
8076:
8038:
7876:
7760:
7645:
7474:
7458:
7422:
7381:
7361:
7330:
7289:
7269:
7246:
7188:
7135:
7080:
7042:
7005:
6973:
6889:
6820:
6762:
6706:
6669:
6579:
6526:
6481:
6385:
6383:{\displaystyle \phi }
6358:
6334:divided by another).
6321:
6296:
6171:
6149:
6118:implicit Euler method
6111:
6085:
6036:
6007:
5973:converges to zero if
5968:
5928:
5806:
5722:
5615:
5579:
5461:
5421:
5252:
5236:
5207:
5172:
5119:
5080:
5078:{\displaystyle k=-15}
5048:
4972:
4931:
4860:
4654:
4613:
4597:
4525:
4438:
4390:
4364:
4327:
4325:{\displaystyle \phi }
4307:
4233:, and, by induction,
4228:
4164:
4115:
4077:
4051:
4006:
3978:
3943:
3941:{\displaystyle y'=ky}
3895:
3862:
3800:
3653:
3633:
3591:
3520:
3446:
3270:
3238:
3212:
3186:
3144:
3015:initial value problem
2978:
2958:
2931:differential equation
2923:
2863:
2831:
2722:
2663:
2588:
2421:
2328:
2269:
2216:
2196:
2137:
2091:steady-state solution
2084:
2049:
1981:
1969:
1946:
1919:
1892:
1843:
1817:
1786:
1755:
1729:
1658:
1574:
1543:
1500:
1412:
1372:
1298:
1233:
1207:
1185:
1124:
1048:
1046:{\displaystyle t\in }
993:
788:
613:
552:
377:
339:
273:
165:
73:initial value problem
66:
29:differential equation
8655:Lawrence F. Shampine
8376:, Englewood Cliffs:
8372:Gear, C. W. (1971),
8354:Ehle, B. L. (1969),
8050:
7891:
7775:
7660:
7486:
7431:
7390:
7370:
7342:
7298:
7278:
7258:
7092:
7051:
7018:
7003:{\displaystyle z=hk}
6985:
6835:
6684:
6504:
6394:
6374:
6345:
6308:
6183:
6158:
6136:
6094:
6049:
6016:
5981:
5940:
5821:
5737:
5627:
5591:
5473:
5433:
5267:
5216:
5181:
5128:
5093:
5060:
4981:
4940:
4872:
4666:
4626:
4534:
4455:
4451:), which is the set
4399:
4373:
4340:
4316:
4237:
4173:
4136:
4086:
4060:
4015:
3987:
3952:
3918:
3871:
3831:
3673:
3642:
3622:
3534:
3471:
3467:and has eigenvalues
3284:
3253:
3221:
3195:
3169:
3025:
2967:
2937:
2927:numerical techniques
2740:
2672:
2613:
2435:
2341:
2278:
2229:
2205:
2146:
2097:
2062:
1978:
1958:
1928:
1901:
1852:
1826:
1795:
1764:
1738:
1683:
1588:
1552:
1525:
1392:
1307:
1242:
1216:
1194:
1149:
1073:
1019:
798:
640:
576:
401:
391:Adams–Moulton method
351:
313:
309:with a step size of
195:
83:
37:numerically unstable
8386:1971nivp.book.....G
8169:, pp. 217–220)
8157:, pp. 216–217)
7012:recurrence relation
7010:. This is a linear
6341:coefficient matrix
4130:Runge–Kutta methods
4125:Runge–Kutta methods
3268:{\displaystyle n=2}
3184:{\displaystyle m=1}
762:
707:
660:
8620:10.1007/BF01932026
8457:Chapman & Hall
8324:10338.dmlcz/103497
8315:10.1007/BF01963532
8299:Dahlquist, Germund
8276:Numerical Analysis
8179:Hirshfelder (1963)
8071:
8033:
7991:
7959:
7936:
7871:
7849:
7817:
7794:
7755:
7744:
7718:
7640:
7586:
7539:
7477:
7453:
7417:
7376:
7356:
7325:
7284:
7264:
7241:
7075:
7037:
7000:
6968:
6815:
6664:
6476:
6380:
6353:
6316:
6291:
6166:
6144:
6132:with coefficients
6130:Runge–Kutta method
6106:
6080:
6031:
6002:
5963:
5923:
5801:
5717:
5610:
5574:
5518:
5456:
5416:
5312:
5259:trapezoidal method
5255:
5231:
5202:
5200:
5167:
5156:
5114:
5112:
5075:
5043:
4967:
4926:
4855:
4649:
4616:
4592:
4520:
4433:
4385:
4359:
4334:stability function
4322:
4302:
4223:
4159:
4110:
4072:
4046:
4001:
3973:
3938:
3890:
3857:
3795:
3648:
3628:
3586:
3515:
3504:
3441:
3432:
3378:
3331:
3265:
3233:
3207:
3181:
3139:
2993:, and problems in
2973:
2953:
2881:in that interval.
2875:initial conditions
2826:
2808:
2717:
2698:
2658:
2583:
2540:
2416:
2365:
2323:
2264:
2211:
2191:
2132:
2079:
2056:transient solution
2044:
1964:
1941:
1914:
1887:
1838:
1822:asymptotically as
1812:
1781:
1750:
1724:
1653:
1569:
1538:
1495:
1367:
1293:
1228:
1202:
1180:
1119:
1043:
988:
986:
783:
781:
608:
547:
446:
387:trapezoidal method
372:
370:
334:
332:
296:numerical solution
268:
160:
69:
59:Motivating example
31:for which certain
8560:978-0-521-88068-8
8528:978-0-471-92990-1
8466:978-0-412-35260-7
8431:978-3-540-60452-5
8204:, pp. 62–68)
8003:
7990:
7958:
7935:
7920:
7912:
7861:
7848:
7816:
7793:
7743:
7717:
7585:
7538:
7379:{\displaystyle w}
7287:{\displaystyle w}
7274:if all solutions
7267:{\displaystyle z}
6490:Multistep methods
6328:rational function
6286:
5899:
5893:
5872:
5850:
5842:
5799:
5793:
5772:
5699:
5690:
5666:
5517:
5311:
5199:
5155:
5111:
4395:is equivalent to
3819:
3818:
3746:
3717:
3651:{\displaystyle c}
3631:{\displaystyle k}
3610:
3609:
3575:
3497:
3482:
3465:
3464:
3163:
3162:
3118:
3058:
3040:
3013:For example, the
2995:chemical kinetics
2850:
2849:
2821:
2801:
2770:
2691:
2639:
2607:
2606:
2533:
2462:
2358:
2352:
2214:{\displaystyle x}
1967:{\displaystyle x}
1677:
1676:
1519:
1518:
1383:) takes the form
1143:
1142:
1013:
1012:
948:
868:
814:
763:
708:
661:
629:chemical reaction
571:
570:
445:
369:
331:
292:
291:
254:
184:
183:
33:numerical methods
8688:
8630:
8601:
8575:
8573:
8572:
8563:. Archived from
8540:
8531:
8508:
8494:
8469:
8446:
8434:
8411:
8388:
8367:
8362:
8349:
8348:
8335:
8326:
8293:
8261:
8258:Dahlquist (1963)
8254:
8248:
8237:
8231:
8224:
8218:
8215:Dahlquist (1963)
8211:
8205:
8199:
8193:
8187:
8181:
8176:
8170:
8164:
8158:
8152:
8146:
8145:
8137:
8111:Condition number
8080:
8078:
8077:
8072:
8042:
8040:
8039:
8034:
8029:
8025:
8024:
8021:
8014:
8010:
8009:
8005:
8004:
8002:
8001:
7992:
7983:
7968:
7960:
7951:
7937:
7928:
7918:
7910:
7909:
7880:
7878:
7877:
7872:
7867:
7863:
7862:
7860:
7859:
7850:
7841:
7826:
7818:
7809:
7795:
7786:
7768:which has roots
7764:
7762:
7761:
7756:
7745:
7736:
7727:
7723:
7719:
7710:
7693:
7692:
7649:
7647:
7646:
7641:
7636:
7632:
7628:
7627:
7609:
7608:
7587:
7578:
7569:
7568:
7556:
7555:
7540:
7531:
7517:
7516:
7504:
7503:
7462:
7460:
7459:
7454:
7446:
7438:
7426:
7424:
7423:
7418:
7385:
7383:
7382:
7377:
7365:
7363:
7362:
7357:
7355:
7334:
7332:
7331:
7326:
7293:
7291:
7290:
7285:
7273:
7271:
7270:
7265:
7250:
7248:
7247:
7242:
7237:
7236:
7221:
7220:
7210:
7205:
7181:
7180:
7165:
7164:
7154:
7149:
7131:
7130:
7084:
7082:
7081:
7076:
7046:
7044:
7043:
7038:
7033:
7032:
7009:
7007:
7006:
7001:
6977:
6975:
6974:
6969:
6961:
6960:
6945:
6941:
6937:
6936:
6924:
6923:
6908:
6903:
6885:
6884:
6869:
6865:
6861:
6860:
6824:
6822:
6821:
6816:
6811:
