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IT Services Circle
IT Services Circle
Jul 2, 2024 · Game Development

A 55‑Year‑Old Bug in the Original Lunar Lander Game: Missing Square‑Root Factor in the Rocket Equation

The article recounts the history of the 1969 text‑based Lunar Lander game, explains how its creator used the rocket equation and Taylor series for physics simulation, and reveals a long‑standing bug—a missing factor in the square‑root denominator—that causes the optimal fuel‑burn strategy to fail, prompting a modern analysis of the error.

Bug AnalysisGame HistoryLunar Lander
0 likes · 11 min read
A 55‑Year‑Old Bug in the Original Lunar Lander Game: Missing Square‑Root Factor in the Rocket Equation
Model Perspective
Model Perspective
Jun 1, 2024 · Fundamentals

Why Do Mathematical Models Miss the Mark? Exploring Three Key Uncertainties

Mathematical modeling helps us understand complex systems, but inherent uncertainties—model, approximation, and input—can cause prediction errors; this article explains each source, illustrates them with climate, weather, and engineering examples, and offers strategies such as validation, higher‑precision numerical methods, and probabilistic analysis to mitigate their impact.

Numerical Methodsmathematical modelingmodel validation
0 likes · 7 min read
Why Do Mathematical Models Miss the Mark? Exploring Three Key Uncertainties
Model Perspective
Model Perspective
Nov 13, 2022 · Fundamentals

Mastering 1D Interpolation in Python with SciPy: Linear, Nearest, Quadratic, Cubic

Learn how to perform one-dimensional interpolation in Python using SciPy's interp1d function, explore various interpolation methods such as linear, nearest, quadratic, and cubic, and visualize the results with Matplotlib to compare their fitting behaviors on sample data.

Data VisualizationMatplotlibNumerical Methods
0 likes · 5 min read
Mastering 1D Interpolation in Python with SciPy: Linear, Nearest, Quadratic, Cubic
Model Perspective
Model Perspective
Aug 19, 2022 · Fundamentals

Mastering SciPy Optimize: From Root Finding to Global Optimization

This guide introduces SciPy's optimize module, covering scalar and multivariate minimization, global optimization algorithms, root finding, linear programming, and assignment problems, complete with clear Python code examples and explanations of each method's usage and output.

Linear ProgrammingNumerical MethodsOptimization
0 likes · 7 min read
Mastering SciPy Optimize: From Root Finding to Global Optimization
Model Perspective
Model Perspective
Jun 22, 2022 · Fundamentals

How to Find the Global Maximum of (1‑x³)·sin(3x) Using Python and SciPy

This article demonstrates how to locate the global maximum of the function f(x) = (1‑x³)·sin(3x) by visualizing it with Matplotlib, applying SciPy’s optimization tools such as fminbound, and comparing deterministic methods with random sampling, highlighting the pitfalls of local optima.

Numerical MethodsOptimizationPython
0 likes · 3 min read
How to Find the Global Maximum of (1‑x³)·sin(3x) Using Python and SciPy
Model Perspective
Model Perspective
Jun 20, 2022 · Fundamentals

How Monte Carlo Integration Quickly Estimates Double Integrals

This article explains how Monte Carlo methods can approximate definite integrals by randomly sampling points inside a bounding box, showing the geometric interpretation, probability reasoning, and providing a Python implementation that yields a fast low‑precision estimate.

IntegrationNumerical MethodsPython
0 likes · 3 min read
How Monte Carlo Integration Quickly Estimates Double Integrals
Model Perspective
Model Perspective
Jun 10, 2022 · Fundamentals

Exploring 1D Interpolation with SciPy: Linear, Nearest, Cubic & More

This article introduces the concept of interpolation for discrete data, demonstrates how to use SciPy's interp1d function with various methods (linear, nearest, nearest‑up, zero, quadratic, cubic), visualizes the resulting curves alongside the original points, and provides complete Python code for reproducing the plots.

Data VisualizationNumerical MethodsPython
0 likes · 5 min read
Exploring 1D Interpolation with SciPy: Linear, Nearest, Cubic & More
Model Perspective
Model Perspective
Jun 5, 2022 · Fundamentals

How to Numerically Solve Differential Equations: Methods & Examples

This article explains why numerical solutions are needed for differential equations, introduces the initial value problem for first‑order ODEs, and details three common discretization techniques—forward difference, numerical integration, and first‑order Taylor expansion—highlighting their formulation and error considerations.

Numerical MethodsTaylor seriesdifferential equations
0 likes · 4 min read
How to Numerically Solve Differential Equations: Methods & Examples
Model Perspective
Model Perspective
Jun 5, 2022 · Fundamentals

How to Solve ODEs Numerically in Python with SciPy’s odeint

This article explains how to obtain numerical solutions for ordinary differential equations in Python using SciPy’s odeint function, demonstrates several example problems including a simple ODE, a system converted from a second‑order equation, and the chaotic Lorenz model, and provides complete code snippets.

Lorenz AttractorNumerical MethodsODE
0 likes · 6 min read
How to Solve ODEs Numerically in Python with SciPy’s odeint
Python Programming Learning Circle
Python Programming Learning Circle
Dec 11, 2019 · Fundamentals

Master Python Optimization: Bisection, Fibonacci, Golden Section & Newton Methods

This article walks through several Python optimization techniques—including the bisection, three‑point division, Fibonacci, golden‑section, quadratic interpolation, and Newton methods—providing clear code examples, explanations of return statements, variable type handling, and debugging tips.

Bisection MethodGolden SectionNewton's Method
0 likes · 12 min read
Master Python Optimization: Bisection, Fibonacci, Golden Section & Newton Methods