Calculates the lower boundaries for futility at each look

1 |

`levels` |
The cumulative beta spending at each look time |

`upperboundary` |
The upper efficacy boundaries at each look |

`t` |
Normalized look times |

`int` |
The numbers of intervals the solution space is partitioned into |

`noncent` |
The noncentrality parameter |

`tol` |
Tolerance of the solution using uniroot |

numeric

1 2 3 4 5 6 7 8 | ```
f<- function(t) 0.025*t^4
g<- function(t) 0.15*t^3
t<-c(0.33,0.67,1)
cum_alphas<-f(t)
cum_betas<-g(t)
noncent<-qnorm(0.975)+qnorm(0.85)
upper_boundaries<-alphaspend(cum_alphas,t,int=rep(500, length(t)),tol=0.005)
lower_boundaries<-betaspend(cum_betas, upper_boundaries, t, int = rep(500,3), noncent, tol = 0.005)
``` |

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