D. Kahaner, C. Moler, and S. Nash. Numerical Methods and Software. Prentice Hall, Englewood Cliffs, NJ, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the fact that f i 1 (t) dt = f i 1 (t 0 x) f i (x) dx dt (4.21) f i 1 (t 0 x) f i (x) dt dx (4.22) F i 1 (a i 1 0 x) f i (x) dx: 4. 23) 111 These expressions can be evaluated using standard numerical integration techniques as long as f(x j H j ) is sufficiently smooth [93]. Since the test must terminate in exactly one of the K stages, the power function consists of K mutually exclusive events. Thus, Pr(accept H 1 at any stage j 2) i (2) 4:24) the false alarm rate for any hypothesized trajectory is ff HT (2) i (2) where i 2 H 0 ; 4:25) ....

C. Moler D. Kahaner and S. Nash. Numerical Methods and Software. Prentice Hall, 1989.

....MAPLE, for example, it is impossible to compute the two solutions where the curves in Fig. 5f are tangent, precisely because they lie inside a region of near singularitiness of D. On the other hand, it is well known that the numerical stability of polynomial root finding is often surprisingly low [15, 7] and that a very small perturbation in just a few coe#cients can yield solutions completely di#erent from the intended ones. Classic solutions to the forward kinematics problem that rely on solving a resultant polynomial must carefully deal with this issue, specially in configurations of the ....

D. Kahaner, C. Moler, S. Nash, Numerical Methods and Software, Prentice Hall, 1989.

....for example, it is impossible to compute the two solutions where the curves in Fig. 5f are tangent, precisely because they lie inside a region of near singularitiness of D. On the other hand, it is well known that the numerical stability of polynomial root finding is often surprisingly low [15] [7] and that a very small perturbation in just a few coefficients can yield solutions completely different from the intended ones. Classic solutions to the forward kinematics problem that rely on solving a resultant polynomial must carefully deal with this issue, specially in configurations of the ....

D. Kahaner, C. Moler, S. Nash, Numerical Methods and Software, Prentice Hall, 1989.

.... known that numerical error accumulation and propagation is an intractable problem for all but the simplest cases [3,34] Therefore, we opt to analyze numerical errors using statistical techniques [29,34] For integration, there is a wide spectrum of applicable techniques such as those presented in [16,32]. In our experiments, we use the relatively simple trapezoidal method. Some alternatives include the Gauss Kronrod and adaptive quadrature algorithms [16] While such techniques reduce calculation errors, they are significantly more computationally intensive. Also, their advantage is most ....

....errors using statistical techniques [29,34] For integration, there is a wide spectrum of applicable techniques such as those presented in [16,32] In our experiments, we use the relatively simple trapezoidal method. Some alternatives include the Gauss Kronrod and adaptive quadrature algorithms [16]. While such techniques reduce calculation errors, they are significantly more computationally intensive. Also, their advantage is most prevalent in cases where functions vary rapidly. Our experiments indicate that this is very rarely the case in calculations of exposure in wireless sensor ....

D. Kahaner, C. Moler and S. Nash, Numerical Methods and Software (Prentice Hall, Englewood Cliffs, NJ, 1989).

.... and propagation is an intractable problem for all but the simplest cases [Act90, Thi88] Therefore, we opt to analyze numerical errors using statistical techniques [Thi86, Ral78] For integration, there is a wide spectrum of applicable techniques such as those presented in [Sto72] and [Kah89]. In our experiments, we use the relatively simple trapezoidal methods. Some alternatives include the Gauss Kronrod and adaptive quadrature algorithms [Kah89] While such techniques reduce calculation errors, they are significantly more intensive computationally. Also, their advantage is most ....

