(498) the numerical solution for the roots polynomials
This study examines the numerical solution of polynomial roots as a fundamental topic in numerical analysis, applied mathematics, and operations research. The study focuses on mathematical techniques used to determine both real and complex roots of polynomial equations of different degrees. It is ba...
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| Aineistotyyppi: | Other |
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the institute of national planning
2018
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| Linkit: | http://repository.inp.edu.eg/handle/123456789/3923 |
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| author | abdel karim, abbas I. |
| author_facet | abdel karim, abbas I. |
| author_sort | abdel karim, abbas I. |
| collection | DSpace |
| description | This study examines the numerical solution of polynomial roots as a fundamental topic in numerical analysis, applied mathematics, and operations research. The study focuses on mathematical techniques used to determine both real and complex roots of polynomial equations of different degrees. It is based on the premise that direct analytical solutions become increasingly difficult or impractical as polynomial degrees increase, making numerical procedures more effective and applicable alternatives.
The study begins by presenting the theoretical framework of polynomial equations and the characteristics of their roots, emphasizing the relationship between equation coefficients and the nature of possible solutions. It explains that root determination is not merely a theoretical mathematical exercise but also an essential component in solving practical problems in engineering, economics, physics, and mathematical programming.
The analysis further addresses several numerical methods used for root estimation through iterative procedures designed to approximate root values progressively until acceptable accuracy levels are achieved. The study explains convergence mechanisms, error estimation criteria, and the influence of initial approximations on computational efficiency and solution speed.
Special attention is given to numerical challenges associated with root calculations, including approximation errors, sensitivity to small coefficient variations, and difficulties related to repeated or closely spaced roots. The study emphasizes that computational efficiency can be improved through algorithms possessing numerical stability and reduced computational requirements.
The study concludes that numerical methods provide effective tools for solving high-degree polynomial equations and that solution accuracy depends substantially on selecting appropriate numerical techniques and model specifications. Such approaches contribute significantly to scientific, engineering, and quantitative applications requiring reliable computational procedures |
| format | Other |
| id | ir-123456789-3923 |
| institution | My University |
| language | other |
| publishDate | 2018 |
| publisher | the institute of national planning |
| record_format | dspace |
| spelling | ir-123456789-39232026-06-24T12:25:34Z (498) the numerical solution for the roots polynomials operations research center abdel karim, abbas I. the numerical solution indeterminate employed This study examines the numerical solution of polynomial roots as a fundamental topic in numerical analysis, applied mathematics, and operations research. The study focuses on mathematical techniques used to determine both real and complex roots of polynomial equations of different degrees. It is based on the premise that direct analytical solutions become increasingly difficult or impractical as polynomial degrees increase, making numerical procedures more effective and applicable alternatives. The study begins by presenting the theoretical framework of polynomial equations and the characteristics of their roots, emphasizing the relationship between equation coefficients and the nature of possible solutions. It explains that root determination is not merely a theoretical mathematical exercise but also an essential component in solving practical problems in engineering, economics, physics, and mathematical programming. The analysis further addresses several numerical methods used for root estimation through iterative procedures designed to approximate root values progressively until acceptable accuracy levels are achieved. The study explains convergence mechanisms, error estimation criteria, and the influence of initial approximations on computational efficiency and solution speed. Special attention is given to numerical challenges associated with root calculations, including approximation errors, sensitivity to small coefficient variations, and difficulties related to repeated or closely spaced roots. The study emphasizes that computational efficiency can be improved through algorithms possessing numerical stability and reduced computational requirements. The study concludes that numerical methods provide effective tools for solving high-degree polynomial equations and that solution accuracy depends substantially on selecting appropriate numerical techniques and model specifications. Such approaches contribute significantly to scientific, engineering, and quantitative applications requiring reliable computational procedures تناقش هذه الدراسة موضوع الحل العددي لجذور كثيرات الحدود بوصفه أحد الموضوعات الأساسية في التحليل العددي والرياضيات التطبيقية وبحوث العمليات، مع التركيز على الأساليب الرياضية المستخدمة في تحديد الجذور الحقيقية والتخيلية للمعادلات متعددة الحدود ذات الدرجات المختلفة. وتنطلق الدراسة من فرضية أساسية مفادها أن الحلول التحليلية المباشرة للمعادلات كثيرة الحدود تصبح أكثر تعقيدًا أو غير عملية كلما ارتفعت درجة المعادلة، مما يجعل الأساليب العددية بديلًا فعالًا للوصول إلى حلول دقيقة وقابلة للتطبيق. تبدأ الدراسة بعرض الإطار النظري للمعادلات كثيرة الحدود وخصائص جذورها، موضحةً العلاقة بين معاملات المعادلة وطبيعة الجذور المحتملة. كما توضح أن إيجاد الجذور لا يمثل هدفًا رياضيًا مجردًا، وإنما يعد مدخلًا ضروريًا لحل العديد من المشكلات التطبيقية في الهندسة والاقتصاد والفيزياء والبرمجة الرياضية. وتتناول الدراسة مجموعة من الطرق العددية المستخدمة في تقدير الجذور، والتي تعتمد على عمليات تكرارية متتابعة تهدف إلى تقريب قيم الجذور بصورة تدريجية حتى الوصول إلى مستوى مقبول من الدقة. وتشرح الدراسة آليات اختبار التقارب ومعايير تحديد الخطأ العددي وتأثير اختيار القيم الابتدائية على سرعة الوصول إلى الحل. ما تركز الدراسة على المشكلات العددية المرتبطة بحساب الجذور، مثل تراكم الأخطاء التقريبية، والحساسية تجاه التغيرات الصغيرة في معاملات المعادلة، وصعوبة التمييز بين الجذور المتقاربة أو المتكررة. وتوضح أن تحسين الكفاءة الحسابية يتطلب استخدام خوارزميات تتمتع بالاستقرار العددي وتقليل عدد العمليات الحسابية المطلوبة. وتخلص الدراسة إلى أن الطرق العددية تمثل أدوات فعالة لمعالجة المعادلات كثيرة الحدود ذات الدرجات المرتفعة، وأن دقة الحل تعتمد بصورة كبيرة على اختيار الطريقة المناسبة وطبيعة النموذج الرياضي المستخدم، بما يسهم في دعم التطبيقات العلمية والهندسية المختلف 2018-08-07T13:26:38Z 2018-08-07T13:26:38Z 1964-10-19 Other http://repository.inp.edu.eg/handle/123456789/3923 other External notes;498 application/pdf the institute of national planning |
| spellingShingle | the numerical solution indeterminate employed abdel karim, abbas I. (498) the numerical solution for the roots polynomials |
| title | (498) the numerical solution for the roots polynomials |
| title_full | (498) the numerical solution for the roots polynomials |
| title_fullStr | (498) the numerical solution for the roots polynomials |
| title_full_unstemmed | (498) the numerical solution for the roots polynomials |
| title_short | (498) the numerical solution for the roots polynomials |
| title_sort | 498 the numerical solution for the roots polynomials |
| topic | the numerical solution indeterminate employed |
| url | http://repository.inp.edu.eg/handle/123456789/3923 |
| work_keys_str_mv | AT abdelkarimabbasi 498thenumericalsolutionfortherootspolynomials AT abdelkarimabbasi operationsresearchcenter |