Relative Integrated Absolute Error (RIAE)

• Relative Integrated Absolute Error (RIAE) is a scale-invariant metric that normalizes the integrated absolute error by the magnitude of a reference solution to yield a dimensionless relative error.
• It is applied in fields such as numerical integration, regression modeling, floating‐point computation, and dynamic systems to ensure precision where signal magnitudes vary widely.
• RIAE supports adaptive quadrature, robust algorithm design, and efficient model order reduction by offering a rigorous measure for balancing error control with computational cost.

Relative Integrated Absolute Error (RIAE) is a performance metric that quantifies the error of a numerical computation, an approximation, or a model by comparing the integrated (global) magnitude of error to that of a reference or true solution, normalized in a relative sense. RIAE is particularly salient when absolute error measures are inadequate for capturing accuracy relative to the scale of the underlying quantity, such as in high-dimensional quadrature, regression, floating-point roundoff analysis, model order reduction, or evolutionary ODEs over long times. This metric enables precision and efficiency trade-offs, and informs algorithmic design for both numerical integration and system identification under conditions where the magnitude of the target quantity is unknown or varies widely.

1. Mathematical Definition and Core Concept

Relative Integrated Absolute Error is formally defined, in a typical time or space integration setting, as

$$\text{RIAE} = \frac{\int_D |A(x) - T(x)| \, dx}{\int_D |T(x)| \, dx},$$

where $A(x)$ is the approximate, computed, or perturbed solution, $T(x)$ is the true or reference solution, and %%%%2%%%% is the domain of integration (often time, space, or a product space). When the domain is discrete, the integral is replaced with a sum.

RIAE thus represents the (integrated) absolute deviation normalized by the total magnitude of the reference, yielding a dimensionless percentage-type accuracy metric. This normalization is essential whenever the magnitude of the true quantity $T(x)$ is variable or subject to significant scaling, and where application performance is meaningful only relative to the underlying signal.

2. RIAE in Numerical Cubature and Integration

In bivariate composite cubature, RIAE emerges as a tight measure of the global error relative to the unknown magnitude of a double integral. The algorithm presented by Lyness and others (Prentice, 2011) for error control in bivariate integrals illustrates the practical implementation of RIAE as follows:

• The domain $$I[G(x, y)] = \int_a^b \int_{l(x)}^{u(x)} G(x, y) \, dy \, dx$$ is mapped to $[0,1]^2$ via affine transformations.
• The resulting integral is rescaled by a maximum-magnitude factor $$M = \max \{1, \max_{w, z} |G(m_1w + a, m_2z + l_1) m_1 m_2| \}$$, ensuring the scaled integral is at most unity in magnitude.
• Cubature is performed on the scaled integrand with an absolute tolerance $\varepsilon$, yielding an upper bound $M\varepsilon$ for the original integral's error.
• When $|I[G(x, y)]|$ is large, relative error control (i.e., enforcing $|I-Q| < \varepsilon |I[G(x, y)]|$) is preferred; when it is small, absolute control suffices.
• The RIAE, defined as the scaled absolute error over the integrated reference, provides a stringent, integrated upper bound on the relative error over the region of integration.

Empirical examples in (Prentice, 2011) demonstrate reduction in computational cost and improved reliability when the RIAE approach is adopted for adaptive quadrature strategies.

3. RIAE and Statistical Models with Relative Error Objectives

In regression modeling for positive-valued responses, particularly single-index models, RIAE is integral in shifting the focus from absolute errors to scale-invariant, distributionally robust estimation (Wang et al., 2016). The Least Product Relative Error (LPRE) estimator minimizes the sum: $$\text{LPRE}(\beta) = \sum_i \left(\frac{Y_i - e^{g(X_i^T\beta)}}{Y_i} \cdot \frac{Y_i - e^{g(X_i^T\beta)}}{e^{g(X_i^T\beta)}}\right),$$ or equivalently,

$$\sum_i \left( Y_i e^{-g(X_i^T\beta)} + \frac{1}{Y_i} e^{g(X_i^T\beta)} - 2 \right),$$

thus directly targeting minimization of a relative error measure integrated over all observations.

This estimation procedure is scale-invariant for both predictors and response, in contrast to least squares or least absolute deviation methods, and yields statistical efficiency improvements particularly in heavy-tailed or heterogeneous error settings. The RIAE interpretation holds whenever the sum/integral of relative errors over the dataset provides a better risk criterion for model adequacy.

