NAG Fortran library Mark 17, routine D03PCF, Numerical Algorithms Group Ltd., Oxford, United Kingdom (1995).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....instabilities that can occur, consider a four spike solution to (1.1) for the exponent set (p, q, m, s) 2, 1, 3, 0) with # = 0.01 (this is experiment 3 of 5) three di#erent parameter sets of D and # we computed the solution to (1. 1) numerically using the routine D03PCF from the NAG library [19]. The equilibrium solution a e and h e is shown in Fig. 1(a) when D = 0.18. In each case, for the initial condition for (1.1) we took a 2 localized perturbation o# of a e and h e , with the perturbation chosen so that it is identical for the first and third, and for the second and fourth, spikes. ....

....confirm numerically our predictions for the onset of an oscillatory instability as # increases past # 0 . We also give numerical results for the large scale oscillations that occur when # is well beyond # 0 . To do so, we solve the GM model (1. 1) numerically using the NAG library routine D03PCF [19] with 2000 uniformly spaced meshpoints and stringent control on the accuracy of local time steps. The initial condition for the GM model (1.1) is taken to be 1 0.02 cos #x e x , h(x, 0) h e (x) 4.1) where a e and h e are the one spike equilibrium solutions given in ....

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NAG Fortran library Mark 17, routine D03PCF, Numerical Algorithms Group Ltd., Oxford, United Kingdom (1995).

.... Elliott and French [9] studied metastable dynamics for the Cahn Hilliard equation by applying the Galerkin nite element method, and Carr and Pego [7] computed the metastable evolution of a transition layer for the Allen Cahn equation using the method of lines subroutine LSODI of the NAG library [20] with 801 grid points. In addition, a standard fourth order nite di erence scheme was employed to compute metastable dynamics for the viscous Cahn Hilliard equation in [23] However, the implementation in [23] was non standard in that quadruple precision arithmetic was used in their numerical ....

NAG Fortran library Mark 17, routine LSODI, Numerical Algorithms Group Ltd. Oxford, United Kingdom (1995). 19

....result is that there is one large steady state hot spot region occupying the interval z 2 (z e1 ; 1 z e1 ) In Fig. 14(a) 14(b) we take = 0:02 and we plot the numerical solution to (3.9) at di erent times for the parameter values of Fig. 13(a) These results were computed using the NAG library [12]. In Fig. 14(a) we show the transient behavior describing the initial formation of the hot spot regions. In Fig. 14(b) we show the evolution of the di usive interfaces, their merger at a later time, and the ultimate steady state solution. This behavior is as predicted by the asymptotic theory. To ....

NAG Fortran library Mark 17, routine D03PCF, Numerical Algorithms Group Ltd. Oxford, United Kingdom (1995).

....of the computational experiments below, we took the initial condition for (1.3) to be a localized perturbation of a e . The initial condition is a(r, 0) 1 0.02e r # cos #r a e (r) h(0) h e . 5.2) With this initial condition, 1. 3) is solved numerically using the NAG library [17] for various exponent sets and dimensions N . The method of lines with 3000 meshpoints across the interval was used. The error tolerance in the time stepping was very strict. In each of the computations below we took # = 0.05. To illustrate the oscillatory behaviour we typically plot am a(0, ....

NAG Fortran library Mark 17, routine D03PCF, Numerical Algorithms Group Ltd., Oxford, United Kingdom (1995).

....new spike. In our one dimensional spatial domain it is possible to largely overcome these diculties by using a very large number of equidistantly spaced spatial meshpoints and stringent tolerances on the adaptive time stepping control for a method of lines based PDE solver from the NAG library [18]. Although, this type of brute force approach is possible in one space dimension, it is computationally inecient and, from practical purposes, infeasible in more than one space dimension, where similar types of dynamical behavior for spike solutions is likely to occur. In this light, the second ....

