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

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

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

### Contents

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

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
2.22. Transfer from accumulator to setting keys or levers
2.23. Multiplication
2.24. Division
2.25. Other calculations
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-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

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
7.54. The Riccati transformation
7.6. Two-point boundary conditions
7.62. Linear equations with two-point boundary conditions
7.63. Factorization method
7.64. Characteristic value problems

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

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
9.52. 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

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

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

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