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      "statement": "# Golub and Kahan: computing singular values through bidiagonalization\n\nGolub and Kahan develop a two-stage scheme that reduces a rectangular matrix to bidiagonal form, then computes its singular values and paired vectors for pseudoinverse and least-squares calculations.\n\nWhen a matrix is nearly singular, looking only at its triangular diagonal can hide the small directions that make a solution unstable. Forming the normal matrix can further damage that information through roundoff. Golub and Kahan first reduce the original matrix with geometry-preserving transformations, then work with the simpler bidiagonal representation. The resulting singular values help a user decide which directions are too uncertain to invert. Such a decision changes the problem by a controlled perturbation; it is not automatic evidence that the discarded directions are exactly zero. The authors also identify an incomplete robustness argument in their deflation step, which this record preserves.\n\n## Scientific scope\n\nThe target is the stated bidiagonalization and singular-value computation for finite matrices. Reduction, iterative diagonalization, convergence/stopping and floating-point behavior are distinct steps. Exact orthogonal transformations do not guarantee accurate small singular values under every numerical condition. Later stable implementations and performance improvements remain separate.\n\n## People and systems\n\nGene Golub; William Kahan\n\n\n\n## Original sources\n\n- [G. Golub and W. Kahan, Calculating the Singular Values and Pseudo-Inverse of a Matrix (1965), author-posted original](https://people.eecs.berkeley.edu/~wkahan/GlbKhnSv.pdf)\n- [SIAM original target bibliography and publication/submission history for Golub-Kahan1965](https://epubs.siam.org/doi/abs/10.1137/0702016?journalCode=sjnaam.1)\n- [Stanford-hosted Trefethen classic numerical-paper list and1993course account](https://web.stanford.edu/class/cme324/classics/)\n- [William Kahan institutional author homepage: target1965publication](https://people.eecs.berkeley.edu/~wkahan/index.htm)\n- [Gene Golub institutional course header and numerical linear algebra topics](https://web.stanford.edu/class/cme324/)\n",
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