Rank-thresholded least squares by the singular value decomposition

Input $A\in {\mathbb{F}}^{m\times n}$, $b\in {\mathbb{F}}^m$, a rank threshold $\tau \ge 0$, and a finite nonnegative integer bidiagonal-QR sweep budget $N$, with $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$; vector residuals use the Euclidean norm of The norm $\parallel v\parallel =\sqrt{⟨v,v⟩}$ induced by a real or complex inner product. First compute the Householder bidiagonal reduction. If it returns range-failure, stop with inconclusive, identify the failed stage as Householder reduction, and return its supplied partial factors and stored transformations, together with any available diagnostics; do not request a bidiagonal matrix or residual reports that the failed call did not supply. Only after successful reduction call the finite-budget bidiagonal QR procedure with deflation tolerance zero and no additional sparsity cleanup. If that call returns range-failure or budget-exhausted, stop with inconclusive, identify the failed stage as bidiagonal QR, and return its supplied partial work matrix, accumulators, transformation lists, deflation record, and available residual reports. No failure or budget-exhausted branch claims a least-squares solution.

On exact-arithmetic convergence, use the exact SVD record $A=U\mathrm{\Sigma }{V}^{\ast }$ from the returned transformation products and diagonal candidates. Set $q=\mathrm{\#}\{i:{\sigma }_i>\tau \}$, form $A_{\tau }=U{\mathrm{\Sigma }}_{\tau }{V}^{\ast }$ and the canonical truncated pseudoinverse $A_{\tau }^{+}$ of The truncated singular-value pseudoinverse obtained by discarding singular values below a declared numerical-rank threshold, and compute $x_{\tau }=A_{\tau }^{+}b$. Return $x_{\tau }$, the declared threshold, $q$, the model residual $\mathrm{\parallel }{A}_{\tau }{x}_{\tau }-b{\mathrm{\parallel }}_2$, the original residual $\mathrm{\parallel }A{x}_{\tau }-b{\mathrm{\parallel }}_2$, and the discarded tail size ${\sigma }_{q+1}$ (defined as $0$ for $q=p=\min (m,n)$). The strict comparison ${\sigma }_i>\tau$ is used, so values equal to the threshold are discarded. In finite precision, use singular-value estimates $(\widehat{U},\widehat{\mathrm{\Sigma }},\widehat{V})$ to form $\widehat{I}=\{i:{\widehat{\sigma }}_i>\tau \}$, ${\widehat{A}}_{\tau }=\widehat{U}{\widehat{\mathrm{\Sigma }}}_{\tau }{\widehat{V}}^{\ast }$, and the coordinate candidate $\widehat{x}=\widehat{V}{\widehat{\mathrm{\Sigma }}}_{\widehat{I}}^{+}{\widehat{U}}^{\ast }b$, where ${\widehat{\mathrm{\Sigma }}}_{\widehat{I}}^{+}$ has diagonal entries $1\mathrm{/}{\widehat{\sigma }}_i$ for $i\in \widehat{I}$ and zero otherwise; compute both residuals and label the threshold count unverified unless a certified error interval separates every compared value from $\tau$. This finite-precision candidate is not called a certified numerical rank or a minimum-norm least-squares solution.