6810:
6795:
6794:
6784:
6779:
6752:
6751:
6736:
6735:
6725:
6720:
6702:
6701:
6673:
6671:
6670:
6665:
6660:
6656:
6655:
6654:
6636:
6635:
6612:
6611:
6601:
6596:
6572:
6571:
6556:
6555:
6545:
6540:
6522:
6521:
6485:
6483:
6482:
6477:
6475:
6474:
6468:
6463:
6462:
6453:
6445:
6428:
6422:
6421:
6414:
6403:
6402:
6389:
6387:
6386:
6381:
6362:
6360:
6359:
6354:
6352:
6325:
6323:
6322:
6317:
6315:
6300:
6298:
6297:
6292:
6287:
6285:
6281:
6270:
6258:
6257:
6253:
6252:
6251:
6250:
6244:
6238:
6227:
6216:
6202:
6175:
6173:
6172:
6167:
6165:
6153:
6151:
6150:
6145:
6143:
6115:
6113:
6112:
6107:
6089:
6087:
6086:
6081:
6073:
6056:
6040:
6038:
6037:
6032:
6011:
6009:
6008:
6003:
5972:
5970:
5969:
5964:
5950:
5932:
5930:
5929:
5924:
5919:
5915:
5914:
5911:
5904:
5900:
5898:
5894:
5886:
5877:
5873:
5865:
5856:
5848:
5840:
5839:
5810:
5808:
5807:
5802:
5800:
5798:
5794:
5786:
5777:
5773:
5765:
5756:
5726:
5724:
5723:
5718:
5713:
5712:
5700:
5698:
5691:
5683:
5674:
5667:
5659:
5650:
5645:
5644:
5619:
5617:
5616:
5611:
5609:
5608:
5583:
5581:
5580:
5575:
5570:
5566:
5565:
5564:
5543:
5542:
5519:
5510:
5504:
5503:
5491:
5490:
5465:
5463:
5462:
5457:
5443:
5425:
5423:
5422:
5417:
5412:
5411:
5402:
5401:
5383:
5382:
5355:
5354:
5342:
5341:
5326:
5325:
5313:
5304:
5298:
5297:
5285:
5284:
5240:
5238:
5237:
5232:
5211:
5209:
5208:
5203:
5201:
5192:
5176:
5174:
5173:
5168:
5157:
5148:
5123:
5121:
5120:
5115:
5113:
5104:
5084:
5082:
5081:
5076:
5052:
5050:
5049:
5044:
5042:
5041:
5029:
5015:
5009:
5008:
5001:
4990:
4989:
4976:
4974:
4973:
4968:
4935:
4933:
4932:
4927:
4925:
4924:
4912:
4911:
4884:
4883:
4864:
4862:
4861:
4856:
4851:
4850:
4817:
4816:
4792:
4791:
4776:
4775:
4751:
4750:
4735:
4734:
4722:
4721:
4697:
4696:
4684:
4683:
4658:
4656:
4655:
4650:
4636:
4601:
4599:
4598:
4593:
4591:
4590:
4562:
4561:
4554:
4543:
4542:
4529:
4527:
4526:
4521:
4519:
4518:
4506:
4489:
4483:
4482:
4475:
4464:
4463:
4449:stability region
4442:
4440:
4439:
4434:
4426:
4406:
4394:
4392:
4391:
4386:
4368:
4366:
4365:
4360:
4352:
4351:
4331:
4329:
4328:
4323:
4311:
4309:
4308:
4303:
4301:
4300:
4288:
4287:
4282:
4281:
4259:
4258:
4249:
4248:
4232:
4230:
4229:
4224:
4222:
4221:
4191:
4190:
4168:
4166:
4165:
4160:
4146:
4119:
4117:
4116:
4111:
4081:
4079:
4078:
4073:
4055:
4053:
4052:
4047:
4045:
4044:
4010:
4008:
4007:
4002:
4000:
3982:
3980:
3979:
3974:
3947:
3945:
3944:
3939:
3928:
3899:
3897:
3896:
3891:
3889:
3888:
3866:
3864:
3863:
3858:
3856:
3855:
3843:
3842:
3813:
3804:
3802:
3801:
3796:
3791:
3790:
3778:
3777:
3765:
3761:
3760:
3759:
3747:
3739:
3734:
3733:
3718:
3710:
3700:
3699:
3667:
3657:
3655:
3654:
3649:
3637:
3635:
3634:
3629:
3604:
3595:
3593:
3592:
3587:
3576:
3574:
3573:
3562:
3556:
3555:
3544:
3538:
3528:
3524:
3522:
3521:
3516:
3505:
3483:
3475:
3459:
3450:
3448:
3447:
3442:
3437:
3436:
3422:
3421:
3392:
3383:
3382:
3345:
3336:
3335:
3291:
3278:
3274:
3272:
3271:
3266:
3242:
3240:
3239:
3234:
3216:
3214:
3213:
3208:
3190:
3188:
3187:
3182:
3157:
3148:
3146:
3145:
3140:
3120:
3119:
3111:
3104:
3103:
3060:
3059:
3051:
3042:
3041:
3033:
3019:
3003:spring constants
2982:
2980:
2979:
2974:
2962:
2960:
2959:
2954:
2952:
2951:
2867:numerical method
2844:
2835:
2833:
2832:
2827:
2822:
2820:
2819:
2818:
2809:
2790:
2789:
2782:
2781:
2780:
2771:
2763:
2752:
2751:
2744:
2734:
2726:
2724:
2723:
2718:
2716:
2715:
2710:
2704:
2703:
2699:
2684:
2683:
2667:
2665:
2664:
2659:
2657:
2656:
2651:
2645:
2644:
2640:
2632:
2625:
2624:
2601:
2592:
2590:
2589:
2584:
2551:
2550:
2541:
2522:
2521:
2512:
2511:
2502:
2501:
2483:
2482:
2473:
2472:
2463:
2455:
2444:
2443:
2429:
2425:
2423:
2422:
2417:
2382:
2381:
2366:
2353:
2345:
2332:
2330:
2329:
2324:
2322:
2321:
2316:
2310:
2309:
2305:
2304:
2290:
2289:
2273:
2271:
2270:
2265:
2263:
2259:
2255:
2254:
2220:
2218:
2217:
2212:
2200:
2198:
2197:
2192:
2190:
2189:
2184:
2178:
2177:
2173:
2172:
2158:
2157:
2141:
2139:
2138:
2133:
2131:
2127:
2123:
2122:
2088:
2086:
2085:
2080:
2069:
2053:
2051:
2050:
2045:
2043:
2042:
2037:
2031:
2030:
2026:
2025:
2011:
2010:
2000:
1995:
1973:
1971:
1970:
1965:
1950:
1948:
1947:
1942:
1940:
1939:
1923:
1921:
1920:
1915:
1913:
1912:
1896:
1894:
1893:
1888:
1886:
1885:
1880:
1874:
1873:
1869:
1868:
1847:
1845:
1844:
1839:
1821:
1819:
1818:
1813:
1802:
1790:
1788:
1787:
1782:
1771:
1759:
1757:
1756:
1751:
1733:
1731:
1730:
1725:
1717:
1716:
1711:
1705:
1704:
1700:
1699:
1671:
1662:
1660:
1659:
1654:
1609:
1608:
1582:
1578:
1576:
1575:
1570:
1559:
1547:
1545:
1544:
1539:
1537:
1536:
1513:
1504:
1502:
1501:
1496:
1482:
1474:
1473:
1468:
1462:
1461:
1457:
1456:
1442:
1441:
1431:
1426:
1399:
1386:
1376:
1374:
1373:
1368:
1336:
1335:
1330:
1321:
1320:
1315:
1302:
1300:
1299:
1294:
1262:
1254:
1253:
1237:
1235:
1234:
1229:
1211:
1209:
1208:
1203:
1201:
1189:
1187:
1186:
1181:
1179:
1178:
1173:
1164:
1156:
1137:
1128:
1126:
1125:
1120:
1106:
1098:
1093:
1085:
1081:
1067:
1052:
1050:
1049:
1044:
1007:
997:
995:
994:
989:
987:
983:
982:
973:
972:
950:
949:
941:
934:
933:
924:
923:
896:
895:
870:
869:
861:
845:
844:
816:
815:
807:
792:
790:
789:
784:
782:
778:
770:
761:
760:
741:
740:
732:
723:
715:
706:
705:
686:
685:
677:
668:
652:
651:
634:
617:
615:
614:
609:
586:
565:
556:
554:
553:
548:
543:
542:
533:
532:
514:
513:
486:
485:
473:
472:
457:
456:
447:
438:
432:
431:
419:
418:
395:
381:
379:
378:
373:
371:
362:
343:
341:
340:
335:
333:
324:
286:
277:
275:
274:
269:
255:
252:
228:
227:
189:
178:
169:
167:
166:
161:
94:
77:
8696:
8695:
8691:
8690:
8689:
8687:
8686:
8685:
8671:
8670:
8643:
8605:
8599:10.1137/1021001
8578:
8570:
8568:
8561:
8544:
8535:
8529:
8512:
8498:
8492:
8473:
8467:
8450:
8438:
8432:
8422:Springer-Verlag
8415:
8409:10.1137/1023002
8392:
8371:
8360:
8353:
8346:
8339:
8297:
8291:
8272:
8269:
8264:
8255:
8251:
8238:
8234:
8225:
8221:
8212:
8208:
8200:
8196:
8188:
8184:
8177:
8173:
8165:
8161:
8153:
8149:
8139:
8138:
8134:
8130:
8101:
8087:
8048:
8047:
7993:
7942:
7938:
7925:
7921:
7917:
7913:
7898:
7894:
7889:
7888:
7851:
7800:
7796:
7773:
7772:
7701:
7697:
7684:
7658:
7657:
7613:
7594:
7560:
7547:
7528:
7524:
7508:
