Function update

Let $m=$number of generations

If (m = 0) Or ( $t_{n+1}>t^{m-1}$)   If NewGeneration is True

Push $t_{n+1}\equiv t^m$ onto $t^{u+1}$ stack

Push $\mathbf{U}(t_{n+1})$ onto ${\mathbf{U}}^{u+1}$ stack

//Implement Sec. 2.3

Update ${\Xi }_m(\mathbf{U}(t_{n+1}))$ using Eq. (2.31)

Push $F({\Xi }_m(\mathbf{U}(t_{n+1})))$

onto $F^{u+1}$ stack

Push $f^{m-1}(t^m)=$

ReformingBondMassFraction

onto $f^{u+1}$ stack

Call CullGenerations

Else if $t_{n+1}=t^m$

Update previously pushed

$\mathbf{U}(t^m)$, $F({\Xi }_m(\mathbf{U}(t^m)))$ and $f^{m-1}(t^m)$

Function ${\Psi }_r$

Let $m=$number of generations

If m = 0 Return ${\Psi }_r={\Psi }_r^e(\mathbf{F}(t_{n+1}))+{\Psi }_0^a(\mathbf{F}(t_{n+1}))$

Let ${\Psi }_r={\Psi }_r^e(\mathbf{F}(t_{n+1}))$

For u = 0 to m – 1

Evaluate $w^{u-1}(t_{n+1})=f^{u-1}(t^u)g(\mathbf{F}(t^u),t_{n+1}-t^u)$

Evaluate ${\mathbf{F}}^{u-1}(t_{n+1})=\mathbf{F}(t_{n+1})·{\mathbf{U}}^{-1}(t^{u-1})$

Evaluate ${\Psi }_r$+= $w^{u-1}(t){\Psi }_0^a({\mathbf{F}}^{u-1}(t_{n+1}))$

Return ${\Psi }_r$