Given a set of age nodes \(0=x_0,x_1,\ldots,x_n\) and survival fractions \(1=l(x_0),l(x_1),\ldots,l(x_n)\) let \(M_i\) be the integrated force of mortality $$ M_i = -\log l(x_i) = \int_0^{x_i} \mu(a) da . $$ Use the \(M_i\) to constructed a curve \(M(x)\) by cubic splines. To complete the spline equations we need to specify boundary conditions. First we impose the condition that \(\mu(x)\) is linear at \(x=0\): $$ M'''(0)=0 $$ If \(T_n\), total person-years lived after \(x_n\), is not specified then we impose the condition that \(\mu(x)\) is linear at \(x_n\): $$ M'''(x_n) = 0. $$ If \(T_n\) is specified then we again impose the condition that \(\mu(x)\) is linear at \(x_n\) but this time the force level \(M'(x_n)\) and slope \( M''(x_n)\) must satisfy the equation $$ \frac{1}{m}= \sqrt{\frac{\pi}{2 M''(x_n)}} e^{\sigma^2} \mathrm{erfc}(\sigma) $$ where \(m = l(x_n)/T_n\) is the open-intervl death rate, \(\sigma = M'(x_n)/\sqrt{2 M''(x_n)}\) and \(\mathrm{erfc}(x)\) is the complementary Error Function. This equation is used together with the spline equation $$ M'(x_n)=A+B M''(x_n) $$ where \(A\) and \(B\) are known to solve for \(M'(x_n)\) and \( M''(x_n)\). The the survival fraction at any age \(x\) is then given by $$ l(x)=l(x_i) e^{-(M(x)-M_i)} $$ where \(x \in [x_i,x_{i+1})\).
Although this method does produce a mortality curve that is smooth is can give large changes in force between nodes that are far apart. Furthermore it can also give negative mortality rates. The Hybrid Force build method avoids these two problems.