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6 Sparse Matrices Function Summary (Continued) Category Function Description gmres lsqr minres pcg qmr symmlq Generalized Minimum Residual Method. LSQR implementation of Conjugate Gradients on the Normal Equations. Minimum Residual Method. Preconditioned Conjugate Gradients Method. Quasi-Minimal Residual Method. Symmetric LQ method Miscellaneous spaugment Form least squares augmented system. spparms symbfact Set parameters for sparse matrix routines. Symbolic factorization analysis. 6-4
Introduction Introduction Sparse matrices are a special class of matrices that contain a significant number of zero-valued elements. This property allows MATLAB to: • Store only the nonzero elements of the matrix, together with their indices. • Reduce computation time by eliminating operations on zero elements. This section provides information about: • Sparse matrix storage • General storage information • Creating sparse matrices • Importing sparse matrices Sparse Matrix Storage For full matrices, MATLAB stores internally every matrix element. Zero-valued elements require the same amount of storage space as any other matrix element. For sparse matrices, however, MATLAB stores only the nonzero elements and their indices. For large matrices with a high percentage of zero-valued elements, this scheme significantly reduces the amount of memory required for data storage. MATLAB uses a compressed column, or Harwell-Boeing, format for storing matrices. This method uses three arrays internally to store sparse matrices with real elements. Consider an m-by-n sparse matrix with nnz nonzero entries stored in arrays of length nzmax: • The first array contains all the nonzero elements of the array in floating-point format. The length of this array is equal to nzmax. • The second array contains the corresponding integer row indices for the nonzero elements stored in the first nnz entries. This array also has length equal to nzmax. • The third array contains n integer pointers to the start of each column in the other arrays and an additional pointer that marks the end of those arrays. The length of the third array is n+1. 6-5
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Index twobvp demo 5-63 two-dimensio
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