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Computed reduced QR least squares is backward stable for full column rank

Statement

Let $A \in R^{m \times n}$ , $m \geq n \geq 1$ , have full column rank. Successful scaled Householder QR gives the exactly orthonormal reduced representative $\hat{Q}$ and computed triangular $\hat{R}$ with $A + Δ_{Q} = \hat{Q} \hat{R}, ∥ Δ_{Q} ∥_{F} \leq β_{Q} ∥ A ∥_{F}, β_{Q} = (1 + κ_{m})^{n} - 1,$ where $κ_{m}$ is the proved local-step coefficient. Apply the stored reflectors to $b$ by rounded rank-one updates and let $\hat{y}$ be the first $n$ coordinates. Set $β_{b} = β_{Q}$ . Alternatively, if an explicitly supplied exactly orthonormal $\hat{Q}$ has entries admissible as arithmetic inputs and $\hat{y} = fl ({\hat{Q}}^{T} b)$ uses sequential rounded dots, set $β_{b} = \sqrt{n} γ_{m}$ . Solve $\hat{R} \hat{x} = \hat{y}$ by successful rounded back substitution. Suppose $m u < 1$ and $γ_{n + 3} < 1 / 8$ . There exist $δ b$ and upper triangular $Δ R$ with $(\hat{R} + Δ R) \hat{x} = {\hat{Q}}^{T} (b + δ b), ∥ δ b ∥_{2} \leq β_{b} ∥ b ∥_{2}, ∥ Δ R ∥_{F} \leq γ_{n + 3} ∥ \hat{R} ∥_{F} .$ If $\hat{R} + Δ R$ is nonsingular, $\hat{x}$ is the unique least-squares solution for $\tilde{A} = \hat{Q} (\hat{R} + Δ R)$ and $\tilde{b} = b + δ b$ . The relative matrix perturbation is at most $β_{Q} + γ_{n + 3} (1 + β_{Q})$ ; the right-hand perturbation is at most $β_{b}$ . For fixed $m, n$ , both perturbation coefficients are $O (u)$ . If additionally $β_{Q} \leq 19 γ_{n + 3} / (1 + γ_{n + 3})$ and $β_{b} \leq 20 γ_{n + 3}$ , both relative data perturbation bounds are at most $20 γ_{n + 3}$ . These two coefficient comparisons are additional checks; they do not follow from $m u < 1$ and $γ_{n + 3} < 1 / 8$ alone. Success and nonsingularity are explicit hypotheses.

Facts & Assumptions

Given: The stated successful QR, right-hand-side evaluation, and triangular solve; all nonzero exact intermediates are normal.

[F1]

The scaled Householder QR implementation is backward stable supplies the proved QR bound and the stored-reflector application bound.

[F2]

Rounded triangular substitution has a componentwise matrix backward error supplies the rounded triangular matrix perturbation, including the zero-solution branch.

[F3]

Reduced QR over the reals solves full-column-rank least squares without squaring the condition number identifies exact full-rank reduced-QR least squares.

[F4]

If |delta_i| <= u and n u < 1, then prod (1+delta_i) = 1 + theta_n with |theta_n| <= gamma_n bounds the coefficient errors of sequential real dots.

[F5]

Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs: for real Euclidean vectors, $∣ ⟨ v, w ⟩ ∣ \leq ∥ v ∥_{2} ∥ w ∥_{2}$ .

Proof

1.1givenF1F4F5algebraconstruct

For stored-reflector evaluation, [F1] bounds $∥ \hat{y} - {\hat{Q}}^{T} b ∥_{2}$ by $β_{Q} ∥ b ∥_{2}$ . In the explicit admissible-input alternative, dot expansion by [F4] gives coordinate errors at most $γ_{m} \sum_{i} ∣ {\hat{q}}_{i j} b_{i} ∣ \leq γ_{m} ∥ b ∥_{2}$ by Cauchy-Schwarz and unit column norms; combining coordinates gives $\sqrt{n} γ_{m} ∥ b ∥_{2}$ . In either case put $z = \hat{y} - {\hat{Q}}^{T} b$ and $δ b = \hat{Q} z$ . Exact orthonormality gives ${\hat{Q}}^{T} δ b = z$ and $∥ δ b ∥_{2} = ∥ z ∥_{2}$ .

2.1F2step 1.1algebra

Apply [F2] to the actual rounded back substitution. It supplies upper triangular $Δ R$ with $(\hat{R} + Δ R) \hat{x} = \hat{y}$ and the displayed Frobenius estimate. This holds also when $\hat{x} = 0$ , since the same row identities force $\hat{y} = 0$ . Combined with step 1.1 this proves the perturbed normal-coordinate equation.

3.1F1F3step 1.1step 2.1algebra∎

Under the stated nonsingularity hypothesis, $\tilde{A} = \hat{Q} (\hat{R} + Δ R)$ has full column rank, and [F3] identifies $\hat{x}$ as its unique least-squares minimizer for $b + δ b$ . Finally $∥ \hat{R} ∥_{F} = ∥ A + Δ_{Q} ∥_{F} \leq (1 + β_{Q}) ∥ A ∥_{F}$ , and $\tilde{A} - A = Δ_{Q} + \hat{Q} Δ R$ gives the stated relative matrix bound. At fixed dimensions, [F1] gives $β_{Q} = O (u)$ , and $γ_{m} = O (u)$ gives $β_{b} = O (u)$ in either right-hand-side branch, so the displayed coefficients are $O (u)$ . Under the additional coefficient checks, write $g = γ_{n + 3}$ ; then $β_{Q} + g (1 + β_{Q}) = (1 + g) β_{Q} + g \leq 19 g + g = 20 g$ , while the assumed $β_{b} \leq 20 g$ gives the right-hand bound. No comparison between $β_{Q}$ and $g$ is inferred from arithmetic success alone.

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Sources

Fundamentals of Numerical Computation, floating-point arithmetic (standard reference, not scraped)
Fundamentals of Numerical Computation, Householder triangularization (standard reference, not scraped)
Fundamentals of Numerical Computation, QR factorization and least squares (standard reference, not scraped)
Stuart and Voss, Matrix Analysis and Algorithms, Warwick lecture notes (standard reference, not scraped)