An algorithm for computing the Hasofer-Lind reliability index
1. Identify the mean and covariance matrix for your random variables (M and Σ)
2. Decompose Σ into a diagonal matrix of standard deviations D and a matrix of correlation coefficients R (Σ = DRD). Also, compute the inverse of D.
3. Compute the Cholesky decomposition of R, (R = LLT, where L is a lower triangular matrix). Also, compute the inverse of L.
4. Use the Hasofer-Lind Rackwitz-Feissler algorithm to find the design point u*. a. Choose an initial guess for the x associated with u* (e.g., x = M). This is x(0) b. Find the u associated with the current value of x, using the transformation 1 1 ( ) ( ) ( ) k k − − u LD x M = − , where ( ) k x is the kth estimate of x*. c. Compute the gradient of g, evaluated at ( ) k x :