We study the numerical stability of polynomial based encoding methods, which has emerged to be a powerful class of techniques for providing straggler and fault tolerance in the area of coded computing. Our contributions are as follows: 1) We construct new codes for matrix multiplication that achieve the same fault/straggler tolerance as the previously constructed MatDot Codes and Polynomial Codes. Unlike previous codes that use polynomials expanded in a monomial basis, our codes use a basis of orthogonal polynomials. 2) We show that the condition number of every m×m sub-matrix of an m×n, n ≥ m Chebyshev-Vandermonde matrix, evaluated on the n-point Chebyshev grid, grows as O(n 2(n−m) ) for n > m. An implication of this result is that, when Chebyshev-Vandermonde matrices are used for coded computing, for a fixed number of redundant nodes s = n − m, the condition number grows at most polynomially in the number of nodes n. 3) By specializing our orthogonal polynomial based constructions to Chebyshev polynomials, and using our condition number bound for Chebyshev-Vandermonde matrices, we construct new numerically stable techniques for coded matrix multiplication. We empirically demonstrate that our constructions have significantly lower numerical errors compared to previous approaches which involve inversion of Vandermonde matrices. We generalize our constructions to explore the trade-off between computation/communication and fault-tolerance. 4) We propose a numerically stable specialization of Lagrange coded computing. Motivated by our condition number bound, our approach involves the choice of evaluation points and a suitable decoding procedure that involves inversion of an appropriate Chebyshev-Vandermonde matrix. Our approach is demonstrated empirically to have lower numerical errors as compared to standard methods.M. Fahim and V. Cadambe are with the 1 For example, [22], reports that "In our experiments we observed large floating point errors when inverting high degree Vandermonde matrices for polynomial interpolation". 2 The master and fusion nodes are logical entities; in practice, they may be the same node, or may be emulated in a decentralized manner by the computation nodes.A simple generalization of the above example, described in Construction 1 in Section IV, leads to a class of codes, we refer to it as OrthoMatDot Codes, with recovery threshold of 2m − 1, the same recovery threshold as MatDot Codes. In general, orthonormal polynomials are defined over arbitrary weight measure 1 −1 · w(x)dx; some well known classes of polynomials corresponding to different weight measures w(x) include Legendre, Chebyshev, Jacobi and Laguerre Polynomials [20], [21] (See Section III for definitions). Our OrthoMatDot Codes in Section IV can use any weight measure, and therefore can be used with different classes of orthonormal polynomials. Of particular interest to our paper are the Chebyshev polynomials ( Fig. 4).With our basic template, the task of developing numerically stable codes boils down to (A) interpolating p A (x)p...
