A convergent algorithm is introduced which solves the exact system relating the Lagrange multipliers and the projections using the successive relaxation method without any approximation. The system gives the minimum-energy solution which is different from that of the convolution algorithm, but it is the same as the ART reconstruction except that it discretises the Lagrange multipliers instead of the image. Its reconstructions are substantially better than those of the convolution algorithm. Although the new algorithm uses the exact matrix of this system, its speed is very high because it utilises the special matrix structure; namely the areas of equal elements. The statistics of the error between the ‘true’ projections and the projections corresponding to the reconstruction can be made the same as the noise statistics in the projection data, but the algorithm does not aim to produce the minimum norm solution to the problem of fitting noisy data. The number of projections N should be related to the number of measurements per projection P by P = vN/2, where v is an integer. Computed results verifying these conclusions are included. The algorithm can also use the object boundaries to improve its reconstruction.