Fitting straight lines and planes with an application to radiometric dating

Conventional practice in geochronology is to fit a straight line or "isochron" to data consisting of two isotopic ratios by a method (e.g. that of York, 1966, 1969, or perhaps the more modern version of Titterington and Haliday, 1979) that takes into account that fact that both ratios are measured with error. In this paper we use matrix algebra to lay out a general method for fitting linear relations between any number of variables, all subject to errors with known variances and covariances and the well-known Newton-Raphson method to do the optimization. This leads to a good computational algorithm which may also be used e.g. to check whether coefficients in several linear relations are the same. In many fields of science one needs to fit linear relations so our method is of wide utility; its use is in no way restricted to radiometric dating.