MATLAB Mathematics - SERC - Index of

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6 Sparse Matrices 0 500 1000 1500 2000 2500 3000 3500 4000 The Laplacian of the mesh. ... . ... .... .... ..... ..... ... ... ... .. ... ...... .... ......... . . .. .. ... . ... ... .. .. . .. . ... . . . . .. .. . . .. . . . . . . .. . . .. .. . ... . . ... .. . . . . ... . ..... . . ...... .... ... . . . ... .. ... . .. .. . .. . . . . ..... . . .... .. .. .. ..... .......... ....... . . . . . . . . . .... . . . .......... . ... ......... . . ....... . ........ . . ..... ... ........ . ........ ...... . ....... ...... . ........ ............... . .... .. . . . ....................... .................. . .. ............ . . ........ ..... ...... ... . . ..... .. . .... . ..... . .......... .... ....... ..... ...... . ... ...... ... . ... ..... .... . ......... . . ....... . . ... . ....... ..... .. .... ...... .. ... .. . . ... .... . .. . . .... ... .. . . . .... . .... . .... . . ...... . ... . .... . . . . . . . . ..... .... . . ........ . ... .. .. ........ . ............ . . ................ ......... .. ......... . . . . . . . . .. ..... ..... . ..... ..... ...... . . . . . . .... .... . ..... . . .. ... . . . . . . . . . ..... . . . .... .. .... . . . . .. . . . . . .. .. .. . . . .. .... .... . . ..... . . . . . ......... ......... . . .......... ....... . ......... ......... ......... ...... . . . . . ...... .. . . . ... ....... . . . . .... ...... .. . . .. ...... ..... ..... .. . . . . . . . . .... .. . ... ... . . . ........ . ... ...... ...... . .... . . . ..... . . . . . . ............ . . . . . .. . .......... ................ . ........... .......... . ........... ...... . ...... .... . .......... ........ . . . .... . . . . . ..... ........ . . ...... ..... . . ............. . ....... . ...... . . . . . ........... ................. .......... ................. ........ ......... . . . ....... .. .... .. .... ..... . ...... ....... ................... . . ......... .................... . . ........ . ............. ..... 0 1000 2000 3000 4000 nz = 28831 6-24
Sparse Matrix Operations Sparse Matrix Operations Most of the MATLAB standard mathematical functions work on sparse matrices just as they do on full matrices. In addition, MATLAB provides a number of functions that perform operations specific to sparse matrices. This section discusses: • “Computational Considerations” on page 6-25 • “Standard Mathematical Operations” on page 6-25 • “Permutation and Reordering” on page 6-26 • “Factorization” on page 6-30 • “Simultaneous Linear Equations” on page 6-36 • “Eigenvalues and Singular Values” on page 6-39 • “Performance Limitations” on page 6-41 Computational Considerations The computational complexity of sparse operations is proportional to nnz, the number of nonzero elements in the matrix. Computational complexity also depends linearly on the row size m and column size n of the matrix, but is independent of the product m*n, the total number of zero and nonzero elements. The complexity of fairly complicated operations, such as the solution of sparse linear equations, involves factors like ordering and fill-in, which are discussed in the previous section. In general, however, the computer time required for a sparse matrix operation is proportional to the number of arithmetic operations on nonzero quantities. Standard Mathematical Operations Sparse matrices propagate through computations according to these rules: • Functions that accept a matrix and return a scalar or vector always produce output in full storage format. For example, the size function always returns a full vector, whether its input is full or sparse. • Functions that accept scalars or vectors and return matrices, such as zeros, ones, rand, and eye, always return full results. This is necessary to avoid introducing sparsity unexpectedly. The sparse analog of zeros(m,n) is 6-25
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