solving systems of linear differential equations in matlab
Tài liệu Integration of Ordinary Differential Equations part 1 doc
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15/12/2013, 04:15
... America).
Chapter 16. 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 ... description of each of these types follows.
1. Runge-Kutta methods propagate a solution over an interval by combining
the information from several Euler-style steps (each involving one evaluation of the
right-hand ... 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...
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Tài liệu Integration of Ordinary Differential Equations part 2 pptx
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... 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
(New
York: Academic ... derive from this basic
712
Chapter 16. Integration of Ordinary Differential Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
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Tài liệu Solution of Linear Algebraic Equations part 1 docx
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15/12/2013, 04:15
... 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 the solution process. In
this ... consists of a particular solution x
p
added 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 involves
ã ... sophistication of complicated linear equation -solving packages”
is devoted to the detection and/or correction of these two pathologies. As you
work with large linear sets of equations, you will...
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Tài liệu Solution of Linear Algebraic Equations part 2 ppt
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... is unchanged and in no way scrambled if we
replace 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 elimination
produces both the solution of the equations for one or more right-hand side vectors
b, 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. The
decomposition...
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Tài liệu Solution of Linear Algebraic Equations part 11 ppt
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... numericalalgorithms involve solvinga successionof linear systems each of which
differs only slightly from its predecessor. Instead of doing O(N
3
) operations each time
to solve the equations from ... 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 RECIPES IN ... 0.0) *sing=1;
}
The next routine, qrsolv, is used to 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.
...
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Tài liệu Integration of Ordinary Differential Equations part 3 doc
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... 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 worthyof
Gilbertand Sullivan’sMikado: ... North America).
including garden-variety ODEs or sets of ODEs, and definite integrals (augmenting
the methods of Chapter 4). For storage of intermediate results (if you desire to
inspect them) we ... + 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)
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Tài liệu Integration of Ordinary Differential Equations part 4 ppt
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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 (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
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... remind you once again that scaling of the variables is often crucial for
successful integration of differential equations. The scaling “trick” suggested in
the discussion following equation (16.2.8) ... 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 0-521-43108-5)
Copyright ... encountered in practice, is discussed in Đ16.7.)
726
Chapter 16. Integration of Ordinary Differential Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
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Tài liệu Integration of Ordinary Differential Equations part 6 pdf
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... class of equations that occurs quite frequently in practice where you can gain
about a factor of two in efficiency by differencing the equations directly. The equations are
second-order systems ... 734
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 (C) ... 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 the second n elements....
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Tài liệu Integration of Ordinary Differential Equations part 7 pptx
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... feature of implicit methods holds only
for linear systems, but even in the general case implicit methods give better stability.
742
Chapter 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:
y
n+1
− y
n−1
=2hf
y
n+1
+ y
n−1
2
(16.6.29)
738
Chapter 16. Integration of Ordinary Differential ... calculates
dydx
.
{
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
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... part of the procedure, and it is desirable to minimize its effect. Therefore, the
integration steps of a predictor-corrector method are overlapping, each one involving
several stepsize intervals ... method .In
functional iteration, we take some initial guess for y
n+1
, insert it into the right-hand
side of (16.7.2) to get an updated value of y
n+1
, insert this updated value back into
the ... been, we think, squeezed
752
Chapter 16. Integration of Ordinary Differential Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
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Tài liệu Solution of Linear Algebraic Equations part 3 pdf
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... backsubstitution.Thecom-
bination of Gaussian elimination and backsubstitution yields a solution to the set
of equations.
The advantage of Gaussian elimination and backsubstitutionover Gauss-Jordan
elimination ... 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 ... 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 (C) 1988-1992...
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