# Faster arbitrary-precision dot product and matrix multiplication

## TL;DR

This paper introduces faster algorithms for arbitrary-precision dot product and matrix multiplication, significantly improving performance for high-precision computations in floating-point and ball arithmetic.

## Contribution

It presents novel algorithms and implementations that enhance the speed of arbitrary-precision linear algebra operations, outperforming previous methods.

## Key findings

- Dot product is about twice as fast as previous code at several hundred bits of precision.
- At up to 128 bits, the algorithms are 3-4 times faster, with low cycle counts per term.
- The methods significantly speed up polynomial operations and linear algebra in Arb.

## Abstract

We present algorithms for real and complex dot product and matrix multiplication in arbitrary-precision floating-point and ball arithmetic. A low-overhead dot product is implemented on the level of GMP limb arrays; it is about twice as fast as previous code in MPFR and Arb at precision up to several hundred bits. Up to 128 bits, it is 3-4 times as fast, costing 20-30 cycles per term for floating-point evaluation and 40-50 cycles per term for balls. We handle large matrix multiplications even more efficiently via blocks of scaled integer matrices. The new methods are implemented in Arb and significantly speed up polynomial operations and linear algebra.