Blog
My thoughts, my ideas and something interesting
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Symmetric Storage Gaxpy
Matrix Computation ·Introduction
A matrix \(\color{black}{A\in\mathcal{R}^{n\times n}}\) is symmetric if \(\color{black}{A^{T}=A}\) and skew-symmetric if \(\color{black}{A^{T}=-A}\). Likewise, a matrix \(\color{black}{A\in\mathcal{C}^{n\times n}}\) is hermitian if \(\color{black}{A^{H}=A}\) and skew-hermitian if \(\color{black}{A^{H}=-A}\).
For such matrices, storage requirements can be halved by simply storing the lower triangle of elements.
For general \(\color{black}{n}\), we set
\[\color{black}{A.vec((n-j/2)(j-1)+i)=a_{ij} 1\le j\le i \le n}\]
for j = 1:n for... -
Band Storage Gaxpy
Matrix Computation ·Introduction
Suppose \(\color{black}{A\in\mathcal{R}^{n\times n}}\) has lower bandwith \(\color{black}{p}\) and upper banwidth \(\color{black}{q}\) and assume that \(\color{black}{p}\) and \(\color{black}{q}\) are much smaller than \(\color{black}{n}\). Such a matrix can be stored in a \(\color{black}{(p+q+1)-\text{ by }-n}\) array \(\color{black}{A.band}\) with the convention that
\[\color{black}{a_{ij}=A.band(i-j+q+1,j)}\]
for all \(\color{black}{(i,j)}\) that fall inside the band.
for j = 1:n alpha_1 = max(1, j-q), alpha_2... -
Vision Perception System of Intelligent Connected Vehicle
Project ·From Setempber 1st, 2018 to January 31st, 2019, I participated in our college project of Intelligent Connected Vehicle (ICV). In this project, I took charge of the vision perception system of the ICV. I finished the whole perception system setup and calibration of multi-sensors. Our recording platform is an electric vehicle, which has been modified with actuators for pedals (acceleration...
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Fast Matrix-Vector Products
Matrix Computation ·The Fast Fourier Transform
The discrete Fourier transform (DFT) of a vector \(\color{black}{x \in C^n}\) is a matrix-vector product \[\color{black}{y = F_nx}\] where the DFT matrix \(\color{black}{F_n = (f_{ij}) \in C^{n\times n}}\) is defined by \[\color{black}{f_{kj} = \omega^{(k-1)(j-1)}_n}\] with \[\color{black}{\omega_n = exp(-2\pi i/n) = cos(2\pi/n) - i\centerdot sin(2\pi/n).(\omega_n^n = 1)}\]
The DFT is ubiquitous throughout computational science and engineering...
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Matrix Multiplication
Matrix Computation ·The Notion of “Level” and the BLAS
The dot product and single alpha x plus y(saxpy) operations are example of level-1 operations. Level-1 operations involve an amount of data and an amount of arithmetic that area linear in the dimension of the operation. An \(\color{black}{m}\)-by-\(\color{black}{n}\) outer product update or a generalized saxpy(gaxpy) operation involves a quadratic amount of data...