system of linear differential equations second order

Ngày tải lên: 21/06/2014, 03:20

... translates into Z L =(r a,b )G L . A proof of Lemma 1.29 finishes the proof of the theorem. ✷ Proof of lemma 1.29. The proof is rather similar to the one of lemma 1.23. The only thing that we have ... as a group of matrices and has the structure of a linear algebraic group, that is, it is a group of invertible matrices defined by the vanishing of a set of polynomials on the entries of these matrices. ... O-submodules N 1 ,N 2 of N with N = N 1 ⊕N 2 and N i = F i for i =1, 2. Proof. The proof is similar to the proof of Proposition 3.17. Let S 1 and S 2 be the set of eigenvalues of E acting on F 1 and...

Ngày tải lên: 31/03/2014, 15:56

Ngày tải lên: 21/06/2014, 05:20

Ngày tải lên: 14/03/2014, 13:20

Ngày tải lên: 14/03/2014, 13:20

Ngày tải lên: 21/06/2014, 20:20

Ngày tải lên: 21/06/2014, 20:20

Ngày tải lên: 22/06/2014, 06:20

Ngày tải lên: 22/06/2014, 19:20

Ngày tải lên: 22/06/2014, 22:20

Ngày tải lên: 23/06/2014, 00:20

Ngày tải lên: 06/07/2014, 15:18

... 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 ... first order, but that have different coefficients of higher -order error terms. Adding up the right combination of these, we can eliminate the error terms order by order. That is the basic idea of...

Ngày tải lên: 15/12/2013, 04:15

... first order, but that have different coefficients of higher -order error terms. Adding up the right combination of these, we can eliminate the error terms order by order. That is the basic idea of ... error term, making the method second order. [A method is conventionally called nth order if its error term is O(h n+1 ).] In fact, (16.1.2) is called the second- order Runge-Kutta or midpoint ... 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...

Ngày tải lên: 15/12/2013, 04:15

... trade@cup.cam.ac.uk (outside North America). Chapter 2. Solution of Linear Algebraic Equations 2.0 Introduction A set of linear algebraic equations looks like this: a 11 x 1 + a 12 x 2 + a 13 x 3 + ... 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 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 machine render them linearly dependent at some stage in...

Ngày tải lên: 15/12/2013, 04:15

... case of N × 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 ... 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 ... 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...

Ngày tải lên: 15/12/2013, 04:15

... is 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 ... solve linear systems. In many applications only the part (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 ... purposes, because of its greater diagnostic capability in pathological cases.) Updating a QR decomposition Some numericalalgorithms involve solvinga successionof linear systems each of which differs...

Ngày tải lên: 15/12/2013, 04:15

... interestingand worthyof Gilbertand Sullivan’sMikado: The routinecan alwaysachieve an apparent zero error by making the stepsize so small that quantities of order hy  add to quantities of order y as if ... may be fifth -order accurate, but we have no way of monitoring its truncation error. Higher order is not always higher accuracy! Use of (16.2.3) rarely does harm, but we have no way of directly ... 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...

Ngày tải lên: 24/12/2013, 12:16

... midpointmethod is a second- order method, like (16.1.2), but with the advantage of requiring(asymptoticallyfor large n) only one derivative evaluation per step h instead of the two required by second- order ... 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 ... our usual tricks of combining steps to knock out higher -order error terms, we can gain two orders at a time! For example, suppose n is even, and let y n/2 denote the result of applying (16.3.1)...

Ngày tải lên: 24/12/2013, 12:16