**1. Introduction**

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Let *A* ∈ R*n*×*n* be a given symmetric positive definite matrix and *x* ∈ R*<sup>n</sup>*. We are interested in estimating the quadratic forms of the type *<sup>x</sup>TA*−*mx*, *m* ∈ N. Our main goal was to find an efficient and cheap approximate evaluation of the desired quadratic form without the direct computation of the matrix *A*−*m*. As such, we revisited the approach for estimating the quadratic form *<sup>x</sup>TA*−1*x*, developed in [1], and extended it to the case of an arbitrary negative power of *A*.

The computation of quadratic forms is a mathematical problem with many applications. Indicatively, we refer to some usual applications.


adjacency matrix of the network, 0 < *a* < 1 *ρ*(*A*), and *ρ*(*A*) is the spectral radius of *A*. This matrix is referred to as a resolvent matrix, see, for example, [4] and the references therein.

Numerical analysis: Quadratic forms arise naturally in the context of the computation of the regularization parameter in Tikhonov regularization for solving ill-posed problems. In this case, the matrix has the form *AA<sup>T</sup>* + *λIn*, *λ* > 0. In the literature, many methods have been proposed for the selection of the regularization parameter *λ*, such

**Citation:** Mitrouli, M.; Polychronou, A.; Roupa, P.; Turek, O. Estimating the Quadratic Form *x<sup>T</sup> A*−*mx* for Symmetric Matrices: Further Progress and Numerical Computations. *Mathematics* **2021**, *9*, 1432. https:// doi.org/10.3390/math9121432

Academic Editor: Mariano Torrisi

Received: 31 May 2021 Accepted: 16 June 2021 Published: 19 June 2021

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as the discrepancy principle, cross-validation, generalized cross-validation (GCV), L-curve, and so forth; see, for an example, [5] (Chapter 15) and references therein. These methods involve quadratic forms of type *x<sup>T</sup>*(*AA<sup>T</sup>* + *<sup>λ</sup>In*)−*mx*, with *m* = 1, 2, 3.

In practice, exact computation of a quadratic form is often replaced using an estimate that is faster to evaluate. Regarding its numerous applications, the estimation of quadratic forms is an important practical problem that has been frequently studied in the literature. Let us indicatively refer to some well-known methods. A widely used method is based on Gaussian quadrature [5] (Chapter 7) and [6]. Moreover, extrapolation procedures have been proposed. Specifically, in [7], families of estimates for the bilinear form *xTA*−<sup>1</sup>*y* for any invertible matrix, and in [8], families of estimates for the bilinear form *y*∗ *f*(*A*)*x* for a Hermitian matrix were developed.

In the present work, we consider alternative approaches to this problem. To begin, notice that the value of the quadratic form (*<sup>x</sup>*, *<sup>A</sup>*−*mx*) is proportional to the second power of the norm of *x*. Therefore, the task of estimating (*<sup>x</sup>*, *<sup>A</sup>*−*mx*) consists of two steps:

1. Finding an *α* such that

$$(\mathbf{x}, A^{-\mathfrak{m}}\mathbf{x}) \approx \mathfrak{a} \left\|\mathbf{x}\right\|^2. \tag{1}$$

2. Assessing the absolute error of the above estimate, i.e., determining a bound for the quantity

$$\left| \left| a \right| \left| \mathbf{x} \right| \right|^2 - \left( \mathbf{x} , A^{-m} \mathbf{x} \right) \Big| . \tag{2}$$

In Section 2, we present the upper bounds for the absolute error (2) for any given *α*. Section 3 is devoted to estimates of the value *α* in (1) using a projection method. In Section 4, we use bounds from Section 2 as a stepping stone for estimating *xTA*−*mx* using the minimization method. A heuristic approach is outlined in Section 5. In Section 6, we briefly describe two methods that were used in previous studies, namely, an extrapolation approach and another one based on Gaussian quadrature. Section 7 is focused on adapting the proposed estimates to the case of the matrix of form *AA<sup>T</sup>* + *λIn*. Numerical examples that illustrate the performance of the derived estimates are found in Section 8. We end this work with several concluding remarks in Section 9.

#### **2. Bounds on the Error**

In Proposition 1 below, we derive an upper bound on the error (2) for a given estimate *αx*<sup>2</sup> of the quadratic form *<sup>x</sup>TA*−*mx*. The first three expressions for the bounds (UB1–UB3) are a direct generalization of a result from [1].

**Proposition 1.** *Let A* ∈ R*n*×*n be a symmetric positive definite matrix and x* ∈ R*n and est* = *αx*<sup>2</sup> *be an estimate of the quadratic form xTA*−*mx. If we denote b* = *αAmx* − *x, the absolute error of the estimate* ""*αx*<sup>2</sup> − (*<sup>x</sup>*, *<sup>A</sup>*−*mx*)"" *is bounded from above by the following expressions:*

$$\begin{aligned} \text{\tiny\tiny\tiny\tiny\tiny\text{IB1.} } & \frac{||\!|\mathbf{x}\|^2||b||}{2||\!|A^m\!\!\mathbf{x}\|||} \left(\kappa^m + \frac{1}{\kappa^m}\right) \\ \text{\tiny\tiny\tiny\tiny\tiny\tiny B2.} & \frac{||\!|\mathbf{x}\|\|\cdot||b||^2}{2||\!|A^m\!\mathbf{b}||} \left(\kappa^m + \frac{1}{\kappa^m}\right) \\ \text{\tiny\tiny\tiny\tiny\tiny B3.} & \frac{||\!|\mathbf{x}\|^2||b||^2}{4\sqrt{\kappa^T A^m \!\!\mathbf{x}} \cdot \sqrt{b^T A^m b}} \left(\kappa^{m/2} + \frac{1}{\kappa^{m/2}}\right)^2 \\ \text{\tiny\tiny\tiny\tiny\tiny B4.} & \frac{||\!|\mathbf{x}\|\|\cdot||b||}{\lambda\_{\text{min}}^m} \end{aligned}$$

*UB5. For estimates satisfying αx*<sup>2</sup> ≤ (*<sup>x</sup>*, *<sup>A</sup>*−*mx*)*, we have also the family of error bounds*

$$\frac{||\mathbf{x}||^2}{2||A^m\mathbf{x}|| \cdot ||A^p\mathbf{x}||} \left(\kappa^m + \frac{1}{\kappa^m}\right) \sqrt{||A^p\mathbf{x}||^2 ||b||^2 - (A^p\mathbf{x}, b)^2},$$

*where p* ≥ 0 *can be chosen as any integer such that* (*<sup>x</sup>*,*Ap x*) (*Amx*,*Ap x*) < *α.*
