First-Order Taylor Expansion

• First-order Taylor expansion is a linear approximation of smooth functions near a point, offering controlled error bounds based on second derivatives.
• It underpins numerical methods such as ODE integration and interpolation, ensuring precise error estimation and enhanced computational performance.
• Recent advancements extend the expansion to matrix functions and Banach spaces using optimal weighted derivatives and integral remainder bounds for improved accuracy.

The first-order Taylor expansion provides a linear approximation of a smooth function near a fixed point, offering critical theoretical and practical utility in numerical analysis, asymptotic approximation, and error analysis across mathematical sciences. For sufficiently differentiable functions, the expansion yields not just an approximation but a controllable bound on the local error. Recent literature extends and refines the classical result by constructing optimal weighted combinations of function values and derivatives, analyzing the resultant remainders, and generalizing to matrices, systems, and Banach spaces.

1. Classical First-Order Taylor Expansion and Remainder

For a scalar function $f$ of a real variable, the first-order Taylor expansion about $a$ at $b$ is: $f(b) \approx f(a) + f'(a)(b - a).$ The classical remainder for $f \in C^2([a, b])$ is given by

$R_1 = \frac{f''(\xi)}{2}(b-a)^2$

for some $\xi \in (a, b)$. Thus,

$|R_1| \le \frac{M_2}{2} (b-a)^2$

where $M_2 = \sup_{x \in [a, b]} |f''(x)|$ (Chaskalovic et al., 2021).

This result generalizes in a vector-valued context: $$f(a+h) = f(a) + \langle Df(a), h \rangle + \int_0^1 (1-t) D^2 f(a+th)(h, h) dt$$ (Chaskalovic et al., 2022), with explicit second-derivative integral remainders.

For ODE systems, as in reactor point kinetics, the first-order expansion is

$$\mathbf{y}(t+h) \approx \mathbf{y}(t) + h \frac{d\mathbf{y}}{dt}(t)$$

with global error $O(h)$, as verified by tightly controlled numerical studies (McMahon et al., 2010).

2. Refined and Optimal First-Order Taylor-like Expansions

Chaskalovic, Assous, and collaborators constructed refined expansions by forming a convex combination of first derivatives at $n+1$ points between $a$ and $b$: $$f(b) = f(a) + (b-a)\sum_{k=0}^n w_k f'(x_k) + (b-a) \varepsilon_{a,n+1}(b), \quad \sum w_k = 1$$ The points $x_k = a + \frac{k}{n}(b-a)$ and the optimal weights are: $$w_0 = w_n = \frac{1}{2n}, \quad w_k = \frac{1}{n}, \quad k=1,\ldots,n-1$$ (Chaskalovic et al., 2021, Chaskalovic et al., 2022, Chaskalovic et al., 2023).

The remainder achieves the reduced bound: $$|\varepsilon_{a,n+1}(b)| \le \frac{M_2 - m_2}{8n}(b-a)$$ where $M_2, m_2$ are upper and lower bounds on $f''$ over $[a, b]$, and is always lower than the classical bound, up to a factor $1/(4n)$ (Chaskalovic et al., 2023).

An optimization-based proof shows that equal spacing of nodes is optimal, and that this structure carries over to higher dimensions and Banach spaces (Chaskalovic et al., 2023, Chaskalovic et al., 2022).

3. Matrix and Banach-Space Taylor Expansions

For matrix functionals, such as the principal square root $f(X) = X^{1/2}$, the first-order Fréchet derivative at $A \succ 0$ in direction $H$ admits several representations:

• Spectral: $Df(A)(H) = Q [M \odot (Q^T H Q)] Q^T$, $$M_{ij} = (\sqrt{\lambda_i} + \sqrt{\lambda_j})^{-1}$$
• Dunford-resolvent:

$$Df(A)(H) = \frac{1}{\pi} \int_0^\infty t^{-1/2} (tI + A)^{-1} H (tI + A)^{-1} dt$$

• Sylvester/Lyapunov form:

$$X Y + Y X = H \Longrightarrow Y = \int_0^\infty e^{-t X} H e^{-t X} dt$$

(Moral et al., 2017).

