QRF, direct rotations
Under construction. This chapter presents a developing rotation-based QR method and identifies numerical and algorithmic questions that still require investigation.
Can one rotation eliminate every subdiagonal entry in an active column?
At each step, a direct plane rotation aligns the active column with its leading coordinate vector. This zeros all entries below the diagonal position at once, after which the completed row and column are set aside and the process repeats on the trailing submatrix. The chapter derives the rotation geometrically, simplifies its block form and develops an update that applies the rotation without explicitly constructing the full matrix.
Key ideas
Instead of eliminating one entry at a time, the method rotates an entire active column directly into its coordinate direction.
- For each active column, construct a rotation in the plane spanned by that column and the first active coordinate vector
- The rotation has determinant 1, is orthogonal and leaves every vector perpendicular to the rotation plane unchanged
- After the rotation, the active column is a scalar multiple of its first coordinate vector, so all subdiagonal entries in that column are zero
- The completed row and column are deflated, and the same construction is repeated on the trailing submatrix
- The rotation can be written as U = I + B(J − I₂)Bᵀ and simplified into a block form determined by a few scalar and vector quantities
- The product UA can be updated directly without explicitly constructing U
- For dense square matrices, the estimated arithmetic cost lies between Givens rotations and Householder reflections under the stated weighting assumptions
- Numerical stability, sparse-matrix efficiency and possible hybrid algorithms remain open questions
Why does one rotation eliminate the whole active part of a column?
The active column and its first coordinate vector determine a two-dimensional plane. Rotating inside that plane can align the column with the coordinate vector while leaving the orthogonal complement fixed. Once aligned, the column has only its leading active entry, so every entry below it is zero simultaneously.
Related chapters
Examples with visualizations
QR factorization by direct plane rotations
One direct rotation aligns each active column with its coordinate direction while leaving the orthogonal complement unchanged
Chapter contents
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How does this method compare with Givens and Householder QR?
Like Householder QR, one step completes the active part of a column; like Givens QR, each step is a proper rotation rather than a reflection. The operation-count comparison in the chapter suggests lower dense arithmetic cost than standard Givens rotations but higher cost than Householder reflections. These are analytical estimates, not measured execution times, and the method’s numerical stability and sparse performance have not yet been established.