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Golub and Kahan: computing singular values through bidiagonalization

Golub 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.

What is recorded here?

Golub 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. When 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.

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Scientific scope

The 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.

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People and systems

Gene Golub; William Kahan

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English catalogue narrative

Golub and Kahan: computing singular values through bidiagonalization

Golub 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.

When 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.

Scientific scope

The 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.

People and systems

Gene Golub; William Kahan

Original sources