Numerical Analysis Using Sage
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Otherwise Sage informs that the matrix must be nonsingular in order to compute the inverse. Sage: A. Sage provides several decompo- sition methods related to solving systems of linear equations e. LU, QR, Cholesky and singular value decomposition and decompositions based on eigenvalues and related concepts e. Schur decomposition, Jordan form.
The availability of these functions depends on the base ring of matrix; for numerical results the use of real double field RDF or complex double field CDF is required. Let us determine the LU decomposition of the matrix A. The result of the function A. In the next example we study the influence of the condition number on the accuracy of the numerical solution of a random matrix with a prescribed condition number.
We see that the condition number appears to depend on cond Ac almost in a linear way. The output of the program used to study the influence of the condition number on the accuracy of the numerical solution of a random matrix with a prescribed condition number. Numerical integration. Numerical integration methods can prove useful if the integrand is known only at certain points or the an- tiderivate is very difficult or even impossible to find.
In education, calculating numerical methods by hand may be useful in some cases, but computer programs are usually better suited in finding patterns and comparing different methods.
The differences between the exact value of integration and the approximation are tabulated by the number of subintervals n Fig 3. The table shows the difference between the exact value of the integral and the approximation using various rules. There are also built-in methods for numerical integration in Sage.
For instance, it is possible to automatically produce piecewise-defined line functions defined by the trapezoidal rule or the midpoint rule. These functions can be used to visualize different geometric interpre- tations of the numerical integration methods. In this program the Jacobian matrix Jf x is computed symbolically and its inverse numerically. As a result, the program produces a table of the iteration steps and an interactive 3D plot that shows the steps in a coordinate system.
Nonlinear fitting of multiparameter functions. An interactive 3D plot shows the iteration steps in a coordinate system. The plot is made with the Jmol application integrated in Sage. In the next example the function minimize uses the Nelder-Mead Method from the scipy. The data points used in this example are generated randomly by deviating the values of the model function. Polynomial Approximation. The algorithm used in the program returns a report on the success of the optimization.
Numerical Analysis Using Sage by George Anastassiou Razvan Mezei
The plot shows the data points and the model function in the same coordinate system. In the code below the integrals on the right hand side are evaluated in terms of the function numerical integral. Concluding remarks During its initial years of development, the Sage project has grown to an environment which offers an attractive alternative for the com- mercial packages in several areas of computational mathematics.
For the purpose of scientific computation teaching, the functionalities of Sage are practically the same as those of commercial packages. A wide selection of ex- tensions and other special libraries are available in the Internet. It is likely that the Sage environment in education will become more popular on all levels of mathematics education from junior high school to graduate level teaching at universities. The support of symbolic computation via Maxima and various numerical packages are notewor- thy in this respect. For purposes of teaching scientific computing, the Sage environment and the modules it contains form an excellent option.
References [ADV] M. Andersen, J. Dahl, and L. Bradshaw, S. Behnel, D. Seljebotn, G. Ewing, et al. Burden and J. Faires: Numerical analysis. Conte and C. Third ed. McGraw-Hill Book Co. Decker, G. Greuel, G. Pfister and H. Heath: Scientific computing: An Introductory Survey. Second ed. McGrawHill Kiusalaas: Numerical methods in engineering with Python. Second edition.
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Cambridge University Press, New York, Mathews and K. Press, S. Teukolsky, W. Vetterling, and B. Flannery: Numerical recipes. The art of scientific computing. Third edi- tion. Cambridge University Press, Cambridge, Prez and B.
Stein et al. Jones, T. Oliphant, P.
Peterson, et al. Tveito, H. Langtangen, B. Nielsen, and X. Cai: Elements of scientific computing. Texts in Computational Science and Engineering, 7. Springer-Verlag, Berlin, ISBN: E-mail address: lauri.
Numerical Analysis Using Sage | George A. Anastassiou | Springer
Related Papers. Python Scientific. By Gaurav Pathak. Introduction to. By Johann J. Python for computational science and engineering. By Fernando Flores. By sofyan casilas. Python for Science. By Luis Guzman. To increase the precision: for mpmath: mpmath. Working with precision , mpmath disagrees with Rivin's answer, though the result seems heavily to depend on the precision. Sign up to join this community. The best answers are voted up and rise to the top.
Home Questions Tags Users Unanswered. Multiprecision numerical evaluation of integral: Sage vs. Asked 4 years, 11 months ago. Active 4 years, 11 months ago. Viewed times. NIntegrate obtained MaviPranav MaviPranav 5 5 bronze badges.
I am not sure any of these can guarantee you bound on the error via numerical methods, though might be wrong. Anyway, I'd think your best bet is to first manipulate the integral in some way to get rid of the singularity, which is causing the numerical instability. If I am not mistaken I am no Mathematica expert , the former is the machine precision that is used for computations, that is, the local error allowed at each computation. Using digit precision mpmath and pari give Having the insight, and the calculus know how, to transform the integrand into something without a singularity, really showcases why we still need "practice integrating".
Nevertheless I largely agree with your comment. Clearly, current software has room for improvement. Igor Rivin Igor Rivin