solving nonlinear systems of equations worksheet
Root Finding and Nonlinear Sets of Equations part 1
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07/11/2013, 19:15
... America).
Chapter 9. Root Finding and
Nonlinear Sets of Equations
9.0 Introduction
We now consider that most basic of tasks, solving equations numerically. While
most equations are born with both a ... (in this context) gives us the hope of satisfying N equations in N unknowns
simultaneously. Note that we have only hope, not certainty. A nonlinear set of
equations may have no (real) solutions ... are the
individual equations to be satisfied simultaneously.
Don’t be fooled by the apparent notational similarity of equations (9.0.2) and
(9.0.1). Simultaneous solution of equations in N dimensions...
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Tài liệu Root Finding and Nonlinear Sets of Equations part 2 docx
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... 352
Chapter 9. Root Finding and Nonlinear Sets of Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
Copyright (C) ... event, with 2
−40
≈ 10
−12
.
354
Chapter 9. Root Finding and Nonlinear Sets of Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
Copyright (C) ... iteration one of
the previous boundary points is discarded in favor of the latest estimate of the root.
The only difference between the methods is that secant retains the most recent
of the prior...
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Tài liệu Root Finding and Nonlinear Sets of Equations part 3 pptx
Ngày tải lên :
15/12/2013, 04:15
... York:
McGraw-Hill),
Đ
8.3.
Ostrowski, A.M. 1966,
Solutions of Equations and Systems of Equations
, 2nd ed. (New York:
Academic Press), Chapter 12.
Ridders, C.J.F. 1979,
IEEE Transactions on Circuits and Systems
, vol. CAS-26, pp. ... converge: Local behavior might send it off
towards infinity.
356
Chapter 9. Root Finding and Nonlinear Sets of Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN ... number of iterations exceeded in rtsec");
return 0.0; Never get here.
}
358
Chapter 9. Root Finding and Nonlinear Sets of Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC...
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Tài liệu Root Finding and Nonlinear Sets of Equations part 4 doc
Ngày tải lên :
15/12/2013, 04:15
... York:
McGraw-Hill),
Đ
8.3.
Ostrowski, A.M. 1966,
Solutions of Equations and Systems of Equations
, 2nd ed. (New York:
Academic Press), Chapter 12.
Ridders, C.J.F. 1979,
IEEE Transactions on Circuits and Systems
, vol. CAS-26, pp. ... superlinear convergence with the
sureness of bisection?
360
Chapter 9. Root Finding and Nonlinear Sets of Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
Copyright ... 362
Chapter 9. Root Finding and Nonlinear Sets of 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 Root Finding and Nonlinear Sets of Equations part 5 doc
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... 366
Chapter 9. Root Finding and Nonlinear Sets of Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
Copyright (C) ... evaluation per iteration.
368
Chapter 9. Root Finding and Nonlinear Sets of Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
Copyright (C) ... is the set of points from which Newton’s method convergesto the root z = 1of the equation
z
3
− 1=0. Its shape is fractal.
of the desired root. But that means that in the neighborhood of an extremum...
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Tài liệu Root Finding and Nonlinear Sets of Equations part 6 docx
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... Newton-Raphson Method for Nonlinear
Systems of Equations
We make an extreme, but wholly defensible, statement: There are no good, gen-
eral methods for solving systems of more than one nonlinear equation. ... as
P
j
(x)=
P
(x)
(x−x
1
)···(x−x
j
)
−
P(x)
(x−x
1
)···(x−x
j
)
j
i=1
(x − x
i
)
−1
(9.5.28)
9.6 Newton-Raphson Method for Nonlinear Systems of Equations
379
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
Copyright ... f’s point of view, or from g’s. In order to find all common points, which are
372
Chapter 9. Root Finding and Nonlinear Sets of Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC...
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Tài liệu Root Finding and Nonlinear Sets of Equations part 7 doc
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... Solution of Nonlinear Equations in Several Vari-
ables
(New York: Academic Press).
9.7 Globally Convergent Methods for Nonlinear
Systems of Equations
We have seen that Newton’s method for solving nonlinear ... Newton-Raphson Method for Nonlinear
Systems of Equations
We make an extreme, but wholly defensible, statement: There are no good, gen-
eral methods for solving systems of more than one nonlinear equation. ... vector of values x
i
and F denote the entire vector of
functions F
i
. In the neighborhood of x, each of the functions F
i
can be expanded
9.6 Newton-Raphson Method for Nonlinear Systems of Equations
379
Sample...
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Tài liệu Root Finding and Nonlinear Sets of Equations part 8 ppt
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15/12/2013, 04:15
... Solution of Nonlinear Equations in Several Vari-
ables
(New York: Academic Press).
9.7 Globally Convergent Methods for Nonlinear
Systems of Equations
We have seen that Newton’s method for solving nonlinear ... corrected edition (Washington: Mathe-
matical Association of America), Chapter 14. [1]
Ostrowski, A.M. 1966,
Solutions of Equations and Systems of Equations
, 2nd ed. (New York:
Academic Press).
