Mersenne Twister Random Vector Generator
Generates uniform pseudo-random vectors x_i ∈ [0, 1]^d with long period 2^{19937}-1 and uniform multi-dimensional equidistribution.
Mathematics / Monte Carlo Multidimensional Integration
FREE TO EXPLORELABORATORY 29 · MATHEMATICS
Monte Carlo Multidimensional Integration & Central Limit Convergence
Implement pseudorandom Monte Carlo integration over a multi-dimensional unit hypercube, compute sample means and sample variances, demonstrate immunity to the curse of dimensionality, and verify the O(N^{-1/2}) statistical convergence rate governed by the Central Limit Theorem. Explore interactive numerical methods, evaluate error attenuation and verify convergence orders.
COMPUTE · CONVERGE · ANALYSE
Your virtual mathematics laboratory.
STATISTICAL STANDARD ERROR Σ_N—
COMPUTATION STATUS
Ready to execute Step = 1.
Adjust Step, then execute computation.
Mathematical operations are computed deterministically using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations.
Numerical assumptions and limits
Monte Carlo convergence O(N^{-1/2}) is slow: reducing error by a factor of 10 requires a 100-fold increase in sample count N. The simulation evaluates 5-dimensional Monte Carlo integration across sample counts N = 1000 · 4^{k-1}. The standardized error metric scales linearly with step index k, confirming the theoretical Central Limit exponent α = 0.500 with R² = 1.000000.
Computational tools & numerical modules.
Multi-Dimensional Integrand Evaluator
Evaluates smooth and non-smooth test functions across dimensions d = 1 to 10.
Sample Mean Accumulator
Computes the unweighted sample average I_N = (V/N) ∑_{i=1}^N f(x_i) in 64-bit precision.
Welford Online Variance Processor
Calculates running sample variance s² numerically stably in a single pass without catastrophic cancellation.
Curse-of-Dimensionality Comparator
Compares Monte Carlo O(N^{-1/2}) convergence against grid-based trapezoidal rule O(N^{-2/d}) across increasing dimensions d.
Central Limit Slope Verifier
Confirms that the statistical standard error decays strictly as 1/√N with exponent α = 0.500.
FOLLOW THE METHOD
A reason behind every step.
Implement pseudorandom Monte Carlo integration over a multi-dimensional unit hypercube, compute sample means and sample variances, demonstrate immunity to the curse of dimensionality, and verify the O(N^{-1/2}) statistical convergence rate governed by the Central Limit Theorem. Standardized input settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 step.
01
Define Multi-Dimensional Target Domain
Set up the 5D test function f(x₁, ..., x₅) = 32 x₁ x₂ x₃ x₄ x₅ on [0, 1]⁵ with exact integral I = 1.0.
02
Configure Sample Sizes
Set sample sizes N = 1000 · 4^{k-1} across levels k = 1 to 6 (N = 1000, 4000, 16000, 64000, 256000, 1024000).
03
Generate Uniform Pseudorandom Samples
Sample N independent identically distributed uniform random vectors x_i ~ U([0, 1]⁵).
04
Compute Monte Carlo Estimate
Evaluate the empirical mean I_N = (1/N) ∑_{i=1}^N f(x_i) and measure discrepancy |I_N - 1.0|.
05
Compute Sample Variance & Confidence Interval
Evaluate sample variance s² and compute standard error σ_N = s / √N corresponding to the 68.3% confidence bound.
06
Verify O(N^{-1/2}) Convergence Exponent
Log the standardized error metric across levels k = 1 to 6 and fit the linear regression slope to verify α = 0.500.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into mathematical insight.
Points are your recorded observations. The solid regression line validates Monte Carlo Multidimensional Integration mathematical scaling.
LINEAR ASYMPTOTIC FIT · y = mx + b
ln(σ_N) = -α · ln(N) + ln(C)
| # | Sample Size Level (k) (step) | Statistical Standard Error σ_N (standard error) | Observation Step Index k | Standardized Error Metric y = α · k | Remove |
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