solving systems of linear differential equations in matlab
Tài liệu Integration of Ordinary Differential Equations part 1 doc
... America).Chapter 16. Integration of Ordinary Differential Equations 16.0 IntroductionProblems involving ordinary differential equations (ODEs) can always bereduced to the study of sets of first-order differential ... description of each of these types follows.1. Runge-Kutta methods propagate a solution over an interval by combiningthe information from several Euler-style steps (each involving one evaluation of theright-hand ... 1973,Computational Methods in Ordinary Differential Equations (New York: Wiley).Lapidus, L., and Seinfeld, J. 1971,Numerical Solution of Ordinary Differential Equations (NewYork: Academic...
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Tài liệu Integration of Ordinary Differential Equations part 2 pptx
... discussion of the pitfalls in constructing a good Runge-Kutta code is given in [3].Here is the routine for carrying out one classical Runge-Kutta step on a set of n differential equations. You input ... 1973,Computational Methods in Ordinary Differential Equations (New York: Wiley).Lapidus, L., and Seinfeld, J. 1971,Numerical Solution of Ordinary Differential Equations (NewYork: Academic ... derive from this basic 712Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C)...
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Tài liệu Solution of Linear Algebraic Equations part 1 docx
... 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 the solution process. In this ... consists of a particular solution xpadded to any linear combination of (typically) N − M vectors (which are said to be in the nullspace of the matrix A).The task of nding the solution space of A involvesã ... sophistication of complicated linear equation -solving packages”is devoted to the detection and/or correction of these two pathologies. As youwork with large linear sets of equations, you will...
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Tài liệu Solution of Linear Algebraic Equations part 2 ppt
... is unchanged and in no way scrambled if wereplace any row in A by a linear combination of itself and any other row,as long as we do the same linear combination of the rows of the b’s and 1(which ... method. For solving sets of linear equations, Gauss-Jordan eliminationproduces both the solution of the equations for one or more right-hand side vectorsb, and also the matrix inverse A−1. ... side vectors in the first instance.For these reasons, Gauss-Jordan elimination should usually not be your method of first choice, either for solving linear equations or for matrix inversion. Thedecomposition...
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Tài liệu Solution of Linear Algebraic Equations part 11 ppt
... numericalalgorithms involve solvinga successionof linear systems each of whichdiffers only slightly from its predecessor. Instead of doing O(N3) operations each timeto solve the equations from ... typical systems of linear equations. However, we willmeet special cases where QR is the method of choice. 100Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN ... 0.0) *sing=1;}The next routine, qrsolv, is used to 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....
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Tài liệu Integration of Ordinary Differential Equations part 3 doc
... methods.free_vector(ytemp,1,n);free_vector(ak6,1,n);free_vector(ak5,1,n);free_vector(ak4,1,n);free_vector(ak3,1,n);free_vector(ak2,1,n);}Noting that the above routines are all in single precision, don’t be too greedy in specifying eps. Thepunishment forexcessive greediness is interestingand worthyofGilbertand Sullivan’sMikado: ... North America).including garden-variety ODEs or sets of ODEs, and definite integrals (augmentingthe methods of Chapter 4). For storage of intermediate results (if you desire toinspect them) we ... + 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 (C)...
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Tài liệu Integration of Ordinary Differential Equations part 4 ppt
... 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 (C) ... hmin) nrerror("Step size too small in odeint");h=hnext;}nrerror("Too many steps in routine odeint");}CITED REFERENCES AND FURTHER READING:Gear, C.W. 1971,Numerical Initial ... Recipes Software. Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machine-readable files (including this...
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Tài liệu Integration of Ordinary Differential Equations part 5 pdf
... remind you once again that scaling of the variables is often crucial forsuccessful integration of differential equations. The scaling “trick” suggested in the discussion following equation (16.2.8) ... 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 0-521-43108-5)Copyright ... encountered in practice, 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...
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Tài liệu Integration of Ordinary Differential Equations part 6 pdf
... class of equations that occurs quite frequently in practice where you can gainabout a factor of two in efficiency by differencing the equations directly. The equations aresecond-order systems ... 734Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) ... 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 the second n elements....
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Tài liệu Integration of Ordinary Differential Equations part 7 pptx
... feature of implicit methods holds onlyfor linear systems, but even in the general case implicit methods give better stability. 742Chapter 16. Integration of Ordinary Differential Equations Sample ... as in the original Bulirsch-Stoer method.The starting point is an implicit form of the midpoint rule:yn+1− yn−1=2hfyn+1+ yn−12(16.6.29) 738Chapter 16. Integration of Ordinary Differential ... calculatesdydx.{void lubksb(float **a, int n, int *indx, float b[]);void ludcmp(float **a, int n, int *indx, float *d);int i,j,nn,*indx;float d,h,x,**a,*del,*ytemp;indx=ivector(1,n);a=matrix(1,n,1,n);del=vector(1,n);ytemp=vector(1,n);h=htot/nstep;...
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Tài liệu Integration of Ordinary Differential Equations part 8 pdf
... part of the procedure, and it is desirable to minimize its effect. Therefore, theintegration steps of a predictor-corrector method are overlapping, each one involvingseveral stepsize intervals ... method .In functional iteration, we take some initial guess for yn+1, insert it into the right-handside of (16.7.2) to get an updated value of yn+1, insert this updated value back intothe ... been, we think, squeezed 752Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C)...
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Tài liệu Solution of Linear Algebraic Equations part 3 pdf
... backsubstitution.Thecom-bination of Gaussian elimination and backsubstitution yields a solution to the set of equations. The advantage of Gaussian elimination and backsubstitutionover Gauss-Jordanelimination ... Recipes Software. Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machine-readable files (including this ... 42Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992...
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Tài liệu Solution of Linear Algebraic Equations part 4 docx
... on whether the number of row interchanges was evenor odd, respectively. This routine is used in combination withlubksbto solve linear equations or invert a matrix.{int i,imax,j,k;float ... the linear setA · x =(L·U)·x=L·(U·x)=b (2.3.3)by first solving for the vector y such thatL · y = b (2.3.4)and then solving U · x = y (2.3.5)What is the advantage of breaking up one linear ... lubksb in their present forms. This scheme is a factor of 2 inefficient in storage, since A and C are stored twice. It is also a factor of 2 inefficient in time, since the complex multiplies in a...
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