#YOK
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Date
2022
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Kırklareli Üniversitesi
Access Rights
info:eu-repo/semantics/openAccess
Abstract
Bu çalışmada katsayılar matrisi Hermitian ve pozitif tanımlı olan AX = B lineer kuaterniyon matris denklemini eşlenik gradyan metodu kullanılarak çözmek amaçlanmıştır. Kuaterniyon matrisler ile çalışmanın başlıca zorluğu kuaterniyonların çarpma işlemine göre değişmeli olmamasıdır. Bir lineer kuaterniyon matris denklemi, lineer reel matris denklemine dönüştürülebilir. Bu dönüşüm sırasında matrislerin boyutları artar. Kuaterniyon Hermitian pozitif tanımlı katsayılar matrisinin boyutunun çok büyük olması durumunda direkt metotları kullanmak uygun değildir. Bu tür problemlere Krylov alt uzay tabanlı yaklaşımları uygulamak fazla depolama alanı gerektirir, fakat bu durumda reel aritmetik üzerinde çalışan hızlı ve kararlı algoritmalardan yararlanabiliriz. Çalışmada, öncelikle lineer kuaterniyon matris denklemini reel matris denklemine dönüştürdük. Daha sonra reel matris denklemine blok eşlenik gradyan metodunu uyguladık. Blok eşlenik gradyan metodu uygulandıktan sonra elde edilen çözüm, orijinal kuaterniyon matris denkleminin çözümünün reel temsilidir. Son olarak bu reel çözümü kuaterniyon çözüme dönüştürdük.
This study aims at the solution of the linear quaternion matrix equation AX = B with a Hermitian and positive definite coefficient matrix A by employing the conjugate gradient method. The main difficulty in studying quaternion matrices is that the multiplication of quaternions is noncommutative. A quaternion linear matrix equation can be transformed into a real linear matrix equation. During this transformation into the real setting, the dimension of the matrices increases. We consider the setting when the quaternion Hermitian positive definite matrix at hand is very large so that direct methods are not applicable. Applications of Krylov subspace-based approaches to these problems require more storage, but then we can benefit from fast and stable algorithms operating on real arithmetic. In the study, we first transform the quaternion linear matrix equation into a real matrix equation. Then a block conjugate gradient method is applied to the real matrix equation. The solution obtained after applying the conjugate gradient method is the real representation of the solution of the original quaternion problem. Thus, a conversion of this real solution to the quaternion setting is performed in the end.
This study aims at the solution of the linear quaternion matrix equation AX = B with a Hermitian and positive definite coefficient matrix A by employing the conjugate gradient method. The main difficulty in studying quaternion matrices is that the multiplication of quaternions is noncommutative. A quaternion linear matrix equation can be transformed into a real linear matrix equation. During this transformation into the real setting, the dimension of the matrices increases. We consider the setting when the quaternion Hermitian positive definite matrix at hand is very large so that direct methods are not applicable. Applications of Krylov subspace-based approaches to these problems require more storage, but then we can benefit from fast and stable algorithms operating on real arithmetic. In the study, we first transform the quaternion linear matrix equation into a real matrix equation. Then a block conjugate gradient method is applied to the real matrix equation. The solution obtained after applying the conjugate gradient method is the real representation of the solution of the original quaternion problem. Thus, a conversion of this real solution to the quaternion setting is performed in the end.
Description
Fen Bilimleri Enstitüsü, Matematik Ana Bilim Dalı, Uygulamalı Matematik Bilim Dalı
Keywords
Matematik, Mathematics