....techniques [Thi86, Ral78] For integration, there is a wide spectrum of applicable techniques such as those presented in [Sto72] and [Kah89] In our experiments, we use the relatively simple trapezoidal methods. Some alternatives include the Gauss Kronrod and adaptive quadrature algorithms [Kah89]. While such techniques reduce calculation errors, they are significantly more intensive computationally. Also, their advantage is most prevalent in cases where functions vary rapidly. Our experiments indicate that this is very rarely the case in calculations of exposure in wireless sensor ....

D. Kahaner, C. Moler, S. Nash. Numerical Methods And Software. Englewood Cliffs, N.J., Prentice Hall, 1989.

....discovery problem as a system of nonlinear polynomial equations and apply one of the the standard numerical analysis techniques to solve it. There are a number of excellent textbooks that discuss techniques for solving systems of nonlinear equations within the context of numerical analysis [10, 20]. It is well known that the numerical stability of polynomial equations with errors, as in location discovery case, is often surprisingly low [10, 20] and that very small perturbations in very few coefficients can result in significantly different solution than the intended one. Furthermore, ....

.... are a number of excellent textbooks that discuss techniques for solving systems of nonlinear equations within the context of numerical analysis [10, 20] It is well known that the numerical stability of polynomial equations with errors, as in location discovery case, is often surprisingly low [10, 20], and that very small perturbations in very few coefficients can result in significantly different solution than the intended one. Furthermore, numerical errors often accumulate quickly [10] and invalidate the final result. 1 2 3 4 d 3 d 4 Figure 1: Multilateration Example However, the ....

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D. Kahaner, C. Moler, and S. Nash. Numerical methods and software. Prentice Hall, Englewood Cliffs, N.J., 1989.

.... Gamma a) n 1) although an adaptive mesh could also be used. The variables in the finite dimensional problem will be x i j x(t i ) The function x(t) will be approximated by a cubic spline s(t) that interpolates the values f x i g at the points f t i g. We use the representation described in [9]. On 62 stephen g. nash and ariela sofer the i th subinterval [x i ; x i 1 ] s(t) x i b1 (t) d i b2 (t) x i 1b3 (t) d i 1b4 (t) j 4 X k=1 ff i;k b k (t) where b1 (t) 2 h 3 (t Gamma t i 1 ) 2 (t Gamma t i h=2) b2 (t) 1 h 2 (t Gamma t i ) t Gamma t i 1 ) 2 b3 (t) ....

....that s(t j ) x j and s 0 (t j ) d j for j = i; i 1. The coefficients f d i g are uniquely determined by the requirements that the function s 00 (t) be continuous, together with two auxiliary conditions (usually involving the derivatives of s(t) at the endpoints of the interval [a; b] [9]. If we define x = x0 ; xn 1 ) T and d = d0 ; dn 1 ) T and use the auxiliary conditions d0 = x1 Gamma x0 h and dn 1 = xn 1 Gamma xn h ; then d can be determined by solving a linear system of the form Td = Sx; where T and S are tridiagonal matrices. With our choice of ....

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D.K. Kahaner, C. Moler, and S.G. Nash, Numerical Methods and Software, Prentice-Hall, Englewood Cliffs, New Jersey, 1989.

....C 002 C 011 C 002 C 022 C 002 C 222 C 022 C 222 C 011 C 112 C 222 C C 112 Fig. 1. Configuration Transition Pairs If we simply use the discontinuities along the edges of T to approximate the shape of the occluder in each of the configurations, some transitions can become ill conditioned (see [8]) in the sense that small changes in the location of the occluder can cause large changes in our approximation of the visible portion. The following transitions are 2 If the fully visible assumption about the C 000 configuration is violated, other transitions are possible. 5 potentially ....

....n iterations, we have evaluated D( f ) at most 2n times, and the expected error is O(1=2 n 1 ) Finally, a word of caution on the MF algorithm. Since MF relies on RSB, and since both are numerical algorithms, additional care must be exerted in controlling the error levels. As a rule of thumb [8], a nested numerical method should be roughly one order of magnitude more precise than the calling method. In our implementation, we have made the RSB routine (as used within MF) ten times as accurate as the MF routine. 8 approximation to occluded domain occluder (a) Linear boundary (b) ....