The first-order Taylor expansion for $A + H$ then reads: $f(A+H) = f(A) + Df(A)(H) + R_2(A,H)$ with explicit integral remainder: $$R_2(A, H) = \int_0^1 (1 - \varepsilon) D^2 f(A + \varepsilon H)[H, H] d\varepsilon,$$ and operator norm estimate

$$\|R_2(A, H)\|_2 \le 2^{-3/2} \lambda_{\min}(A)^{-5/2} \|H\|_2^2$$

provided $$A + \varepsilon H \succ 0, \forall \varepsilon \in [0,1]$$ (Moral et al., 2017).

4. Quantitative Bounds for Special Functions

Shevtsova analyzed $e^{ix}$ and its Taylor expansion, obtaining sharp uniform bounds: $$e^{ix} = 1 + i x + R_1(x), \quad |R_1(x)| \le \theta |x|, \quad \theta \approx 0.7246$$ and for the real part,

$|\cos x - 1| \le \theta |x|$

with applications to control of characteristic functions and moment bounds in probability (Shevtsova, 2013).

5. Applications in Numerical Analysis and Error Estimation

First-order Taylor expansions underpin well-posed stepwise integration for stiff systems, e.g., reactor point kinetics:

• Explicit updates for neutron density $n$ and precursors $C_i$:

$$n_{m+1} = n_m + h D_n^m,\quad C_{i,m+1} = C_{i,m} + h D_{C_i}^m$$

• Demonstrated $O(h)$ accuracy, stability for suitable step size, and performance comparable to higher-order or implicit schemes even for stiff regimes (McMahon et al., 2010).

Refined Taylor-like formulas lead to sharper interpolation and quadrature bounds:

• Modified Lagrange interpolation with error $\sim (b-a)^2 (M_2-m_2)/32$ versus classical $(b-a)^2 M_2 / 4$ (Chaskalovic et al., 2021).
• Corrected trapezoid rule with error $(b-a)^3 (M_2 - m_2)/48$, halving the classical bound (Chaskalovic et al., 2021).

In finite elements, the refined expansions yield smaller a priori and a posteriori error estimates, enabling the use of coarser meshes for comparable accuracy (Chaskalovic et al., 2022).

Table: Remainder Bounds for First-Order Taylor Schemes

Formula Variant	Remainder Bound	Reference
Classical (scalar, $f$)	$\frac{M_2}{2}(b-a)^2$	(Chaskalovic et al., 2021)
Optimized ($n$-point)	$\frac{M_2 - m_2}{8n}(b-a)^2$	(Chaskalovic et al., 2023)
Matrix Square Root ($f(X)$)	$2^{-3/2} \lambda_{\min}(A)^{-5/2} \|H\|_2^2$	(Moral et al., 2017)
Reactor point-kinetics (step)	$O(h)$ global error, $O(h^2)$ local truncation error	(McMahon et al., 2010)

6. Significance, Limitations, and Generalizations

The emergence of weighted, multipoint, and matrix-functional Taylor-like formulas significantly sharpens error control, particularly in interpolation, quadrature, and large-scale simulation. Equally spaced nodes with optimal weights minimize remainders for $C^2$ functions. The affine structure of weights and points ensures that improvements generalize readily to higher-order schemes and multidimensional domains (Chaskalovic et al., 2023, Chaskalovic et al., 2022). For matrix functionals, integral representations and resolvent techniques yield computable derivatives and remainders under spectral or operator-norm control (Moral et al., 2017).

A key limitation is strict smoothness: the most refined bounds require $f \in C^2$, known upper/lower bounds on $f''$, or positivity constraints for matrix expansions. In functional and operator contexts, maintaining these constraints may require step-size or perturbation-norm control.

7. Research Directions and Extensions

Recent research continues to generalize the first-order Taylor framework:

• Higher-order optimized Taylor-like expansions with similarly minimized remainders (see (Chaskalovic et al., 2023)).
• Multi-dimensional analogs for smooth functions on $\mathbb{R}^n$ and their numerical applications (Chaskalovic et al., 2022).
• Automated error estimation and certified numerics in PDEs, ODEs, and stochastic simulation.
• Advanced operator-functional calculus, including classes beyond the principal square root (spectral, holomorphic, and Lyapunov-based approaches) (Moral et al., 2017).

A plausible implication is widespread adoption of these optimized expansions in numerical libraries to further reduce global error bounds and computational cost in finite element and quadrature schemes.