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

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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.