power series solution of coupled differential equations

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... Integration of Ordinary Differential Equations 16.0 Introduction Problems involving ordinary differential equations (ODEs) can always be reduced to the study of sets of first-order differential equations. ... auxiliary variables. The generic problem in ordinary differential equations is thus reduced to the study of a set of N coupled first-order differential equations for the functions y i ,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 (New York: Academic Press). 16.1...

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... 1973, Computational Methods in Ordinary Differential Equations (New York: Wiley). Lapidus, L., and Seinfeld, J. 1971, Numerical Solution of Ordinary Differential Equations (New York: Academic Press). 16.1 ... 710 Chapter 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 verify by expansion in power series) that the step’s error is only one power of h smaller than the correction, i.e O(h 2 ) added to (16.1.1). There...

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... as many equations as unknowns, and there is a good chance of solving for a unique solution set of x j ’s. Analytically, there can fail to be 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 (A T Ã A) Ã x =(A T Ãb)(2.0.4) where A T denotes the transpose of the matrix A. Equations (2.0.4) are called the normal 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 are effectively fewer equations...

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... of N ì N matrices, with M sets of right-hand side vectors, in completely analogous fashion. The routine implemented below is, of course, general. 38 Chapter 2. Solution of Linear Algebraic Equations Sample ... rows of A and the corresponding rows of the b’s and of 1, does not change (or scramble in any way) the solution x’s and Y. Rather, it just corresponds to writing the same set of linear equations in ... of this procedure, however, is thatthechoice of pivotwilldepend on the originalscaling of the equations. If we take the third linear equation in our original set and multiply it by a factor of...

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... (2.10.4) of the algorithm is needed, so we separate it off into its own routine rsolv. 98 Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC ... not used for typical systems of linear equations. However, we will meet special cases where QR is the method of choice. 100 Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL ... America). x[i]=sum/p[i]; } } A typicaluseof choldcand cholslis in theinversionof covariancematrices describing the fit of data to a model; see, e.g., Đ15.6. In this, and many other applications,one often needs L −1 ....

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... 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 ... ,n−1 y(x+H)≈y n ≡ 1 2 [z n +z n−1 +hf(x + H, z n )] (16.3.2) 714 Chapter 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 accuracy may be demanded not directly in the solution itself, but in...

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... 722 Chapter 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 turns out that the error of (16.3.2), expressed as a power ... power series in h, the stepsize, contains only even powers of h, y n − y(x + H)= ∞  i=1 α i h 2i (16.3.3) where H is held constant, but h changes by varying n in (16.3.1). The importance of this...

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... extrapolate each component of a vector of quantities. 728 Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN ... method a degree of robustness for problems with discontinuities. Let us remind you once again that scaling of the variables is often crucial for successful integration of differential equations. The ... method does an excellent job of feeling its way through rocky or discontinuous terrain. It is also an excellent choice for quick-and-dirty, low-accuracy solution of a set of equations. A second warning...

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... vol. 27, pp. 505–535. 16.6 Stiff Sets of Equations As soon as one deals with more than one first-order differential equation, the possibility of a stiff set of equations arises. Stiffness occurs ... Second-Order Conservative Equations Usually when you have a system of high-order differential equations to solve it is best to reformulate them as a system of rst-order equations, as discussed ... compatibility with bsstep the arrays y and d2y are of length 2n for a system of n second-order equations. The values of y arestoredinthefirstnelements of y, while the first derivatives are stored in...

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... in fact the correct solution of 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, the possibility of a stiff set of equations arises. Stiffness occurs ... form of the midpoint rule: y n+1 − y n−1 =2hf  y n+1 + y n−1 2  (16.6.29) 738 Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC...

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... Problems in Ordinary Differential Equations (Englewood Cliffs, NJ: Prentice-Hall), Chapter 9. [1] Shampine, L.F., and Gordon, M.K. 1975, Computer Solution of Ordinary Differential Equations. The Initial ... adjusting the stepsize is difficult. 748 Chapter 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 we will describe later. Nevertheless, these methods have had a long historical run. Textbooks are full of information...

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... 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 ... 42 Chapter 2. Solution of 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 Solution of Linear Equations (New York: Wiley). Ralston, A., and Rabinowitz, P. 1978, A...

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... 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 Solution of 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 Solution of 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 solution of a triangular set of equations is quite trivial, as we have...

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... 1967, Computer Solution of 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 Solution of 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, has nonzero elements only on the ... limitations of bandec, and the above routine does take advantage of the opportunity. In general, when TINY is returned as a diagonal element of U, then the original matrix (perhaps as modified by roundoff...

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... 104 Chapter 2. Solution of 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 order N =2 m , recursively by partitions in half. At each step, halving the order doubles the number of inverse operations. But this ... complicated nature of the recursive Strassen algorithm, you will find that LU decomposition is in no immediate danger of becoming obsolete. If, on the other hand, you like this kind of fun, then try...

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