Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Factorisation — where it appears

Writing a matrix as a product of matrices with structure, which is what makes a solve cheap and what a residual is printed to check. It is what makes a solve cheap, and on this site it is never drawn without the residual that says how far the product of its factors is from the matrix.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

024681012141610⁻¹1index kλthe transformthe eigensolverC = F* Λ F is a factorisationworst relative disagreement2.6·10⁻¹⁵‖Cx − b‖/‖b‖ from the transform solve4.7·10⁻¹⁶imaginary part of a real spectrum1.2·10⁻¹⁶n = 16, and the whole matrix is 16 numberseigenvectors known in advance

The matrix that is one row

A circulant of size 16 is sixteen numbers, has no zero entry anywhere, and hands over its entire spectrum in closed form — the discrete Fourier transform of its first column, exactly. An eigensolver spends a sweep of Jacobi rotations over 256 entries arriving at the same answer, and agrees to 1.2·10⁻¹⁵.

structure · Circulant 10¹10³10⁵10⁷10⁹10¹¹110¹10²10³10⁴condition number of the matrixgrowth factorbound 2^11partial pivotingCholeskyno pivot to gain fromCholesky growth, every κ1Cholesky interchanges0partial pivoting, worst9the bound, 2^112048both eliminations reach the same growthand only one of them had to swap to get there

A factorisation with nothing to pivot for

Cholesky's growth factor is not bounded by one. It is equal to one, at every size and every condition number, and the two-line reason is why the algorithm needs no pivoting at all — not "usually gets away without it". Its only failure is the square root of a non-positive number, which is exactly the test for definiteness, and in floating point that test moves with the precision.

elimination · Cholesky

Named alongside it

The objects these essays reach for when they reach for this one.

CholeskyCirculant matrixCondition numberDiscrete fourier transformEigenvaluesFast fourier transformGaussian eliminationGrowth factorLDLᵀ factorisationMatrix structurePartial pivotingPositive definite

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