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The Shooting Method in Software Print PDF417 in Software The Shooting Method




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18.1 The Shooting Method generate, create pdf417 2d barcode none on software projects Microsoft Office Excel Website minimum stepsize Software pdf417 2d barcode hmin. At x2 it calls the user-supplied routine score and returns the n2 functions that ought to be zero. newt uses a globally convergent Newton s method to adjust the values of v until the returned functions are in fact zero.

h1=(x2-x1)/100.0; y=load(x1,v); Output out; No output generated by Odeint. Odeint<StepperDopr853<R> > integ(y,x1,x2,atol,rtol,h1,hmin,out,d); integ.

integrate(); return score(x2,y); } };. Note that Shoot is templated on the load, right-hand side for Odeint, and score routines. In practice, you will almost always want to write these as functors rather than functions. This makes communicating the various parameters in the problem easy just pass them as parameters in the constructors.

For some problems the initial stepsize V might depend sensitively upon the initial conditions. It is straightforward to alter load to compute a suggested stepsize h1 as a member variable and feed it st to Shoot and hence to NRfdjac when the Shoot object is passed to newt. A complete cycle of the shooting method thus requires n2 C 1 integrations of the N coupled ODEs: one integration to evaluate the current degree of mismatch, and n2 for the partial derivatives.

Each new cycle requires a new round of n2 C 1 integrations. This illustrates the enormous extra effort involved in solving two-point boundary value problems compared with initial value problems. If the differential equations are linear, then only one complete cycle is required, since (18.

1.3) (18.1.

4) should take us right to the solution. A second round can be useful, however, in mopping up some (never all) of the roundoff error. As given here, Shoot uses the high-ef ciency eighth-order Runge-Kutta method of 17.

2 to integrate the ODEs, but any of the other methods of 17 could just as well be used. You, the user, must supply Shoot with: (i) a function or functor load(x1,v) that returns the n-vector y[0..

n-1] (satisfying the starting boundary conditions, of course), given the freely speci able variables of v[0..n2-1] at the initial point x1; (ii) a function or functor score(x2,y) that returns the discrepancy vector f[0.

. n2-1] of the ending boundary conditions, given the vector y[0..

n-1] at the endpoint x2; (iii) a starting vector v[0..n2-1]; (iv) a function or functor, called d in the routine, for the ODE integration; and other obvious parameters as described in the header comment above.

In 18.4 we give a sample program illustrating how to use Shoot..

CITED REFERENCES Software barcode pdf417 AND FURTHER READING: Acton, F.S. 1970, Numerical Methods That Work; 1990, corrected edition (Washington, DC: Mathematical Association of America).

Keller, H.B. 1968, Numerical Methods for Two-Point Boundary-Value Problems; reprinted 1991 (New York: Dover).

. 18. Two-Point Boundary Value Problems 18.2 Shooting to a Fitting Point The shooting met hod described in 18.1 tacitly assumed that the shots would be able to traverse the entire domain of integration, even at the early stages of convergence to a correct solution. In some problems it can happen that, for very wrong starting conditions, an initial solution can t even get from x1 to x2 without encountering some incalculable, or catastrophic, result.

For example, the argument of a square root might go negative, causing the numerical code to crash. Simple shooting would be stymied. A different, but related, case is where the endpoints are both singular points of the set of ODEs.

One frequently needs to use special methods to integrate near the singular points, analytic asymptotic expansions, for example. In such cases it is feasible to integrate in the direction away from a singular point, using the special method to get through the rst little bit and then reading off initial values for further numerical integration. However, it is generally not feasible to integrate into a singular point.

Usually the desired boundary condition is that one wants a regular solution at the singular point, but integrating into a singularity is guaranteed to pick out a singular solution, which by de nition is growing as one integrates inward. Any small numerical inaccuracy will include some admixture of the wrong solution, which grows and swamps the desired solution. The solution to the above-mentioned dif culties is shooting to a tting point.

Instead of integrating from x1 to x2 , we integrate rst from x1 to some point xf that is between x1 and x2 ; and second from x2 (in the opposite direction) to xf . If (as before) the number of boundary conditions imposed at x1 is n1 , and the number imposed at x2 is n2 , then there are n2 freely speci able starting values at x1 and n1 freely speci able starting values at x2 . (If you are confused by this, go back to 18.

1.) We can therefore de ne an n2 -vector V .1/ of starting parameters at x1 and a prescription load1(x1,v1) for mapping V .

1/ into a y that satis es the boundary conditions at x1 :. .1/ yi .x1 / D y barcode pdf417 for None i .

x1 I V0.1/ ; : : : ; Vn2 1 /.
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