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skills/julia-numerical/examples.jl

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+# Julia Numerical Calculation Examples
+# This file contains common numerical computation patterns
+
+# ============================================================================
+# Linear Algebra Examples
+# ============================================================================
+
+function linear_algebra_examples()
+    using LinearAlgebra
+
+    println("=== Linear Algebra Examples ===")
+
+    # Matrix creation and basic operations
+    A = [1 2 3; 4 5 6; 7 8 10]
+    b = [1, 2, 3]
+
+    println("Matrix A:")
+    println(A)
+
+    # Solve linear system Ax = b
+    x = A \ b
+    println("\nSolution to Ax = b:")
+    println(x)
+
+    # Eigenvalues
+    eigenvals, eigenvecs = eigen(A)
+    println("\nEigenvalues:")
+    println(eigenvals)
+
+    # Singular value decomposition
+    U, S, V = svd(A)
+    println("\nSingular values:")
+    println(S)
+
+    # Determinant and norm
+    println("\nDeterminant: ", det(A))
+    println("Frobenius norm: ", norm(A))
+end
+
+
+# ============================================================================
+# Numerical Integration Examples
+# ============================================================================
+
+function integration_examples()
+    using QuadGK
+
+    println("\n=== Numerical Integration Examples ===")
+
+    # Integrate sin(x) from 0 to π
+    f1(x) = sin(x)
+    result1, error1 = quadgk(f1, 0, π)
+    println("∫sin(x)dx from 0 to π = ", result1)
+    println("Estimated error: ", error1)
+
+    # Integrate exp(-x^2) from -∞ to ∞ (Gaussian)
+    f2(x) = exp(-x^2)
+    result2, error2 = quadgk(f2, -Inf, Inf)
+    println("\n∫exp(-x²)dx from -∞ to ∞ = ", result2)
+    println("Theoretical value: ", sqrt(π))
+
+    # Integrate 1/(1+x^2) from 0 to 1
+    f3(x) = 1/(1 + x^2)
+    result3, error3 = quadgk(f3, 0, 1)
+    println("\n∫1/(1+x²)dx from 0 to 1 = ", result3)
+    println("Theoretical value (π/4): ", π/4)
+end
+
+# ============================================================================
+# Optimization Examples
+# ============================================================================
+
+function optimization_examples()
+    using Optim
+
+    println("\n=== Optimization Examples ===")
+
+    # Simple quadratic function
+    f(x) = (x[1] - 2)^2 + (x[2] - 3)^2
+
+    result = optimize(f, [0.0, 0.0])
+    println("Minimize f(x,y) = (x-2)² + (y-3)²")
+    println("Minimum found at: ", Optim.minimizer(result))
+    println("Minimum value: ", Optim.minimum(result))
+
+    # Rosenbrock function (more challenging)
+    rosenbrock(x) = (1 - x[1])^2 + 100(x[2] - x[1]^2)^2
+
+    result2 = optimize(rosenbrock, [0.0, 0.0])
+    println("\nMinimize Rosenbrock function")
+    println("Minimum found at: ", Optim.minimizer(result2))
+    println("Minimum value: ", Optim.minimum(result2))
+end
+
+# ============================================================================
+# Statistics Examples
+# ============================================================================
+
+function statistics_examples()
+    using Statistics
+
+    println("\n=== Statistics Examples ===")
+
+    data = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 20]
+
+    println("Data: ", data)
+    println("\nStatistical measures:")
+    println("Mean: ", mean(data))
+    println("Median: ", median(data))
+    println("Standard deviation: ", std(data))
+    println("Variance: ", var(data))
+    println("Minimum: ", minimum(data))
+    println("Maximum: ", maximum(data))
+    println("Range: ", maximum(data) - minimum(data))
+
+    # Quantiles
+    println("\nQuantiles:")
+    println("25th percentile: ", quantile(data, 0.25))
+    println("50th percentile: ", quantile(data, 0.50))
+    println("75th percentile: ", quantile(data, 0.75))
+end
+
+# ============================================================================
+# Root Finding Examples
+# ============================================================================
+
+function root_finding_examples()
+    using Roots
+
+    println("\n=== Root Finding Examples ===")
+
+    # Find root of f(x) = x^3 - 2
+    f(x) = x^3 - 2
+    root = find_zero(f, 1.0)
+    println("Root of x³ - 2 = 0: ", root)
+    println("Verification: f(root) = ", f(root))
+
+    # Find root of f(x) = sin(x) - 0.5
+    f2(x) = sin(x) - 0.5
+    root2 = find_zero(f2, 0.5)
+    println("\nRoot of sin(x) - 0.5 = 0: ", root2)
+    println("Verification: f(root) = ", f2(root2))
+end
+
+# ============================================================================
+# Main execution
+# ============================================================================
+
+if abspath(PROGRAM_FILE) == @__FILE__
+    linear_algebra_examples()
+    integration_examples()
+    optimization_examples()
+    statistics_examples()
+    root_finding_examples()
+end