7489:
7484:
7483:
7469:
7429:
7428:
7388:
7387:
7368:
7367:
7340:
7339:
7296:
7295:
7276:
7275:
7256:
7255:
7222:
7212:
7166:
7156:
7116:
7090:
7089:
7049:
7048:
7024:
7016:
7015:
6983:
6982:
6946:
6928:
6915:
6914:
6910:
6870:
6849:
6842:
6838:
6833:
6832:
6796:
6786:
6737:
6727:
6687:
6682:
6681:
6640:
6621:
6620:
6616:
6603:
6557:
6547:
6507:
6502:
6501:
6492:
6454:
6392:
6391:
6372:
6371:
6343:
6342:
6306:
6305:
6259:
6239:
6211:
6207:
6203:
6181:
6180:
6156:
6155:
6134:
6133:
6126:
6092:
6091:
6047:
6046:
6014:
6013:
5979:
5978:
5943:
5938:
5937:
5878:
5857:
5851:
5847:
5843:
5828:
5824:
5819:
5818:
5778:
5757:
5735:
5734:
5704:
5675:
5651:
5630:
5625:
5624:
5594:
5589:
5588:
5550:
5534:
5530:
5526:
5495:
5476:
5471:
5470:
5436:
5431:
5430:
5387:
5368:
5346:
5333:
5289:
5270:
5265:
5264:
5247:
5214:
5213:
5179:
5178:
5126:
5125:
5091:
5090:
5085:. The value of
5058:
5057:
4979:
4978:
4938:
4937:
4916:
4903:
4875:
4870:
4869:
4842:
4808:
4783:
4767:
4742:
4726:
4713:
4688:
4669:
4664:
4663:
4629:
4624:
4623:
4608:
4532:
4531:
4453:
4452:
4397:
4396:
4371:
4370:
4343:
4338:
4337:
4314:
4313:
4312:. The function
4292:
4275:
4240:
4235:
4234:
4213:
4176:
4171:
4170:
4139:
4134:
4133:
4127:
4084:
4083:
4058:
4057:
4033:
4013:
4012:
3985:
3984:
3950:
3949:
3921:
3916:
3915:
3912:
3874:
3869:
3868:
3844:
3834:
3829:
3828:
3811:
3779:
3769:
3748:
3719:
3705:
3701:
3691:
3671:
3670:
3640:
3639:
3620:
3619:
3602:
3557:
3539:
3532:
3531:
3469:
3468:
3457:
3431:
3430:
3424:
3423:
3413:
3406:
3377:
3376:
3370:
3369:
3359:
3330:
3329:
3321:
3312:
3311:
3306:
3296:
3282:
3281:
3251:
3250:
3219:
3218:
3193:
3192:
3167:
3166:
3155:
3095:
3023:
3022:
2991:control systems
2965:
2964:
2940:
2935:
2934:
2919:servomechanisms
2907:
2855:
2842:
2783:
2745:
2738:
2737:
2729:stiffness ratio
2705:
2685:
2675:
2670:
2669:
2646:
2626:
2616:
2611:
2610:
2599:
2493:
2433:
2432:
2373:
2339:
2338:
2311:
2296:
2291:
2281:
2276:
2275:
2246:
2236:
2232:
2227:
2226:
2203:
2202:
2179:
2164:
2159:
2149:
2144:
2143:
2114:
2104:
2100:
2095:
2094:
2060:
2059:
2032:
2017:
2012:
2002:
1976:
1975:
1956:
1955:
1931:
1926:
1925:
1904:
1899:
1898:
1875:
1860:
1855:
1850:
1849:
1824:
1823:
1793:
1792:
1762:
1761:
1736:
1735:
1706:
1691:
1686:
1681:
1680:
1669:
1600:
1586:
1585:
1550:
1549:
1528:
1523:
1522:
1511:
1463:
1448:
1443:
1433:
1390:
1389:
1325:
1310:
1305:
1304:
1245:
1240:
1239:
1214:
1213:
1192:
1191:
1168:
1147:
1146:
1135:
1076:
1071:
1070:
1059:
1057:Stiffness ratio
1017:
1016:
1005:
985:
984:
974:
964:
951:
936:
935:
925:
915:
887:
871:
856:
855:
836:
817:
796:
795:
794:
780:
779:
764:
752:
725:
724:
709:
697:
670:
669:
662:
638:
637:
621:
579:
574:
573:
563:
518:
499:
477:
464:
423:
404:
399:
398:
349:
348:
311:
310:
284:
213:
193:
192:
176:
87:
81:
80:
61:
17:
12:
11:
5:
8694:
8692:
8684:
8683:
8673:
8672:
8669:
8668:
8649:
8642:
8641:External links
8639:
8638:
8637:
8632:
8614:(4): 475–489,
8603:
8576:
8559:
8542:
8533:
8527:
8510:
8505:Academic Press
8496:
8490:
8471:
8465:
8448:
8436:
8430:
8413:
8390:
8369:
8351:
8337:
8295:
8289:
8268:
8265:
8263:
8262:
8249:
8232:
8219:
8206:
8202:Kreyszig (1972
8194:
8192:, p. 314)
8182:
8171:
8159:
8147:
8131:
8129:
8126:
8125:
8124:
8119:
8113:
8108:
8100:
8097:
8086:
8085:General theory
8083:
8070:
8067:
8064:
8061:
8058:
8055:
8044:
8043:
8032:
8028:
8023:
8020:
8017:
8013:
8008:
8000:
7996:
7989:
7986:
7980:
7977:
7974:
7971:
7966:
7963:
7957:
7954:
7948:
7945:
7941:
7934:
7931:
7924:
7916:
7908:
7904:
7901:
7897:
7882:
7881:
7870:
7866:
7858:
7854:
7847:
7844:
7838:
7835:
7832:
7829:
7824:
7821:
7815:
7812:
7806:
7803:
7799:
7792:
7789:
7783:
7780:
7766:
7765:
7754:
7751:
7748:
7742:
7739:
7733:
7730:
7726:
7722:
7716:
7713:
7707:
7704:
7700:
7696:
7691:
7687:
7683:
7680:
7677:
7674:
7671:
7668:
7665:
7651:
7650:
7639:
7635:
7631:
7626:
7623:
7620:
7616:
7612:
7607:
7604:
7601:
7597:
7593:
7590:
7584:
7581:
7575:
7572:
7567:
7563:
7559:
7554:
7550:
7546:
7543:
7537:
7534:
7527:
7523:
7520:
7515:
7511:
7507:
7502:
7499:
7496:
7492:
7468:
7465:
7452:
7449:
7445:
7441:
7437:
7416:
7413:
7410:
7407:
7404:
7401:
7398:
7395:
7375:
7366:for which all
7354:
7350:
7347:
7324:
7321:
7318:
7315:
7312:
7309:
7306:
7303:
7283:
7263:
7252:
7251:
7240:
7235:
7232:
7229:
7225:
7219:
7215:
7209:
7204:
7201:
7198:
7195:
7191:
7187:
7184:
7179:
7176:
7173:
7169:
7163:
7159:
7153:
7148:
7145:
7142:
7138:
7134:
7129:
7126:
7123:
7119:
7115:
7112:
7109:
7106:
7103:
7100:
7097:
7074:
7071:
7068:
7065:
7062:
7059:
7056:
7036:
7031:
7027:
7023:
6999:
6996:
6993:
6990:
6979:
6978:
6967:
6964:
6959:
6956:
6953:
6949:
6944:
6940:
6935:
6931:
6927:
6922:
6918:
6913:
6907:
6902:
6899:
6896:
6892:
6888:
6883:
6880:
6877:
6873:
6868:
6864:
6859:
6856:
6852:
6848:
6845:
6841:
6826:
6825:
6814:
6809:
6806:
6803:
6799:
6793:
6789:
6783:
6778:
6775:
6772:
6769:
6765:
6761:
6758:
6755:
6750:
6747:
6744:
6740:
6734:
6730:
6724:
6719:
6716:
6713:
6709:
6705:
6700:
6697:
6694:
6690:
6675:
6674:
6663:
6659:
6653:
6650:
6647:
6643:
6639:
6634:
6631:
6628:
6624:
6619:
6615:
6610:
6606:
6600:
6595:
6592:
6589:
6586:
6582:
6578:
6575:
6570:
6567:
6564:
6560:
6554:
6550:
6544:
6539:
6536:
6533:
6529:
6525:
6520:
6517:
6514:
6510:
6497:have the form
6491:
6488:
6473:
6467:
6461:
6457:
6452:
6448:
6444:
6440:
6437:
6434:
6431:
6427:
6420:
6413:
6409:
6406:
6401:
6379:
6351:
6314:
6302:
6301:
6290:
6284:
6280:
6276:
6273:
6269:
6265:
6262:
6256:
6249:
6243:
6237:
6233:
6230:
6226:
6222:
6219:
6215:
6210:
6206:
6200:
6197:
6194:
6191:
6188:
6164:
6142:
6125:
6124:General theory
6122:
6105:
6102:
6099:
6079:
6076:
6072:
6068:
6065:
6062:
6059:
6055:
6030:
6027:
6024:
6021:
6001:
5998:
5995:
5992:
5989:
5986:
5962:
5959:
5956:
5953:
5949:
5946:
5934:
5933:
5922:
5918:
5913:
5910:
5907:
5903:
5897:
5892:
5889:
5884:
5881:
5876:
5871:
5868:
5863:
5860:
5854:
5846:
5838:
5834:
5831:
5827:
5812:
5811:
5797:
5792:
5789:
5784:
5781:
5776:
5771:
5768:
5763:
5760:
5754:
5751:
5748:
5745:
5742:
5728:
5727:
5716:
5711:
5707:
5703:
5697:
5694:
5689:
5686:
5681:
5678:
5673:
5670:
5665:
5662:
