May 29th, 2015
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Applications of linear algebraLinear algebra.Computer science applications: wavelets, transformations in computer graphics, computer vision, Google's PageRank algorithm, linear programming, linear regression, Markov chains. Other applications: linear and nonlinear optimization, control theory, combinatorial optimization, numerical solutions to ODEs, analysis of electrical networks, portfolio optimization, qunatum mechanics. Desire to solve such problems has driven the development of computing technology. BLAS.Matrix.In numerical linear algebra, amatrixis a rectangular table of real or complex numbers. Given a matrix A, we use the notation Aijto represent the entry in the ith row and the jth column. We can implement a matrix in Java by using a two dimensional array. We access AijusingA[i][j]. We begin indexing at 0 to conform to Java indexing conventions.Matrix multiplication.The product of two N-by-N matrices A and B is an N-by-N matrix C defined byThe following code fragment computes C = AB.for (int i = 0; i < N; i++) for (int j = 0; j < N; j++) for (int k = 0; k < N; k++) C[i][j] += A[i][k] * B[k][j];Row major order vs. column major orderCan have a huge impact on caching and performance. Better to iterate over a row than a column in Java. We can rearrange the matrix multiplication triple loop in any of 3! = 6 ways to achieve the same answer. Each possibility has different memory access patterns may perform very differently

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