Two-point approximations are developed by utilizing both the function
and gradient information of two data points. The objective of this wor
k is to build a high quality approximation to realize computational sa
vings in solving complex optimization and reliability analysis problem
s. Two developments are proposed in the new two-point approximations:
1) calculation of a correction term by matching with the previous know
n function value and supplementing it to the first order approximation
for including the effects of higher order terms and 2) development of
a second order approximation without the actual calculation of second
order derivatives by using an approximate Hessian matrix. Several hig
hly nonlinear functions and structural examples are used for demonstra
ting the new two-point approximations that improved the accuracy of ex
isting first order methods.