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Download e-book for iPad: A Survey of Numerical Mathematics [Vol I] by David M. Young, Robert Todd Gregory, Mathematics

By David M. Young, Robert Todd Gregory, Mathematics

Designed as a textual content for a three-semester direction in numerical research, this very hot paintings provides a huge evaluation of computer-oriented numerical algorithms for fixing numerous varieties of mathematical difficulties. The booklet is meant not just to impart a operating wisdom of sensible recommendations for fixing actual difficulties yet to arrange the coed for complicated stories on the graduate level.
Topics include:
Evaluation of uncomplicated functions
Solution of a unmarried nonlinear equation with particular connection with polynomial equations
Interpolation and approximation
Numerical differentiation and quadrature
Ordinary differential equations
Computational difficulties in linear algebra
Numerical answer of elliptic and parabolic partial differential equations via finite distinction methods
Solution of huge linear platforms via iterative methods
In addition to thorough insurance of the basics, those wide-ranging volumes comprise such targeted beneficial properties as an creation to laptop mathematics, together with an blunders research of a approach of linear algebraic equations with rational coefficients, and an emphasis on computations in addition to mathematical features of varied problems.
Geared towards senior-level undergraduates and first-year graduate scholars, the ebook assumes a few wisdom of complicated calculus, uncomplicated complicated research, matrix conception, and traditional and partial differential equations. even though, the paintings is essentially self-contained, with easy fabric summarized in an appendix, making it an ideal source for self-study.
Ideal as a path textual content in numerical research or as a supplementary textual content in numerical tools, A Survey of Numerical Mathematics judiciously blends arithmetic, numerical research, and computation. the result's an surprisingly useful reference and studying software for contemporary mathematicians, machine scientists, programmers, engineers, and actual scientists.

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Extra info for A Survey of Numerical Mathematics [Vol I]

Example text

Then repeat Problem 8 for Problem 1 (b)-(d). 11. Consider the initial value problem Y'(t)=ata-1, Y(0)=0, where a > 0. The true solution is Y(t) = ta. When a ^ integer, the true solution is not infinitely differentiable. In particular, to have Y twice continuously differentiable, we need a > 2. 05. Compute the solution errors at the nodes, and determine numerically the convergence orders of the Euler method for these problems. 12. The solution of Y'{t) = \Y(t) + cos(«) - Asin(t), Y(0) = 0 is Y(t) = sin(t).

The solution of Y'{t) = \Y(t) + cos(«) - Asin(t), Y(0) = 0 is Y(t) = sin(t). 36) in this case. 05, and A = 1, —1. Compare the true errors with those obtained from the asymptotic estimate Y(tn) - yn « hD(tn). 13. RepeatProblem 12for Problem 1(d). 05. 14. 05. Also, produce the Richardson extrapolate yii(tn) and compute its error. Do this for t„ = 1,2,3,4,5,6. 36 EULER'S METHOD 15. Repeat Problem 14 for Problems 1 (a)-(d). 16. Use Taylor's theorem to show the standard numerical differentiation method Y'(tn+1) = Y{tn+l) n Y{tn) + 0(h).

Then the equivalent initial value problem for a system of first-order equations is Y[(t)=Y2(t), Yl(to)=Y0, Y^_,{t)=Ym{t), Ym^{t0)=Y^m-2\ y ^ ( t ) = / ( « , r , ( t ) , • • •, Ym(t)), (3 17) - Ym(t0)=Y^m-x). 17). 20) is reformulated as Y{(t)=Y2(t), Vi(0)=l, Yi(t)=Y3(t), y2(0)=l, r 3 '(t)=-Ki(*) - 3*2(0 - 3*3(0 - 4sin(0, y3(0)=-l. 21) can be generated from it. This system will be solved numerically later in this chapter. 2 NUMERICAL METHODS FOR SYSTEMS Euler's method and the numerical methods discussed in later chapters can be applied without change to the solution of systems of first-order differential equations.

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A Survey of Numerical Mathematics [Vol I] by David M. Young, Robert Todd Gregory, Mathematics


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