Mathematical Methods for Physics and Engineering : A Comprehensive Guide

27.1 ALGEBRAIC AND TRANSCENDENTAL EQUATIONS nxn f(xn) 1 1.7 5.42 2 1.545 01 1.03 3 1.498 87 7. 20 × 10 −^2 4 1.495 13 4. 49 × 10 ...

NUMERICAL METHODS Of the four methods mentioned, no single one is ideal, and, in practice, some mixture of them is usually to be ...

27.2 CONVERGENCE OF ITERATION SCHEMES xn xn+1xn+2 y=x y=F(x) ξ x y Figure 27.3 Illustration of the convergence of the iteration ...

NUMERICAL METHODS nxn+1 n 18.5 4.5 2 5.191 1.19 3 4.137 1. 4 × 10 −^1 4 4.002 257 2. 3 × 10 −^3 5 4.000 000 637 6. 4 × 10 −^7 6 ...

27.3 SIMULTANEOUS LINEAR EQUATIONS variables (unknowns),xi,i=1, 2 ,...,N. The equations take the general form A 11 x 1 +A 12 x 2 ...

NUMERICAL METHODS appreciate how this would apply in (say) a computer program for a 1000-variable case, perhaps with unforseeabl ...

27.3 SIMULTANEOUS LINEAR EQUATIONS the case then an iterative method may produce a satisfactory degree of precision with less ca ...

NUMERICAL METHODS nx 1 x 2 x 3 12 2 2 24 0.1 1.34 3 12.76 1.381 2.323 4 9.008 0.867 1.881 5 10.321 1.042 2.039 6 9.902 0.987 1.9 ...

27.3 SIMULTANEOUS LINEAR EQUATIONS contain non-zero entries. Such matrices are known as tridiagonal matrices. They may also be u ...

NUMERICAL METHODS
Solve the following tridiagonal matrix equation, in which only non-zero elements are shown:       12 − ...

27.4 NUMERICAL INTEGRATION xi xi+1/ 2 xi+1 xi xi+1 xi− 1 xi xi+1 h hhh f(x) f(x) (a) (b) (c) fi fi fi+1 fi+1 fi+1 fi− 1 Figure 2 ...

NUMERICAL METHODS other exact expressions are possible, e.g. the integral off(xi+y) over the range 0 ≤y≤h, but we will find (27. ...

27.4 NUMERICAL INTEGRATION This provides a very simple expression for estimating integral (27.34); its accuracy is limited only ...

NUMERICAL METHODS The difference between the estimate of the integral and the exact answer is 1/12. Equation (27.38) estimates t ...

27.4 NUMERICAL INTEGRATION 27.4.3 Gaussian integration In the cases considered in the previous two subsections, the functionf wa ...

NUMERICAL METHODS and orthogonal over the interval− 1 ≤x≤1, as discussed in subsection 18.1.2. Therefore, in order to use their ...

27.4 NUMERICAL INTEGRATION so, providedg(x) is a reasonably smooth function, the approximation is a good one. Taking 3-point int ...

NUMERICAL METHODS Gauss–Legendre integration ∫ 1 − 1 f(x)dx= ∑n i=1 wif(xi) ±xi wi ±xi wi n=2 n=9 0.57735 02692 1.00000 00000 0. ...

27.4 NUMERICAL INTEGRATION factor is treated accurately in Gauss–Chebyshev integration. Thus ∫ 1 − 1 f(x) √ 1 −x^2 dx≈ ∑n i=1 wi ...

NUMERICAL METHODS Gauss–Laguerre and Gauss–Hermite integration ∫∞ 0 e−xf(x)dx= ∑n i=1 wif(xi) ∫∞ −∞ e−x 2 f(x)dx= ∑n i=1 wif(xi) ...