(424) Interpolation Formulas صيغ الاستيفاء
This study examines the theoretical and mathematical foundations of interpolation formulas as essential quantitative tools used in mathematical, statistical, and economic analysis for estimating unknown values located between known observations within a dataset. The main objective of the study is to...
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معهد التخطيط القومى
2024
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| Urunga tuihono: | http://repository.inp.edu.eg//handle/123456789/5579 |
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| _version_ | 1869359356070330368 |
|---|---|
| author | Abdel Karim, Abbas I. |
| author_facet | Abdel Karim, Abbas I. |
| author_sort | Abdel Karim, Abbas I. |
| collection | DSpace |
| description | This study examines the theoretical and mathematical foundations of interpolation formulas as essential quantitative tools used in mathematical, statistical, and economic analysis for estimating unknown values located between known observations within a dataset. The main objective of the study is to present the principal interpolation formulas and evaluate their theoretical and practical characteristics while demonstrating their role in solving estimation and approximation problems. The paper begins by introducing interpolation as a mathematical process intended to construct a functional relationship capable of estimating values of a variable based on previously observed data points. The study emphasizes the importance of interpolation techniques because of their wide range of applications in economics, statistics, engineering, and applied sciences, particularly in situations where data are incomplete or where intermediate values cannot be directly observed. The study further reviews several general interpolation approaches, including interpolation polynomials, the Lagrange interpolation method, and Newton's general interpolation formula. Particular attention is given to the concept of interpolation error and the factors affecting the accuracy and reliability of estimated values. The research explains that the choice of an appropriate interpolation technique depends on the characteristics of the available data, the number of observations, and the regularity of intervals among observations. The paper also discusses special interpolation formulas such as Newton–Gregory formulas, Gaussian interpolation methods, Everett–Laplace formulas, linear interpolation techniques, inverse interpolation procedures, and Aitken's repeated process. These methods are compared in terms of computational efficiency, accuracy, and suitability for various numerical applications. The academic significance of the study lies in its development of a systematic mathematical framework for understanding interpolation techniques and their practical applications. The study therefore contributes to improving quantitative estimation accuracy and supports analytical and decision-making processes across scientific and applied disciplines. |
| format | Book |
| id | ir-123456789-5579 |
| institution | My University |
| publishDate | 2024 |
| publisher | معهد التخطيط القومى |
| record_format | dspace |
| spelling | ir-123456789-55792026-06-25T11:19:45Z (424) Interpolation Formulas صيغ الاستيفاء Abdel Karim, Abbas I. Interpolation Formulas Lagrange method of Interpolation errors of Interpolation Formulas Linear of Interpolation This study examines the theoretical and mathematical foundations of interpolation formulas as essential quantitative tools used in mathematical, statistical, and economic analysis for estimating unknown values located between known observations within a dataset. The main objective of the study is to present the principal interpolation formulas and evaluate their theoretical and practical characteristics while demonstrating their role in solving estimation and approximation problems. The paper begins by introducing interpolation as a mathematical process intended to construct a functional relationship capable of estimating values of a variable