... 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 Solutionof 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 Solutionof Ordinary Differential Equations (NewYork: Academic Press).16.1 ... 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 ... (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, i.e O(h2) added to (16.1.1).There...
... as many 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 ... set of equationsto be solved can bewritten as the N ìN set of equations (ATÃ A) Ã x =(ATÃb)(2.0.4)where ATdenotes the transpose of the matrix A. Equations (2.0.4) are called thenormal equations ... columns of the matrix inverse of A (Đ2.1 and Đ2.3).ã Calculation of the determinant of a square matrix A (Đ2.3).If M<N,orifM=Nbut the equations are degenerate, then there areeffectively fewer equations...
... 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. Solutionof Linear Algebraic Equations Sample ... 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 of linear equations in ... 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...
... (2.10.4) of the algorithm is needed, so we separate it off into its own routine rsolv. 98Chapter 2. Solutionof Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC ... not used for typical systems of linear equations. However, we willmeet special cases where QR is the method of choice. 100Chapter 2. Solutionof Linear Algebraic Equations Sample page from NUMERICAL ... America).x[i]=sum/p[i];}}A typicaluseof choldcand cholslis in theinversionof covariancematrices describingthe fit of data to a model; see, e.g., Đ15.6. In this, and many other applications,one often needsL−1....
... 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 ... efficiency are not mere tens of percents or factors of two;they can sometimes be factors of ten, a hundred, or more. Sometimes accuracymay be demanded not directly in the solution itself, but in...
... 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 ... method does an excellent job of feeling its way through rocky or discontinuousterrain. It is also an excellent choice for quick-and-dirty, low-accuracy solution of a set of equations. A second warning...
... 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...
... in fact thecorrect solutionof the differential equation. If we think of x as representing time,then the implicit method converges to the true equilibriumsolution(i.e., the solution at late times) ... 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 ... form of the midpoint rule:yn+1− yn−1=2hfyn+1+ yn−12(16.6.29) 738Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC...
... Problems in Ordinary Differential Equations (EnglewoodCliffs, NJ: Prentice-Hall), Chapter 9. [1]Shampine, L.F., and Gordon, M.K. 1975,Computer Solutionof Ordinary Differential Equations. The Initial ... adjustingthe stepsize is difficult. 748Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright ... high-precision solution of very smooth equations with very complicated right-hand sides, as wewill describe later.Nevertheless, these methods have had a long historical run. Textbooks arefull of information...
... is called backsubstitution.Thecom-bination of Gaussian elimination and backsubstitution yields a solution to the set of equations. The advantage of Gaussian elimination and backsubstitutionover ... 42Chapter 2. Solutionof Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright ... York:McGraw-Hill), Program B-2, p. 298.Westlake, J.R. 1968,A Handbook of Numerical Matrix Inversion and Solutionof Linear Equations (New York: Wiley).Ralston, A., and Rabinowitz, P. 1978,A...
... modify the loop of the above fragment and (e.g.) divide by powers of ten,to keep track of the scale separately, or (e.g.) accumulate the sum of logarithms of the absolute values of the factors ... 1967,Computer Solutionof Linear Algebraic Systems(Engle-wood Cliffs, NJ: Prentice-Hall), Chapters 9, 16, and 18.Westlake, J.R. 1968,A Handbook of Numerical Matrix Inversion and Solutionof Linear Equations (New ... · x = y (2.3.5)What is the advantage of breaking up one linear set into two successive ones?The advantage is that the solutionof a triangular set ofequations is quite trivial, aswe have...
... 1967,Computer Solutionof Linear Algebraic Systems(Engle-wood Cliffs, NJ: Prentice-Hall), Chapters 9, 16, and 18.Westlake, J.R. 1968,A Handbook of Numerical Matrix Inversion and Solutionof Linear Equations (New ... Cambridge University Press).2.4 Tridiagonal and Band Diagonal Systems of Equations The special case of a system of linear equations that is tridiagonal, that is, hasnonzero elements only on the ... limitations of bandec, and the aboveroutine does take advantage of the opportunity. In general, when TINY is returned as adiagonal element of U, then the original matrix (perhaps as modified by roundoff...
... 104Chapter 2. Solutionof Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright ... submatrices. Imagine doing the inversionof a very large matrix, of orderN =2m, recursively by partitions in half. At each step, halving the order doublesthe number of inverse operations. But this ... complicated nature of the recursive Strassen algorithm, you will find that LU decomposition is in noimmediate danger of becoming obsolete.If, on the other hand, you like this kind of fun, then try...