A Block Conjugate Gradient Method for Quaternion Linear

Yıl 2023, Cilt: 28 Sayı: 2, 394 - 403, 31.08.2023

Sinem Şimşek , Ayça Körükçü

https://doi.org/10.53433/yyufbed.1168844

Cited By: 1

Öz

This study aims at the simultaneous solution of several quaternion linear systems with the same Hermitian and positive definite coefficient matrix by employing the conjugate gradient method. We consider the setting when the quaternion Hermitian positive definite coefficient matrix at hand is very large so that direct methods are not applicable. In the study, we first transform linear quaternion systems into real linear systems. The transformed real linear systems have special structure due to the fact that they are real representations of quaternion systems. Benefitting from the special structure, we further reduce the size of these linear systems. Then a block conjugate gradient method is applied to the resulting reduced real linear systems. The solution obtained after applying the conjugate gradient method is a 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.

Anahtar Kelimeler

Block conjugate gradient method, Krylov subspace, Linear quaternion systems

Kuaterniyon Lineer Sistemler için Blok Eşlenik Gradyan Metot

Öz

Bu çalışmada katsayılar matrisi aynı ve Hermitian, pozitif tanımlı olan bir takım lineer kuaterniyon sistemlerinin eşlenik gradyan metodu kullanılarak eş zamanlı çözümü amaçlanmıştır. Kuaterniyon Hermitian pozitif tanımlı katsayılar matrisinin boyutunun çok büyük olması durumunda, lineer sistemlerin çözümü için direk metotlar kullanıma uygun değildir. Çalışmada, öncelikle lineer kuaterniyon sistemlerini reel lineer sistemlere dönüştürdük. Dönüştürülen reel lineer sistemler, kuaterniyon sistemlerin reel temsilleri olmalarından ötürü özel yapıya sahiplerdir. Bu özel yapıyı kullanarak reel lineer sistemlerin boyutlarını indirgedik. Daha sonra elde edilen indirgenmiş reel sistemlere yinelemeli bir yöntem olan blok eşlenik gradyan metodunu uyguladık. Blok eşlenik gradyan metodu uygulandıktan sonra elde edilen çözümler, orijinal kuaterniyon lineer sistemlerinin çözümlerinin bir reel temsilidir. Son olarak bu reel çözümleri original sistemin kuaterniyon çözümlerine dönüştürdük.

Anahtar Kelimeler

Blok eşlenik gradyan metot, Krylov altuzay, Lineer kuaterniyon sistemler

Kaynakça

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