solving systems of linear differential equations by elimination
Tài liệu Integration of Ordinary Differential Equations part 1 doc
Ngày tải lên :
15/12/2013, 04:15
... 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 ... 708
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) 1988-1992 by Cambridge University...
- 4
- 361
- 0
Tài liệu Integration of Ordinary Differential Equations part 2 pptx
Ngày tải lên :
15/12/2013, 04:15
... 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 (C) 1988-1992 by Cambridge University ... 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 ... 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)
Copyright (C) 1988-1992 by Cambridge University...
- 5
- 448
- 0
Tài liệu Solution of Linear Algebraic Equations part 1 docx
Ngày tải lên :
15/12/2013, 04:15
... 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 ... 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
+ ... 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...
- 5
- 461
- 0
Tài liệu Solution of Linear Algebraic Equations part 2 ppt
Ngày tải lên :
15/12/2013, 04:15
... 9.
2.1 Gauss-Jordan Elimination
For inverting a matrix, Gauss-Jordan elimination is about as efficient as any
other method. For solving sets of linear equations, Gauss-Jordan elimination
produces ... 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 ... 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 then is no longer the identity matrix, of course).
ã...
- 6
- 410
- 0
Tài liệu Solution of Linear Algebraic Equations part 11 ppt
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 ... 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 scratch, one can often ... 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...
- 5
- 357
- 0
Tài liệu Integration of Ordinary Differential Equations part 3 doc
Ngày tải lên :
24/12/2013, 12:16
... 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) 1988-1992 by Cambridge University ... 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 (C) 1988-1992 by Cambridge University ... 720
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) 1988-1992 by Cambridge University...
- 9
- 436
- 0
Tài liệu Integration of Ordinary Differential Equations part 4 ppt
Ngày tải lên :
24/12/2013, 12:16
... 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) 1988-1992 by Cambridge University ... modified midpoint method, which advances a vector
of dependent variables y(x) from a point x to a point x + H by a sequence of n
substeps each of size h,
h = H/n (16.3.1)
In principle, one could ... RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.
Permission...
- 3
- 461
- 0
Tài liệu Integration of Ordinary Differential Equations part 5 pdf
Ngày tải lên :
24/12/2013, 12:16
... 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 ... integration of differential equations. The scaling “trick” suggested in
the discussion following equation (16.2.8) is a good general purpose choice, but
not foolproof. Scaling by the maximum values of ... 724
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) 1988-1992 by Cambridge University...
- 9
- 486
- 0
Tài liệu Integration of Ordinary Differential Equations part 6 pdf
Ngày tải lên :
24/12/2013, 12:16
... 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 ... 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 ... 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...
- 3
- 395
- 0
Tài liệu Integration of Ordinary Differential Equations part 7 pptx
Ngày tải lên :
24/12/2013, 12:16
... 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 ... 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...
- 14
- 407
- 0
Tài liệu Integration of Ordinary Differential Equations part 8 pdf
Ngày tải lên :
24/12/2013, 12:16
... 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 (C) 1988-1992 by Cambridge University ... 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)
Copyright (C) 1988-1992 by Cambridge University ... 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...
- 6
- 457
- 0
Tài liệu Solution of Linear Algebraic Equations part 3 pdf
Ngày tải lên :
24/12/2013, 12:16
... defined by equation (2.2.4) is called backsubstitution.Thecom-
bination of Gaussian elimination and backsubstitution yields a solution to the set
of equations.
The advantage of Gaussian elimination ... 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 by Cambridge University ... 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...
- 3
- 404
- 0
Tài liệu Solution of Linear Algebraic Equations part 4 docx
Ngày tải lên :
24/12/2013, 12:16
... 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 ... 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 ... Analysis
(Cambridge: 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...
- 8
- 464
- 0