5657:
5654:
5648:
5643:
5640:
5637:
5633:
5607:
5604:
5601:
5597:
5585:
5584:
5573:
5569:
5563:
5560:
5557:
5553:
5549:
5546:
5541:
5537:
5533:
5529:
5525:
5522:
5516:
5513:
5507:
5502:
5498:
5494:
5489:
5486:
5483:
5479:
5455:
5452:
5449:
5446:
5442:
5439:
5427:
5426:
5415:
5410:
5405:
5400:
5397:
5394:
5390:
5386:
5381:
5378:
5375:
5371:
5367:
5364:
5361:
5358:
5353:
5349:
5345:
5340:
5336:
5332:
5329:
5324:
5319:
5316:
5310:
5307:
5301:
5296:
5292:
5288:
5283:
5280:
5277:
5273:
5246:
5243:
5230:
5227:
5224:
5221:
5198:
5195:
5189:
5186:
5166:
5163:
5160:
5154:
5151:
5145:
5142:
5139:
5136:
5133:
5110:
5107:
5101:
5098:
5074:
5071:
5068:
5065:
5040:
5035:
5032:
5028:
5024:
5021:
5018:
5014:
5007:
5000:
4996:
4993:
4988:
4966:
4963:
4960:
4957:
4954:
4951:
4948:
4945:
4923:
4919:
4915:
4910:
4906:
4902:
4899:
4896:
4893:
4890:
4887:
4882:
4878:
4866:
4865:
4854:
4849:
4845:
4841:
4838:
4835:
4832:
4829:
4826:
4823:
4820:
4815:
4811:
4807:
4804:
4801:
4798:
4795:
4790:
4786:
4782:
4779:
4774:
4770:
4766:
4763:
4760:
4757:
4754:
4749:
4745:
4741:
4738:
4733:
4729:
4725:
4720:
4716:
4712:
4709:
4706:
4703:
4700:
4695:
4691:
4687:
4682:
4679:
4676:
4672:
4648:
4645:
4642:
4639:
4635:
4632:
4607:
4604:
4589:
4584:
4581:
4578:
4575:
4572:
4569:
4566:
4560:
4553:
4549:
4546:
4541:
4517:
4512:
4509:
4505:
4501:
4498:
4495:
4492:
4488:
4481:
4474:
4470:
4467:
4462:
4432:
4429:
4425:
4421:
4418:
4415:
4412:
4409:
4405:
4384:
4381:
4378:
4358:
4355:
4350:
4346:
4332:is called the
4321:
4299:
4295:
4291:
4286:
4280:
4274:
4271:
4268:
4265:
4262:
4257:
4252:
4247:
4243:
4220:
4216:
4212:
4209:
4206:
4203:
4200:
4197:
4194:
4189:
4186:
4183:
4179:
4169:take the form
4158:
4155:
4152:
4149:
4145:
4142:
4126:
4123:
4109:
4106:
4103:
4100:
4097:
4094:
4091:
4071:
4068:
4065:
4043:
4040:
4036:
4032:
4029:
4026:
4023:
4020:
3999:
3995:
3992:
3972:
3969:
3966:
3963:
3960:
3957:
3937:
3934:
3931:
3927:
3924:
3911:
3908:
3887:
3884:
3881:
3877:
3854:
3851:
3847:
3841:
3837:
3817:
3816:
3807:
3805:
3794:
3789:
3786:
3782:
3776:
3772:
3768:
3764:
3758:
3755:
3751:
3745:
3742:
3737:
3732:
3729:
3726:
3722:
3716:
3713:
3708:
3704:
3698:
3694:
3690:
3687:
3684:
3681:
3678:
3647:
3627:
3608:
3607:
3598:
3596:
3585:
3582:
3579:
3572:
3568:
3565:
3561:
3554:
3550:
3547:
3543:
3514:
3511:
3508:
3503:
3500:
3495:
3492:
3489:
3486:
3481:
3478:
3463:
3462:
3453:
3451:
3440:
3435:
3429:
3426:
3425:
3420:
3416:
3412:
3411:
3409:
3404:
3401:
3398:
3395:
3391:
3386:
3381:
3375:
3372:
3371:
3368:
3365:
3364:
3362:
3357:
3354:
3351:
3348:
3344:
3339:
3334:
3328:
3325:
3322:
3320:
3317:
3314:
3313:
3310:
3307:
3305:
3302:
3301:
3299:
3294:
3290:
3264:
3261:
3258:
3232:
3229:
3226:
3206:
3203:
3200:
3180:
3177:
3174:
3161:
3160:
3151:
3149:
3138:
3135:
3132:
3129:
3126:
3123:
3117:
3114:
3107:
3102:
3098:
3094:
3091:
3088:
3085:
3082:
3078:
3075:
3072:
3069:
3066:
3063:
3057:
3054:
3048:
3045:
3039:
3036:
3030:
2972:
2950:
2947:
2943:
2906:
2903:
2902:
2901:
2898:
2895:
2854:
2851:
2848:
2847:
2838:
2836:
2825:
2817:
2812:
2807:
2804:
2799:
2796:
2793:
2788:
2779:
2774:
2769:
2766:
2761:
2758:
2755:
2750:
2714:
2709:
2702:
2697:
2694:
2688:
2682:
2678:
2655:
2650:
2643:
2638:
2635:
2629:
2623:
2619:
2605:
2604:
2595:
2593:
2582:
2579:
2576:
2573:
2570:
2567:
2564:
2561:
2558:
2554:
2549:
2544:
2539:
2536:
2531:
2528:
2525:
2520:
2515:
2510:
2505:
2500:
2496:
2492:
2489:
2486:
2481:
2476:
2471:
2466:
2461:
2458:
2453:
2450:
2447:
2442:
2426:be defined by
2415:
2412:
2409:
2406:
2403:
2400:
2397:
2394:
2391:
2388:
2385:
2380:
2376:
2372:
2369:
2364:
2361:
2356:
2351:
2348:
2335:slow transient
2320:
2315:
2308:
2303:
2299:
2294:
2288:
2284:
2262:
2258:
2253:
2249:
2245:
2242:
2239:
2235:
2223:fast transient
2210:
2188:
2183:
2176:
2171:
2167:
2162:
2156:
2152:
2130:
2126:
2121:
2117:
2113:
2110:
2107:
2103:
2078:
2075:
2072:
2068:
2054:is called the
2041:
2036:
2029:
2024:
2020:
2015:
2009:
2005:
1999:
1994:
1991:
1988:
1984:
1963:
1938:
1934:
1911:
1907:
1884:
1879:
1872:
1867:
1863:
1858:
1837:
1834:
1831:
1811:
1808:
1805:
1801:
1780:
1777:
1774:
1770:
1749:
1746:
1743:
1723:
1720:
1715:
1710:
1703:
1698:
1694:
1689:
1675:
1674:
1665:
1663:
1652:
1649:
1646:
1643:
1640:
1637:
1634:
1631:
1628:
1625:
1621:
1618:
1615:
1612:
1607:
1603:
1599:
1596:
1593:
1568:
1565:
1562:
1558:
1535:
1531:
1517:
1516:
1507:
1505:
1494:
1491:
1488:
1485:
1481:
1477:
1472:
1467:
1460:
1455:
1451:
1446:
1440:
1436:
1430:
1425:
1422:
1419:
1415:
1411:
1408:
1405:
1402:
1398:
1366:
1363:
1360:
1357:
1354:
1351:
1348:
1345:
1342:
1339:
1334:
1329:
1324:
1319:
1314:
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1261:
1257:
1252:
1248:
1227:
1224:
1221:
1200:
1177:
1172:
1167:
1163:
1159:
1155:
1141:
1140:
1131:
1129:
1118:
1115:
1112:
1109:
1105:
1101:
1097:
1092:
1088:
1084:
1080:
1058:
1055:
1042:
1039:
1036:
1033:
1030:
1027:
1024:
1011:
1010:
1001:
999:
981:
977:
971:
967:
963:
960:
957:
954:
952:
947:
944:
938:
937:
932:
928:
922:
918:
914:
911:
908:
905:
902:
899:
894:
890:
886:
883:
880:
877:
874:
872:
867:
864:
858:
857:
854:
851:
848:
843:
839:
835:
832:
829:
826:
823:
820:
818:
813:
810:
804:
803:
777:
773:
769:
765:
759:
755:
751:
748:
744:
739:
735:
731:
727:
726:
722:
718:
714:
710:
704:
700:
696:
693:
689:
684:
680:
676:
672:
671:
667:
663:
659:
655:
650:
646:
645:
631:of Robertson:
620:
619:
607:
604:
601:
598:
595:
592:
589:
585:
582:
569:
568:
559:
557:
546:
541:
536:
531:
528:
525:
521:
517:
512:
509:
506:
502:
498:
495:
492:
489:
484:
480:
476:
471:
467:
463:
460:
455:
450:
444:
441:
435:
430:
426:
422:
417:
414:
411:
407:
393:) is given by
383:
368:
365:
359:
356:
345:
330:
327:
321:
318:
307:Euler's method
303:
290:
289:
280:
278:
267:
264:
261:
258:
253: as
250:
247:
244:
241:
238:
235:
231:
226:
223:
220:
216:
212:
209:
206:
203:
200:
182:
181:
172:
170:
159:
156:
153:
150:
147:
144:
140:
137:
134:
131:
127:
124:
121:
118:
115:
112:
109:
106:
103:
100:
97:
93:
90:
60:
57:
44:solution curve
25:stiff equation
15:
13:
10:
9:
6:
4:
3:
2:
8693:
8682:
8679:
8678:
8676:
8667:
8664:, 2(3):2855.
