Verify Jacobian Against Finite Differences

Compares the analytically computed Jacobian (via score identities) against numerical finite differences to validate the implementation.

Usage

verify_jacobian(
  J,
  a,
  b,
  eps = 1e-06,
  M = .QUAD_NODES_VERIFICATION,
  verbose = TRUE
)

Arguments

J

Integer; sample size.

a

Numeric; shape parameter.

b

Numeric; rate parameter.

eps

Numeric; step size for finite differences (default: 1e-6).

M

Integer; number of quadrature nodes (default: 200 for verification).

verbose

Logical; if TRUE, print detailed comparison.

Value

A named list with components:

analytic

The analytically computed Jacobian

numeric

The numerically computed Jacobian (finite differences)

abs_error

Matrix of absolute errors

rel_error

Matrix of relative errors

max_rel_error

Maximum relative error across all entries

pass

Logical; TRUE if max relative error < 0.01

Details

Uses central finite differences: $$\frac{\partial f}{\partial a} \approx \frac{f(a+\epsilon) - f(a-\epsilon)}{2\epsilon}$$

Important: This is a SECONDARY verification method because both the analytical Jacobian and the finite differences use the same quadrature layer. For independent verification, compare against adaptive integration (scipy.integrate.quad in Python).

Examples

# Verify Jacobian for a specific case
result <- verify_jacobian(J = 50, a = 2.0, b = 1.0, verbose = TRUE)
#> Jacobian Verification (J=50, a=2.00, b=1.00, M=200)
#> ------------------------------------------------------------ 
#> 
#> Analytic Jacobian (score-based):
#>   dM1/da =   2.24553449  dM1/db =  -4.13558517
#>   dV/da  =   2.94445788  dV/db  = -13.03822850
#> 
#> Numeric Jacobian (finite diff):
#>   dM1/da =   2.24552030  dM1/db =  -4.13558517
#>   dV/da  =   2.94463271  dV/db  = -13.03822849
#> 
#> Relative Errors:
#>   dM1/da: 6.32e-06  dM1/db: 1.29e-11
#>   dV/da:  5.94e-05  dV/db:  6.13e-10
#> 
#> Max Relative Error: 5.94e-05 [PASS]