Many numerical methods for evaluating matrix functions can be naturally viewed as computational graphs. Rephrasing these methods as directed acyclic graphs (DAGs) is a particularly effective approach to study existing techniques, improve them, and eventually derive new ones. The accuracy of these matrix techniques can be characterized by the accuracy of their scalar counterparts, thus designing algorithms for matrix functions can be regarded as a scalar-valued optimization problem. The derivatives needed during the optimization can be calculated automatically by exploiting the structure of the DAG in a fashion analogous to backpropagation. This article describes GraphMatFun.jl , a Julia package that offers the means to generate and manipulate computational graphs, optimize their coefficients, and generate Julia, MATLAB, and C code to evaluate them efficiently at a matrix argument. The software also provides tools to estimate the accuracy of a graph-based algorithm and thus obtain numerically reliable methods. For the exponential, for example, using a particular form (degree-optimal) of polynomials produces implementations that in many cases are cheaper, in terms of computational cost, than the Padé-based techniques typically used in mathematical software. The optimized graphs and the corresponding generated code are available online.

In this paper, we consider communication over the bandwidth limited analog white Gaussian noise channel using non-orthogonal pulses. In particular, we consider non-orthogonal transmission by signaling samples at a rate higher than the Nyquist rate. Using the faster-than- Nyquist (FTN) framework, Mazo showed that one may transmit symbols carried by sinc pulses at a higher rate than that dictated by Nyquist without loosing bit error rate. However, as we will show in this paper, such pulses are not necessarily well localized in time. In fact, assuming that signals in the FTN framework are well localized in time, one can construct a signaling scheme that violates the Shannon capacity bound. We also show directly that FTN signals are in general not well localized in time. We also consider FTN signaling in the case of pulses that are different from the sinc pulses. We show that one may use a precoding scheme of low complexity, in order to remove the intersymbol interference. This leads to the possibility of increasing the number of transmitted samples per time unit and compensate for spectral inefficiency due to signaling at the Nyquist rate of the non sinc pulses. We demonstrate the power of the precoding scheme by simulations.