8663:
8660:
8659:Skip Thompson
8656:
8653:
8652:Stiff systems
8650:
8648:
8645:
8644:
8640:
8636:
8633:
8629:
8625:
8621:
8617:
8613:
8609:
8604:
8600:
8596:
8592:
8588:
8587:
8582:
8577:
8567:on 2011-08-11
8566:
8562:
8556:
8552:
8548:
8543:
8539:
8534:
8530:
8524:
8520:
8516:
8511:
8506:
8502:
8497:
8493:
8491:0-471-50728-8
8487:
8483:
8479:
8478:
8472:
8468:
8462:
8458:
8454:
8449:
8444:
8443:
8437:
8433:
8427:
8423:
8419:
8414:
8410:
8406:
8402:
8398:
8397:
8391:
8387:
8383:
8379:
8378:Prentice Hall
8375:
8370:
8366:
8359:
8358:
8352:
8345:
8344:
8338:
8334:
8330:
8325:
8320:
8316:
8312:
8308:
8304:
8300:
8296:
8292:
8290:0-534-93219-3
8286:
8282:
8278:
8277:
8271:
8270:
8266:
8259:
8253:
8250:
8246:
8242:
8236:
8233:
8229:
8223:
8220:
8216:
8210:
8207:
8203:
8198:
8195:
8191:
8186:
8183:
8180:
8175:
8172:
8168:
8167:Lambert (1992
8163:
8160:
8156:
8155:Lambert (1992
8151:
8148:
8143:
8136:
8133:
8127:
8123:
8120:
8117:
8114:
8112:
8109:
8106:
8103:
8102:
8098:
8096:
8093:
8084:
8082:
8068:
8065:
8062:
8059:
8056:
8053:
8030:
8026:
8018:
8015:
8011:
8006:
7998:
7994:
7987:
7984:
7978:
7975:
7972:
7969:
7964:
7961:
7955:
7952:
7946:
7943:
7939:
7932:
7929:
7922:
7914:
7902:
7899:
7895:
7887:
7886:
7885:
7868:
7864:
7856:
7852:
7845:
7842:
7836:
7833:
7830:
7827:
7822:
7819:
7813:
7810:
7804:
7801:
7797:
7790:
7787:
7781:
7778:
7771:
7770:
7769:
7752:
7749:
7746:
7740:
7737:
7731:
7728:
7724:
7720:
7714:
7711:
7705:
7702:
7698:
7694:
7689:
7685:
7681:
7675:
7672:
7669:
7656:
7655:
7654:
7637:
7633:
7624:
7621:
7618:
7614:
7610:
7605:
7602:
7599:
7595:
7588:
7582:
7579:
7573:
7565:
7561:
7557:
7552:
7548:
7541:
7535:
7532:
7525:
7521:
7518:
7513:
7509:
7505:
7500:
7497:
7494:
7490:
7482:
7481:
7480:
7473:
7466:
7464:
7450:
7447:
7439:
7414:
7411:
7405:
7402:
7399:
7373:
7348:
7345:
7336:
7322:
7319:
7313:
7310:
7307:
7281:
7261:
7238:
7233:
7230:
7227:
7223:
7217:
7213:
7207:
7202:
7199:
7196:
7193:
7189:
7185:
7182:
7177:
7174:
7171:
7167:
7161:
7157:
7151:
7146:
7143:
7140:
7136:
7132:
7127:
7124:
7121:
7117:
7113:
7107:
7104:
7101:
7088:
7087:
7086:
7072:
7069:
7063:
7057:
7054:
7029:
7025:
7013:
6997:
6994:
6991:
6988:
6965:
6962:
6957:
6954:
6951:
6947:
6942:
6938:
6933:
6929:
6925:
6920:
6916:
6911:
6905:
6900:
6897:
6894:
6890:
6886:
6881:
6878:
6875:
6871:
6866:
6862:
6857:
6854:
6850:
6846:
6843:
6839:
6831:
6830:
6829:
6812:
6807:
6804:
6801:
6797:
6791:
6787:
6781:
6776:
6773:
6770:
6767:
6763:
6759:
6756:
6753:
6748:
6745:
6742:
6738:
6732:
6728:
6722:
6717:
6714:
6711:
6707:
6703:
6698:
6695:
6692:
6688:
6680:
6679:
6678:
6661:
6657:
6651:
6648:
6645:
6641:
6637:
6632:
6629:
6626:
6622:
6617:
6613:
6608:
6604:
6598:
6593:
6590:
6587:
6584:
6580:
6576:
6573:
6568:
6565:
6562:
6558:
6552:
6548:
6542:
6537:
6534:
6531:
6527:
6523:
6518:
6515:
6512:
6508:
6500:
6499:
6498:
6496:
6489:
6487:
6459:
6455:
6446:
6435:
6429:
6407:
6404:
6377:
6369:
6364:
6340:
6335:
6333:
6329:
6288:
6274:
6271:
6254:
6231:
6228:
6220:
6217:
6208:
6198:
6192:
6186:
6179:
6178:
6177:
6131:
6123:
6121:
6119:
6097:
6077:
6063:
6057:
6044:
6025:
6019:
5999:
5990:
5984:
5976:
5960:
5957:
5954:
5951:
5947:
5944:
5920:
5916:
5908:
5905:
5901:
5895:
5890:
5887:
5882:
5879:
5874:
5869:
5866:
5861:
5858:
5852:
5844:
5832:
5829:
5825:
5817:
5816:
5815:
5795:
5790:
5787:
5782:
5779:
5774:
5769:
5766:
5761:
5758:
5752:
5746:
5740:
5733:
5732:
5731:
5714:
5709:
5705:
5701:
5695:
5692:
5687:
5684:
5679:
5676:
5671:
5668:
5663:
5660:
5655:
5652:
5646:
5641:
5638:
5635:
5631:
5623:
5622:
5621:
5605:
5602:
5599:
5595:
5571:
5567:
5561:
5558:
5555:
5551:
5547:
5544:
5539:
5535:
5531:
5527:
5523:
5520:
5514:
5511:
5505:
5500:
5496:
5492:
5487:
5484:
5481:
5477:
5469:
5468:
5467:
5453:
5450:
5447:
5444:
5440:
5437:
5413:
5398:
5395:
5392:
5388:
5384:
5379:
5376:
5373:
5369:
5362:
5359:
5351:
5347:
5343:
5338:
5334:
5327:
5317:
5314:
5308:
5305:
5299:
5294:
5290:
5286:
5281:
5278:
5275:
5271:
5263:
5262:
5261:
5260:
5257:Consider the
5251:
5244:
5242:
5228:
5225:
5222:
5219:
5196:
5193:
5187:
5184:
5164:
5161:
5158:
5152:
5149:
5143:
5140:
5137:
5134:
5131:
5108:
5105:
5099:
5096:
5088:
5072:
5069:
5066:
5063:
5054:
5033:
5030:
5022:
5019:
5016:
4994:
4991:
4964:
4961:
4958:
4955:
4949:
4943:
4921:
4917:
4913:
4908:
4900:
4897:
4894:
4891:
4885:
4880:
4876:
4852:
4847:
4843:
4836:
4833:
4830:
4827:
4824:
4818:
4813:
4809:
4805:
4802:
4799:
4796:
4793:
4788:
4784:
4780:
4772:
4768:
4764:
4758:
4755:
4752:
4747:
4743:
4739:
4731:
4727:
4723:
4718:
4714:
4707:
4704:
4701:
4698:
4693:
4689:
4685:
4680:
4677:
4674:
4670:
4662:
4661:
4660:
4646:
4643:
4640:
4637:
4633:
4630:
4621:
4612:
4605:
4603:
4582:
4579:
4573:
4567:
4564:
4547:
4544:
4510:
4507:
4496:
4490:
