Mathematical Modeling for the Solution of Equations and Systems of Equations with Applications. Volume IV



Series: Theoretical and Applied Mathematics
BISAC: MAT003000
DOI: 10.52305/EQOT3361

The exponential growth of technology forces all disciplines to adjust accordingly, so they can meet the demands of a very dynamic world that heavily depends upon it. Therefore, mathematics cannot be an exception. In fact, mathematics should be the first to adjust and in fact it is. In this volume, which is a continuation of the previous three under the same title, we present state-of-the-art iterative methods for solving equations related to concrete problems from diverse areas such as applied mathematics, mathematical: biology, chemistry, economics, physics and also engineering to mention a few. Most of these methods are new and a few are old but still very popular.

One major problem with iterative methods is that the convergence domain is small in general. We have introduced a technique that finds a smaller set than before containing the iterates leading to tighter Lipschitz functions than before. This way and under the same computational effort, we derive: weaker sufficient convergence criteria (leading to a wider choice of initial points); tighter error bounds on the distances involved (i.e., fewer iterates are needed to obtain a desired predetermined accuracy), and a more precise information on the location of the solution. These advantages are considered major achievements in computational disciplines. The volume requires knowledge of linear algebra, numerical functional analysis and familiarity with contemporary computing programing. It can be used by researchers, practitioners, senior undergraduate and graduate students as a source material or as a required textbook in the classroom.
(Imprint: Nova)

Table of Contents

Table of Contents



Chapter 1. A Family of Cubically Convergent Methods

Chapter 2. Characterization of Some Newton-Like Methods

Chapter 3. Newton’s Method Defined on Not Necessarily Bounded Domains

Chapter 4. Local Convergence of a Class of Multi-Point Super–Halley Methods

Chapter 5. ULM’s-Like Method under Weak Convergence Conditions

Chapter 6. Newton like Method Free of Bilinear Operators

Chapter 7. On the Local Convergence of a Secant like Method

Chapter 8. Extending the Applicability of Newton’s Method

Chapter 9. A Derivative Free Solver in Banach Space

Chapter 10. Convergence for Two Optimal Eighth-Order Methods

Chapter 11. Local Convergence of an Eighth-Order Method

Chapter 12. An Efficient Class of Fourth-Order Jarratt-Type Methods

Chapter 13. Unified Local Convergence for a Certain Family of Methods

Chapter 14. Ball Convergence for EI15th-Order Variants of Hansenpatrick’s Family

Chapter 15. Extended Local Convergence of Newton-Type Methods

Chapter 16. Local Analysis of an Ostrowski-Like Method

Chapter 17. Local Convergence for Multipoint Methods

Chapter 18. Choosing Good Starting Points for the Convergence of Newton’s Method

Chapter 19. Local Convergence of a Tri-Parametric Eighth 18DER Method

Chapter 20. Ball Convergence for Ostrowski-Like Method with Accelerated Eighth Order Convergence

Chapter 21. A Traub-Steffensen-Like Composition for Banach Space Valued Operators

Chapter 22. A Novel Traub-Steffensen Three Step Iterative Method Free Of Derivatives

Chapter 23. Extended Semilocal Convergence of the NHSS Method under Generalized Lipschitz Condition

Chapter 24. A Novel Traub-Steffensen Three Step Iterative Method Free Of Derivatives

Chapter 25. Extended Local Convergence of an Efficient Sixth Order Method

Chapter 26. Traub-Steffensen-Type Solvers for Nonlinear Equations

Chapter 27. Unified Local Convergence for Third Order Methods

Chapter 28. Improved Error Bounds for Newton-Type Solvers-I

Chapter 29. Improved Error Bounds for Newton-Type Solvers-II

Chapter 30. Comparing the Extended Yamamoto’s Error Bounds for Newton’s Solver

Chapter 31. Extended Newton’s Method for Nonsmooth Operators

Chapter 32. Extended and Unified Convergence Theory for Iterative Processes

Chapter 33. Comparison of Two Sixth Order Solvers Using the First Derivative

Chapter 34. Efficient Third Convergence Order Method for Solving Nonlinear Systems