4. RIAE in Finite-Precision Floating-Point Computation

Verification of numerical software under finite-precision arithmetic, especially for safety- and mission-critical code, increasingly relies on integrated relative error measures (Izycheva et al., 2017). The RIAE in this context is generally expressed as: $$\max_{x\in I} \left| \frac{f(x) - \hat{f}(x)}{f(x)} \right|,$$ or, for an integrated variant over the input domain,

$$\text{RIAE} = \frac{\int |\hat{f}(x) - f(x)| \, dx}{\int |f(x)| \, dx}.$$

Novel static analysis tools now directly optimize this metric, yielding bounds several orders of magnitude tighter than those by naive normalization of absolute errors. The papers have established that such direct optimization-based RIAE analysis not only avoids spurious over-approximations but also provides more discriminative guarantees when function values pass near or through zero, provided care is taken to isolate problematic domains via interval subdivision.

This direct control of RIAE informs the design and certification of control, signal-processing, and scientific computation where the error as a fraction of the signal is the relevant quantity.

5. Model Order Reduction and RIAE in Dynamic Systems

When reducing large-scale dynamical systems to lower-order surrogates, the RIAE is closely related to the $\mathcal{H}_2$ norm of the relative error system, particularly over finite time or frequency domains (Zulfiqar et al., 2022, Zulfiqar et al., 2022). The error system is expressed in either additive or relative (multiplicative) form: $$H(s) = \hat{H}(s) + \Delta_{\text{add}}(s), \quad H(s) = \hat{H}(s) (I + \Delta_{\text{rel}}(s)),$$ with the relative error

$$\Delta_{\text{rel}}(s) = \hat{H}^{-1}(s) (H(s) - \hat{H}(s)).$$

Minimizing the time-limited or frequency-limited $\mathcal{H}_2$ norm of $\Delta_{\text{rel}}(s)$ controls the RIAE across the signal domain: $$\text{RIAE}_\omega = \frac{ \int |\hat{H}(j\omega) - H(j\omega)| d\omega }{ \int |H(j\omega)| d\omega }.$$ Oblique projection algorithms are developed to iteratively satisfy optimality conditions for the relative error criterion, avoiding the need for large-scale Lyapunov or Riccati solvers and yielding reduced models with certified low RIAE, as demonstrated in empirical benchmarks.

6. RIAE and Long-Term Propagation in Linear Dynamical Systems

The propagation of initial uncertainties in linear ODEs is governed by the evolution of the relative error, and thus the RIAE, which offers a global integral perspective distinct from absolute error amplification (Maset, 11 Jul 2025). For $y'(t) = Ay(t)$ under perturbed initial data,

$$\text{RIAE} = \frac{\int_0^T \|\tilde{y}(t) - y(t)\| dt}{\int_0^T \|y(t)\| dt}.$$

The analysis reveals that under generic conditions,

$$\text{RIAE} \approx \varepsilon \cdot K_\infty(y_0, \tilde{z}_0),$$

where $K_\infty$ is an asymptotic condition number reflecting the projection of initial data onto dominant eigenspaces of $A$. This shows that, even as the absolute error may grow or shrink exponentially due to non-normality or instability, the RIAE stabilizes to a value indicating the true sensitivity of the system to data uncertainty over the evolution interval.

This property is essential for determining the relative conditioning of simulations in contexts where long-run behavior, reliability, and trustworthiness are at stake, and where absolute error may be misleading due to signal amplitude decay or growth.

7. Significance and Limitations

RIAE provides a universal, scale-invariant metric that is applicable across numerous computational fields—numerical analysis, statistics, control theory, and dynamical systems. Its widespread adoption is motivated by the following:

• It reflects practical significance in applications where only the error relative to a potentially small underlying quantity is meaningful.
• It allows for robust algorithm design without a priori knowledge of the solution's scale.
• It informs model reduction, uncertainty quantification, and system identification in domains where large dynamic range or near-zero crossings would render absolute metrics ineffective.

However, limitations include the need for careful treatment near zeros of the reference function, as division-by-zero or near-singular behavior can skew RIAE interpretation or require fallback to absolute error controls. Interval subdivision, adaptive strategies, or domain restriction are typical remedies. Sensitivity to the projection of signals onto dominant subspaces also means that RIAE must be interpreted in the context of the actual system structure and conditioning.

In summary, RIAE is a fundamental tool in quantitative error analysis, providing rigorous and interpretable bounds for the quality of numerical and model-based approximations in scientific computing, statistics, and engineering.