4468:
4465:
4450:
4446:
4430:
4427:
4416:
4413:
4407:
4376:
4356:
4348:
4344:
4335:
4319:
4297:
4293:
4289:
4284:
4269:
4266:
4260:
4250:
4245:
4241:
4218:
4214:
4210:
4204:
4201:
4195:
4192:
4187:
4184:
4181:
4177:
4156:
4153:
4150:
4147:
4143:
4140:
4131:
4124:
4122:
4107:
4104:
4098:
4092:
4089:
4063:
4041:
4038:
4034:
4030:
4024:
4018:
3993:
3990:
3970:
3967:
3961:
3955:
3935:
3932:
3929:
3925:
3922:
3909:
3907:
3905:
3904:
3885:
3882:
3879:
3875:
3852:
3849:
3845:
3839:
3835:
3826:
3825:
3815:
3808:
3806:
3792:
3787:
3784:
3780:
3774:
3770:
3766:
3762:
3756:
3753:
3749:
3743:
3740:
3735:
3730:
3727:
3724:
3720:
3714:
3711:
3706:
3702:
3696:
3692:
3688:
3682:
3676:
3669:
3668:
3665:
3663:
3662:
3645:
3625:
3617:
3616:
3606:
3599:
3597:
3583:
3580:
3577:
3566:
3563:
3548:
3545:
3530:
3529:
3526:
3512:
3509:
3506:
3501:
3498:
3493:
3490:
3487:
3484:
3476:
3461:
3454:
3452:
3438:
3433:
3427:
3418:
3414:
3407:
3402:
3396:
3384:
3379:
3373:
3366:
3360:
3355:
3349:
3337:
3332:
3326:
3323:
3318:
3315:
3308:
3303:
3297:
3292:
3280:
3279:
3276:
3262:
3259:
3256:
3248:
3247:
3230:
3227:
3224:
3204:
3201:
3198:
3178:
3175:
3172:
3159:
3152:
3150:
3136:
3133:
3130:
3124:
3115:
3112:
3105:
3100:
3096:
3092:
3086:
3080:
3076:
3073:
3070:
3067:
3064:
3061:
3055:
3052:
3046:
3043:
3037:
3034:
3028:
3021:
3020:
3017:
3016:
3010:
3008:
3004:
3001:having large
3000:
2996:
2992:
2987:
2984:
2970:
2948:
2945:
2941:
2932:
2928:
2922:
2920:
2916:
2912:
2904:
2899:
2896:
2893:
2889:
2888:
2887:
2882:
2880:
2876:
2872:
2868:
2862:
2859:
2852:
2846:
2839:
2837:
2823:
2805:
2802:
2794:
2791:
2764:
2756:
2753:
2736:
2735:
2732:
2730:
2712:
2700:
2695:
2692:
2686:
2680:
2676:
2653:
2641:
2633:
2627:
2621:
2617:
2603:
2596:
2594:
2580:
2577:
2574:
2571:
2568:
2565:
2562:
2559:
2556:
2552:
2537:
2534:
2526:
2523:
2513:
2498:
2494:
2487:
2484:
2474:
2456:
2448:
2445:
2431:
2430:
2427:
2410:
2407:
2404:
2401:
2398:
2395:
2392:
2389:
2386:
2383:
2378:
2374:
2367:
2362:
2359:
2354:
2346:
2336:
2318:
2306:
2301:
2297:
2292:
2286:
2282:
2260:
2251:
2247:
2240:
2237:
2233:
2224:
2208:
2186:
2174:
2169:
2165:
2160:
2154:
2150:
2128:
2119:
2115:
2108:
2105:
2101:
2092:
2073:
2057:
2039:
2027:
2022:
2018:
2013:
2007:
2003:
1997:
1992:
1989:
1986:
1982:
1961:
1954:Interpreting
1952:
1936:
1932:
1909:
1905:
1882:
1870:
1865:
1861:
1856:
1829:
1806:
1775:
1741:
1721:
1713:
1701:
1696:
1692:
1687:
1673:
1666:
1664:
1650:
1647:
1644:
1641:
1638:
1635:
1632:
1629:
1626:
1623:
1619:
1616:
1613:
1605:
1601:
1594:
1591:
1584:
1583:
1580:
1563:
1533:
1529:
1515:
1508:
1506:
1492:
1486:
1475:
1470:
1458:
1453:
1449:
1444:
1438:
1434:
1428:
1423:
1420:
1417:
1413:
1409:
1403:
1388:
1387:
1384:
1382:
1381:
1364:
1361:
1358:
1355:
1352:
1349:
1346:
1343:
1340:
1337:
1332:
1322:
1317:
1290:
1287:
1284:
1281:
1278:
1275:
1272:
1269:
1266:
1263:
1255:
1250:
1246:
1225:
1222:
1219:
1175:
1165:
1157:
1139:
1132:
1130:
1116:
1110:
1099:
1086:
1082:
1069:
1068:
1065:
1064:
1061:Consider the
1056:
1054:
1037:
1034:
1031:
1025:
1022:
1009:
1002:
1000:
998:
979:
975:
969:
965:
961:
958:
955:
953:
945:
942:
930:
926:
920:
916:
912:
909:
906:
903:
900:
897:
892:
888:
884:
881:
878:
875:
873:
865:
862:
852:
849:
846:
841:
837:
833:
830:
827:
824:
821:
819:
811:
808:
771:
757:
753:
749:
746:
742:
733:
716:
702:
698:
694:
691:
687:
678:
657:
653:
636:
635:
632:
630:
626:
602:
599:
596:
590:
587:
583:
580:
567:
560:
558:
544:
529:
526:
523:
519:
515:
510:
507:
504:
500:
493:
490:
482:
478:
474:
469:
465:
458:
448:
442:
439:
433:
428:
424:
420:
415:
412:
409:
405:
397:
396:
392:
388:
384:
366:
363:
357:
354:
346:
328:
325:
319:
316:
308:
305:
304:
302:
299:
297:
288:
281:
279:
265:
256:
248:
239:
233:
229:
224:
221:
218:
214:
210:
204:
198:
191:
190:
187:
180:
173:
171:
157:
154:
148:
142:
138:
135:
132:
129:
125:
119:
113:
110:
107:
104:
98:
91:
88:
79:
78:
75:
74:
71:Consider the
65:
58:
56:
54:
53:stiff systems
50:
45:
40:
38:
34:
30:
26:
22:
8662:Scholarpedia
8611:
8607:
8590:
8584:
8569:. Retrieved
8565:the original
8550:
8537:
8517:, New York:
8514:
8503:, New York:
8500:
8476:
8452:
8440:
8417:
8403:(1): 10–24,
8400:
8394:
8373:
8356:
8342:
8309:(1): 27–43,
8306:
8302:
8275:
8252:
8235:
8222:
8209:
8197:
8185:
8174:
8162:
8150:
8141:
8135:
8088:
8045:
7883:
7767:
7652:
7478:
7337:
7253:
6980:
6827:
6676:
6493:
6365:
6336:
6303:
6176:is given by
6127:
5974:
5935:
5813:
5729:
5587:Solving for
5586:
5428:
5256:
5086:
5055:
4867:
4620:Euler method
4617:
4448:
4444:
4333:
4128:
3913:
3901:
3822:
3820:
3809:
3659:
3613:
3611:
3600:
3466:
3455:
3244:
3164:
3153:
3012:
2988:
2985:
2924:
2908:
2884:
2878:
2864:
2860:
2856:
2840:
2728:
2608:
2597:
2334:
2222:
2090:
2055:
1953:
1951:is complex.