....[12] In x3.2 we derive an analogous result for the dynamics of a one spike solution to (1. 3) The asymptotic results for the location of the spike as a function of time are compared with corresponding numerical results computed using the moving mesh method of x2 and the NAG library routine D03PCF [18]. 3.1 The GM Model For 1, the method of matched asymptotic expansions was used in [12] to derive the following asymptotic result for the dynamics of a one spike solution to (1.1) Proposition 3.1(From [12] For 0 1 and = 0, the dynamics of a one spike solution to (1.1) is ....

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NAG Fortran library Mark 17, routine D03PCF, Numerical Algorithms Group Ltd., Oxford, United Kingdom (1995).

....spikes n, and of the initial spike locations x j (0) for j = 1; n. The time variable given in the plots below correspond to the slow time variable de ned in (2.1) by = t. With = 02, we get t = 2500 . To compute the full numerical results from (1. 1) we use the NAG library code [11] with 2001 equidistant meshpoints. For given values of D, n, and x j (0) for j = 1; n, we take the quasiequilibrium solution (2.20) to be the initial condition for a and h. To compute a(x; 0) and h(x; 0) we must determine the initial values h j (0) for j = 1; n, from the nonlinear ....

....This failure, which is a result of the non invertibility of the linearized system for the h j at certain speci c parameter values, is explained below. Once the nonlinear system for the h j in (2.27) is solved, the initial condition for a(x; 0) and h(x; 0) is known from (2. 20) and the NAG solver [11] is used to compute the solution to (1.1) at later times . The locations of the spikes from these numerical results were obtained by a local quadratic interpolation. The asymptotic results were obtained by solving the di erential algebraic system (2.27) for x j ( and h j ( using the ODE ....

NAG Fortran library Mark 17, routine D03PCF, Numerical Algorithms Group Ltd., Oxford, United Kingdom (1995).

....new spike. In our one dimensional spatial domain it is possible to largely overcome these difficulties by using a very large number of equidistantly spaced spatial meshpoints and stringent tolerances on the adaptive time stepping control for a method of lines based PDE solver from the NAG library [17]. Although, this type of brute force approach is possible in one space dimension, it is computationally inefficient and, from practical purposes, infeasible in more than one space dimension, where similar types of dynamical behavior for spike solutions is likely to occur. In this light, the second ....

....[11] In x3.2 we derive an analogous result for the dynamics of a one spike solution to (1. 3) The asymptotic results for the location of the spike as a function of time are compared with corresponding numerical results computed using the moving mesh method of x2 and the NAG library routine D03PCF [17]. 3.1 The GM Model For 1, the method of matched asymptotic expansions was used in [11] to derive the following asymptotic result for the dynamics of a one spike solution to (1.1) Proposition 3.1(From [11] For 0 1 and = 0, the dynamics of a one spike solution to (1.1) is ....

[Article contains additional citation context not shown here]

NAG Fortran library Mark 17, routine D03PCF, Numerical Algorithms Group Ltd., Oxford, United Kingdom (1995).

....is the default provided for our CRAY YMP8 832) and a user is calling one of these libsci routines, parallelism is introduced automatically. The strategic direction is obvious: Routines from libsci are frequently called from user codes and even more often from third party libraries like NAG ([16]) and IMSL ( 12] 13] therefore, there are a reasonable number of parallel programs in our production environment all the time, hopefully causing only negligible overhead. If memory limits will prevent the machine from being fully utilized, for instance, these programs may use multiple CPUs, ....

....party libraries are pro moted: NAG and IMSL. Both libraries have their strong points, therefore they cannot be replaced by each other. Whenever a user is interested to run his code on different computer systems, he has to rely on third party libraries. 2.3. 1 NAG library The NAG Fortran Library ([16]) is a comprehensive collection of Fortran 77 routines for the solution of numerical and statistical problems. NAG Mark 15 was tested. Compared to the former version NAG Mark 14 a new chapter containing LAPACK routines [1] for solving linear equations was included. For a number of existing ....

NAG Fortran library manual, Mark 15, The Numerical Algorithms Group Ltd., Oxford, 1991.

