Abstract
Linear systems with tridiagonal structures are very common in problems related not only to engineering, but chemistry, biomedical or finance, for example, real time cubic B-Spline interpolation of ND-images, real time processing of Electrocardiography (ECG) and hand drawing recognition. In those problems which the matrix is positive definite, it is possible to optimize the solution in O(n) time. This paper describes such systems whose size grows over time and proposes an approximation in O(1) time of such systems based on a series of previous approximations. In addition, it is described the development of the method and is proved that the proposed solution converges linearly to the optimal. A real-time cubic B-Spline interpolation of an ECG is computed with this proposal, for this application the proposed method shows a global relative error near to 10-6 and its computation is faster than traditional methods, as shown in the experiments.
Highlights
Linear systems with tridiagonal structures are very common in problems related to engineering, but chemistry, biomedical or finance, for example, real time cubic B-Spline interpolation of ND-images, real time processing of Electrocardiography (ECG) and hand drawing recognition
The system A+x+ = b+ appears in problems that grow over time, such as: real time interpolation of biomedical data (ND Images or ECG), filtering data, handwriting recognition and real time image processing
This paper proposes a new method that computes an approximation of A+x+ = b+, based on the solution of Ax = b in O(1) time, it is demonstrated that the proposed method has linear convergence and it requires only 6 steps to reach a relative error about 10−4
Summary
We show the computation of an approximation of the (n + 1) × (n + 1) linear system, based on the solution of a similar system of size n × n. This paper proposes a new method that computes an approximation of A+x+ = b+, based on the solution of Ax = b in O(1) time, it is demonstrated that the proposed method has linear convergence and it requires only 6 steps to reach a relative error about 10−4. Theorem 2 and corollary 1 show that the error between x+n−j and xn−j decreases linearly with respect to j, as a consequence, the approximation of the expanded system needs few elements and the computation can be done in constant time
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