numerical-methods（数值方法）

Numerical Methods

You are an expert in numerical stability and computational aspects of statistical methods.

Floating-Point Fundamentals

IEEE 754 Double Precision

• Precision: ~15-17 significant decimal digits
• Range: ~10⁻³⁰⁸ to 10³⁰⁸
• Machine epsilon: ε ≈ 2.2 × 10⁻¹⁶
• Special values: Inf, -Inf, NaN

Key Constants in R

.Machine$double.eps      # ~2.22e-16 (machine epsilon)
.Machine$double.xmax     # ~1.80e+308 (max finite)
.Machine$double.xmin     # ~2.23e-308 (min positive normalized)
.Machine$double.neg.eps  # ~1.11e-16 (negative epsilon)

Common Numerical Issues

1. Catastrophic Cancellation

When subtracting nearly equal numbers:

# BAD: loses precision
x <- 1e10 + 1
y <- 1e10
result <- x - y  # Should be 1, may have errors

# BETTER: reformulate to avoid subtraction
# Example: Computing variance
var_bad <- mean(x^2) - mean(x)^2   # Can be negative!
var_good <- sum((x - mean(x))^2) / (n-1)  # Always non-negative

2. Overflow/Underflow

# BAD: overflow
prod(1:200)  # Inf

# GOOD: work on log scale
sum(log(1:200))  # Then exp() if needed

# BAD: underflow in probabilities
prod(dnorm(x))  # 0 for large x

# GOOD: sum log probabilities
sum(dnorm(x, log = TRUE))

3. Log-Sum-Exp Trick

Essential for working with log probabilities:

log_sum_exp <- function(log_x) {
  max_log <- max(log_x)
  if (is.infinite(max_log)) return(max_log)
  max_log + log(sum(exp(log_x - max_log)))
}

# Example: log(exp(-1000) + exp(-1001))
log_sum_exp(c(-1000, -1001))  # Correct: ~-999.69
log(exp(-1000) + exp(-1001))   # Wrong: -Inf

4. Softmax Stability

# BAD
softmax_bad <- function(x) exp(x) / sum(exp(x))

# GOOD
softmax <- function(x) {
  x_max <- max(x)
  exp_x <- exp(x - x_max)
  exp_x / sum(exp_x)
}

Matrix Computations

Conditioning

The condition number κ(A) measures sensitivity to perturbation:

• κ(A) = ‖A‖ · ‖A⁻¹‖
• Rule: Expect to lose log₁₀(κ) digits of accuracy
• κ > 10¹⁵ means matrix is numerically singular

# Check condition number
kappa(X, exact = TRUE)

# For regression: check X'X conditioning
kappa(crossprod(X))

Solving Linear Systems

Prefer: Decomposition methods over explicit inversion

# BAD: explicit inverse
beta <- solve(t(X) %*% X) %*% t(X) %*% y

# GOOD: QR decomposition
beta <- qr.coef(qr(X), y)

# BETTER for positive definite: Cholesky
R <- chol(crossprod(X))
beta <- backsolve(R, forwardsolve(t(R), crossprod(X, y)))

# For ill-conditioned: SVD/pseudoinverse
beta <- MASS::ginv(X) %*% y

Symmetric Positive Definite Matrices

Always use specialized methods:

# Cholesky for SPD
L <- chol(Sigma)

# Eigendecomposition
eig <- eigen(Sigma, symmetric = TRUE)

# Check positive definiteness
all(eigen(Sigma, symmetric = TRUE, only.values = TRUE)$values > 0)

Optimization Stability

Gradient Computation

# Numerical gradient (for verification)
numerical_grad <- function(f, x, h = sqrt(.Machine$double.eps)) {
  sapply(seq_along(x), function(i) {
    x_plus <- x_minus <- x
    x_plus[i] <- x[i] + h
    x_minus[i] <- x[i] - h
    (f(x_plus) - f(x_minus)) / (2 * h)
  })
}

# Central difference is O(h²) accurate
# Forward difference is O(h) accurate

Hessian Stability

# Check Hessian is positive definite at optimum
check_hessian <- function(H, tol = 1e-8) {
  eigs <- eigen(H, symmetric = TRUE, only.values = TRUE)$values
  min_eig <- min(eigs)

  list(
    positive_definite = min_eig > tol,
    min_eigenvalue = min_eig,
    condition_number = max(eigs) / min_eig
  )
}