1848:; the term
1678:
1667:
1520:
1509:
1378:
1144:
1133:
1060:
1014:
1003:
793:
622:
561:
300:
293:
282:
185:
174:
70:
52:
48:
41:
24:
18:
8593:(1): 1–17,
8586:SIAM Review
8453:Order Stars
8396:SIAM Review
8243:; see also
8228:Ehle (1969)
6368:order stars
6043:L-stability
5975:and only if
3910:A-stability
2892:eigenvalues
1791:approaches
21:mathematics
8571:2011-08-17
8267:References
7386:such that
6332:polynomial
5212:, we have
3005:(physical
1521:where the
294:We seek a
8507:: 451–501
8445:: 367–376
8333:120241743
8092:Dahlquist
8066:≤
8060:≤
8054:−
7965:±
7903:∈
7823:±
7695:−
7664:Φ
7622:−
7603:−
7574:−
7394:Φ
7349:∈
7302:Φ
7231:−
7200:−
7190:∑
7183:−
7175:−
7137:∑
7133:−
7096:Φ
7058:
6955:−
6891:∑
6887:−
6855:−
6847:−
6805:−
6774:−
6764:∑
6746:−
6708:∑
6649:−
6630:−
6591:−
6581:∑
6566:−
6528:∑
6430:ϕ
6408:∈
6378:ϕ
6272:−
6218:−
6187:ϕ
6104:∞
6101:→
6075:→
6058:ϕ
6029:∞
6026:−
6023:→
5997:→
5985:ϕ
5958:⋅
5883:−
5833:∈
5783:−
5741:ϕ
5702:⋅
5680:−
5524:⋅
5451:⋅
5318:⋅
5226:−
5162:−
5144:×
5138:−
5070:−
4995:∈
4944:ϕ
4914:⋅
4834:⋅
4806:⋅
4800:⋅
4759:⋅
4705:⋅
4644:⋅
4568:
4548:∈
4491:ϕ
4469:∈
4408:ϕ
4383:∞
4380:→
4354:→
4320:ϕ
4290:⋅
4261:ϕ
4211:⋅
4196:ϕ
4154:⋅
4093:
4070:∞
4067:→
3994:∈
3880:−
3850:−
3821:Equation
3785:−
3767:≈
3754:−
3725:−
3707:−
3564:−
3546:−
3510:−
3502:_
3499:λ
3488:−
3480:¯
3477:λ
3324:−
3316:−
3116:˙
3056:˙
3038:¨
3007:stiffness
2971:λ
2946:λ
2905:Etymology
2871:stability
2806:_
2803:λ
2795:
2768:¯
2765:λ
2757:
2696:_
2693:λ
2677:κ
2637:¯
2634:λ
2618:κ
2575:…
2538:_
2535:λ
2527:
2514:≥
2495:λ
2488:
2475:≥
2460:¯
2457:λ
2449:
2405:…
2375:λ
2368:∈
2363:_
2360:λ
2350:¯
2347:λ
2298:λ
2283:κ
2248:λ
2241:
2166:λ
2151:κ
2116:λ
2109:
2019:λ
2004:κ
1983:∑
1933:λ
1906:λ
1862:λ
1836:∞
1833:→
1748:∞
1745:→
1719:→
1693:λ
1642:…
1602:λ
1595:
1530:κ
1450:λ
1435:κ
1414:∑
1359:…
1323:∈
1285:…
1256:∈
1247:λ
1223:×
1166:∈
1026:∈
962:⋅
946:˙
913:⋅
907:−
901:⋅
885:−
866:˙
850:⋅
825:−
812:˙
750:⋅
695:⋅
263:∞
260:→
246:→
219:−
133:≥
108:−
49:stiffness
8675:Category
8099:See also
7427:satisfy
5948:′
5441:′
4634:′
4144:′
3926:′
2963:, where
2609:so that
1083:′
743:→
688:→
654:→
584:′
92:′
8628:8824105
8382:Bibcode
5620:yields
4868:Hence,
3249:) with
2999:systems
2337:. Let
8626:
8557:
8525:
8488:
8463:
8428:
8331:
8287:
7919:
7911:
6981:where
6304:where
5849:
5841:
2986:. . .
2915:driven
1145:where
572:where
8624:S2CID
8519:Wiley
8482:Wiley
8361:(PDF)
8347:(PDF)
8329:S2CID
8128:Notes
6330:(one
5466:, is
5229:1.875
4936:with
4082:when
3983:with
3664:) is
3165:with
2879:stiff
2865:If a
2225:; if
2093:. If
27:is a
8657:and
8555:ISBN
8523:ISBN
8486:ISBN
8461:ISBN
8426:ISBN
8285:ISBN
8256:See
8016:<
7448:<
7070:<
6447:>
6154:and
5906:<
5165:3.75
5031:<
4580:<
4508:<
4428:<
4105:<
3883:1000
3741:1000
3728:1000
3581:1000
3549:1000
3491:1000
3327:1001
3319:1000
3275:and
3231:1000
3205:1001
2089:the
2058:and
1614:<
1190:and
879:0.04
828:0.04
658:0.04
385:The
23:, a
8616:doi
8608:BIT
8595:doi
8405:doi
8319:hdl
8311:doi
8303:BIT
7294:of
6261:det
6205:det
6090:as
6012:as
5124:is
4659:is
4369:as
3744:999
3715:999
3009:).
2917:in
2731:as
1734:as
19:In
8677::
8622:,
8612:18
8610:,
8591:21
8589:,
8583:,
8549:.
8521:,
8484:,
8459:,
8455:,
8424:,
8401:23
8399:,
8380:,
8363:,
8327:,
8317:,
8305:,
8283:,
7055:Re
5141:15
5073:15
4565:Re
4108:0.
4090:Re
3903:10
3824:13
3812:13
3661:10
3615:10
3603:12
3458:11
3217:,
3191:,
3156:10
2792:Re
2754:Re
2524:Re
2485:Re
2446:Re
2238:Re
2106:Re
1592:Re
1038:40
966:10
917:10
889:10
838:10
754:10
699:10
222:15
158:1.
111:15
55:.
8631:.
8618::
8602:.
8597::
8574:.
8541:.
8532:.
8509:.
8495:.
8470:.
8447:.
8435:.
8412:.
8407::
8389:.
8384::
8368:.
8350:.
8336:.
8321::
8313::
8307:3
8294:.
8260:.
8247:.
8230:.
8217:.
8069:0
8063:z
8057:1
8031:.
8027:}
8019:1
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8007:)
7999:2
7995:z
7988:4
7985:9
7979:+
7976:z
7973:+
7970:1
7962:z
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7944:1
7940:(
7933:2
7930:1
7923:|
7915:|
7907:C
7900:z
7896:{
7869:,
7865:)
7857:2
7853:z
7846:4
7843:9
7837:+
7834:z
7831:+
7828:1
7820:z
7814:2
7811:3
7805:+
7802:1
7798:(
7791:2
7788:1
7782:=
7779:w
7753:0
7750:=
7747:z
7741:2
7738:1
7732:+
7729:w
7725:)
7721:z
7715:2
7712:3
7706:+
7703:1
7699:(
7690:2
7686:w
7682:=
7679:)
7676:z
7673:,
7670:w
7667:(
7638:.
7634:)
7630:)
7625:1
7619:n
7615:y
7611:,
7606:1
7600:n
7596:t
7592:(
7589:f
7583:2
7580:1
7571:)
7566:n
7562:y
7558:,
7553:n
7549:t
7545:(
7542:f
7536:2
7533:3
7526:(
7522:h
7519:+
7514:n
7510:y
7506:=
7501:1
7498:+
7495:n
7491:y
7451:1
7444:|
7440:w
7436:|
7415:0
7412:=
7409:)
7406:w
7403:,
7400:z
7397:(
7374:w
7353:C
7346:z
7323:0
7320:=
7317:)
7314:w
7311:,
7308:z
7305:(
7282:w
7262:z
7239:.
7234:j
7228:s
7224:w
7218:j
7214:b
7208:s
7203:1
7197:=
7194:j
7186:z
7178:i
7172:s
7168:w
7162:i
7158:a
7152:s
7147:0
7144:=
7141:i
7128:1
7125:+
7122:s
7118:w
7114:=
7111:)
7108:w
7105:,
7102:z
7099:(
7073:0
7067:)
7064:z
7061:(
7035:}
7030:n
7026:y
7022:{
6998:k
6995:h
6992:=
6989:z
6966:0
6963:=
6958:j
6952:n
6948:y
6943:)
6939:z
6934:j
6930:b
6926:+
6921:j
6917:a
6912:(
6906:s
6901:0
6898:=
6895:j
6882:1
6879:+
6876:n
6872:y
6867:)
6863:z
6858:1
6851:b
6844:1
6840:(
6813:,
6808:j
6802:n
6798:y
6792:j
6788:b
6782:s
6777:1
6771:=
6768:j
6760:k
6757:h
6754:+
6749:i
6743:n
6739:y
6733:i
6729:a
6723:s
6718:0
6715:=
6712:i
6704:=
6699:1
6696:+
6693:n
6689:y
6662:.
6658:)
6652:j
6646:n
6642:y
6638:,
6633:j
6627:n
6623:t
6618:(
6614:f
6609:j
6605:b
6599:s
6594:1
6588:=
6585:j
6577:h
6574:+
6569:i
6563:n
6559:y
6553:i
6549:a
6543:s
6538:0
6535:=
6532:i
6524:=
6519:1
6516:+
6513:n
6509:y
6472:}
6466:|
6460:z
6456:e
6451:|
6443:|
6439:)
6436:z
6433:(
6426:|
6419:|
6412:C
6405:z
6400:{
6350:A
6313:e
6289:,
6283:)
6279:A
6275:z
6268:I
6264:(
6255:)
6248:T
6242:b
6236:e
6232:z
6229:+
6225:A
6221:z
6214:I
6209:(
6199:=
6196:)
6193:z
6190:(
6163:b
6141:A
6098:z
6078:0
6071:|
6067:)
6064:z
6061:(
6054:|
6020:z
6000:1
5994:)
5991:z
5988:(
5961:y
5955:k
5952:=
5945:y
5921:.
