======== Discrete ======== The ``discrete`` module in SymPy implements methods to compute discrete transforms and convolutions of finite sequences. .. automodule:: sympy.discrete Since the discrete transforms can be used to reduce the computational complexity of the discrete convolutions, the ``convolutions`` module makes use of the ``transforms`` module for efficient computation (notable for long input sequences). Transforms ---------- .. module:: sympy.discrete.transforms This section lists the methods which implement the basic transforms for discrete sequences. Fast Fourier Transform ^^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: fft .. autofunction:: ifft Number Theoretic Transform ^^^^^^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: ntt .. autofunction:: intt Fast Walsh Hadamard Transform ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: fwht .. autofunction:: ifwht MÃ¶bius Transform ^^^^^^^^^^^^^^^^ .. autofunction:: mobius_transform .. autofunction:: inverse_mobius_transform Convolutions ------------ .. module:: sympy.discrete.convolutions This section lists the methods which implement the basic convolutions for discrete sequences. Convolution ^^^^^^^^^^^ This is a general method for calculating the convolution of discrete sequences, which internally calls one of the methods ``convolution_fft``, ``convolution_ntt``, ``convolution_fwht``, or ``convolution_subset``. .. autofunction:: convolution Convolution using Fast Fourier Transform ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: convolution_fft Convolution using Number Theoretic Transform ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: convolution_ntt Convolution using Fast Walsh Hadamard Transform ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. autofunction:: convolution_fwht Subset Convolution ^^^^^^^^^^^^^^^^^^ .. autofunction:: convolution_subset Covering Product ^^^^^^^^^^^^^^^^ .. autofunction:: covering_product Intersecting Product ^^^^^^^^^^^^^^^^^^^^ .. autofunction:: intersecting_product