Solving Tridiagonal Symmetric Systems of Equations Using Circuit Theory Approach

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We propose a novel solution method for symmetric tridiagonal systems. The main idea is to construct a specific electric circuit with the same node-voltage equations as the original system. This circuit has a specific “ladder” structure and it is efficiently solved using a methodology know as the admittance summation method [1]. The method avoids possible zero divisions exploiting the specific structure of the circuit. This specific property is an equivalent to a pivoting strategies used in other methods.

The test have shown that the performance of the proposed method is comparable to the Thomas algorithm [2] and Gaussian elimination adapted for tridiagonal systems [3], as well as the Matlab backslash operator.

Fig. 1. A Simple Electric Circuit

For the circuit from Fig. 1 we assume that all resistors are with resistance of 1 Ω and the independent current sources are i₁ = 0 A, i₂ = 2 A, i₃ = 3 A and i₄ = 5 A. In this case the following system of equations solves for circuit voltages.

Fig. 2. Tridiagonal System of Equations

Of course, we do not store the full n × n matrix, but only its nonzero elements stored as two row vectors a and b for the main diagonal and diagonals below and above the main diagonal. They are a = [2 3 3 2] and b = [-1 -1 -1]. Right-hand side vector is r = [0 2 3 5].

The input data is given in ex_1.m as a function returning a, b and r.

function [a, b, r] = ex_1()
a = [2 3 3 2];
b = [-1 -1 -1];
r = [0 2 3 5];

The solution is obtained with

u = tridiag_ysum(a, b, r)

The method consist of the following steps

Initialization

Calculate vector Y

Y = a + [0 b] + [b 0], where [0 b] is a concatenation of 0 and row vector b, while [b 0] is another concatenation where 0 is added after b.

Simple Algorithm

May fail for systems that are ill-conditioned or have zero diagonal/off-diagonal elements.

Complete Algorithm

Detects possible divisions by zero and handles them without the need for pivoting.

Backward sweep

Forward sweep

References

[1] D. Rajičić, R. Taleski, Two novel methods for radial and weakly meshed network analysis, Electric Power Systems Research 48 (2) (1998) 79 – 87

[2] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes – The Art of Scientific Computing, 3rd Edition, Cambridge University Press, 2007, Ch. 2.4, pp. 56–57.

[3] D. Kincaid, W. Cheney, Numerical Analysis – Mathematics of Scientific Computing, American Mathematical Society, 2002, Ch. 4.3, pp. 179–180.