... Integration of Ordinary Differential Equations 16.0 IntroductionProblems involving ordinary differentialequations (ODEs) can always bereduced to the study of sets of first-order differential equations. ... auxiliary variables.The generic problem in ordinary differentialequations is thus reduced to thestudy of a set of N coupled first-order differentialequations for the functionsyi,i=1,2, ,N, having ... 1973,Computational Methods in Ordinary Differential Equations (New York: Wiley).Lapidus, L., and Seinfeld, J. 1971,Numerical Solution of Ordinary Differential Equations (NewYork: Academic Press).16.1...
... 1973,Computational Methods in Ordinary Differential Equations (New York: Wiley).Lapidus, L., and Seinfeld, J. 1971,Numerical Solution of Ordinary Differential Equations (NewYork: Academic Press).16.1 ... information onlyat the beginning of that interval (see Figure 16.1.1). That means (and you can verifyby expansion in power series) that the step’s error is only one powerof h smallerthan the correction, ... 710Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright...
... trade@cup.cam.ac.uk (outside North America).Chapter 2. Solution of Linear Algebraic Equations 2.0 IntroductionA set oflinear algebraic equations looks like this:a11x1+ a12x2+ a13x3+ ... equations as unknowns, and there is a goodchance of solving for a unique solution set of xj’s. Analytically, there can fail tobe a unique solution if one or more of the M equations is a linear ... While not exact linear combinations of each other, some of the equations may be so close to linearly dependent that roundoff errors in the machinerender them linearly dependent at some stage in...
... of N ì N matrices, with M sets of right-handside vectors, in completely analogous fashion. The routine implemented belowis, of course, general. 38Chapter 2. Solution ofLinear Algebraic Equations Sample ... of this procedure, however, isthatthechoice of pivotwilldepend on the originalscaling of the equations. If we takethe third linear equation in our original set and multiply it by a factor of ... rows of A and the corresponding rows of the b’sand of 1, does not change (or scramble in any way) the solution x’s andY. Rather, it just corresponds to writing the same set oflinear equations in...
... not used for typical systems oflinear equations. However, we willmeet special cases where QR is the method of choice. 100Chapter 2. Solution ofLinear Algebraic Equations Sample page from NUMERICAL ... solve linear systems. In many applications only thepart (2.10.4) of the algorithm is needed, so we separate it off into its own routine rsolv. 98Chapter 2. Solution ofLinear Algebraic Equations Sample ... solve the next set oflinear equations. The LUdecomposition is complicated to update because of pivoting. However, QR turns out to bequite simple for a very common kind of update,A → A + s...
... generally useful stepperroutine 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].Letus require ... ,n−1y(x+H)≈yn≡12[zn+zn−1+hf(x + H, zn)](16.3.2) 714Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright ... informationcan beobtained. Obviously, 720Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright...
... 722Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright ... usefulness ofthe modied midpointmethod tothe Bulirsch-Stoertechnique(Đ16.4) derives from a deep result about equations (16.3.2), due to Gragg. It turnsout that the error of (16.3.2), expressed as a power ... powerseries in h, the stepsize, containsonly even powers of h,yn− y(x + H)=∞i=1αih2i(16.3.3)where H is held constant, but h changes by varying n in (16.3.1). The importance of this...
... extrapolate eachcomponent of a vector of quantities. 728Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN ... methoda degree of robustness for problems with discontinuities.Let us remind you once again that scaling of the variables is often crucial forsuccessful integration ofdifferential equations. The ... is discussed in Đ16.7.) 726Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright...
... vol. 27, pp. 505–535.16.6 Stiff Sets of Equations As soon as one deals with more than one first-order differential equation, thepossibility of a stiff set ofequations arises. Stiffness occurs ... Second-Order Conservative Equations Usually when you have a system of high-order differentialequations to solve it is bestto reformulate them as a system of rst-order equations, as discussed ... compatibility with bsstep the arrays y and d2y are of length 2n for asystem of n second-order equations. The values of y arestoredinthefirstnelements of y,while the first derivatives are stored in...