Book:D.R. Hartree/Numerical Analysis/Second Edition

Subject Matter

 * Numerical Analysis

Contents

 * Preface to the Second Edition (Cavendish Laboratory Cambridge October $1957$)
 * Preface to the First Edition (Cavendish Laboratory Cambridge May $1952$)


 * . INTRODUCTION


 * 1.1. What numerical analysis is about
 * 1.2. The main types of problems in numerical analysis
 * 1.3. Errors, mistakes and checking
 * 1.4. Arrangement of work
 * 1.5. Accuracy and precision


 * . THE TOOLS OF NUMERICAL WORK AND HOW TO USE THEM


 * 2.1. The main tools of numerical work
 * 2.2. Desk machines
 * 2.21. Addition and subtraction
 * 2.22. Transfer from accumulator to setting keys or levers
 * 2.23. Multiplication
 * 2.24. Division
 * 2.25. Other calculations
 * 2.26. Adding machines
 * 2.3. Mathematical tables
 * 2.4. Slide rule
 * 2.5. Graph paper
 * 2.6. Other machines


 * . EVALUATION OF FORMULAE


 * 3.1. The significance of formulae in numerical work
 * 3.2. Evaluation of polynomials
 * 3.3. Evaluation of power series
 * 3.4. Kinds of formulae to avoid
 * 3.5. Evaluation of a function in the neighbourhood of a value of the argument at which it becomes indeterminate


 * . FINITE DIFFERENCES


 * 4.1. Functions of a continuous variable in numerical analysis
 * 4.2. Finite differences
 * 4.21. Notation for finite differences
 * 4.3. Finite differences in terms of function values
 * 4.4. Simple applications of differences
 * 4.41. Differences of a polynomial
 * 4.42. Building up of polynomials
 * 4.43. Checking by differences
 * 4.44. Effect of rounding errors on differences
 * 4.45. Direct evaluation of second differences
 * 4.46. Building up from second differences
 * 4.5. Differences and derivatives
 * 4.6. Finite difference operators
 * 4.7. Examples of the use of finite difference operators
 * 4.71. Derivatives in terms of differences
 * 4.72. Negative powers of $\paren {U / \delta}$
 * 4.73. $\delta^2 f$ in terms of $f''$ and its differences
 * 4.74. $\delta f_{\frac 1 2}$ symmetrically in terms of $f'$ and its differences at $x_0$ and $x_1$
 * 4.75. $\mu \delta f_0$ in terms of $f'$ and its differences at $x = x_0$


 * . INTERPOLATION


 * 5.1. Linear and non-linear interpolation
 * 5.11. Linear interpolation
 * 5.2. Non-lienar interpolation
 * 5.21. Half-way interpolation
 * 5.22. Newton's forward-difference formula
 * 5.3. Some expansions
 * 5.4. Everett's interpolation formula
 * 5.41. Bessel's interpolation formula
 * 5.42. Use of Bessel's and Everett's formulae
 * 5.43. Practical details in non-linear interpolation
 * 5.5. Lagrange's formula
 * 5.51. Special interpolation methods for particular functions
 * 5.6. Subtabulation
 * 5.61. End-figure method for subtabulation
 * 5.7. Interpolation of a function given at unequal intervals of the argument
 * 5.71. Evaluation of Lagrange's interpolation formula by a sequence of linear cross-means
 * 5.72. Divided differences
 * 5.8. Inverse interpolation
 * 5.81. How not to do inverse interpolation
 * 5.9. Truncation errors in interpolation formulae
 * 5.91. Whittaker's cardinal function


 * . INTEGRATION (QUADRATURE) AND DIFFERENTIATION


 * 6.1. Definite and indefinite integrals, and the integration of differential equations
 * 6.2. Integration formula in terms of integrand and its differences
 * 6.21. An alternative derivation
 * 6.22. Integration formula in terms of the integrand and the differences of its derivative
 * 6.23. Integration formula in terms of the integrand and its derivatives (Euler-Maclaurin formula)
 * 6.3. Integration over more than one interval
 * 6.4. Evaluation of an integral as a function of its upper limit
 * 6.41. Change of interval length in an integration
 * 6.42. Integration in the neighbourhood of a singularity of the integrand
 * 6.43. Integration when the integrand increases 'exponentially'
 * 6.44. Two-fold integration
 * 6.5. Integrals between fixed limits
 * 6.51. Gregory's formula
 * 6.52. Integral in terms of function values
 * 6.53. Use of Simpson's or Weddle's rules
 * 6.54. Integrations of functions for which $\map {f^{\paren {2 n + 1} } } x = 0$ at both ends of the range of integration
 * 6.55. Evaluation of a definite integral when the integrand has a singularity
 * 6.56. Definite integrals which are functions of a parameter
 * 6.6. Use of unequal intervals of the independent variables
 * 6.61. Gaussian integration formulae
 * 6.62. Gaussian formulae for $\int \limits_0^\infty e^{-k x} \map {p_{2 n + 1} } x \, dx$
 * 6.7. Numerical differentiation
 * 6.71. Differential formulae
 * 6.72. Graphical differentiation
 * 6.8. Errors of interpolation and integration formulae
 * 6.81. Use of formulae for the error


