Press, W. H., B. P. Flannery, S. A., Teukolsky, and W. T. Vetterling, 1986: Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, 818 pp.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....cache blocks. Section 3, however, discusses how a conservative marking strategy can be used when the block size is larger than a single word. Using the trace driven simulation methodology described in Section 4, Section 5 presents the performance of a few programs from the Numerical Recipes [32] that are hand marked following the compiler algorithm described in the paper. Additionally, trace level marking is performed on memory traces from the Perfect Club benchmark programs [3] to give an optimistic estimate of the performance of this coherence strategy for programs that are too large ....

....the fork synchronization signal sent by p 0 . After executing the tasks in a parallel region, all processors synchronize at the join point. Processor p 0 then continues to execute the code in the next sequential region. Figure 3 shows an example of a fork join program taken from Numerical Recipes [32]. 13 jmax = nr = nits = 100 nr DO kits = 1, nits DO n = 1, nr DO j = 2, jmax 1 uu(1) s1) DO l = 2, jmax 1 uu(l) uu(l 1) s2) ENDDO DOALL l = 2, jmax 1 (s3) uu(l 1) s4) psi(j,l) psi(j,l) uu(l 1) s5) ENDDOALL ENDDO ENDDO ENDDO ....

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W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. Numerical Recipes: The Art of Scientific Computing (Fortran Version). Cambridge University Press, 1989.

....allows for straightforward, accurate determination of the Hellmann Feynman forces [17] which can be used for ab initio relaxation of the ionic positions to the lowest energy configuration. This ionic relaxation was carried out with a BFGS (Broyden Fletcher Goldfarb Shanno) Hessian update scheme [18], found to give convergence in a significantly smaller number of steps than conventional conjugate gradients. The relaxation continued until all the residual forces on ions were less than 0.1 eV A. The grain boundary structure was relaxed using a supercell having the form shown in Fig. 1(a) It ....

W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, (1989).

....equation components. 3.1 Discretization of the Integral Term Straightforward discretization of equation (2. 5) could use standard numerical discretization methods [26, 34, 35] for the di#erential operator combined with numerical integration methods such as Simpson s rule or Gaussian quadrature [24] for the integral term. However, this straightforward approach is computationally expensive [26] Instead we transform the integral in equation (2.5) into a correlation integral. This allows e#cient Fast Fourier Transform (FFT) methods to be used to evaluate the integral for all values of S. ....

W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling. Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, Cambridge (UK) and New York, 2nd edition, 1992.

....different eigenvector responses, i.e. temporal responses of spatially decorrelated channels. Fig. 4 shows the 20 eigenvectors (time series) with the largest eigenvalues (i.e. the greatest signal powers) The eigenvectors and eigenvalues are solved using the singular value decomposition method [12]. The eigenvector in the top left graph (R 1 ) has the largest eigenvalue and the eigenvalues decrease from left to right and top to bottom. In fact, the top twenty traces account for 70 of the signal variance. The lowest traces are mostly experimental noise. The noise variance is estimated to be ....

Press WH, Teukolsky SA, Vetterling WT, Flannery BP, 1992. Numerical Recipes: The Art of Scientific Computing, Second Edition, Cambridge Press, Cambridge)

....the centres of mass of different molecules was calculated and used as the structural feature to be reproduced. All potential parameters of this section have been optimised automatically by minimising penalty functions of the kind of Eq. 3.1. This was done using a standard amoeba simplex method. [24, 25] For details of the procedure, see the respective references for every system. Figure 6 shows the success of various approximations. It is possible to reproduce the centre of mass RDF with a 3 site model, where one super atom is placed on the carbonate group of DPC, the other two on the phenyl ....

W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetter, Numerical Recipes - The Art of Scientific Computing, Cambridge University Press, Cambridge 1986.

....of the solutions in order to attach an uncertainty to the result. In order to predict the variance, we examine the shape of the ssd surface around the minimum point. A sharp or shallow depression corresponds to a low or high variance, respectively. We follow the recipe in Numerical Recipes[8] for predicting the variance of our results based on the ssd surface. We designate (p, q ) as the location of the minimum. The Hessian of the ssd surface at this point is [O q ssd (p , q ) qSSd(p , q ) J (5) If the errors in the power spectra are normally distributed, then the covariance ....

Press, William H., Brian P. Flannery, Saul A. Teukolsky, and William T. Vetterling. Numerical Recipes: The Art of Scientific Computing, First Edition, Cambridge Univer- sity Press, 1986.

