High-contrast homogenization: Consistency.PrintedForm #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The intrinsic contrast and Λ_0 in exactly their printed form #
This module checks the definitions blockContrast and bigLambdaRef, together
with the derived constants lambdaRef and aspectRatio: the development's
encodings of the intrinsic contrast, the reference constant Λ_0, the reciprocal
λ_0, and the aspect ratio Π of e.Theta.m and e.reference.aspect.ratio.
If that check failed, those encodings would be infima of Loewner scalings with
nothing tying them to the objects the reference text prints, and the mismatch
would propagate through the development. The intrinsic contrast blockContrast
could read 0 for a reason unrelated to the block, so the annealed contrast
annealedContrast of e.Theta.m would not be the paper's Θ_m, and the
conclusion annealedContrast P mEnt - 1 ≤ δ of p.response.transfer would
constrain the wrong quantity and could hold vacuously; the aspect ratio
aspectRatio would be the quotient of two quantities the paper does not name, so
the Π-bounds of t.polynomial.entry would bound the wrong constant.
This module is a consistency check of those definitions and is not a result of
the paper. The reference text defines the intrinsic contrast and the reference
constant Λ_0 as minima over skew matrices of spectral norms, while the
encodings are infima of Loewner scalings. Three separate facts are needed to
identify the two readings, and all three are available on a symmetric positive
definite doubled block matrix:
- the Loewner-scaling infimum is the infimum of the family of conjugated norms;
- that infimum is attained, so it is a minimum;
- the members of the family are spectral norms, because the conjugated skew-corrected Schur form is symmetric positive semidefinite.
This file composes them, so that the encodings are stated to be the printed objects with nothing left implicit.
Λ_0 is exactly the printed object: the least value of
|σ_0 + (k_0 - h)ᵗ σ_{*,0}^{-1} (k_0 - h)| over the skew matrices, the bars
being the spectral norm.
The intrinsic contrast is exactly the printed object: the least value of
|σ_*^{-1/2} (σ + (k - h)ᵗ σ_*⁻¹ (k - h)) σ_*^{-1/2}| over the skew matrices,
the bars being the spectral norm.
The printed value of Λ_0 is realized at some skew matrix.
The printed value of the intrinsic contrast is realized at some skew matrix.
The lower reference constant λ_0 = |σ_{*,0}^{-1}|^{-1} of
e.reference.aspect.ratio reads its spectral norm on the lower-right block, which
is positive semidefinite on a symmetric positive definite reference block.
The reference aspect ratio Π is exactly the printed quotient
Λ_0 / λ_0 of e.reference.aspect.ratio: both constants are read as printed, the
numerator being a minimum of spectral norms realized at a skew matrix and the
denominator the reciprocal spectral norm of the lower-right block.