Plots for computing the constant \(A_k\)

The goal of this notebook is to plot the function \((\lambda,\gamma)\mapsto A_k(\lambda,\gamma)\) defined in Lemma 4 of https://arxiv.org/abs/1906.08530

functionA <- function(k,lambda,gamma)
{
  require(pracma)
  library(pracma)
  A <- sqrt(lambda - 1)/lambda
  A <- A * (2*sqrt(lambda)/log(lambda-1))^k * k
  gamma_arg <- 0.5*sqrt(gamma)*log(lambda-1)
  A <- A * gammainc(gamma_arg,k)[[2]]
  A <- A + (k * (gamma*lambda)^(0.5* k - 1) - 2)/(k-2)
}

We will now compute the values of \(A_4(\cdot,\cdot)\) on a sufficiently fine grid of the rectangle \((\lambda,\gamma)\in [12,18]\times [4,6]\).

m <- 30
gamma_vec <- 4 + (6-4)*((1:m)/m)
lambda_vec <- 12 + (18 - 12) * ((1:m)/m)
for (ncol in 1:m)
{
  gamma = gamma_vec[ncol] 
  for (nrow in 1:m)
  {
    lambda = lambda_vec[nrow] 
    Matrix_A4[nrow,ncol] <- functionA(4, lambda, gamma)
  }
}

We will now plot the corresponding surface.

lambda <- lambda_vec
gamma <- gamma_vec
A4 <- Matrix_A4
fig <- plot_ly(x = ~lambda, y = ~gamma, z = ~A4) %>% add_surface(
  contours = list(
    z = list(
      show=TRUE,
      start = 441, 
      end = 443, size = 0.3,
      usecolormap=TRUE,
      highlightcolor="#ff0000",
      project=list(z=TRUE)
    )
  )
)
fig <- fig %>% layout(
  scene = list(
    camera=list(
      eye = list(x=2, y=0.98, z=0.24)
    )
  )
)

fig

#print(functionA(4,15,5))

We do now the same for \(k=3\).

m <- 30
Matrix_A3 = matrix(0,m,m)
gamma_vec <- 4 + (6-4)*((1:m)/m)
lambda_vec <- 12 + (18 - 12) * ((1:m)/m)
for (ncol in 1:m)
{
  gam = gamma_vec[ncol] 
  for (nrow in 1:m)
  {
    lamb = lambda_vec[nrow] 
    Matrix_A3[nrow,ncol] <- functionA(3, lamb, gam)
  }
}
A3 <- Matrix_A3
fig <- plot_ly(x = ~lambda, y = ~gamma, z = ~A3) %>% add_surface(
  contours = list(
    z = list(
      show=TRUE,
      start = 40, 
      end = 41, size = 0.1,
      usecolormap=TRUE,
      highlightcolor="#ff0000",
      project=list(z=TRUE)
    )
  )
)
fig <- fig %>% layout(
  scene = list(
    camera=list(
      eye = list(x=2, y=0.98, z=0.24)
    )
  )
)
fig

The graphically determined nearly optimal values are computed here.

print(functionA(3,13,5))

[1] 40.73317

print(functionA(4,15,5))

[1] 441.4962