PROBABILISTIC ERROR ANALYSIS FOR INNER PRODUCTS.

Area of Science:

• Numerical Analysis
• Scientific Computing

Background:

• Numerical computation of inner products is fundamental in many scientific fields.
• Existing deterministic error bounds can be overly conservative, especially for high-dimensional vectors.
• Understanding and quantifying numerical errors is crucial for reliable scientific results.

Purpose of the Study:

• To develop novel probabilistic models for bounding the forward error in numerically computed inner products.
• To derive explicit, non-asymptotic probabilistic perturbation and roundoff error bounds.
• To establish a clear relationship between failure probability and relative error in inner product computations.

Main Methods:

• Probabilistic modeling using Azuma's inequality and martingales.
• Derivation of bounds from first principles, yielding customized condition numbers.
• Representation of roundoff errors as bounded, zero-mean random variables.

Main Results:

• Probabilistic perturbation and roundoff error bounds for sequential inner product accumulation.
• Bounds demonstrate a clear relationship between failure probability and relative error.
• Numerical experiments show probabilistic bounds are significantly more informative than deterministic ones, even for small dimensions.

Conclusions:

• The proposed probabilistic models provide more accurate and informative error bounds for inner product computations.
• Probabilistic roundoff error bounds depend on the number of operations, not vector dimension, confirming prior intuition.
• The probabilistic approach offers a superior alternative to traditional deterministic bounds for assessing numerical accuracy.