based on previously observed data points. The study emphasizes the importance of interpolation techniques because of their wide range of applications in economics, statistics, engineering, and applied sciences, particularly in situations where data are incomplete or where intermediate values cannot be directly observed. The study further reviews several general interpolation approaches, including interpolation polynomials, the Lagrange interpolation method, and Newton's general interpolation formula. Particular attention is given to the concept of interpolation error and the factors affecting the accuracy and reliability of estimated values. The research explains that the choice of an appropriate interpolation technique depends on the characteristics of the available data, the number of observations, and the regularity of intervals among observations. The paper also discusses special interpolation formulas such as Newton–Gregory formulas, Gaussian interpolation methods, Everett–Laplace formulas, linear interpolation techniques, inverse interpolation procedures, and Aitken's repeated process. These methods are compared in terms of computational efficiency, accuracy, and suitability for various numerical applications. The academic significance of the study lies in its development of a systematic mathematical framework for understanding interpolation techniques and their practical applications. The study therefore contributes to improving quantitative estimation accuracy and supports analytical and decision-making processes across scientific and applied disciplines. يتناول هذا البحث الأسس النظرية والرياضية لصيغ الاستيفاء بوصفها إحدى الأدوات الكمية الأساسية المستخدمة في التحليل الرياضي والإحصائي والاقتصادي لتقدير القيم المجهولة الواقعة بين قيم معروفة ضمن مجموعة من البيانات. وتهدف الدراسة إلى عرض أهم الصيغ الرياضية المستخدمة في الاستيفاء وتحليل خصائصها النظرية والتطبيقية، مع بيان دورها في معالجة المشكلات المرتبطة بتقدير البيانات وإجراء الحسابات التقريبية. بناء دالة أو علاقة رياضية تسمح بتقدير قيمة متغير معين اعتماداً على مجموعة من النقاط أو المشاهدات المعروفة مسبقاً. وتوضح الدراسة أن أهمية الاستيفاء تنبع من استخداماته الواسعة في العلوم التطبيقية والاقتصاد والإحصاء والهندسة، خاصة عندما تكون البيانات غير مكتملة أو عند الحاجة إلى تقدير قيم وسيطة غير متاحة بصورة مباشرة. كما تستعرض الدراسة عدداً من الأساليب العامة للاستيفاء، بما في ذلك كثيرات الحدود الخاصة بالاستيفاء، وطريقة لاغرانج، وصيغة نيوتن العامة، إضافة إلى مناقشة مفهوم خطأ الاستيفاء والعوامل المؤثرة في دقة النتائج. وتوضح الدراسة أن اختيار الطريقة المناسبة يعتمد على طبيعة البيانات وعدد المشاهدات ومدى انتظامها. وتتناول الدراسة كذلك بعض الصيغ الخاصة للاستيفاء مثل صيغ نيوتن–جريجوري، وصيغ جاوس، وصيغة إيفريت–لابلاس، إضافة إلى الاستيفاء الخطي، والاستيفاء العكسي، وطريقة آيتكن التكرارية. وتوضح أن هذه الأساليب تختلف من حيث الكفاءة الحسابية والدقة وإمكانية استخدامها في المشكلات العددية المختلفة. وتتمثل الأهمية العلمية للدراسة في تقديم إطار رياضي متكامل لفهم تقنيات الاستيفاء وتطبيقاتها العملية، بما يسهم في تحسين دقة التقديرات الكمية ودعم عمليات التحليل واتخاذ القرار في المجالات العلمية والتطبيقية المختلفة. 2024-12-15T10:42:47Z 2024-12-15T10:42:47Z 1964-04-01 Book http://repository.inp.edu.eg//handle/123456789/5579 memo 424;45 p application/pdf معهد التخطيط القومى |
| spellingShingle | Interpolation Formulas Lagrange method of Interpolation errors of Interpolation Formulas Linear of Interpolation Abdel Karim, Abbas I. (424) Interpolation Formulas صيغ الاستيفاء |
| title | (424) Interpolation Formulas
صيغ الاستيفاء |
| title_full | (424) Interpolation Formulas
صيغ الاستيفاء |
| title_fullStr | (424) Interpolation Formulas
صيغ الاستيفاء |
| title_full_unstemmed | (424) Interpolation Formulas
صيغ الاستيفاء |
| title_short | (424) Interpolation Formulas
صيغ الاستيفاء |
| title_sort | 424 interpolation formulas صيغ الاستيفاء |
| topic | Interpolation Formulas Lagrange method of Interpolation errors of Interpolation Formulas Linear of Interpolation |
| url | http://repository.inp.edu.eg//handle/123456789/5579 |
| work_keys_str_mv | AT abdelkarimabbasi 424interpolationformulasṣygẖạlạstyfạʾ |