Chapter 35. On the Local Convergence of an Efficient Third Convergence Order Method

Chapter 36. Extended Convergence for Newton-Like Solvers

Chapter 37. Ball Convergence of Schröder-Like Methods for Multiple Roots

Chapter 38. Ball Convergence of Eight Order Methods for Multiple Roots

Chapter 39. Fourth Order Newton-Type Methods for Roots of Multiplicity

Chapter 40. Gauss-Newton Solvers with Projections for Solving Least Squares Problems

Chapter 41. Extended Ball Convergence Results for Newton’s Solver under Hölder-Like Conditions

Chapter 42. Ball Convergence for Traub-Steffensen-Chebyshev Solver

Chapter 43. Extended Ball Convergence of the Gauss-Newton Solver for Injective-Overdetermined Systems of Equations

Chapter 44. Extended Ball Convergence for Newton Conditional Gradient Solver

Chapter 45. Extended Ball Convergence of the Gauss- Newton-Like Solver for Injective-Verdetermined Systems of Equations

Chapter 46. Extended Ball Convergence for Inexact Newton-Like Conditional Gradient Solver

Chapter 47. Extended Ball Convergence of the Gauss-Newton Solver

Chapter 48. Extended Semi-Local Convergence of the Gauss-Newton Solver for Convex Composite Optimization

Chapter 49. Extended Local Convergence Analysis of a Proximal Gauss-Newton Procedure

Chapter 50. Extended Local Convergence of the Gauss-Newton Method

Chapter 51. Solvers for Problems with Small Divisors

Chapter 52. A Two Step Iterative Scheme with a Free Parameter

Chapter 53. A Family of Unified High Convergence Order Methods

Chapter 54. Three Step Seventh Order Method for Solving Equations

Chapter 55. A Unified Family of Jarratt-Like Iterative Methods in Banach Space

Chapter 56. Extended Local Convergence for Newton Simpson’s 3/8th Solver

Chapter 57. On the Harmonic Mean and Midpoint Newton’s Solver

Chapter 58. Comparing the Local Convergence of Three Newtonmean-Type Solvers

Chapter 59. Comparing Two High Convergent Order Methods

Chapter 60. Generalized Solvers for Equations

Chapter 61. On the Harmonic Mean and Midpoint Newton’s Solver-II

Chapter 62. Extended Jarratt-Type Solver

Chapter 63. Extended Chebyshev-Type Solver without Second Derivatives

Chapter 64. Newton-Type Solvers Using Fifth Order Quadrature Formulas

Chapter 65. Extending the Local Convergence of an Efficient Sixth Order Method

Chapter 66. Increased and Extended Local Convergence for Some Iterative Methods

Chapter 67. Extended Newton-Traub-Type Methods

Chapter 68. Extended Newton-Traub Methods

Chapter 69. Solvers of Convergence Three and Four with a Free Parameter

Chapter 70. Extended Fourth Order Weighted Newton Solver

Chapter 71. Extended Sixth Order Schemes with Parameters

Chapter 72. Extended Newton-Jarratt Scheme

Chapter 73. Chebyshev-Halley-Type Methods with Parameters

Chapter 74. Ostrowski-Chun Like Schemes with Parametrs

Chapter 75. Extended Seventh Order Method with Divided Differences

Chapter 76. Derivative-Free Methods 1: Order Six

Chapter 77. Derivative-Free Methods II: Order Seven

Chapter 78. High-Order and Efficient Solvers in Banach Spaces

Chapter 79. Derivative Free Method III: Order Seven

Chapter 80. Derivative Free Methods IV: Order Six

Chapter 81. Extended Semilocal Convergence of a Sixth Order Jarratt-Type Method in Banach Space

Chapter 82. Extended Inexact Gauss-Newton-Like Schemes for Injective Overdetermined Equations


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