Reference · Matrices

rainbow.math.linear.matrix.affine-residual-decay

Matricessrc/rainbow/math/linear/matrix/affine_residual_decay.clj0 definitions, 4 theorems

Geometric decay and finite-horizon error bounds from computable residuals. Iterating from `F(x0)` for n steps is equivalent to the successor iterate from `x0`. Applying paired-trajectory contraction to those two initial states therefore proves d(x_n,x_(n+1)) <= (1/2)^n d(x_0,x_1). Combining this with the a posteriori residual certificate bounds the error of `x_n` using only the initial residual and the chosen horizon. Practical corollaries accept actual-row NNRAT bounds gamma <= 1/2. Native, kernel-checked, and axiom-free on D63.

Depends onmath.linear.matrix.affine-residual-certificate

matrix/affine-half-iterate-successive-residual-rate :
  (rows : List (List ℚ)) (offset initial : List ℚ)
  (step-shape : vector/same-shape initial (matrix/affine-action rows offset initial))
  (offset-shape : vector/same-shape offset (matrix/vector-multiply rows initial))
  (positive-rows : ∀ (row : List ℚ), ∀ (member : list/membership (List ℚ) rows row), vector/at-most (vector/zero-like row) row)
  (bounded-sums : ∀ (row : List ℚ), ∀ (member : list/membership (List ℚ) rows row), vector/sum row ≤ signed/from-nnrat nnrat/half)
  (steps : ℕ)
  → vector/sup-distance (iterate₂) (iterate) ≤ signed/half-power-scale steps (distance)
  where distance := vector/sup-distance initial (matrix/affine-action rows offset initial)
  where iterate := matrix/affine-iterate rows offset (data/succ steps) initial
  where iterate₂ := matrix/affine-iterate rows offset steps initial

matrix/affine-nnrat-strong-row-successive-residual-rate :
  (rows : List (List ℚ)) (offset initial : List ℚ) (gamma : ℚ≥0)
  (gamma-at-most-half : gamma ≤ nnrat/half)
  (step-shape : vector/same-shape initial (matrix/affine-action rows offset initial))
  (offset-shape : vector/same-shape offset (matrix/vector-multiply rows initial))
  (positive-rows : ∀ (row : List ℚ), ∀ (member : list/membership (List ℚ) rows row), vector/at-most (vector/zero-like row) row)
  (bounded-sums : ∀ (row : List ℚ), ∀ (member : list/membership (List ℚ) rows row), vector/sum row ≤ signed/from-nnrat gamma)
  (steps : ℕ)
  → vector/sup-distance (iterate₂) (iterate) ≤ signed/half-power-scale steps (distance)
  where distance := vector/sup-distance initial (matrix/affine-action rows offset initial)
  where iterate := matrix/affine-iterate rows offset (data/succ steps) initial
  where iterate₂ := matrix/affine-iterate rows offset steps initial

matrix/affine-nnrat-strong-row-iterate-error-from-initial-residual :
  (rows : List (List ℚ)) (offset initial fixed-point : List ℚ) (gamma : ℚ≥0)
  (gamma-at-most-half : gamma ≤ nnrat/half)
  (shape : vector/same-shape initial fixed-point)
  (offset-shape : vector/same-shape offset (matrix/vector-multiply rows initial))
  (positive-rows : ∀ (row : List ℚ), ∀ (member : list/membership (List ℚ) rows row), vector/at-most (vector/zero-like row) row)
  (bounded-sums : ∀ (row : List ℚ), ∀ (member : list/membership (List ℚ) rows row), vector/sum row ≤ signed/from-nnrat gamma)
  (fixed : vector/equivalent (action) fixed-point) (steps : ℕ)
  → vector/sup-distance (iterate) fixed-point ≤ scale + scale
  where action := matrix/affine-action rows offset fixed-point
  where iterate := matrix/affine-iterate rows offset steps initial
  where scale := signed/half-power-scale steps (vector/sup-distance initial (matrix/affine-action rows offset initial))

matrix/affine-nnrat-strong-row-iterate-stopping-certificate :
  (rows : List (List ℚ)) (offset initial fixed-point : List ℚ) (gamma : ℚ≥0)
  (gamma-at-most-half : gamma ≤ nnrat/half)
  (shape : vector/same-shape initial fixed-point)
  (offset-shape : vector/same-shape offset (matrix/vector-multiply rows initial))
  (positive-rows : ∀ (row : List ℚ), ∀ (member : list/membership (List ℚ) rows row), vector/at-most (vector/zero-like row) row)
  (bounded-sums : ∀ (row : List ℚ), ∀ (member : list/membership (List ℚ) rows row), vector/sum row ≤ signed/from-nnrat gamma)
  (fixed : vector/equivalent (action) fixed-point) (steps : ℕ) (threshold : ℚ)
  (envelope-at-most-threshold : scale + scale ≤ threshold)
  → vector/sup-distance (iterate) fixed-point ≤ threshold
  where action := matrix/affine-action rows offset fixed-point
  where iterate := matrix/affine-iterate rows offset steps initial
  where scale := signed/half-power-scale steps (vector/sup-distance initial (matrix/affine-action rows offset initial))