Understanding Tridiagonal and Band Diagonal Systems of Equations
Tridiagonal and band diagonal matrices play a key role in solving systems of equations efficiently. A tridiagonal matrix has non-zero elements on the main diagonal, superdiagonal, and subdiagonal, while a band diagonal matrix allows non-zero elements anywhere around the main diagonal. The Thomas Algorithm is commonly used to solve tridiagonal systems of equations, involving forward sweep computations for efficient solutions.
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Tridiagonal and Band Diagonal Systems of Equations
Overview Definition Tridiagonal Matrix: a band matrix that has non-zero elements on the main diagonal, the first diagonal above the main diagonal (superdiagonal) and the first diagonal below the main diagonal (subdiagonal) only.
Tridiagonal Example Matrix The only non-zero elements in a tridiagonal matrix or on the diagonal, superdiagonal, and subdiagonal. The Main Diagonal elements are B11 - B66 The superdiagonal has elements B12 - B56 The subdiagonal has elements B21 - B65
Overview Definition Band Diagonal Matrix: a band matrix that has non-zero elements on the main diagonal, but is more relaxed in definition vs tridiagonal. Non-zero elements are allowed anywhere below or above the main diagonal.
Band Diagonal Example Matrix The only non-zero elements in a Band diagonal matrix or on the main diagonal, and either below (to the left of) the main diagonal m1 0 or above (to the right of) the main m2 0. The Main Diagonal elements are A11 - A66 The m1 0 has no listed elements The m2 0 has elements A12 - A56 and A13 - A46
Tridiagonal Matrix Algorithm (Thomas Algorithm) Given a Tridiag Matrix, work the Forward Sweep Computations:
Tridiagonal Matrix Algorithm (Thomas Algorithm) Given a Tridag Matrix, work the Forward Sweep Computations:
Tridiagonal Matrix Algorithm (Thomas Algorithm) Given a Tridag Matrix, work the Forward Sweep Computations:
Tridiagonal Matrix Algorithm (Thomas Algorithm) After the Forward Sweep, the Solution is gained by Back Substitution:
Tridiagonal Matrix Algorithm (Thomas Algorithm) Example Problem: