K-R Defect Framework — Python Library

Based on the published paper:

"A K-R Defect Framework for Classical Inequalities: Curvature Recovery, Uniqueness, Stability, and Applications to Nonlinear Boundary Value Problems" RamaKrishna Pasupuleti, Boundary Value Problems, SpringerOpen, 2026.

Quick Start

import math
from kr_defect import kr_second_derivative, kr_both_derivatives

# Estimate f''(0.5) for f(x) = e^x
est = kr_second_derivative(math.exp, x=0.5, h=0.01)
print(f"K-R estimate: {est:.4f}")   # ~1.6487
print(f"True value:   {math.exp(0.5):.4f}")

# Get BOTH f'(x) and f''(x) from three evaluations
fp, fpp = kr_both_derivatives(math.exp, x=1.0, h=0.01)

Functions

Function	Description
kr_defect(f,x,y,K)	Raw K-R defect D_f
kr_normalised(f,x,y,K)	Normalised defect Φ_f = (1/2)f″(ξ)
kr_second_derivative(f,x,h,K)	Estimate f″(x), optional averaging
kr_first_derivative(f,x,h,K)	Estimate f′(x)
kr_both_derivatives(f,x,h,K)	Both f′ and f″ from 3 evaluations
kr_reconstruct(data,grid,K,f0,fp0)	Recover f from defect measurements
kr_inverse_bvp(u_data,x_grid,K)	Recover f in -u″=f(u) from u data
kr_convexity(f,x_grid,h,K)	Classify convex/concave at each point
kr_stability_bound(eps,a,b)	Stability Theorem bound

Key Advantages Over Finite Differences

• Works near domain boundaries (all evaluation points stay interior)
• Parameter averaging over K ∈ (0,1) reduces noise sensitivity
• Uniqueness Principle guarantees unique reconstruction
• Stability Theorem gives explicit error bound

Run Benchmarks

python3 benchmarks.py
python3 generate_plots.py