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## Local Storage Schemes for Tridiagonal Matrices

A global tridiagonal matrix A , represented as three vectors (DL, D, DU), should be distributed over a one-dimensional process grid assuming a block-column data distribution. We assume that the coefficient tridiagonal matrix A is of size () and is represented by the following.

If we first assume that the matrix A is nonsymmetric (diagonally dominant like), and it is known a priori that no pivoting is required for numerical stability, the user may choose to perform no pivoting during the factorization (PxDTTRF ). If we distribute this matrix (assuming no pivoting) onto a process grid with a block size of , the processes would contain the local arrays found in figure 4.13.

Figure 4.13: Mapping of local arrays for nonsymmetric tridiagonal matrix A

Finally, a global symmetric positive definite tridiagonal matrix A , represented as two vectors (D and E), should be distributed over a one-dimensional process grid assuming a block-column data distribution.

Let us now assume that this matrix A is symmetric positive definite and that we distribute this matrix assuming lower triangular storage (UPLO='L') onto a process grid with a block size . The processes would contain the local arrays found in figure 4.14. We would then call the routine PxPTTRF to perform the factorization, for example.

Figure 4.14: Mapping of local arrays for symmetric positive definite tridiagonal matrix A (UPLO='L')

If we then distributed this same matrix assuming upper triangular storage (UPLO='U') onto a process grid with a block size , the processes would contain the local arrays found in figure 4.15.

Figure 4.15: Mapping of local arrays for symmetric positive definite tridiagonal matrix A (UPLO='U')

Note that in the tridiagonal cases, it is not necessary to maintain the empty storage positions as designated by in the narrow band routines.

The matrix of right-hand-side vectors B (for example, used in PxDTTRS and PxPTTRS ) is assumed to be a dense matrix distributed block-row across the process grid. Thus, consecutive blocks of rows of the matrix B are assigned to successive processes in the process grid, as described in section 4.4.1.

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