Substitution algorithms for rational matrix equations

Abstract

We study equations of the form r(X) = A, where r is a rational function and A and X are square matrices of the same size. We develop two techniques for solving these equations by inverting (through a substitution strategy) two schemes for the evaluation of rational functions of matrices. For triangular matrices, the new methods yield the same computational cost as the evaluation schemes from which they are obtained. A general equation can be reduced to upper triangular form by exploiting the Schur decomposition of the given matrix. For real data, the algorithms rely on the real Schur decomposition in order to compute real solutions using only real arithmetic. Numerical experiments show that our implementations are faster than existing alternatives without sacrificing accuracy.

The dual inverse scaling and squaring algorithm for the matrix logarithm

“…Proof. The proof is analogous to that of [3,Thm. 3.4] once one observes that the matrix in (18), which determines the applicability of Algorithm 1, is the same as that in [3, Eq.…”

“…The applicability of the algorithms depends on the existence of solutions to the equations in (9) and to the nonsingularity of the operators in (10). It has been shown in [3,Thm. 3.4] that once the diagonal blocks are computed, the three algorithms are applicable if and only if for 1 ≤ < ≤ one has that [ , ] ≠ 0, where and denote an eigenvalue of and , respectively.…”

“…1], or [3, Alg. 2] can be used to solve (8), but by exploiting the special structure of the numerator and denominator of this rational equation we can develop tailored algorithms that require fewer arithmetic operations than a straightforward application of the algorithms in [5] or [3].…”

“…As we shall see, this is not the case for the matrix function at hand. In previous work [5], [3], it has been shown that if is a quasi-triangular matrix and is a rational function, then computing a quasi-triangular solution to ( ) = has the same asymptotic computational cost as evaluating ( ). The use of quasi-triangular matrices is not a restriction: if a quasi-triangular matrix satisfies ( ) = , where = * is upper quasi-triangular and is unitary, then the matrix = * is a solution to the matrix equation ( ) = , and conversely, any solution to ( ) = that is a polynomial of can be obtained in this way [5].…”

“…In the next section we recall some key definitions and results that will be of used in later sections. In section 3 we briefly review existing substitution algorithms for rational equations and introduce our new algorithm for rational functions that can be written in the form (3). In section 4 we discuss how this technique can be used to compute the matrix logarithm in an inverse scaling and squaring fashion, and evaluate the performance of the resulting algorithm in section 5.…”