....all possible cases of 0, fl; fl 0 2 R. 2 In order to determine local minima of F , we applied the NAG routine E04UPF with the nonlinear constraint 0 G( 0 ; fl; fl 0 ) and the bounds 0 , Gamma fl 0 . E04UPF is an implementation of the sequential quadratic programming algorithm (see [NAG]) As the reliability of the routine increases with the availability of the partial derivatives of f j and G, we set G= 0 = 1 and G= g( fl; fl 0 ) The approximation of the derivatives of G with respect to ; fl; fl 0 by finite differences was left to E04UPF. The FORTRAN code 17 0 200 ....

NAG Fortran Library Manual, Mark 14, The Numerical Algorithms Group Ltd, Oxford 1990. 19

....four range blocks R k , hence 8 fractal parameters (four scaling and four o set values) As in [9] for each test image we rst used collage coding to determine a fractal code p c that minimizes the collage error. We then used this code as a starting point for a gradient descent method. The NAG [23] subroutine E04DKF, which performs a quasi Newton conjugate gradient minimization, was used. It was also desirable to compare these results with the non gradient calculations of [9] However, since some of our collage error results di ered from those of [9] we have independently carried out ....

NAG Fortran Library, The Numerical Algorithms Group Ltd, Oxford, UK.

....solution for (1.1) Notice that x 0 0 as t 1 for any initial condition x 0 (0) x 0 0 2 ( 1; 1) To validate (3.16) we compared it with the corresponding full numerical result computed from (1. 1) for the parameter set (p; q; r; s) 2; 1; 2; 0) using the routine D03PCF from the NAG library [8]. The initial condition was taken to be of the form (3.14a) and (3.14b) with x 0 (0) 0:6 and = 03, 1:0, and D = 1:0. In Fig. 2 and in Table 1 we show the favorable comparison between the numerical result for x 0 with the corresponding asymptotic result obtained by solving the di erential ....

NAG Fortran library Mark 17, routine D03PCF, Numerical Algorithms Group Ltd. Oxford, United Kingdom (1995).

....(4.31) when N = j = 1. To verify (5.10c) for the parameter set (p; q; r; s) 2; 1; 2; 0) we compared the asymptotic result (5.10c) for x 0 (t) with the corresponding full numerical result computed from (1.4) The problem (1. 4) was solved numerically using the routine D03PCF from the NAG library [11]. The initial condition was taken to be of the form (5.10a) and (5.10b) with x 0 (0) 0:6 and = 03, 1:0, and D = 1:0. An interpolation scheme was then used to locate the position of the maximum of a on the computational grid. In Fig. 3 and in Table 1 we compare this numerical result for x ....

NAG Fortran library Mark 17, routine D03PCF, Numerical Algorithms Group Ltd. Oxford, United Kingdom (1995).

....an 8 Theta 8 pixel image with four range blocks R k , hence 8 fractal parameters. As in [6] for each test image we first used collage coding to determine the fractal code col that minimizes the collage error. We then used this code as a starting point for a gradient descent method. The NAG [13] subroutine E04DKF, which performs a quasi Newton conjugate gradient minimization, was used. It was also desirable to compare these results with the non gradient calculations of [6] However, since some of our collage error results differed from those of [6] we have independently carried out ....

NAG Fortran Library, The Numerical Algorithms Group Ltd, Oxford, UK.

....of the global error. Hence, whatever parallelism is employed, a global, shared data structure must be used: all participating processors must be able to access and update this data structure. D01DAF and D01AUF are available in both the NAG Fortran 77 Library [3] and NAG Numerical PVM Library [4], a parallel library for distributed memory machines. In the latter, parallelism is achieved via a master slave paradigm, where global data structures are processed by the master , a specified processor, which assigns and distributes parcels of work to the slaves , all other processors. The ....

NAG Numerical PVM Library Manual, Release 1. The Numerical Algorithms Group Ltd, Oxford, 1995.

....a number of guidelines to be adhered to at all times in the course of this work. ffl All codes were written in Fortran 77. ffl All codes were encapsulated in user callable routines of identical interface to the existing corresponding serial versions, as available in the NAG Fortran Library [3]. ffl All parallelism was entirely contained within these routines. ffl All optimization of Fortran 77 code was carried out by using software tools: under no circumstances was any generated assembler or machine code handcrafted to achieve better performance. 5 ffl All explicit parallelism, ....

