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Matrix Solvers Direct Methods

This document explores matrix structures and their mathematical formulations, particularly focusing on numerical methods including Finite Elements, Finite Differences, and Boundary Element Methods. It provides detailed discussions on various matrix types—Penta-Diagonal, Tri-Diagonal, and Three-Dimensional problems. The text covers strategies for Gaussian processes, LU factorization, and the importance of row operations. Additionally, it introduces open-source libraries such as Atlas, PlaPack, Scalapack, and PETSc. Ultimately, it emphasizes optimizing computational performance for large, dense systems.

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Matrix Solvers Direct Methods

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  1. Matrix SolversDirect Methods Phil Bording Memorial University

  2. What makes the matrix structure? The mathematical formulation!! Finite Elements Finite Differences Boundary Element Methods

  3. I, j+1 i-1,j-1 I,j i+1,j I,j-1 Penta-Diagonal - FIVE Star Pattern Two Dimensional Problems

  4. I, j+1 i-1,j-1 I,j i+1,j I,j-1 i+1,j-1 Tri – tri Diagonal - Nine Star Pattern Two Dimensional Problems

  5. I, j+1 i-1,j-1 I,j i+1,j I,j-1 i+1,j-1 Add k to index list, I,j,,k Tri – tri – tri Diagonal - 15 Star Pattern Three Dimensional Problems

  6. Boundary Element No diagonals – totally dense matrix

  7. Discussion Time

  8. Gaussian Processes Use row operations to make a triangle matrix. Only does right had sides at time of solve Factor Matrix into A = L U ( two triangular matrices) Why do row operations ? – because the simple high school problems were too small

  9. Gaussian Processes How about using matrix multiply? N^3 work with N^2 I/O – Fastest thing we can do on a computer – 2-4 gigaflops on PC 33 gigaflops on Cell Processor Factor Matrix into A = L U ( two triangular matrices) But L = L 0 So we have block matrix operations A L Inside the LU factorization

  10. Open Source Codes • Atlas – uses automatic learning to set matrix block sizes to maximize performance of system cache, etc. • PlaPack – written to make solving large dense systems easy. High speed matrix multipy • Scalapack – sparse matrix solvers • PetSc – family of solvers • And many others – do a web search

  11. Open Source Codes • Mesh generators • 2D triangles • 3D tetrahedral - hex • And many others – do a web search

  12. Programs • First Session Run Gauss.f on system – in your directory Build code for matrix - vector multiplication. • Second Session Modify MXM to do Gaussian elimination

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