Stored inverseH4−1
0.744211-0.189431-0.000055-0.135110-0.1894310.798436-0.2541680.052016-0.000055-0.2541680.801594-0.195230-0.1351100.052016-0.1952300.736350
The calculation begins from this stored inverse.
Frozen-state inverse downdate
Remove one observation without refitting the others
Select a point to trace its stored fields out of the model, retain the other robust weights, and verify the reduced state.
Stored inverse
Use the stored inverse as the input to the principal-submatrix downdate.
Removal ledger for x₄
No inner iteration is required; retained u and C values stay fixed.
| State | Observation | u | C | System | Inverse | α | Verification |
|---|---|---|---|---|---|---|---|
| Before | x₁, x₂, x₃, x₄ | [-1.000000, -1.000000, -0.993071, -0.976401] | [1.000000, 1.000000, 0.993071, 0.976401] | H₄ | stored H₄⁻¹ | [0.256389, 0.248592, 0.215161, 0.279858] | converged |
| Remove | x₄ | -0.976401 | 0.976401 | delete row/column 4 | rank-one downdate | recompute on retained set | ✓ |
| After | x₁, x₂, x₃ | [-1.000000, -1.000000, -0.993071] | [1.000000, 1.000000, 0.993071] | H−4 | H−4−1 | [0.372600, 0.277051, 0.350349] | 2.78 × 10⁻17 |
Selected downdate
What changes when x4 is removed?
Only the selected row, column, and coefficient entry are removed. All retained u and C values remain unchanged.
Retained systemH−4
1.5000000.4000000.2000000.4000001.5000000.5000000.2000000.5000001.503488
Rows and columns correspond to x₁, x₂, x₃.
Downdated inverseH−4−1
0.719420-0.179886-0.035877-0.1798860.794762-0.240377-0.035877-0.2403770.749832
Matches direct inversion to floating-point precision.
Retained-fit cross-check
Removal versus adding the retained observations with the fit
The reference starts empty and adds x₁, x₂, x₃ in their original order using the same stored converged u and C values.
Fit step 1Add x1u = -1.000000 · C = 1.000000
Fit step 2Add x2u = -1.000000 · C = 1.000000
Fit step 3Add x3u = -0.993071 · C = 0.993071
The removal and retained-point fit states match
The comparison includes every stored or recomputed model variable, not only the inverse.
| Variable | After removal | Fit by adding retained terms | Maximum difference |
|---|---|---|---|
| u | [-1.000000, -1.000000, -0.993071] | [-1.000000, -1.000000, -0.993071] | 0 |
| C | [1.000000, 1.000000, 0.993071] | [1.000000, 1.000000, 0.993071] | 0 |
| H | Retained principal matrix H−4 | H built along the 3-term fit path | 0 |
| H−1 | Rank-one downdated inverse | Direct inverse of the fitted H | 2.78 × 10⁻17 |
| α | [0.372600, 0.277051, 0.350349] | [0.372600, 0.277051, 0.350349] | 1.11 × 10⁻16 |
Removal H−4
1.5000000.4000000.2000000.4000001.5000000.5000000.2000000.5000001.503488
Added-terms fit H
1.5000000.4000000.2000000.4000001.5000000.5000000.2000000.5000001.503488
Removal H−4−1
0.719420-0.179886-0.035877-0.1798860.794762-0.240377-0.035877-0.2403770.749832
Added-terms fit H−1
0.719420-0.179886-0.035877-0.1798860.794762-0.240377-0.035877-0.2403770.749832
This is a frozen-state fit comparison: the retained u and C values are added back exactly as stored. It does not re-optimize historical robust weights.
Principal-submatrix downdate
The removal method in three steps
A removal introduces no new robust variable and therefore has no inner convergence loop.
Partition around the removed point
H−1 = [ E f ; f⊤ g ]
Conceptually move the selected observation to the final row and column.
Downdate the retained inverse
H−r−1 = E − ff⊤/g
The rank-one correction produces the inverse of the retained principal submatrix.
Refresh retained coefficients
α−r = H−r−1e / (e⊤H−r−1e)
Here every kernel diagonal is one, so this normalized form applies and e⊤α = 1.
Removal verification
Every selectable downdate matches the retained fit
The comparison is recomputed for the selected observation using unrounded matrix values.
Maximum inverse difference
Rank-one downdate versus direct inversion of the fitted retained system.
u retainedC retainedH exact
Retained α entries sum to one
The equality constraint is preserved after recomputing the coefficient vector.
3 entriesnormalized
Historical u and C changes
Every retained observation’s robust state is copied forward without another inner loop.
0 iterations3 retained