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A method for multivariate linear predictive spectral analysis, employing weighted forward and backward averaging, is presented and programmed in FORTRAN. The method constitutes a generalization of Burg's univariate algorithm to the multivariate case. The essential analytical procedure is to minimize the trace of the sum of the weighted forward and backward error matrices by choice of the partial correlation coefficients, subject to a linear matrix constraint which guarantees that the forward-extrapolated and backward- extrapolated correlation matrix estimates are Hermitians of each other. The choice of error weighting is important and is discussed. Solution of a bilinear matrix equation is required in the algorithm.