Ortega, ... rate of decrease of f.(Forexamplesof
such sequences, see
[1]
, p. 117.)
A simple way to fix the first problem is to require the average rate of decrease of f to
be at least some fraction α of the...
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Enclosing solutions of systems of equations
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12/01/2014, 22:20
... figure is reduced for the visualization.
Enclosing solutions of systems of equations
involving ODE
Aurelien Lejeune
National Institute of Informatics
2-1-2 Hitotsubashi, Chyoda-ku
Tokyo 101-8430 ... part.
Mots-clefs : syst`emes hybrides, equations differentielles ordinaires, anal-
yse par intervalles.
References
1. Hansen, E. and Sengupta, S.: Bounding solutions of systems of equations using
interval ... of Hybrid Systems Using Interval Arith-
metic Constraints. Hybrid Systems: Computation and Control, 139–142 (2004)
9. Ishii D., Ueda K., Hosobe H., and Goldsztejn A.: Interval-based Solving of...
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Tài liệu Đề tài " Isomonodromy transformations of linear systems of difference equations" pptx
Ngày tải lên :
14/02/2014, 17:20
... equal
to I. The proof of uniqueness is complete.
To prove the existence we note, first of all, that it suffices to provide a
proof if one of the κ
i
’s is equal to ±1 and one of the δ
j
’s is equal ... solutions of
isomonodromy problems for such systems of difference equations. In the case
of one-interval gap probability this has been done (in a different language) in
[Bor], [BB]. One example of the ... ALEXEI BORODIN
Most of the results of the present paper can be carried over to the case
of q-difference equations of the form Y (qz)=A(z)Y (z). The q-difference
Schlesinger equations are, cf. (3)–(6),
(3q)
(4q)
(5q)
(6q)
B
i
(...
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kosmann-schwarzbach y., grammaticos b., tamizhmani k.m. (eds.) integrability of nonlinear systems (springer lnp 638, 2004)(342s)
Ngày tải lên :
24/04/2014, 16:50
... excellence a
domain of application of differential equations. The nonlinearity inherent in most
classical equations of motion makes the question of stability and the prediction
of long-term behaviour ... components of angular velocity, (α, β, γ) the directions
cosines of the direction of gravity, (A, B, C) the moments of inertia, (x
0
,y
0
,z
0
)
the position of the centre of mass of the system, ... for studying them are those of q-analysis.
At the center of the theory of integrable systems lies the notion of a Lax pair,
describing the isospectral deformation of a linear operator, a matrix...
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Báo cáo sinh học: "Generalized Models for high-throughput analysis of uncer- tain nonlinear systems" doc
Ngày tải lên :
18/06/2014, 18:20
... levels of modeling.
,2
1
Generalized Models for high-throughput analysis of uncer-
tain nonlinear systems
Thilo Gross , Stefan Siegmund
∗ 2
1
Max-Planck Institute for the Physics of Complex Systems, ... instrumental, cp.
Table 1. However, application of the traditional
modeling paradigm, based on systems of specific
equations, faces some principal difficulties in these
systems. Insights from modeling are most ... will be made available soon.
Generalized Models for high-throughput analysis of uncer- tain nonlinear
systems
Journal of Mathematics in Industry 2011, 1:9 doi:10.1186/2190-5983-1-9
Thilo Gross...
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Báo cáo hóa học: " Generalized Models for high-throughput analysis of uncer- tain nonlinear systems" pdf
Ngày tải lên :
20/06/2014, 01:20
... levels of modeling.
,2
1
Generalized Models for high-throughput analysis of uncer-
tain nonlinear systems
Thilo Gross , Stefan Siegmund
∗ 2
1
Max-Planck Institute for the Physics of Complex Systems, ... instrumental, cp.
Table 1. However, application of the traditional
modeling paradigm, based on systems of specific
equations, faces some principal difficulties in these
systems. Insights from modeling are most ... for the analysis of uncertain models, e.g. in biological research.
Methods — Generalized modeling for conceptual analysis of large classes of models.
Results — Local dynamics of uncertain networks...
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Báo cáo toán học: " Generalized Models for high-throughput analysis of uncer- tain nonlinear systems" ppt
Ngày tải lên :
20/06/2014, 21:20
... ,2
1
Generalized Models for high-throughput analysis of uncer-
tain nonlinear systems
Thilo Gross , Stefan Siegmund
∗ 2
1
Max-Planck Institute for the Physics of Complex Systems, Năothnitzer Str. 38, 01187 Dresden, ... instrumental, cp.
Table 1. However, application of the traditional
modeling paradigm, based on systems of specific
equations, faces some principal difficulties in these
systems. Insights from modeling are most ... steady
state that exists in the class of systems. For deter-
mining the stability of X
∗
one can use dynamical
systems theory and evaluate the Jacobian of (1) at
X
∗
J
∗
=
∂G
∂X
X=X∗
−
∂L
∂X
X=X∗
.
For...
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