Mathematical Functions and Scientific Computing
This guide demonstrates mathematical and scientific computing operations using DotCompute's GPU acceleration.
🚧 Documentation In Progress - Mathematical computing examples are being developed.
Overview
DotCompute supports:
- Numerical integration algorithms
- Differential equation solvers
- Statistical computations
- Monte Carlo simulations
- Mathematical function evaluation
Numerical Integration {#numerical-integration}
Numerical integration approximates definite integrals using GPU-accelerated algorithms for fast computation over large intervals or complex functions.
Trapezoidal Rule
The trapezoidal rule approximates the area under a curve using trapezoids.
// TODO: Provide trapezoid integration example
// - Parallel computation of trapezoid areas
// - Adaptive interval refinement
// - Error estimation and convergence
GPU Advantages: Compute millions of trapezoids in parallel for high accuracy.
Simpson's Rule
Simpson's rule uses quadratic approximations for higher accuracy than the trapezoidal rule.
// TODO: Document Simpson's rule implementation
// - Parabolic curve fitting
// - Error bounds analysis
// - Composite Simpson's rule for multiple intervals
Gauss Quadrature
Gauss quadrature achieves high accuracy with fewer function evaluations by using optimally chosen sample points.
// TODO: Explain Gaussian quadrature integration
// - Legendre polynomial roots as sample points
// - Weight computation
// - Multi-dimensional integration
Differential Equations {#differential-equations}
Differential equation solvers compute numerical solutions to ordinary (ODE) and partial differential equations (PDE) using GPU parallelization.
ODE Solvers
Ordinary differential equations describe systems with a single independent variable (typically time).
// TODO: Cover ordinary differential equation solving:
// - Euler method (first-order explicit)
// - Runge-Kutta methods (RK4 for accuracy)
// - Implicit methods (stability for stiff equations)
// - Adaptive step-size control
GPU Benefits: Solve thousands of coupled ODEs simultaneously for parameter sweeps or ensemble simulations.
PDE Solvers
Partial differential equations describe systems with multiple independent variables (space and time).
// TODO: Document partial differential equations:
// - Finite difference methods (spatial discretization)
// - Heat equation (diffusion processes)
// - Wave equation (oscillatory systems)
// - Boundary condition handling
Applications: Fluid dynamics, heat transfer, electromagnetic fields, quantum mechanics.
Monte Carlo Methods {#monte-carlo}
Monte Carlo methods use random sampling to solve mathematical problems that are difficult or impossible to solve analytically.
Monte Carlo Integration
Monte Carlo integration estimates integrals by random sampling, particularly effective for high-dimensional problems.
// TODO: Document integration using Monte Carlo:
// - Uniform sampling over integration domain
// - Variance reduction techniques (importance sampling, stratified sampling)
// - Error estimation (standard error decreases as 1/sqrt(N))
// - Parallel random number generation on GPU
Performance: GPU generates and processes millions of samples per second for accurate estimates.
Monte Carlo Simulation
Monte Carlo simulation models complex systems by repeated random sampling.
// TODO: Explain Monte Carlo simulation patterns:
// - Financial option pricing (Black-Scholes, binomial trees)
// - Risk analysis and portfolio optimization
// - Physical system simulation (particle transport, Ising model)
// - Convergence diagnostics and confidence intervals
GPU Advantages: Run thousands of independent simulations concurrently for statistical analysis.
Statistics {#statistics}
Statistical computations analyze and summarize large datasets using GPU acceleration for real-time insights.
Descriptive Statistics
Descriptive statistics summarize central tendencies and variability in data.
// TODO: Provide statistical computation examples:
// - Mean, variance, standard deviation (parallel reduction)
// - Percentiles and quantiles (parallel sorting and selection)
// - Histograms (atomic operations for binning)
// - Correlation and covariance matrices
Performance: Compute statistics on billions of data points in milliseconds.
Hypothesis Testing
Hypothesis testing determines statistical significance of observed patterns.
// TODO: Document hypothesis testing operations:
// - t-tests (one-sample, two-sample, paired)
// - Chi-square tests (goodness-of-fit, independence)
// - ANOVA (analysis of variance)
// - Non-parametric tests (Mann-Whitney, Kruskal-Wallis)
GPU Acceleration: Bootstrap resampling and permutation tests with millions of iterations.
Probability Distributions
Probability distributions model random variables and generate samples.
// TODO: Cover probability distribution sampling and evaluation:
// - Normal (Gaussian) distribution (Box-Muller transform)
// - Uniform, exponential, gamma distributions
// - Discrete distributions (binomial, Poisson)
// - Probability density/mass function evaluation
Applications: Monte Carlo simulation, statistical modeling, machine learning.
Advanced Computations
Linear Algebra Operations
TODO: Explain linear algebra on GPU
Matrix Operations
TODO: Document matrix computations
Performance Considerations
TODO: List optimization techniques for mathematical workloads
Examples
TODO: Provide complete mathematical computing examples