....calculations is IEEE double precision, like in MATLAB. However, even using this, the expression (4.1.4.3.6) is numerically ine#cient, and imprecise. Instead of the calculation of x 0 = D i D i ) 1 (D i y) 1) one needs to use rather matrix factorization algorithms to solve y = D i x i . [7] The result is theoretically the same, however, in extreme cases the explicit solution may give erroneous results while the numerical solution still works. 2.3 Stop criterion An iterative algorithm needs to perform a finite number of iterations. The problem is in general that the number of ....

Press, W. H., B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, Cambridge, 1986.

....technique which appears to be simpler and more natural than some elaborate multistep schemes (c.f. e.g. 5, 6] 1. 1 Background Numerical analysts have demonstrated the benefits of spatially (and temporally) adaptive approximation schemes for the solution of differential equations (see, e.g. [7] 70 Published in the Proceedings of the IEEE Computer Vision and Pattern Recognition Conference (CVPR 91) Lahaina, HI, June, 1991, 70 75. Sec. 16.5) The primary motivation for our approach comes from the field of numerical grid generation which has evolved powerful tools for solving field ....

....for each node i and numerically integrate the equations of motion forward though time until the mesh stabilizes: # # # # # # #. At each time step t we evaluate the current nodal forces and accelerations, the new velocities, and the new positions using the explicit Euler time integration procedure [7] # # # # # # # # ##t # # ##t# # ; 4) for i ##; N and t ##;#t;##t; A convenient way to compute the net nodal forces is to think of the # variables as nodal force accumulators. Each spring in the mesh makes the appropriate force contributions into the variables ....

W. H. Press, B. P. Fannery, S. A. Teukolsky, and W. T. Vetterling. Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, Cambridge, UK, 1986.

....but focuses instead on the technical aspects of finding non negative matrix factorizations. Of course, other types of matrix factorizations have been extensively studied in numerical linear algebra, but the nonnegativity constraint makes much of this previous work inapplicable to the present case [8]. Here we discuss two algorithms for NMF based on iterative updates of W and H . Because these algorithms are easy to implement and their convergence properties are guaranteed, we have found them very useful in practical applications. Other algorithms may possibly be more efficient in overall ....

....be applied to find local minima. Gradient descent is perhaps the simplest technique to implement, but convergence can be slow. Other methods such as conjugate gradient have faster convergence, at least in the vicinity of local minima, but are more complicated to implement than gradient descent [8]. The convergence of gradient based methods also have the disadvantage of being very sensitive to the choice of step size, which can be very inconvenient for large applications. 4 Multiplicative update rules We have found that the following multiplicative update rules are a good compromise ....

Press, WH, Teukolsky, SA, Vetterling, WT & Flannery, BP (1993). Numerical recipes: the art of scientific computing. (Cambridge University Press, Cambridge, England).

....variable. These settings will be used throughout in this chapter to clearly illustrate how the GA code process. While using these operators, the GA generates random numbers, which is the essence of the method. In this application, random numbers were generated using 80 Knuth s subtractive method [36]. Knuth s algorithm is regarded as one of the best random number generators. In this thesis, as mentioned before, each variable set defines a cascade configuration with a corresponding chromosome. The real value of each design variable is expressed as a string of binary digits, which is called as ....

Press W. H., Flannery B. P., Teukolsky S. A., and Vetterling W. T., Numerical Recipes: The Art of Scientific Computing (FORTRAN Version), Cambridge Univ. Press, Cambridge, 1989. 135

....N 2 equations turn out to have the form: 1) ln(D 0 ) 2) ln(D 1 ) 1 1) ln(D 2 ) for i = 3 . N (13) Inserting these N 2 equations into (7) 8) and (10) we can numerically solve for D 0 , D 1 , and D 2 using the Newton Raphson method for nonlinear systems of equations [22]. The maximal entropy distribution, thus defined, depends on only two parameters, the firing probability f 1 , and the pair wise correlation #. Figure 2 illustrates the shape of the N cluster distribution with maximal entropy for a network of 150 neurons, and its dependence on f 1 and #. Figure 2 ....

W.H. Press, B.P. Flanney, S.A. Teukolsky, and W.T. Vetterling, editors. Numerical Recipes: the art of scientific computing. Cambridge University Press, Cambridge, 1986.

