By David M. Young, Robert Todd Gregory, Mathematics

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.

**Read or Download A Survey of Numerical Mathematics [Vol I] PDF**

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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.

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

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