Numerical matheatical l analysis/
James B.Scarborough
- New Delhi: Oxford & IBH,
- 600 p.
1. Introduction 2. Approximate Numbers and Significant Figures 2 3. Bounding of Numbers 4. Absolute, Relative, and Percentage Errors 6. Relation between Relative Error and the Number of Significant Figures ^ 6. The General Formula for Errors • , 8 7. Application of the Error Formulas to the Fundamental Opera tions of Arithmetic and to Logarithms V10 10 8. The Impossibili^, in General, of Obtaining a Result More Accurate than the Data Used 9. Further Considerations on the Accuracy of a Computed Result 23 10. Accuracy in the Evaluation of a Formula or Complex Ex 11. Accuracy in the Determination of Arguments from a Tabulated Function 12. Accuracy of Series Approximations 82 13. Errors in Determinants 14. A Final Remark Exercises I CHAPTER H INTERPOLATION DIFFERENCES. NEWTON'S FORMULAS OF INTERPOLATION 15. Introduction 16. Differences 17. Effect of an Error in a Tabular Value 52 18. Relation between Differences and Derivatives 64 19. Differences of a Polynomial. 20. Newton's Formula for Forward Interpolation 66 21. Newton's Formula for Backward Interpolation 59 Exercises II xiv CONTENTS CHAPTER III INTERPOLATION WITH UNEQUAL INTERVALS OF THE ARGUMENT ARTICLE PAGE 22. Divided Differences 66 23. Tables of Divided Differences 66 24. Symmetry of Divided Differences 67 25. Relation between Divided Differences and Simple Differences.. 68 26. Newton's General Interpolation Formula 70 27. Lagrange's Interpolation Formula 74 Exercises III 77 CHAPTER IV CENTRAL-DIFFERENCE INTERPOLATION FORMULAS 28. Introduction 79 29. Gauss's Central-Difference Formulas 79 30. Stirling's Interpolation Formula 82 31. Bessel's Interpolation Formulas 84 Exercises IV 90 CHAPTER V INVERSE INTERPOLATION 32. Definition 93 33. By Lagrange's Formula 93 34. By Successive Approximations 9J 36. By Reversion of Series 96 Exercises-V CHAPTER VI THE ACCURACY OF INTERPOLATION FORMULAS 36. Introduction 102 37. Remainder Term in Newton's Formula (I) and in Lagrange's Formula • 102 38. Remainder Term in Newton's Formula (II) 104 39. Remainder Term in Stirling's Formula 105 40. Remainder Terms in Bessel's Formulas 106 ARTICLE CONTENTS 41. Recapitulation of Formulas for the Remainder 42. Accuracy of Linear Interpolation from Tables Exercises ^ CHAPTER VII INTERPOLATION WITH TWO INDEPENDENT VARIABLES TRIGONOMETRIC INTERPOLATION 43. Introduction t i * 44. Double Interpolation by a Double Application of Single Inter polation 45. Double or Two-Way Differences 46. A General Formula for Double Interpolation ' 47. Trigonometric Interpolation ^ Exercises CHAPTER VII NUMERICAL DIFFERENTIATION AND INTEGRATION I, NUMERICAL DIFFERENTIATION 48. Numerical Differentiation II. NUMERICAL INTEGRATION 49. Introduction 50. A General Quadrature Formula for Equidistant Ordinates.... 136 51. Simpson's Rule 52. Weddle's Rule 52A. The Trapezoidal Rule 53. Central-Difference Quadrature Formulas 144 54. Gauss's Quadrature Formula 55. Lobatto's Formula 159 56. Tchebycheff's Formula 57. Euler's Formula of Summation and Quadrature 165 58. Caution in the Use of Quadrature Formulas 168 59. Mechanical Cubature 60. Prismoids and the Prismoidal Formula. 176