.... b a) n) let loop ( i 1) sum ( 2) let (if ( f a) f b) x ( a ( i ( i n) loop ( i 1) sum (f x) sum h) We use this to estimate r: define (f x) 4 ( 1 ( x x) define pi estimator (trapezoid f 0 1) It is shown in standard texts (for example [2] or [3]) that, for f in U2[a,b] as we ve assumed, the truncation error involves only even powers of h. Hence we proceed: define (pi estimator sequence n) cons stream (pi estimator n) pi estimator sequence ( 2 n) 15 (print stream (richardson sequence (pi estimator sequence 10) 2 2) ....

W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling. Numerical Recipes: The Art of Scientific Computing. Canbridge University Press, 1986.

....allowable error yields set point heat outputs U ui, u2, minimizing the deviation from T and ensuring that it is less than . Control parameter design requires simultaneous optimization of many parame ters (the heat source values) While algorithms for multi parameter optimization exist [28], they are computationally expensive for large problems and difficult to parallelize for distributed applications. This section demonstrates that structural knowledge, in the form of the influence graph, significantly improves the performance of a basic decentralized optimization algorithm. A ....

....closed form analytical solutions are often impossible, engineers typically use techniques such as finite differences and finite elements [19] to represent a system s governing partial differential equations in terms of matrices on an appropriate discretization. They then apply iterative algorithms [28] to solve the resulting sets of equations. Advanced techniques such as domain decomposition [4] and multigrid methods [5] achieve additional efficiency in convergence or parallelizability of computation. Our approach differs from these traditional techniques in a number of ways. We provide ....

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes: the Art of Scientific Computing, Cambridge University Press, 1986.

....calculations is IEEE double precision, like in MATLAB. However, even using this, the expression (4.1.4.3.6) is numerically inefficient, and imprecise. Instead of the calculation of ( y D D D x i i T i 1 = one needs to use rather matrix factorization algorithms to solve i i x D y = [10] The result is theoretically the same, however, in extreme cases the explicit solution may give erroneous results while the numerical solution still works. Stop criterion An iterative algorithm needs to perform a finite number of iterations. The problem is in general that the number of necessary ....

Press, W. H., B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, Cambridge, 1986.

....5=2 (L OE z;t L OE z;r r Gamma1 L OE t ) 2 Z r 1=2 e 4 ;r dt (68) W = r Gamma1=2 e 2 ;r dt (69) 5. NUMERICAL ISSUES In differencing the hyperbolic equations, i. e (8) 10) 32) and (33) we use an explicit, second order (in space and time) leapfrog technique [16]. The grids have a regular spacing of Deltar in the Cauchy region and Deltay on the characteristic side. For convenience the interface is set at a radial position of r I = y I = 1 (see figure 3) The coordinate timesteps in the two regions can be related using (24) Thus u n 1 = u n ....

Press W. H., Flannery B. P., Teukolsky S. A., Vetterling W. T., Numerical Recipes: The Art of Scientific Computing, (Cambridge University Press, 1986).

....spectral aliasing effects. The high order spectral method [13] for hydrodynamic evolution of a surface is an equivalent hydrodynamic technique based on a similar expansion, but uses a different method to avoid aliasing problems. A fourth order Adams Bashforth predictor corrector algorithm [14] is used to time step (1) 2) and is initialized again with a realization of a Pierson Moskowitz spectrum which is assumed to be propagating linearly at previous time steps according to (3) 5) To avoid introducing discontinuities into the time evolution, an initial ramp up period is ....

....CROSS SECTIONS Fig. 3. Comparison of Doppler spectra for 0 ffi incidence (a) VV and (b) HH. primarily at peaks and troughs of waves, with no horizontal shift in the profile as expected. Normalized spectra in plots (b) and (d) were estimated using a 256 point periodogram method as described in [14] and show somewhat larger deviations from linear results with the West model; the vertical lines in these plots indicate the Bragg wavenumbers at 40 and 80 incidence. Statistics of the set of 82 final profiles can also be compared for the three models. Standard deviations averaged over all 82 ....

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 2nd ed. New York: Cambridge Univ. Press, 1992.

....density functions, which may be represented as analytic functions or as parametric arrays of sample values. B. Calculation of Element Matrices The integral expressions for the mass, damping, and stiffness matrices associated with each element are evaluated numerically using Gaussian quadrature [27]. We shall explain the computation of the element mass matrix; the computation of the damping and stiffness matrices follow suit. Assuming the parametric domain of the element is Omega Gamma the expression for entry m ij of the mass matrix takes the integral form m ij = Z Omega (u; v)f ij ....