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Kahaner, D., Moler, C., Nash, S., Numerical Methods and Software, Prentice Hall, Englewood Cliffs, New Jersey, 1989.

....in 1873 and Jordan in 1874 for the case of square matrices. The technique was extended to rectangular matrices by Eckart and Young in the 1930s. Not until the 1960s, it has been used as a computational tool. The computation of the SVD requires a variety of sophisticated numerical techniques [18]. The idea of adapting the symmetric QR algorithm to compute the SVD first appeared in [10] and the first working code was reported in [4] The Algol code in [11] was the basis for SVD subroutines in LINPACK [6] In 1990, Demmel and Kahan [5] modified the LINPACK SVD with zero shift QR algorithm ....

D. Kahaner, C. Moler, S. Nash, Numerical Methods and Software, Prentice Hall, Englewood Cliffs, New Jersey (1989).

....quadrature algorithm takes as inputs a function f , an interval [a; b] and an accuracy request ffl and produces a result Q and an error estimate E. The algorithm recursively divides all subintervals until it meets the accuracy request ffl. A sequential, globally adaptive quadrature technique [71] saves computation time by starting integral evaluation from coarser subintervals and repeatedly dividing the subinterval with the largest local error estimate until the accuracy request is met. PEs Random D = 7 D = 6 D = 5 1 105.4 91.23 90.70 90.35 2 N Ay 59.02 53.96 52.37 4 118.2 33.09 27.91 ....

D. Kahaner, C. Moler, and S. Nash. Numerical Methods and Software. Prentice Hall, 1989.

....the problem. For some classes of systems (for example, systems with tight feedback) small amounts of parametric uncertainty do not affect the results. For systems with larger amounts of uncertainty or nonlinearities, however, the model device mismatch may be large. ffl Use Monte Carlo analysis [Kahaner et al. 1989]. By running repeated simulations of the ODE system using different combinations of parameter values, it is possible to get some idea of the behavior of the models in the model space. Unfortunately, in addition to being slow, this approach may miss certain key combinations of parameters that ....

....robustly handling noise. Guided by this observation, we will suggest certain enhancements that will enable us to reliably use MSQUID in the context of semiquantitative identification. 3.5. 1 Quality of the Estimation Function To determine the quality of the estimate, we ran both MSQUID and UNCMND [Kahaner et al. 1989], a fast, readily available, unconstrained optimizer, using the same neural network structure on the following series of test functions: ffl linear (y = 0:5x 5) on x 2 [0; 100] FLOW(CC SEC) LEVEL(CM) 0.00 5.00 10.00 15.00 20.00 25.00 30.00 0.00 2.00 4.00 6.00 8.00 Figure 3.8: 95 envelope for ....

David Kahaner, Cleve Moler, and Stephen Nash. Numerical Methods and Software. Prentice-Hall, Englewood Cliffs, 1989.

....and partial differential equations sometimes augmented by algebraic equations. Under transient conditions one gets mixed systems of differential and algebraic equations (DAEs) Numerical solutions of initial and boundary problems of ordinary differential equations are given in many books (see e.g. [30, 31, 33 35]) Numerical solutions of partial differential equations (PDEs) are e.g. given in books by Lapidus and Pinder [36] or Ames [37] A recent compilation of literature on DAEs is given by Unger et al. 38] Models for unit operations are scattered over many journals. Books that refer to special types ....

Kahaner, D., Moler, C. and Nash, S., 1989, Numerical Methods and Software. Prentice Hall, Englewood Cliffs, N.Y.

