BookDOI
A History of Numerical Analysis from the 16th through the 19th Century
Herman H. Goldstine
- Vol. 2
TLDR
In this article, Jacobi et al. introduce the Euler-Maclaurin Formula and the Laplace Summation Formula, which is used by Laplace, Legendre, and Gauss in the Sixteenth and Early Seventeenth Centuries.Abstract:
1. The Sixteenth and Early Seventeenth Centuries.- 1.1. Introduction.- 1.2. Napier and Logarithms.- 1.3. Briggs and His Logarithms.- 1.4. Burgi and His Antilogarithms.- 1.5. Interpolation.- 1.6. Vieta and Briggs.- 1.7. Kepler.- 2. The Age of Newton.- 2.1. Introduction.- 2.2. Logarithms and Finite Differences.- 2.3. Trigonometric Tables.- 2.4. The Newton-Raphson and Other Iterative Methods.- 2.5. Finite Differences and Interpolation.- 2.6. Maclaurin on the Euler-Maclaurin Formula.- 2.7. Stirling.- 2.8. Leibniz.- 3. Euler and Lagrange.- 3.1. Introduction.- 3.2. Summation of Series.- 3.3. Euler on the Euler-Maclaurin Formula.- 3.4. Applications of the Summation Formula.- 3.5. Euler on Interpolation.- 3.6. Lunar Theory.- 3.7. Lagrange on Difference Equations.- 3.8. Lagrange on Functional Equations.- 3.9. Lagrange on Fourier Series.- 3.10. Lagrange on Partial Difference Equations.- 3.11. Lagrange on Finite Differences and Interpolation.- 3.12. Lagrange on Hidden Periodicities.- 3.13. Lagrange on Trigonometric Interpolation.- 4. Laplace, Legendre, and Gauss.- 4.1. Introduction.- 4.2. Laplace on Interpolation.- 4.3. Laplace on Finite Differences.- 4.4. Laplace Summation Formula.- 4.5. Laplace on Functional Equations.- 4.6. Laplace on Finite Sums and Integrals.- 4.7. Laplace on Difference Equations.- 4.8. Laplace Transforms.- 4.9. Method of Least Squares.- 4.10. Gauss on Least Squares.- 4.11. Gauss on Numerical Integration.- 4.12. Gauss on Interpolation.- 4.13. Gauss on Rounding Errors.- 5. Other Nineteenth Century Figures.- 5.1. Introduction.- 5.2. Jacobi on Numerical Integration.- 5.3. Jacobi on the Euler-Maclaurin Formula.- 5.4. Jacobi on Linear Equations.- 5.5. Cauchy on Interpolation.- 5.6. Cauchy on the Newton-Raphson Method.- 5.7. Cauchy on Operational Methods.- 5.8. Other Nineteenth Century Results.- 5.9. Integration of Differential Equations.- 5.10. Successive Approximation Methods.- 5.11. Hermite.- 5.12. Sums.read more
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Historical development of the Newton-Raphson method
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