nonlinear systems of equations worksheet

... 348 Chapter 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) ... scaled axes, can save you a lot of grief as you enter the world of root finding 350 Chapter Root Finding and Nonlinear Sets of Equations for (i=1;i

Ngày tải lên: 07/11/2013, 19:15

... within an interval of size 354 Chapter Root Finding and Nonlinear Sets of Equations #include #define JMAX 40 Maximum allowed number of bisections f=(*func)(x1); fmid=(*func)(x2); if (f*fmid ... 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 ... decrease by a factor of two If after n iterations the root is known to be within an interval of size n , then after the next iteration it will be bracketed within an interval of size 354 Chapter...

Ngày tải lên: 15/12/2013, 04:15

... 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 979–980 ... rtf; Convergence } nrerror("Maximum number of iterations exceeded in rtflsp"); return 0.0; Never get here 358 Chapter Root Finding and Nonlinear Sets of Equations Ridders’ Method f(x1 ) − 2f(x3 ... method never jumps out of its brackets Second, the convergence of successive applications of equation (9.2.4) is quadratic, that is, m = in equation (9.1.4) Since each application of (9.2.4) requires...

Ngày tải lên: 15/12/2013, 04:15

... a quadratic function of y) whose value at y = is taken as the next estimate of the root x Of course one must have contingency plans for what to if the root falls outside of the brackets Brent’s ... estimate of the root and P/Q ought to be a “small” correction Quadratic methods work well only when the function behaves smoothly; they run the serious risk of giving very bad estimates of the ... guarantee at least linear convergence This kind of super-strategy requires attention to bookkeeping detail, and also careful consideration of how roundoff errors can affect the guiding strategy Also,...

Ngày tải lên: 15/12/2013, 04:15

... and Nonlinear Sets of Equations of the desired root But that means that in the neighborhood of an extremum there must be a tiny, perhaps distorted, copy of the basin of convergence — a kind of ... is the set of points from which Newton’s method converges to the root z = of the equation z3 − = Its shape is fractal 9.5 Roots of Polynomials 369 9.5 Roots of Polynomials Deflation of Polynomials ... f(x) 364 Chapter Root Finding and Nonlinear Sets of Equations f(x) Figure 9.4.3 Unfortunate case where Newton’s method enters a nonconvergent cycle This behavior is often encountered when the function...

Ngày tải lên: 15/12/2013, 04:15

... Newton-Raphson Method for Nonlinear Systems of Equations We make an extreme, but wholly defensible, statement: There are no good, general methods for solving systems of more than one nonlinear equation ... must be off the real axis, otherwise you will never get off that axis — and may get shot off to infinity by a minimum or maximum of the polynomial For real polynomials, the alternative means of polishing ... sc,sb,s,rc,rb,r,dv,delc,delb; float *q,*qq,*rem; float d[3]; 379 9.6 Newton-Raphson Method for Nonlinear Systems of Equations Hence one step of Newton-Raphson, taking a guess xk into a new guess xk+1 , can be written...

Ngày tải lên: 15/12/2013, 04:15

... Solution of Nonlinear Equations in Several Variables (New York: Academic Press) 9.7 Globally Convergent Methods for Nonlinear Systems of Equations We have seen that Newton’s method for solving nonlinear ... sums of squares of the individual functions Fi to get a master function F which (i) is positive definite, and (ii) has a global minimum of zero exactly at all solutions of the original set of nonlinear ... 9.7 Globally Convergent Methods for Nonlinear Systems of Equations 383 such methods can still occasionally fail by coming to rest on a local minimum of F , they often succeed where a direct attack...

Ngày tải lên: 15/12/2013, 04:15

... 389 9.7 Globally Convergent Methods for Nonlinear Systems of Equations The routine newt assumes that typical values of all components of x and of F are of order unity, and it can fail if this ... rate of decrease of f (For examples of 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 ... of f, this is quite rare in practice The routine newt below will warn you if this happens The remedy is to try a new starting point 9.7 Globally Convergent Methods for Nonlinear Systems of Equations...

Ngày tải lên: 15/12/2013, 04:15

... solutions of systems of equations using interval analysis BIT 21, 203–211 (1981) Neumaier, A.: Interval Methods for Systems of Equations Cambridge Univ Press, (1990) Goldsztejn, A.: A Comparison of ... integrations on each of them as initial conditions 3 Rigorous enclosure of discrete change during a hybrid system simulation Hybrid dynamic systems (HDSs) are systems described by a mix of discrete and ... Courcoubetis, et al Discrete abstractions of hybrid systems Proceeding of IEEE 88, 970–983 (2000) Nedialkov, N.S., Mohrenschildt, M.v.: Rigorous Simulation of Hybrid Dynamic Systems with Symbolic and Interval...

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Ngày tải lên: 20/06/2014, 22:20

... Annals of Mathematics, 160 (2004), 1141–1182 Isomonodromy transformations of linear systems of difference equations By Alexei Borodin Abstract We introduce and study “isomonodromy” transformations of ... result is construction of an isomonodromy action of Zm(n+1)−1 on the space of coefficients A(z) (here m is the size of matrices and n is the degree of A(z)) The (birational) action of certain rank n ... 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...

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... theory of nonlinear systems, and in particular of integrable systems, is related to several very active fields of theoretical physics For instance, the role played in the theory of integrable systems ... 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 ... between the role of q-analysis in the theory of discrete integrable sytems and that of q-deformations of algebras of functions on Lie groups and of universal enveloping algebras of Lie algebras...

Ngày tải lên: 24/04/2014, 16:50

... exploration of large parameter spaces The ongoing revolution in systems biology is revealing the structure of important systems For understanding the functioning and failure of these systems, mathematical ... Models for high-throughput analysis of uncertain nonlinear systems Thilo Gross1,2 , Stefan Siegmund∗2 Max-Planck Technical Institute for the Physics of Complex Systems, N¨thnitzer Str 38, 01187 ... instrumental, cp Table 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...

Ngày tải lên: 18/06/2014, 18:20

... exploration of large parameter spaces The ongoing revolution in systems biology is revealing the structure of important systems For understanding the functioning and failure of these systems, mathematical ... Models for high-throughput analysis of uncertain nonlinear systems Thilo Gross1,2 , Stefan Siegmund∗2 Max-Planck Technical Institute for the Physics of Complex Systems, N¨thnitzer Str 38, 01187 ... instrumental, cp Table 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...

Ngày tải lên: 20/06/2014, 01:20

... exploration of large parameter spaces The ongoing revolution in systems biology is revealing the structure of important systems For understanding the functioning and failure of these systems, mathematical ... Models for high-throughput analysis of uncertain nonlinear systems Thilo Gross1,2 , Stefan Siegmund∗2 Max-Planck Technical Institute for the Physics of Complex Systems, N¨thnitzer Str 38, 01187 ... instrumental, cp Table 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...

Ngày tải lên: 20/06/2014, 21:20

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