Implementasi Algoritma Sherman-Morrison untuk Penyelesaian Sistem Persamaan Linier Menggunakan Matlab
DOI:
https://doi.org/10.59632/leibniz.v5i02.556Keywords:
MATLAB, Sherman-Morrison, Sistem Persamaan LinierAbstract
Artikel ini membahas implementasi algoritma Sherman-Morrison untuk menyelesaikan sistem persamaan linier, di mana A merupakan matriks nonsingular berukuran n×n sehingga sistem memiliki solusi tunggal. Penelitian ini dilatarbelakangi oleh kebutuhan pendekatan penyelesaian yang adaptif dan efisien bagi sistem linier yang mengalami perubahan lokal secara berulang. Algoritma Sherman-Morrison merupakan bentuk khusus dari rumus Sherman–Morrison–Woodbury yang berlaku untuk modifikasi rank-satu, sehingga memungkinkan pembaruan invers matriks tanpa perhitungan ulang secara menyeluruh. Kajian dilakukan melalui pembuktian teoritis rumus tersebut dan implementasi komputasional menggunakan MATLAB R2013a. Implementasi disusun sebagai fungsi khusus yang dikembangkan secara modular untuk mengeksekusi langkah-langkah algoritma. Pengujian dilakukan terhadap empat jenis matriks: acak penuh, simetris positif definit, jarang (sparse), dan hampir singular. Evaluasi mencakup akurasi solusi, error residual, serta waktu eksekusi, kemudian dibandingkan dengan metode standar (A\b). Hasil implementasi menunjukkan bahwa algoritma memberikan solusi yang akurat dan efisien pada sistem dengan skala kecil serta matriks sparse, namun cenderung tidak stabil bila diterapkan pada matriks yang hampir singular. Dengan demikian, algoritma ini layak digunakan untuk kasus pembaruan rank-satu dengan kondisi yang baik, serta relevan untuk penerapan dalam konteks pembelajaran dan komputasi numerik.
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