A workstation sustains about flops per second on dense arithmetic and has 64 GB of memory

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Error, Chance and the Machine, Ch Linear systems, least squares and the singular value decomposition

Problem

A workstation sustains about 5×1095\times10^{9} flops per second on dense arithmetic and has 64 GB of memory. A diffusion problem on a fine grid gives a tridiagonal, strictly diagonally dominant system Ax=b\mathsf{A}\mathbf{x} = \mathbf{b} with n=2×105n = 2\times10^{5} unknowns, stored in binary64 (8 bytes per number). ((a)) Estimate the time for dense LU factorisation, using the cost 23n3\tfrac{2}{3}n^3 flops, and the memory needed to hold A\mathsf{A} as a dense array. Can the workstation do it? ((b)) Estimate the time for the tridiagonal solver, using the cost 8n8n flops, and the memory for its three diagonals and the right-hand side. ((c)) Find the largest nn for which dense LU would finish within one hour on this machine, ignoring memory.

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