From 8e8ac3e4347dc8f46f7781d4da4c834913c67309 Mon Sep 17 00:00:00 2001 From: Thomas Guillod Date: Tue, 21 Jan 2025 21:46:16 -0500 Subject: [PATCH] FFT matvec doc --- docs/content/method.rst | 9 +- docs/method/peec_7.png | Bin 0 -> 125549 bytes docs/method/peec_7.svg | 5133 +++++++++++++++++++++++++++++++++++++++ 3 files changed, 5140 insertions(+), 2 deletions(-) create mode 100644 docs/method/peec_7.png create mode 100644 docs/method/peec_7.svg diff --git a/docs/content/method.rst b/docs/content/method.rst index 88326d15..985747bf 100644 --- a/docs/content/method.rst +++ b/docs/content/method.rst @@ -90,8 +90,13 @@ Hence, the computational cost and memory requirement for generating and storing Fig. 6 - Illustration of the remapping of the inductance matrix coefficients. With all the repeated coefficients, the inductance matrix is a a block-Toeplitz Toeplitz-block matrix. -For such matrices, the matrix-vector multiplications can be done with Fast Fourier Transforms. -Hence, the computational complexity of multiplications is reduced from O(n^2) to O(n*log(n)). +For such matrices, the matrix-vector multiplications can be done with Fast Fourier Transforms (:ref:`Fig. 7 `). +Hence, the Fast Fourier Transforms are reducing the computational complexity of multiplications from O(n^2) to O(n*log(n)). + +.. _fig_7: +.. figure:: ../method/peec_7.png + + Fig. 7 - Toeplitz matrix-vector multiplication with FFT. 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coefficientsStep 3. - Compute the Circulant matrix-vector multiplication with FFTStep 4. - Extract the Toeplitz matrix-vector multiplication result