Line Search

For gradient descent stability:

backtracking_line_search <- function(f, x, d, grad, alpha = 1, rho = 0.5, c = 1e-4) {
  # Armijo condition
  while (f(x + alpha * d) > f(x) + c * alpha * sum(grad * d)) {
    alpha <- rho * alpha
    if (alpha < 1e-10) break
  }
  alpha
}

Integration and Quadrature

Numerical Integration Guidelines

# Adaptive quadrature (default choice)
integrate(f, lower, upper)

# For infinite limits
integrate(f, -Inf, Inf)

# For highly oscillatory or peaked functions
# Increase subdivisions
integrate(f, lower, upper, subdivisions = 1000)

# For known singularities, split the domain

Monte Carlo Integration

mc_integrate <- function(f, n, lower, upper) {
  x <- runif(n, lower, upper)
  fx <- sapply(x, f)

  estimate <- (upper - lower) * mean(fx)
  se <- (upper - lower) * sd(fx) / sqrt(n)

  list(value = estimate, se = se)
}

Root Finding

Newton-Raphson Stability

newton_raphson <- function(f, df, x0, tol = 1e-8, max_iter = 100) {
  x <- x0
  for (i in 1:max_iter) {
    fx <- f(x)
    dfx <- df(x)

    # Check for near-zero derivative
    if (abs(dfx) < .Machine$double.eps * 100) {
      warning("Near-zero derivative")
      break
    }

    x_new <- x - fx / dfx

    if (abs(x_new - x) < tol) break
    x <- x_new
  }
  x
}

Brent's Method

For robust root finding without derivatives:

uniroot(f, interval = c(lower, upper), tol = .Machine$double.eps^0.5)

Statistical Computing Patterns

Safe Likelihood Computation

# Always work with log-likelihood
log_lik <- function(theta, data) {
  # Compute log-likelihood, not likelihood
  sum(dnorm(data, mean = theta[1], sd = theta[2], log = TRUE))
}

Robust Standard Errors

# Sandwich estimator with numerical stability
sandwich_se <- function(score, hessian) {
  # Check Hessian conditioning
  H_inv <- tryCatch(
    solve(hessian),
    error = function(e) MASS::ginv(hessian)
  )

  meat <- crossprod(score)
  V <- H_inv %*% meat %*% H_inv

  sqrt(diag(V))
}

Bootstrap with Error Handling

safe_bootstrap <- function(data, statistic, R = 1000) {
  results <- numeric(R)
  failures <- 0

  for (i in 1:R) {
    boot_data <- data[sample(nrow(data), replace = TRUE), ]
    result <- tryCatch(
      statistic(boot_data),
      error = function(e) NA
    )
    results[i] <- result
    if (is.na(result)) failures <- failures + 1
  }

  if (failures > 0.1 * R) {
    warning(sprintf("%.1f%% bootstrap failures", 100 * failures / R))
  }

  list(
    estimate = mean(results, na.rm = TRUE),
    se = sd(results, na.rm = TRUE),
    failures = failures
  )
}

Debugging Numerical Issues

Diagnostic Checklist

1. Check for NaN/Inf: any(is.nan(x)), any(is.infinite(x))
2. Check conditioning: kappa(matrix)
3. Check eigenvalues: For PD matrices
4. Check gradients: Numerically vs analytically
5. Check scale: Variables on similar scales?

Debugging Functions

# Trace NaN/Inf sources
debug_numeric <- function(x, name = "x") {
  cat(sprintf("%s: range [%.3g, %.3g], ", name, min(x), max(x)))
  cat(sprintf("NaN: %d, Inf: %d, -Inf: %d\n",
              sum(is.nan(x)), sum(x == Inf), sum(x == -Inf)))
}

# Check relative error
rel_error <- function(computed, true) {
  abs(computed - true) / max(abs(true), 1)
}

Best Practices Summary

1. Always work on log scale for products of probabilities
2. Use QR or Cholesky instead of matrix inversion
3. Check conditioning before solving linear systems
4. Center and scale predictors in regression
5. Handle edge cases (empty data, singular matrices)
6. Use existing implementations (LAPACK, BLAS) when possible
7. Test with extreme values (very small, very large, near-zero)
8. Compare analytical and numerical gradients
9. Monitor convergence in iterative algorithms
10. Document numerical assumptions and limitations

Key References

• Higham
• Golub & Van Loan