Tests for the ScaLAPACK LLT routines

The Cholesky test program generates random symmetric test matrices with values in the interval [-1,1] and then modifies these matrices to be diagonally dominant with positive diagonal elements thus creating symmetric positive-definite matrices. It then calls the ScaLAPACK routines to factor and solve the system, and computes a solve and/or factorization residual error check to verify that each operation has performed correctly. Condition estimation and iterative refinement routines are included and optionally tested.

Specifically, each test matrix is subjected to the following tests:

• Compute the LLT factorization using PxPOTRF
• Solve the system $A X = B$ using PxPOTRS, and compute the ratio
  • SRESID $= \vert\vert AX - B\vert\vert / ( n \vert\vert A\vert\vert \, \vert\vert X\vert\vert \varepsilon )$
• IF SRESID $>$ THRESH, then compute the ratio
  • FRESID $= \vert\vert L L^T - A \vert\vert / ( n \vert\vert A \vert\vert \varepsilon )$

The expert driver (PxPOSVX) performs condition estimation and iterative refinement and thus incorporates the following additional tests:

• Compute the reciprocal condition number RCOND using PxPOCON.
• Use iterative refinement (PxPORFS) to improve the solution, and recompute the ratio
  • SRESID $= \vert\vert AX - B\vert\vert / ( n \vert\vert A\vert\vert \, \vert\vert X\vert\vert \varepsilon )$