5917:}
5909:1
5902:|
5896:z
5891:2
5888:1
5880:1
5875:z
5870:2
5867:1
5862:+
5859:1
5853:|
5845:|
5837:C
5830:z
5826:{
5796:z
5791:2
5788:1
5780:1
5775:z
5770:2
5767:1
5762:+
5759:1
5753:=
5750:)
5747:z
5744:(
5715:.
5710:n
5706:y
5696:k
5693:h
5688:2
5685:1
5677:1
5672:k
5669:h
5664:2
5661:1
5656:+
5653:1
5647:=
5642:1
5639:+
5636:n
5632:y
5606:1
5603:+
5600:n
5596:y
5572:.
5568:)
5562:1
5559:+
5556:n
5552:y
5548:k
5545:+
5540:n
5536:y
5532:k
5528:(
5521:h
5515:2
5512:1
5506:+
5501:n
5497:y
5493:=
5488:1
5485:+
5482:n
5478:y
5454:y
5448:k
5445:=
5438:y
5414:,
5409:)
5404:)
5399:1
5396:+
5393:n
5389:y
5385:,
5380:1
5377:+
5374:n
5370:t
5366:(
5363:f
5360:+
5357:)
5352:n
5348:y
5344:,
5339:n
5335:t
5331:(
5328:f
5323:(
5315:h
5309:2
5306:1
5300:+
5295:n
5291:y
5287:=
5282:1
5279:+
5276:n
5272:y
5223:=
5220:z
5197:8
5194:1
5188:=
5185:h
5159:=
5153:4
5150:1
5135:=
5132:z
5109:4
5106:1
5100:=
5097:h
5087:z
5067:=
5064:k
5039:}
5034:1
5027:|
5023:z
5020:+
5017:1
5013:|
5006:|
4999:C
4992:z
4987:{
4965:z
4962:+
4959:1
4956:=
4953:)
4950:z
4947:(
4922:0
4918:y
4909:n
4905:)
4901:k
4898:h
4895:+
4892:1
4889:(
4886:=
4881:n
4877:y
4853:.
4848:n
4844:y
4840:)
4837:k
4831:h
4828:+
4825:1
4822:(
4819:=
4814:n
4810:y
4803:k
4797:h
4794:+
4789:n
4785:y
4781:=
4778:)
4773:n
4769:y
4765:k
4762:(
4756:h
4753:+
4748:n
4744:y
4740:=
4737:)
4732:n
4728:y
4724:,
4719:n
4715:t
4711:(
4708:f
4702:h
4699:+
4694:n
4690:y
4686:=
4681:1
4678:+
4675:n
4671:y
4647:y
4641:k
4638:=
4631:y
4588:}
4583:0
4577:)
4574:z
4571:(
4559:|
4552:C
4545:z
4540:{
4516:}
4511:1
4504:|
4500:)
4497:z
4494:(
4487:|
4480:|
4473:C
4466:z
4461:{
4431:1
4424:|
4420:)
4417:k
4414:h
4411:(
4404:|
4377:n
4357:0
4349:n
4345:y
4298:0
4294:y
4285:n
4279:)
4273:)
4270:k
4267:h
4264:(
4256:(
4251:=
4246:n
4242:y
4219:n
4215:y
4208:)
4205:k
4202:h
4199:(
4193:=
4188:1
4185:+
4182:n
4178:y
4157:y
4151:k
4148:=
4141:y
4102:)
4099:k
4096:(
4064:t
4042:t
4039:k
4035:e
4031:=
4028:)
4025:t
4022:(
4019:y
3998:C
3991:k
3971:1
3968:=
3965:)
3962:0
3959:(
3956:y
3936:y
3933:k
3930:=
3923:y
3886:t
3876:e
3853:t
3846:e
3840:0
3836:x
3814:)
3810:(
3793:.
3788:t
3781:e
3775:0
3771:x
3763:)
3757:t
3750:e
3736:+
3731:t
3721:e
3712:1
3703:(
3697:0
3693:x
3689:=
3686:)
3683:t
3680:(
3677:x
3646:c
3626:k
3605:)
3601:(
3584:,
3578:=
3571:|
3567:1
3560:|
3553:|
3542:|
3513:1
3507:=
3494:,
3485:=
3460:)
3456:(
3439:,
3434:)
3428:0
3419:0
3415:x
3408:(
3403:=
3400:)
3397:0
3394:(
3390:x
3385:,
3380:)
3374:0
3367:0
3361:(
3356:=
3353:)
3350:t
3347:(
3343:f
3338:,
3333:)
3309:1
3304:0
3298:(
3293:=
3289:A
3263:2
3260:=
3257:n
3246:5
3228:=
3225:k
3202:=
3199:c
3179:1
3176:=
3173:m
3158:)
3154:(
3137:,
3134:0
3131:=
3128:)
3125:0
3122:(
3113:x
3106:,
3101:0
3097:x
3093:=
3090:)
3087:0
3084:(
3081:x
3077:,
3074:0
3071:=
3068:x
3065:k
3062:+
3053:x
3047:c
3044:+
3035:x
3029:m
2949:t
2942:e
2845:)
2843:9
2841:(
2824:.
2816:|
2811:)
2798:(
2787:|
2778:|
2773:)
2760:(
2749:|
2713:t
2708:c
2701:x
2687:e
2681:t
2654:t
2649:c
2642:x
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2622:t
2602:)
2600:8
2598:(
2581:n
2578:,
2572:,
2569:2
2566:,
2563:1
2560:=
2557:t
2553:,
2548:|
2543:)
2530:(
2519:|
2509:|
2504:)
2499:t
2491:(
2480:|
2470:|
2465:)
2452:(
2441:|
2414:}
2411:n
2408:,
2402:,
2399:2
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2390:=
2387:t
2384:,
2379:t
2371:{
2355:,
2319:t
2314:c
2307:x
2302:t
2293:e
2287:t
2261:|
2257:)
2252:t
2244:(
2234:|
2209:x
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2155:t
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2120:t
2112:(
2102:|
2077:)
2074:x
2071:(
2067:g
2040:t
2035:c
2028:x
2023:t
2014:e
2008:t
1998:n
1993:1
1990:=
1987:t
1962:x
1937:t
1910:t
1883:t
1878:c
1871:x
1866:t
1857:e
1830:x
1810:)
1807:x
1804:(
1800:g
1779:)
1776:x
1773:(
1769:y
1742:x
1722:0
1714:t
1709:c
1702:x
1697:t
1688:e
1672:)
1670:7
1668:(
1651:,
1648:n
1645:,
1639:,
1636:2
1633:,
1630:1
1627:=
1624:t
1620:,
1617:0
1611:)
1606:t
1598:(
1567:)
1564:x
1561:(
1557:g
1534:t
1514:)
1512:6
1510:(
1493:,
1490:)
1487:x
1484:(
1480:g
1476:+
1471:t
1466:c
1459:x
1454:t
1445:e
1439:t
1429:n
1424:1
1421:=
1418:t
1410:=
1407:)
1404:x
1401:(
1397:y
1380:5
1365:n
1362:,
1356:,
1353:2
1350:,
1347:1
1344:=
1341:t
1338:,
1333:n
1328:C
1318:t
1313:c
1291:n
1288:,
1282:,
1279:2
1276:,
1273:1
1270:=
1267:t
1264:,
1260:C
1251:t
1226:n
1220:n
1199:A
1176:n
1171:R
1162:f
1158:,
1154:y
1138:)
1136:5
1134:(
1117:,
1114:)
1111:x
1108:(
1104:f
1100:+
1096:y
1091:A
1087:=
1079:y
1041:]
1035:,
1032:0
1029:[
1023:t
1008:)
1006:4
1004:(
980:2
976:y
970:7
959:3
956:=
943:z
931:2
927:y
921:7
910:3
904:z
898:y
893:4
882:x
876:=
863:y
853:z
847:y
842:4
834:+
831:x
822:=
809:x
776:C
772:+
768:A
758:4
747:1
738:C
734:+
730:B
721:B
717:+
713:C
703:7
692:3
683:B
679:+
675:B
666:B
649:A
606:)
603:y
600:,
597:t
594:(
591:f
588:=
581:y
566:)
564:3
562:(
545:,
540:)
535:)
530:1
527:+
524:n
520:y
516:,
511:1
508:+
505:n
501:t
497:(
494:f
491:+
488:)
483:n
479:y
475:,
470:n
466:t
462:(
459:f
454:(
449:h
443:2
440:1
434:+
429:n
425:y
421:=
416:1
413:+
410:n
406:y
367:8
364:1
358:=
355:h
329:4
326:1
320:=
317:h
287:)
285:2
283:(
266:.
257:t
249:0
243:)
240:t
237:(
234:y
230:,
225:t
215:e
211:=
208:)
205:t
202:(
199:y
179:)
177:1
175:(
155:=
152:)
149:0
146:(
143:y
139:,
136:0
130:t
126:,
123:)
120:t
117:(
114:y
105:=
102:)
99:t
96:(
89:y
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