.... obtain the fundamental quantity # k 1 y n 1 as the limit of d (i) y (i) n 1 y (0) n 1 computed from d (i 1) d (i) # (i) y (i 1) n 1 = y (0) n 1 d (i 1) Many of the tactics adopted in the code resemble those found in the well known codes DIFSUB [17] DDRIV2 [24], LSODE [22] and VODE [7] In particular, local extrapolation is not done. The selection of the initial step size follows Curtis [10] who observes that by forming partial derivatives of F (t, y)att 0 , it is possible to estimate the optimal initial step size. In the context of Matlab it is ....

....hb3ex 766 690 9.9 19.79 17.77 10.2 vdpex 708 573 19.1 20.75 19.33 6.9 and for all problems, the code is faster when using the NDFs (an average of 8. 2 faster) To verify that the performance of ode15s is comparable to that of a modern BDF code, we have compared ode15s using the NDFs to DDRIV2 [24] on some relatively di#cult problems. DDRIV2 is an easy to use driver for a more complex code with an appearance not too di#erent from ode15s. It is a quasi constant step size implementation of the BDFs of orders 1 to 5 that computes answers at specified points by interpolation. It approximates ....

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D. KAHANER,C .MOLER, AND S. NASH, Numerical Methods and Software, Prentice-Hall, Englewood Cli#s, NJ, 1989.

....the evaluation of the expression for each input whereas the join continuation based implementation is able to pipeline the execution of evaluation of the expression for different inputs. Example: Adaptive Quadrature The adaptive quadrature technique computes the integral of a given function [8] by dividing the given interval into subintervals. It recursively divides subintervals in which the local estimated error is greater than the given error bound and uses smaller step size for those subintervals. The algorithm can be implemented using a manager worker structure [10] such that the ....

D. Kahaner, C. Moler, and S. Nash. Numerical Methods and Software. Prentice Hall, 1989.

....that the given interval of integration is divided into smaller subintervals and given to several workers. The workers perform the integration in their subintervals in parallel and return the values to the master where they are combined to obtained the final result. Adaptive integration techniques [12] can vary the step size used for integrating in a region based on the error present locally. Thus the work available for each worker may dynamically increase as the computation proceeds. We allow the workers to create newer workers, thereby keeping the amount of work allocated to each worker same ....

D. Kahaner, C. Moler, and S. Nash. Numerical Methods and Software. Prentice Hall, 1989.

....algorithm uses any where the inverse transform exists. For general inputs of , numerical software can estimate the invertibility of to caution the user if the transform is nearly singular. A singular matrix does not have an inverse. A condition number estimate could be calculated for the matrix [KAHA89], similar to what Matlab or Mathematica [WOLF88] uses for matrix calculations. The inverse is analytically solvable and typically stable for affine and orthogonal transforms without projections [FOLE90] To compare filters the number of linear interpolations required was given in (EQ 13) ....

D. Kahaner, C. Moler, S. Nash, Numerical Methods and Software. Prentice Hall, Englewood Cliffs, New Jersey 1989.

....the applicability of an algorithm, for example that a particular meshing algorithm is only valid for piecewise smooth 2 manifolds. And finally, reference books have been developed that precisely document the standard library algorithms in terms of the communal database of mathematical knowledge [48, 49, 61] . Indeed it is the elaborate knowledge base that characterizes the domain in which we work. The emergence of applied logic and formal methods has opened new opportunities for computers to help in more effectively employing the methodology of mathematics. We now know how to specify algorithms ....

D. Kahaner, C. Moler, and S. Nash. Numerical Methods and Software. Prentice Hall, Englewod Cliffs, NJ, 1989.

....14, 20] Application to Legacy Systems Applying reverse engineering to legacy systems is a growth business. According to [27] there are hundreds of billions of lines of source code in the world, and 70 of it is COBOL. Within the realm of scientific computing, the code is almost entirely FORTRAN [13]. It takes little imagination to speculate on the condition and style of the code, the presence and quality of documentation, and lack of corporate memory of the understanding how these codes actually work. In some cases, organizations consider software systems capital assets because they are ....

