The Chemistry Maths Book, Second Edition

570 Chapter 20Numerical methods EXAMPLE 20.9Newton’s method of divided differences fore x Five points on the graph ofe x are giv ...

20.4 Interpolation 571 Forx 1 = 1 0.832the sequence of approximations fore 0.832 is p 1 (x) 1 = 1 2.29820177 p 2 (x) 1 = 1 2.297 ...

572 Chapter 20Numerical methods is, with continuous first and second (and possibly higher) derivatives at the nodes. In the simp ...

20.5 Numerical integration 573 Figure 20.8. This demonstrates that curve fitting with splines produces a continuous curve that c ...

574 Chapter 20Numerical methods The area of the trapezium betweenx j andx j+ 1 is(f j 1 + 1 f j+ 1 )h 22 , and the total area is ...

20.5 Numerical integration 575 The derivative of the integrand is f′(x) 1 = 1 e −x 2 (1 1 − 12 x 2 ), so thatf 0 ′ 1 = 1 f′(0) 1 ...

576 Chapter 20Numerical methods EXAMPLE 20.11Find estimates of the integral by means of Simpson’s rule (20.35) for 2 n 1 = 1 2, ...

20.5 Numerical integration 577 wheref 0 ′,f n ′,f 0 ′′′, f n ′′′, =are the odd-order derivatives off(x)evaluated at the endpoint ...

578 Chapter 20Numerical methods Formula (20.37) is used in the following examples to derive two results that are important in st ...

20.5 Numerical integration 579 The largest value of θfor a molecule is θ 1 = 1 87.5 1 K for H 2 but most molecules have rotation ...

580 Chapter 20Numerical methods There are a number of Gaussian quadrature formulas appropriate to several kinds of integrand. Th ...

20.7 Gauss elimination for the solution of linear equations 581 Gaussian quadrature formulas are used in applications in which v ...

582 Chapter 20Numerical methods Step 2. Elimination ofy Equation (2′) is the new pivot equation, and yis eliminated from subsequ ...

20.7 Gauss elimination for the solution of linear equations 583 Whenλ 1 = 123 , equation (3′′) is redundant, and back substituti ...

584 Chapter 20Numerical methods (2). The latter is therefore chosen as pivot equation. Similarly forx 2 in step 2, and so on. A ...

20.9 First-order differential equations 585 The first stage is to reduce Ato upper triangular form by Gauss elimination: The sec ...

586 Chapter 20Numerical methods because no appropriate method exists or because there is no analytical solution. It is then nece ...

20.9 First-order differential equations 587 A numerical method for solving a differential equation is equivalent to starting at ...

588 Chapter 20Numerical methods Euler’s method is called a first-order method because only the first power of his retained in th ...

20.9 First-order differential equations 589 The computed values ofy(1) are 3.9766 for step sizeh 1 = 1 0.2and 4.1875 forh 1 = 1 ...