The Chemistry Maths Book, Second Edition

550 Chapter 19The matrix eigenvalue problem then corresponds to a rotation of the coordinate system, from (x 1 ,x 2 )to(y 1 ,y 2 ...

19.6 Complex matrices 551 or, in terms of components, l x 1 = 1 I xx ω x 1 + 1 I xy ω y 1 + 1 I xz ω z l y 1 = 1 I yx ω x 1 + 1 ...

552 Chapter 19The matrix eigenvalue problem The Hermitian conjugate matrix A † The Hermitian conjugate of Ais the transpose of t ...

19.6 Complex matrices 553 The inner product is complex in general, but the quantity (19.55) is a positive real number, and defin ...

554 Chapter 19The matrix eigenvalue problem EXAMPLE 19.18Show that the following matrix is Hermitian: We have 0 Exercise 40 Unit ...

19.7 Exercises 555 A real unitary matrix is an orthogonal matrix. Hermitian and unitary matrices play the same roles for complex ...

556 Chapter 19The matrix eigenvalue problem 20.The Hückel Hamiltonian matrix of butadiene is Find (i)the eigenvalues (in terms o ...

19.7 Exercises 557 Repeat Exercise 25 for: of Exercise 4 27. of Exercise 8 of Exercise 9 Section 19.5 Express in matrix form: ...

20 Numerical methods 20.1 Concepts A numerical method is a method for obtaining the solution of a mathematical problem in the fo ...

20.2 Errors 559 1.2335 and 1.2345. The number ais said to have a maximum absolute error or (absolute) error boundε a 1 = 1 0.000 ...

560 Chapter 20Numerical methods The quantityε a ε b can be ignored for small enough errors, and the error bound of the product i ...

20.2 Errors 561 so that, in the present case, andx 2 has only one significant figure. However, because the roots of a quadratic ...

562 Chapter 20Numerical methods 20.3 Solution of ordinary equations An ordinary equation (an equation not involving derivatives ...

20.3 Solution of ordinary equations 563 Then, iff(x 3 ) 1 > 10 , so that the zero lies betweenx 1 andx 3 as in the figure, re ...

564 Chapter 20Numerical methods The Newton–Raphson method The Newton–Raphson method, also called simply Newton’s method, is the ...

20.3 Solution of ordinary equations 565 Table 20.3 The square root of 5 nx n f(x n ) 2 f′(x n ) x n+ 1 0 3 0.66666 6666 2.33333 ...

566 Chapter 20Numerical methods 20.4 Interpolation Interpolation is the process of finding a function whose graph goes through a ...

20.4 Interpolation 567 Linear interpolation The straight liney 1 = 1 p 1 (x) 1 = 1 a 0 1 + 1 a 1 xthrough the two points (x 0 , ...

568 Chapter 20Numerical methods Quadratic interpolation The quadratic function that passes through three successive points, or n ...

20.4 Interpolation 569 When quadratic interpolation is used for each triple of neighbouring points we have piecewise quadratic i ...