D. Kahaner, C. Moler, and S. Nash, "Numerical Methods and Software," Prentical Hall, Englewood Cliffs, NJ, 1989.

....values can be searched in the image. Conversely when given four corresponding pairs we can use these equations to solve for the unknown global parameters. When even this information is not available we use the integration constraint to solve a global optimization problem using numerical methods [15]. This method is demonstrated in our experiments. 5 Cylindrical Objects Before turning to the reconstruction of general bilaterally symmetric objects we will first consider the simpler case of cylindrical objects. Such objects are characterized by a surface z = f(x) For this class, neither the ....

D. Kahaner, C. Moler, and S. Nash. Numerical Methods and Software. Prentice Hall, Englewood Cliffs, NJ, 1989.

....such that the given interval is divided into subintervals with each interval passed to a worker. The workers perform the integration in their subintervals in parallel and return the values to the master where they are combined to obtained the final result. Adaptive integration techniques [8] vary the step size used for the integration in a region based on the local error estimate. Thus the work available for each worker may dynamically increase as the computation proceeds. We allow the workers to create new workers, thereby dynamically changing the number of workers and keeping ....

D. Kahaner, C. Moler, and S. Nash. Numerical Methods and Software. Prentice Hall, 1989.

....are accumulated through the loop of evaluating m components of the error norm, where each component corresponds to a curve sample point. 5 Optimization Process The algorithm we apply to minimize the objective function uses a secant method with BFGS updates to the approximated Hessian matrix [14][15]. The algorithm combines line search to ensure that it finds a point with a lower value in the objective function at every step. The rate of convergence of the secant method is super linear. We supply a subroutine to evaluate the objective function and the gradient, and we provide an initial ....

D. Kahaner, C. Moler & S. Nash, Numerical Methods and Software, Prentice Hall, New Jersey, 1989. a b c d

....function defined by [5] C(x) lim T 1 Z T GammaT A(t)B(x Gamma t) dt For convenient numerical calculation, the discrete convolution is typically used. The discrete convolution of two sequences A and B with equal length N is the set of numbers C i ; i = 0; 1; N Gamma 1; defined by [9] as C i = i X j=0 A j B i Gammaj The algorithm can be written as: for (i=0 to N 1) do C i = 0; for (j=0 to i) do C i = C i A j B i Gammaj ; Actually there are only two simple addressing sequences, sequential addressing and sequential with offset addressing, used for sequences A, B, and C in ....

D.Kahaner, C.Moler, and S.Nash, Numerical Methods and Software, Prentice Hall, Englewood Cliffs, N.J. 1989.

....valid.Theorems thatcharacterizetheapplicabilityofan algorithm,for example thata particularmeshing algorithmisonlyvalidforpiecewisesmooth 2 manifolds. And finally,referencebookshavebeendevelopedthatpreciselydocument thestandardlibrary algorithmsintermsofthecommunal databaseofmathematicalknowledge[34,35,40].Indeed itistheelaborateknowledgebasethatcharacterizesthedomain inwhich we work. The emergence of appliedlogicand formalmethods has opened new opportunitiesfor computerstohelpinmore effectivelyemployingthemethodologyofmathematics.We now know how tospecifyalgorithmsrigorouslyand how ....

D.Kahaner,C.Moler,and S.Nash.Numerical Methods and Software .PrenticeHall,EnglewodCliffs, NJ,1989.

....possible way of searching for a minimum is to sample the search space. Stochastic methods (such as simulated annealing) may also be applied to minimize the non smooth distance function (such as [3] These methods are usually computationally expensive. We use the search by golden section method [14] to find the minimum along the rotation dimension in the distance function. We choose this method because of its efficiency in terms of required function evaluations. The search by the golden section method is stated as follows. Assume that the global minimum is enclosed in an interval [ 1 ; 2 ....

