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

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## D.R. Hartree:

## D.R. Hartree: *Numerical Analysis (2nd Edition)*

Published $\text {1958}$, **Oxford -- At the Clarendon Press**.

### Subject Matter

### Contents

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

- Preface to the Second Edition (

- Chapter $\text {I}$. 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

- Chapter $\text {II}$. 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

- Chapter $\text {III}$. 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

- Chapter $\text {IV}$. 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 $x = x_0$

- Chapter $\text {V}$. INTERPOLATION

- 5.1. Linear and non-linear interpolation
- 5.11. Linear interpolation

- 5.2. Non-linear 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

- 5.1. Linear and non-linear interpolation

- Chapter $\text {VI}$. 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

- Chapter $\text {VII}$. 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

- 7.1. Step-by-step methods

- Chapter $\text {VIII}$. 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

- 8.1. Direct and indirect methods for simultaneous linear equations

- Chapter $\text {IX}$. 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

- Chapter $\text {X}$. 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

- 10.1. Functions of a complex variable and functions of two variables

- Chapter $\text {XI}$. 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

- 11.1. Summation of series

- Chapter $\text {XII}$. 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

## Further Editions

- 1952: D.R. Hartree:
*Numerical Analysis*

## Source work progress

- 1958: D.R. Hartree:
*Numerical Analysis*(2nd ed.) ... (next): Chapter $\text {I}$: Introduction: $1.1$. What numerical analysis is about