....to (16) f ij (u; v) j i j j : Here, the j i and j j are the columns of the Jacobian matrix for the D NURBS surface element. Given integers N g , we can find Gauss weights a g , and abscissas u g , v g in the two parametric directions of Omega such that m ij can be approximated by [27] m ij Ng X g=1 a g (u g ; v g )f ij (u g ; v g ) 8 Element DOFs Element DOFs Element DOFs Element Structure Element Structure Element Structure Local Geometric Data Local Geometric Data Local Geometric Data Element Matrices Physical Quantities Element Matrices Physical Quantities Element ....

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W. Press, B. Flanney, S. Teukolsky, and W. Verttering. Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, Cambridge, 1986.

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Press, W. H., B. P. Flannery, S. A., Teukolsky, and W. T. Vetterling, 1986: Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, 818 pp.

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Press, W. H., Teukolsky, S. A., Vetterling, W. T. & Flannery, B. P. 1992 Numerical recipes: the art of scientific computing , pp. 747--752. Cambridge University Press.

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Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T. (1986), Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, Cambridge, UK).

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W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, 1986.

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Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T. 1989 Numerical recipes: the art of scientific computing. Cambridge University Press.

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W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, 1986.

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W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. Numerical recipes: the art of scientific computing. Cambridge University Press, 1989.

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W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, Numerical Recipes: The Art of Scientific Computing. Cambridge, U.K.: Cambridge Univ. Press, 1986.

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W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling. Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, Cambridge, UK, second edition, 1992.

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W. H. Press et al., Numerical Recipes: The Art of Scientific Computing. New York: Cambridge Univ. Press, 1986.

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Press,W.H.etal.(1992) Numerical Recipes: The Art of Scientific Computing. Cambridge University Press.

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) Press, W.H., et. al., Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, 1988.

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W. H. Press, B. P. Flannery, S. A. Teukolksy, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, 1989.

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W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, Cambridge (UK) and New York, 2nd edition, 1992.

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Press, W., Flannery, B., Teukolsky, S., Vetterling, W.: Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, Cambridge (UK) and New York (1992)

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W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing, 1st ed. Cambridge (UK) and New York: Cambridge University Press, 1986.

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W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling. Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, Cambridge (UK) and New York, 2nd edition, 1992.

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W. H. Press, B. P. Flannery, S. A. Teukolsky andW. T. Vetterling, Numerical Recipes: The Art of Scientific Computing, Cambidge Unviersity Press, 1986.

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W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, Cambridge (UK) and New York, 2nd edition, 1992.

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W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing, 1st Edition, Cambridge University Press, Cambridge (UK) and New York, 1986.

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W. Press, B. Flannery, S. Teukolski, and W. Vettering. Numerical recipes : the art of scientific computing, chapter 13, Fourier and Spectral Applications, pages 572--575. Cambridge University Press, Cambridge, 1988.

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W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes---The Art of Scientific Computing. Cambridge, U.K.: Cambridge Univ. Press, 1986, p. 197.

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W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling. Numerical Recipes: the Art of Scientific Computing. Cambridge University Press, 1986.

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W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterlin, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, 1986.

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W. H. PRESS, B. P. FLANNERY, S. A. TEUKOLSKY, AND W. T. VETTERLING, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, New York, 1986.

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W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery. Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, different years.

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W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. Numerical Recipes: The Art of Scientific Computing, chapter 10.9, pages 444--455. Cambridge University Press, Cambridge (UK) and New York, 2nd edition, 1992.

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W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, Numerical Recipes---The Art of Scientific Computing, Cambridge University Press, Cambridge 1986.

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W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery. Numerical Recipes: The art of scientific computing. Cambridge University Press, different years.

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W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, "Numerical Recipes: The Art of Scientific Computing," Cambridge University Press, Cambridge, 1986.

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Press W H, Flannery B P, Teukolsky S A, and Vetterling W T Numerical Recipes: The Art of Scientific Computing. Cambridge University Press,1986.

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W. Press et al, Numerical Recipes - the art of scientific computing, (Cambridge University Press, Cambridge, 1992).

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W. H. Press, B. P. Flannery, S. A. Teukolski, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing , Cambridge University Press, Cambridge, 1989.

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