Transformasi Error-Free Untuk Perkalian Matriks Kompleks

Kazal, Nurul Yakim (2021) Transformasi Error-Free Untuk Perkalian Matriks Kompleks. Masters thesis, Institut Teknologi Sepuluh Nopember.

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Abstract

Dalam sistem bilangan floating-point yang memiliki jumlah digit atau presisi berhingga, operasi perkalian matriks sering menyebabkan terjadinya rounding error. Untuk memperkecil nilai error yang diperoleh, beberapa algoritma perkalian matriks dikembangkan berdasarkan konsep transformasi error-free. Namun demikian, algoritma transformasi error-free yang telah dikembangkan sebelumnya hanya dimaksudkan untuk perkalian matriks floating-point real. Dalam penelitian ini, tiga algoritma transformasi error-free untuk perkalian matriks floating-point kompleks diajukan. Hal ini dilakukan dengan memanfaatkan konsep transformasi error-free untuk perkalian matriks floating-point real, matriks blok dan manipulasi aljabar. Algoritma-algoritma ini mentransformasi bagian real dan imajiner dari perkalian matriks floating-point kompleks ke dalam bentuk jumlahan dari matriks-matriks floating-point real. Selanjutnya, hasil perkalian matriks floating-point kompleks diperoleh melalui implementasi salah satu dari algoritma penjumlahan akurat atau operasi penjumlahan floating-point biasa pada jumlahan matriks-matriks tersebut. Berdasarkan hasil simulasi numerik, algoritma yang dikembangkan dengan memanfaatkan konsep matriks blok memberikan hasil evaluasi perkalian matriks floating-point kompleks paling akurat. Selain itu, hasil ini juga diperoleh dengan perbedaan waktu komputasi yang tidak terlalu signifikan jika dibandingkan dengan algoritma-algoritma yang diajukan lainnya.

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The presence of rounding error is frequently inevitable when floating-point matrix multiplication is computed in finite precision. Consequently, the development of matrix multiplication algorithm based on the notion of error-free transformation is proposed as one of the solutions. However, such algorithm is only available for the real one. In this research, three error-free transformation algorithms for complex floating-point matrix multiplication are constructed. This is done by utilizing the concepts of error-free transformation for the real counterpart, block matrices and algebraic manipulation. These proposed algorithms transform the real and imaginary parts of the complex floating-point matrix multiplication into unevaluated sums of real floating-point matrices. If these sums are evaluated by applying either the accurate summation algorithm or pure floating-point addition, then the accurately computed result of complex floating-point matrix multiplication is obtained. Based on the findings in the numerical experiments, it is concluded that the algorithm developed by taking advantage of block matrix notion yields the most accurate results. Additionally, these results are achieved with relatively comparable computing times to those of other proposed algorithms.

Item Type: Thesis (Masters)
Uncontrolled Keywords: aritmetika sistem bilangan floating-point, perkalian matriks, komputasi numerik akurat, floating-point arithmetic, matrix multiplication, accurate numerical computations
Subjects: Q Science > QA Mathematics > QA184 Algebra, Linear
Q Science > QA Mathematics > QA9.58 Algorithms
Divisions: Faculty of Science and Data Analytics (SCIENTICS) > Mathematics > 44101-(S2) Master Thesis
Depositing User: Nurul Yakim Kazal
Date Deposited: 08 Mar 2021 02:03
Last Modified: 08 Mar 2021 02:03
URI: http://repository.its.ac.id/id/eprint/83716

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