D. Kahaner, C. Moler, and S. Nash. Numerical Methods and Software. Prentice Hall, Englewood Cliffs, NJ, 1989.

....in 1873 and Jordan in 1874 for the case of square matrices. The technique was extended to rectangular matrices by Eckart and Young in the 1930s. Not until the 1960s, it has been used as a computational tool. The computation of the SVD requires a variety of sophisticated numerical techniques [18]. The idea of adapting the symmetric QR algorithm to compute the SVD first appeared in [10] and the first working code was reported in [4] The Algol code in [11] was the basis for SVD subroutines in LINPACK [6] In 1990, Demmel and Kahan [5] modified the LINPACK SVD with zero shift QR algorithm ....

Kahaner, D., Moler, C., Nash, S. (1989): Numerical Methods and Software. Prentice Hall, Englewood Cliffs, New Jersey

....the residual vector, r = b Gamma Ax, possesses a minimal squared Euclidean norm r T r. Qualitatively, this solution is a close fit to the experimental data, and tends to smooths out the effects of errors. To compute least squares solutions, we use the standard technique of QR Factorization [20]. Unfortunately, we cannot simply apply QR Factorization to the full system Ax b to obtain a high quality solution. This is because the system is poorly conditioned due to being poorly scaled: the coefficients of t b span a wide range of magnitudes. In addition, the system contains errors from ....

D. Kahaner, C. Moler, and S. Nash. Numerical Methods and Software. Prentice Hall, 1989.

....in curvature at the knot points. Cubic polynomials that have a continuous first and secondderivative at each knot point could be fitted to the series of N points. Solving for the cubic polynomials requires inverting an NxN matrix. The matrix is tridiagonal so the inversion is straight forward [5], though computationally time consuming. If the navigation data is noisy, B splines [5] could be used because they are not forced to pass through each knot point as are the cubicpolynomials. Computing the B splines also involves inverting a banded matrix. The added complexity of fitting splines to ....

....secondderivative at each knot point could be fitted to the series of N points. Solving for the cubic polynomials requires inverting an NxN matrix. The matrix is tridiagonal so the inversion is straight forward [5] though computationally time consuming. If the navigation data is noisy, B splines [5] could be used because they are not forced to pass through each knot point as are the cubicpolynomials. Computing the B splines also involves inverting a banded matrix. The added complexity of fitting splines to the path points should be undertaken only if the selected steering method requires a ....

D. Kahaner, C. Moler, and S. Nash, Numerical Methods and Software, New Jersey; Prentice Hall, 1989, ch. 4.

....relies on numerical techniques (Newton Raphson iteration method) to provide the solution for a cycle of constraints. However, in order to provide a more efficient solving, the algorithm tries to reduce the size of the Jacobian matrix which is decomposed during each iteration of Newton Raphson [Kahaner et al. 89] First, the algorithm traverses the ancestor subgraph and eliminates those constraints which fix the value of a variable (FixValue constraint, c i : v = ff; ff 2 ) and keeps their attached numbers. These constraints are represented as smaller dashed ellipses in Fig. 12. The number attached to ....

Kahaner, D., Moler, C., Nash. S., 1989 - Numerical Methods and Software. Prentice Hall, Englewood Cliffs, NJ.

....Inc. Computing environments like MATLAB, Maple, and Mathematica contain special libraries that we ll discuss more in a moment. Some quality software is available by means of books. Indeed, a couple come with small libraries that include a quality code for IVPs, viz. Kahaner, Moler, and Nash [28] includes SDRIV2 and Forsythe, Malcolm, and Moler [16] includes RKF45. As might be expected, specialized texts on the numerical solution of IVPs such as Hairer, N rsett, and Wanner [21, 22] and Shampine and Gordon [37] include codes, but even the introductory text on ODEs by Sanchez, Allen, and ....

