power series solution of coupled differential equations
Tài liệu Integration of Ordinary Differential Equations part 1 doc
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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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Tài liệu Integration of Ordinary Differential Equations part 2 pptx
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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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Tài liệu Solution of Linear Algebraic Equations part 1 docx
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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
à A) à 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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Tài liệu Solution of Linear Algebraic Equations part 2 ppt
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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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Tài liệu Solution of Linear Algebraic Equations part 11 ppt
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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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Tài liệu Integration of Ordinary Differential Equations part 3 doc
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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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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 ... 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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Tài liệu Integration of Ordinary Differential Equations part 5 pdf
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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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Tài liệu Integration of Ordinary Differential Equations part 6 pdf
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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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Tài liệu Integration of Ordinary Differential Equations part 7 pptx
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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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Tài liệu Integration of Ordinary Differential Equations part 8 pdf
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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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Tài liệu Solution of Linear Algebraic Equations part 3 pdf
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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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Tài liệu Solution of Linear Algebraic Equations part 4 docx
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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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Tài liệu Solution of Linear Algebraic Equations part 5 docx
Ngày tải lên :
24/12/2013, 12:16
... 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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Tài liệu Solution of Linear Algebraic Equations part 12 pdf
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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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