 * . INTEGRATION OF ORDINARY DIFFERENTIAL EQUATIONS


 * 7.1. Step-by-step methods
 * 7.11. One-point and two-point boundary conditions
 * 7.2. Second-order equation with first derivative absent
 * 7.21. Change of the interval of integration
 * 7.22. Variants of the method
 * 7.23. Numerov's method
 * 7.3. First-order differential equations
 * 7.31. Another method for a first-order equation
 * 7.32. First-order linear equations
 * 7.33. Second-order equation with the first derivative present
 * 7.34. Equations of order higher than the second
 * 7.4. Taylor series method
 * 7.5. Other procedures
 * 7.51. Richardson's 'deferred approach to the limit'
 * 7.52. Iterative processes
 * 7.53. The Madelung transformation
 * 7.54. The Riccati transformation
 * 7.6. Two-point boundary conditions
 * 7.61. Iterative quadrature
 * 7.62. Linear equations with two-point boundary conditions
 * 7.63. Factorization method
 * 7.64. Characteristic value problems


 * . SIMULTANEOUS LINEAR ALGEBRAIC EQUATIONS AND MATRICES


 * 8.1. Direct and indirect methods for simultaneous linear equations
 * 8.11. Matrices
 * 8.12. Ill-conditioned equations
 * 8.13. Normal equations
 * 8.2. Elimination
 * 8.21. General elimination process
 * 8.22. Evaluation of a solution by elimination
 * 8.23. Alternative arrangement of the elimination process
 * 8.3. Inverse of a matrix by elimination
 * 8.4. Choleski's method
 * 8.41. Inverse of a matrix by Choleski's method
 * 8.5. Relaxation method
 * 8.51. Group relaxations
 * 8.52. Use and limitations of the relaxation method
 * 8.6. Linear differential equations and linear simultaneous equations
 * 8.7. Characteristic values and vectors of a matrix
 * 8.71. Iterative method for evaluation of characteristic values and characteristic vectors of a symmetrical matrix
 * 8.72. Richardson's purification process for characteristic vectors
 * 8.73. Relaxation process for characteristic vectors


 * . NON-LINEAR ALGEBRAIC EQUATIONS


 * 9.1. Solution of algebraic equations
 * 9.2. Graphical methods
 * 9.3. Iterative processes
 * 9.31. Examples of iterative processes
 * 9.32. Derivation of a second-order process from a first-order process
 * 9.4. Multiple roots and neighbouring roots
 * 9.5. Special processes for special types of equations
 * 9.51. Quadratic equations
 * 9.52. Cubic and quartic equations
 * 9.53. Polynomial equations
 * 9.54. Repeated roots
 * 9.55. Division of a polynomial by a quadratic
 * 9.56. Real quadratic factors of a polynomial
 * 9.57. Second-order process for improving the approximation to a quadratic factor
 * 9.6. Simultaneous non-linear equations
 * 9.7. Three or more variables


 * . FUNCTIONS OF TWO OR MORE VARIABLES


 * 10.1. Functions of a complex variable and functions of two variables
 * .10.11. Numerical calculations with complex numbers
 * 10.2. Finite differences in two dimensions; square grid
 * 10.3. The operator $\partial^2 / \partial x^2 + \partial^2 / \partial y^2$
 * 10.31. Special relations when $\partial^2 f / \partial x^2 + \partial^2 f / \partial y^2 = 0$
 * 10.4. Finite differences in cylindrical coordinates
 * 10.5. Partial differential equations
 * 10.6. Elliptic equations
 * 10.61. Relaxation process
 * 10.62. Reducing the mesh size
 * 10.63. Further notes on the relaxation process
 * 10.64. Richardson-Liebmann process for Laplace's equation
 * 10.7. Parabolic equations
 * 10.71. Replacement of the second-order (space) derivative by a finite difference
 * 10.72. Replacement of the first-order (time) derivative by a finite difference
 * 10.73. Replacement of both derivatives by finite differences
 * 10.74. Note on methods for parabolic equations
 * 10.8. Hyperbolic equations. Characteristics
 * 10.81. Finite differences between characteristics
 * 10.82. Use of given intervals in one independent variable
 * 10.83. Two simultaneous first-order equations


 * . MISCELLANEOUS PROCESSES


 * 11.1. Summation of series
 * 11.11. Euler's transformation for a slowly converging series of terms of alternate signs
 * 11.12. Use of the Euler-Maclaurin integration formula in the summation of series
 * 11.2. Harmonic analysis
 * 11.3. Recurrence relations for a sequence of functions
 * 11.4. Smoothing
 * 11.41. Automatic methods of smoothing
 * 11.42. Smoothing by use of an auxiliary function


 * . ORGANIZATION OF CALCULATIONS FOR AN AUTOMATIC MACHINE


 * 12.1. Automatic digital calculating machines
 * 12.2. Preparation of calculations for an automatic digital calculating machine
 * 12.3. Hand and automatic calculation


 * EXAMPLES


 * BIBLIOGRAPHY


 * INDEX