D. Kahaner, C.B. Moler, and S. Nash, Numerical Methods and Software, Prentice-Hall, Englewood Cliffs, NJ, 1989.

....and reconstruction algorithm was implemented in a global illumination testbed. The environment is built in C , including a set of classes that implement sampling and reconstruction. 3.6. 1 Numerical Issues To find the maximum and the critical points, Brent s minimisation algorithm was used [36]. For a given interval , it is first determined whether is in the interval, by examining the values of the derivative at a and b. If the maximum exists, it is found by maximising the illumination function. It must be noted that this expense can be avoided when the maximum can be found ....

Kahaner David, Cleve Moler and Stephen Nash, "Numerical Methods and Software", Prentice Hall Series in Computational Mathematics, New Jersey, 1989. 186

....body, these dynamic equations are predetermined and calculated off line, symbolically using a program based on Kane s formulation[8] These equations are integrated to obtain position and velocity information. We have used an AdamsBashforth adaptive step size predictor corrector ODE integrator [7]. Figure 5: Graphical User Interface We have also developed a graphical user interface (GUI) to make our simulator more accessible to the user. Figure 5 shows our GUI that controls camera views, collision detection, movie record playback, and simulation. 7 Results We have developed a modeling ....

D. Kahaner, et. al., "Numerical Methods and Software, " Prentice Hall, Englewood Cliffs, NJ, 1989.

....of the mesh are indicative of the condition number of the matrix used in the finite element method. The condition number is a measure of how close the matrix is to being singular. The condition number is the reciprocal of the relative distance from the matrix to the set of singular matrices [12]. Matrices with large condition numbers are said to be ill conditioned [13] The ideal problem conditioning occurs for orthogonal matrices that have (A) 1, while an ill conditioned matrix will have (A) AE 1. When one inverts a matrix that has a very large condition number (i.e. solves the ....

....one of several norms, including the 1 norm, 2 norm, and 1 Gamma norm. The 1 Gamma norm is defined as kAk1= max 1in n X j=1 j a ij j : 2) The size of the condition number is an inherent property of the matrix and is proportional to the size of the matrix or the number of degrees of freedom [12]. Additionally, the number of degrees of freedom is proportional to the mesh refinement, which is based on the volume of the elements [14] While a quantitative relationship between the Figure 4: Interface for the interactive histogram. The vertical bars are movable and allow the user to specify ....

D. Kahaner, C. Moler, and S. Nash, Numerical Methods and Software, Prentice Hall, 1989.

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D. Kahaner, C. Moler, and S. Nash. Numerical Methods and Software. Prentice Hall, Englewood Cliffs, NJ, 1989.

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D. Kahaner, C. Moler, and S. Nash, Numerical Methods and Software, Prentice-Hall, 1989.

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D. Kahaner, C. Moler, and S. Nash, Numerical Methods and Software, Prentice Hall, 1989.

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Kahaner, D. , C. Moler, and S. Nash, Numerical Methods and Software, Prentice-Hall, New Jersey, 1989.

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D. Kahaner, et. al., "Numerical Methods and Software, " Prentice Hall, Englewood Cliffs, NJ, 1989.

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D. Kahaner, C. Moler and S. Nash, Numerical Methods and Software, Prentice Hall, Englewood Cliffs, May 1989.

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D. Kahaner, C. Moler & S. Nash, Numerical methods and Software, Prentice-Hall, 1989.

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D. Kahaner, C. Moler and S. Nash, Numerical Methods and Software, Prentice-Hall, Englewood Cliffs, NJ, 1989.

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D. Kahaner, C. Moler, and S. Nash, Numerical Methods and Software, Prentice Hall, Englewood Cliffs, NJ, 1989.

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D. Kahaner, Numerical Methods and Software, Prentice Hall, Englewood Cliffs, NJ, 1988.

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D. Kahaner, C. Moler, and S. Nash, Numerical Methods and Software, Prentice Hall, New Jersey, 1989.

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D. Kahaner, C. Moler, and S. Nash, Numerical Methods and Software, Prentice-Hall, 1989.

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