....and D01AUF assess convergence in terms of the global error. Hence, whatever parallelism is employed, a global, shared data structure must be used: all participating processors must be able to access and update this data structure. D01DAF and D01AUF are available in both the NAG Fortran 77 Library [3] and NAG Numerical PVM Library [4] a parallel library for distributed memory machines. In the latter, parallelism is achieved via a master slave paradigm, where global data structures are processed by the master , a specified processor, which assigns and distributes parcels of work to the ....

NAG Fortran Library Manual, Mark 17. The Numerical Algorithms Group Ltd, Oxford, 1995. 19

....using MPI. Other areas covered by existing parallel libraries include optimization and PDE solvers. All of the libraries mentioned above are available from Netlib. Commercial products are provided by many of the machine vendors and NAG provides a commercial, supported, general Parallel Library [12] based on MPI and also an SMP library [13,14] based on OpenMP. The libraries that have been mentioned so far all have a traditional library interface. One of the packages that offer parallel functionality in the setting of an environment is PETSc [15,16] PETSc provides an object based interface ....

The NAG Parallel Library Manual, Release 2, Numerical Algorithms Group Ltd, Oxford (1997).

....is the default provided for our CRAY YMP8 832) and a user is calling one of these libsci routines, parallelism is introduced automatically. The strategic direction is obvious: Routines from libsci are frequently called from user codes and even more often from third party libraries like NAG ([16]) and IMSL ( 12] 13] therefore, there are a reasonable number of parallel programs in our production environment all the time, hopefully causing only negligible overhead. If memory limits will prevent the machine from being fully utilized, for instance, these programs may use multiple CPUs, ....

....party libraries are promoted: NAG and IMSL. Both libraries have their strong points, therefore they cannot be replaced by each other. Whenever a user is interested to run his code on different computer systems, he has to rely on third party libraries. 2.3. 1 NAG library The NAG Fortran Library ([16]) is a comprehensive collection of Fortran 77 routines for the solution of numerical and statistical problems. NAG Mark 15 was tested. Compared to the former version NAG Mark 14 a new chapter containing LAPACK routines [1] for solving linear equations was included. For a number of existing ....

NAG Fortran library manual, Mark 15, The Numerical Algorithms Group Ltd., Oxford, 1991.

.... the work which has been carried out at NAG Ltd to update the iterative solvers for sparse systems of linear equations, both symmetric and unsymmetric, in the NAG Fortran 77 Library [7] Our current plans to extend this work and include it in our other numerical libraries (Fortran 90 [8] C [9], Numerical PVM [10] in our range are also briefly mentioned. We have added to the Library the new Chapter F11, entirely dedicated to sparse linear algebra. At Mark 17 (released in early 1996) the F11 Chapter includes sparse iterative solvers, preconditioners, utilities and black box routines ....

NAG (1994) NAG C Library Manual, Mark 3, The Numerical Algorithms Group Ltd, Oxford, UK.

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NAG Fortran library Mark 17, routine D03PCF, Numerical Algorithms Group Ltd., Oxford, United Kingdom (1995).

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NAG Fortran library Mark 17, routine D03PCF, Numerical Algorithms Group Ltd., Oxford, United Kingdom (1995).

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NAG Fortran library Mark 17, routine D03PCF, Numerical Algorithms Group Ltd. Oxford, United Kingdom (1995).

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NAg E04UCF. NAg Mark 14 Reference Manual. Numerical Algorithms Group Ltd., Oxford, 1990.

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NAG Fortran library Mark 17, routine D03PCF, Numerical Algorithms Group Ltd. Oxford, United Kingdom (1995).

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NAg E04UCF Nag Mark 14 Reference Manual. Numerical Algorithms Group Ltds., Oxgord,1990.

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NAG Fortran library Mark 17, routine D03PCF, Numerical Algorithms Group Ltd. Oxford, United Kingdom (1995).

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