Concept

Modified Gram–Schmidt — where it appears

2 essays name this object, across 2 fields. What follows is each of them, and the objects they name alongside it.

Two Gram–Schmidts

One argument changes. Classical Gram–Schmidt projects the original column onto each previous direction; modified projects what is left of it. In exact arithmetic the coefficients are identical. In floating point they differ by eight orders of magnitude in the thing that matters.

orthogonality · gram schmidt

The number that is re-derived

GMRES prints a residual it never computes from its answer either. On the matrix that sends a conjugate gradient recurrence 7.3·10¹⁰ wrong, and on two others chosen to be worse, its number is never more than a factor of 2.86 out — while the basis it is computed from has lost orthogonality entirely. The disease is not iterative methods, and it is not floating point.

iterative · residual gap

Named alongside it

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

Orthogonality Arnoldi iteration Backward error Givens rotation GMRES Gram–Schmidt Hilbert matrix Householder reflection Krylov subspace Loss of orthogonality Non-normality Orthogonal projection

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