On the propagation of errors in the inversion of certain tridiagonal matrices

Abstract

When the differential equation of heat conduction is replaced by the implicit difference analog, one is led to the solution of Ay = b where A is a tridiagonal matrix whose elements on the principal diagonal are = 2 + 2r and whose elements off the principal diagonal are = -r. The system of equations may be solved by the following algorithm: \[ β k = u = r 2 β k − 1 − 1 , β 1 = u 1 ; γ k = − r β − 1 ; z k = ( b k + r z k − 1 ) β k − 1 , z 1 = b 1 u − 1 ; y k = z k − γ k y k + 1 , y M = z M . {\beta _k}\, = \,u\, = \,{r^2}\beta _{k - 1}^{ - 1},\,{\beta _1}\, = \,{u_1}\,;\qquad {\gamma _k}\, = \, - r{\beta ^{ - 1}};\qquad {z_k}\, = \,({b_k}\, + \,r{z_{k - 1}})\beta _k^{ - 1},\,{z_1}\, = \,{b_1}{u^{ - 1}};\qquad {y_k}\, = \,{z_k}\, - \,{\gamma _k}{y_{k + 1}},\qquad {y_M}\,=\,{z_M}. \] An upper bound of the round-off errors in the computed values of the y k {y_k} ’s is obtained. An actual test case showed that the theoretical upper bound is about four times larger than the true round-off error. Moreover, the theoretical upper bound does not seem to vary appreciably with r.