Functions
	load (path)

	ldl_pivots (S)

	analyse_operator (path)

	analyse (path)

	main (dirs)

Variables
int	MAGIC = 0x50434D47444E5331

Detailed Description

WO-M — spectrum of the densely assembled multigrid preconditioner (float vs double storage).

Reads the matrices written by `mg_dense_precond` and answers, for each configuration, the ONE
question WO-M was opened on: is the V-cycle preconditioner M indefinite because the coarse operator
is a bad approximation (VOF_PLAN S3, coefficient-aware coarsening), or because the operator is
STORED in float (mac_cutcell_mg.hpp MReal) and float rounding breaks A*1 = 0 at eps_f32 x contrast?

Both builds run the identical code path on the identical problem; only MReal differs.

Method note. sym(M) is singular BY CONSTRUCTION: M's inputs are mean-removed and its output is
mean-removed at the end of the level-0 V-cycle, so the constant vector is an exact null direction.
An UNPIVOTED LDL^T on such a matrix produces one pivot at round-off whose SIGN is noise — which is
why WO-H's "1 negative pivot (-1.1e-12)" at ratio 1e2 must not be read as indefiniteness. This
script therefore reports both:
  * the raw unpivoted LDL^T pivots, for direct comparability with the WO-H table, and
  * the eigenvalues of sym(M) deflated onto the mean-free subspace (P sym(M) P, P = I - 11^T/n),
    where a negative eigenvalue that is not at round-off is unambiguous indefiniteness.

Usage: python mg_precond_analyze.py <dir-with-*.bin> [more dirs...]

Function Documentation

◆ load()

mg_precond_analyze.load ( path )

Definition at line 32 of file mg_precond_analyze.py.

Referenced by analyse(), and analyse_operator().

◆ ldl_pivots()

mg_precond_analyze.ldl_pivots ( S )

Unpivoted LDL^T pivots of a symmetric matrix (the WO-H instrument, reproduced).

Definition at line 42 of file mg_precond_analyze.py.

Referenced by analyse().

◆ analyse_operator()

mg_precond_analyze.analyse_operator ( path )

kappa(A) of the FINE operator on the mean-free subspace, and the attainable-accuracy floor it
implies. Classical result (Greenbaum; Higham NLA ch.19): a CG-type iteration cannot drive the
TRUE relative residual below O(eps * kappa(A)) whatever the preconditioner, because round-off in
the recurrence enters at that level. So this column says what a solve on this operator could
ever reach in each storage precision — the ceiling any precision policy is measured against, and
the reason a fixed rtol of 1e-8 stops being meaningful once kappa exceeds ~1e8 in fp64.

Definition at line 57 of file mg_precond_analyze.py.

References load().

Referenced by main().

◆ analyse()

mg_precond_analyze.analyse ( path )

Definition at line 79 of file mg_precond_analyze.py.

References ldl_pivots(), and load().

Referenced by main().

◆ main()

mg_precond_analyze.main ( dirs )

Definition at line 119 of file mg_precond_analyze.py.

References analyse(), analyse_operator(), and main().

Variable Documentation

◆ MAGIC

int mg_precond_analyze.MAGIC = 0x50434D47444E5331

Definition at line 29 of file mg_precond_analyze.py.

Functions

Variables