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A new method to treat the inherent instability of Lanczos-type algorithms is introduced. It enables us to capture the properties of the sequence of iterates generated by a Lanczos-type algorithm by interpolating on this sequence of points. The interpolation model found is then used to generate a point that is outside the range. It is expected that this new point will link up the rest of the sequence of points generated by the Lanczos-type algorithm if breakdown does not occur. However, becausedoi:10.5614/j.math.fund.sci.2018.50.2.4 fatcat:73hhowm7w5a7bgdtdt35kbgfd4