SAME RESIDUAL
Same residual size. Different error.
The equation misses by the same amount. Turn the direction of that miss, and the solution can move a hundred times farther from the answer.
A circle of misses. A narrow ellipse of errors.
Fixed axes · Euclidean norms · direct algebra
Drag around the residual circle, or use the direction slider and its arrow keys. The error point follows through A⁻¹.
Circle · ‖r‖₂ = 1Equal scale on both axes
Ellipse · horizontal radius 0.01Equal scale on both axes
The bound stays. The error moves.
At 45°, the relative residual is 1% and the relative solution error is 70.714%. The condition-number bound remains 100%.
A condition number bounds relative amplification. It does not say that every residual direction attains that bound.
r = b − A x̂ · e = x − x̂
‖e‖₂ / ‖x‖₂ ≤ κ₂(A) · ‖r‖₂ / ‖b‖₂
- Relative residual · fixed
- 1.000%
- Relative error · actual
- 70.714%
- Relative amplification
- 70.714×
- Condition-number bound
- 100.000%
Relative amplification = relative error / relative residual. It ranges from 1× to 100× in this specimen.
- Residual r
- (0.707107, 0.707107)
- Error e
- (0.007071, 0.707107)
- Approximation x̂
- (0.992929, −0.707107)
- Exact answer x · fixed
- (1, 0)
- Residual norm ‖r‖₂ · fixed
- 1.000000
- Error norm ‖e‖₂
- 0.707142
Read the study
One equation, one fixed scale
A = diag(100, 1), b = (100, 0), and the exact solution x = (1, 0) never change. The singular values of A are 100 and 1, so its Euclidean condition number κ₂(A) is 100. Coordinates and norms are fixed throughout.
The residual r = (cos θ, sin θ) has norm 1. Since ‖b‖₂ = 100, its relative size is always 1%. Each panel uses equal horizontal and vertical units and never rescales. The panels represent different spaces: residuals in equation space, errors in solution space.
A residual is not the solution error
The residual is what remains when an approximate solution is substituted into the equation: r = b − A x̂. The error is the difference from the exact solution: e = x − x̂. Because Ax = b, Ae = r and e = A⁻¹r.
Here A⁻¹ = diag(0.01, 1). The residual circle maps to an error ellipse with horizontal radius 0.01 and vertical radius 1. It is drawn at that true aspect ratio, even though it looks almost like a line. Point markers are enlarged for visibility.
What the condition number promises
For an invertible A and nonzero b, ‖e‖₂ / ‖x‖₂ ≤ κ₂(A) ‖r‖₂ / ‖b‖₂. This follows from e = A⁻¹r, b = Ax, and the two operator-norm bounds. The inequality holds for every residual direction.
This specimen attains the upper bound on the second axis. On the first axis, the relative error is 1%, a hundred times smaller. The measured relative amplification is direction-dependent; κ₂ is the uniform bound. Absolute error/residual ratios here instead range from 0.01 to 1.
What this exhibit does not test
The page constructs x̂ directly as x − A⁻¹r. It does not run an iterative solver, inject rounding errors, compare algorithms, or measure their stability. The varying error is a property of this fixed equation and these residual directions.
All states follow the stated algebra. Ordinary floating-point trigonometry and rounded readouts are used for display. Equality of residual norms is mathematical; the displayed values agree to the shown precision.